The basic idea in this paper is to separate the standard rationality requirements embedded in the rational expectations hypothesis into internal and external components. Internal rationality means that the agents make fully optimal decisions given some well-defined subjective beliefs about payoff relevant variables. External rationality requires that the probability distribution generated by agent subjective beliefs matches the true distribution of underlying variables.

This paper will maintain the assumption of internal rationality, but not impose external rationality. To demonstrate the implications of this change, the authors use a simple Lucas asset pricing model to describe how the evolution of prices is different with and without extra rationality.

Model

Time is discrete. There are I infinitely-lived investor types, each with unit mass. Each agent is endowed with an equal share to an infinitely lived Lucas tree that stochastically yields consumable dividends Dt each period.

Agents have risk-neutral, time separable preferences over streams of consumption. The discount factor and probability measure used by agents is type-specific. Each agent chooses a sequence of consumption and asset holdings to maximize the expected discounted value of consumption subject to budget constraint and that asset holdings are between 0 and some (large) upper bound each period. The budget constraint requires that consumption plus the cost of asset purchases is less than the sum of asset sales; dividend receipts; and an fixed, exogenous endowment of the consumption good. Each period the price of the asset is Pt.

The non-standard part of the setup is that agents form beliefs over both realizations of asset prices and dividends. Under the REH, we typically assume that beliefs are over only dividend realizations and that agents know a mapping between dividends and prices when computing expectations.

In this model, internal rationality is that each agent chooses consumption and asset holdings to maximize expected discounted utility subject to the constraints, taking their type’s probability measure as given.

Equilibrium

Agent’s optimality conditions are standard. An interior solution equates the current price with the discounted expected price plus dividends tomorrow. Without external rationality, agents have joint beliefs over prices and dividends, so they use this first order condition to derive the equilibrium price. Because discounting and expectations are formed type be type, the equilibrium price will be the maximum of this discounted expectation over all types.

With external rationality (i.e. under standard rational expectations conditions) agents only have beliefs over the dividend process. They would use this first order condition together with the law of iterated expectations write the price today as the present discounted value of all future dividend payments.

Let’s take a step back and think about this…

Note that we can consider external rationality as a special case of the model without external rationality. Specifically with external rationality agents are given a probability measure over dividends and, implicitly, a mapping from histories of dividend realizations to prices. Knowledge of this function is not an outcome of agent maximization (i.e. internal rationality), but rather the impact of a set of assumptions the modeler makes about what agents know about how the market operates. Given that economists haven’t found a mapping from dividend streams to prices, it seems reasonable to assume that agents don’t have this mapping either.

Internal only to REH

We now consider which assumptions are needed to go from the internal rationality only model to the model with both internal and external rationality. That is, we consider assumptions that allow our agents to write the current price as a expected discounted present value of future dividend payments.

1. It is common knowledge that a single agent “sets the price” each period. I call this agent the marginal agent. This allows each agent to use their own beliefs about who is marginal each period to write today’s price as the present discounted value of dividends.
2. It is common knowledge that the last term in the infinite sum is zero, when the marginal agent’s beliefs and discount factor are applied each period. This gives agents information about the market in that all agents expect all future marginal agents to expect (and so on…) that prices grow slower than the marginal discount factors. This is a no rational bubbles condition.
3. All agents know which agent is marginal each period and what the marginal agent’s discount factor and probability measure are. This allows all agents to write down the same infinite sum, that coincides with the equilibrium part.

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Builds on the internal rationality framework from last week to build a model of asset pricing that can explain 5 facts that have puzzled the literature at one time or another.

Facts

1. Standard deviation of price dividend ratio is very high (about ½ the mean of the PD ratio)
2. First-order quarterly autocorrelation of PD ratio is vary high
3. Standard deviation of stock returns is almost 4 times as large as standard deviation of dividend growth
4. PD ratio is good long run predictor of stock returns
5. Equity premium puzzle: return on risky stocks is too high relative to bonds for standard models

Model

There is a unit mass of infinitely lived investors, endowed with one unit of a stock that can be traded in a competitive market and that pays a per period dividend D in units of a perishable consumption good. The log growth rate of dividends follows an AR(1).

Agents also receive a stochastic endowment of the consumption good. The feasibility constraint requires that consumption be equal to dividends plus the endowment. Following the consumption-based asset pricing literature they choose to model the consumption process directly instead of the income process. The log growth of consumption is also an AR(1). The innovations in consumption growth and dividend growth are correlated with coefficient equal to 0.2 (estimated in the data).

In addition to trading shares of the dividend yielding risky asset, agents can also trade in a one period risk free bond.

All agents discount the future at the same, constant rate.

The objective of each consumer is to choose sequences of functions that map histories of observed prices, dividends, and endowments into consumption, stock holdings, and bond holdings to maximize the expected discounted utility of consumption; subject to a budget constraint that equates expenditure on consumption and stock and bond purchases to dividends, the endowment and bond returns.

Each agent is allowed to have subjective beliefs over the joint evolution of prices, dividends, and endowments. Agents behave fully rationally, given these beliefs. The only difference between the setup here and the classic setup is that agents form beliefs over prices instead of being assumed to have a mapping between histories of dividends and endowments into current prices. These subjective beliefs will be updated over time and are crucial for the model to explain the facts in the data.

Solution

The agents first order necessary conditions for stocks and bonds are standard.

A key equilibrium result is the pricing function that relates current dividends and expected price and dividend growth into current prices.

This function is that prices today are equal to risk adjusted expected dividend growth, divided by one minus risk adjusted expected price growth, times the discount factor, times dividends. It is crucial that todays price is increasing the divided flow, risk-adjusted expectations about dividend growth, and risk-adjusted expectations about price growth.

Under the rational expectations hypothesis, the expectation of both of these growth rates is constant. This results in growth rate of prices to exactly equal the growth rate of dividends. It is for this reason that the standard rational expectations model fails to match the first 4 facts I gave before. The model also generates a low risk premium, so it misses all 5 facts.

However, when beliefs are subjective the expected risk-adjusted growth rates are not constant over time. The authors argue that the important component of their model is fluctuations in the expectation of price growth. To test this, they assume that investors use the true probability measure for divided growth, but hold subjective beliefs over price growth. This is the same as assuming agents still cannot map perfectly from dividend and endowment realizations into prices.

Results

The baseline parameterization of the proposed model is able to quantitatively match all the facts except the risk premium. In this model the CRRA parameter is set to 5. If they allow this to float to 80, they are also able to match the risk premium fact.

The question is, how does it happen?

The key mechanism is a feedback loop between expectations about price growth and the realization of price growth. Here’s a rough sketch of how it works

• Expected price growth exhibits both momentum and mean reversion to the rational expectants “fundamental level” of prices
• This momentum means that beliefs have a tendency to increase further following an initial increase whenever beliefs are at or below their fundamental value.
• Mean reversion means that beliefs can’t stay away from the rational expectations level forever.

These facts together cause the price dividend ratio to fluctuate around its rational expectation value. This allows the model to explain the observed volatility and serial correlation of the PD ratio (facts 1 and 2). This behavior also causes stock returns to be more volatile than dividend growth; which explains fact 3. Finally, serial correlation and mean-reversion in the PD ratio give rise to excess return predictability – fact 4 (NOTE: the notion of predictability is explained in another paper by Cochrane that I didn’t study).

The model still fails, however to generate a high enough risk premium. In order to match this fact, they authors have to adjust risk aversion. They do so by cranking up the CRRA parameter. I believe that using recursive preferences and adjusting the inter temporal elasticity of substitution is another way to generate excess returns without resorting to very high levels of risk aversion.

Model

This is a general equilibrium trade model that focuses on producer export, entry, expansion, and exit. The structure of the model nests other standard models such as Krugman, Melitz and others.

In the model there are two symmetric countries. Within each country there are consumers, final goods producers, intermediate goods producers, and a government.

Consumers

• A unit mass of identical consumers live in each country and inelastically supple one unit of labor
• Choose sequences of capital, risk free one period bond holdings, and consumption to maximize expected present discounted value of CRRA utility over consumption.

Final Goods Producers

• The final goods producers combine home and foreign intermediate goods to produce a non-traded consumption good

Intermediate Goods Producers

Intermediate goods producers are the most interesting agents in the economy. Each producer is characterized by three state variables: technology level (z), iceberg costs (ξ), and fixed costs (f) and produces intermediate goods using capital, labor, and materials using a nested Cobb Douglass technology subject to TFP shocks.

The cost structure is discrete: fixed costs can be either low or high while iceberg costs can be one of three levels: low, high, or infinite.

A producer with infinite iceberg costs is a non-exporter. He can choose to pay no fixed cost and remain a non-exporter, or pay the high fixed cost and start the next period as an exporter with a high iceberg cost.

If a producer has either low or high iceberg costs, they can pay the low fixed cost to draw an iceberg cost for the next period. In the parameterization, the distribution is symmetric and places about 90% probability on staying with the same finite iceberg cost.

This structure nests other common models. If f_L < f_H, then there is a sunk cost to being an exporter. If f_L = f_H and ξ_L = ξ_H the export decision is static. If in addition the fixed costs are zero, then there is no export decision and the model is a generalized version of Krugman’s monopolistic competition.

Entry

Entrants pay a fixed cost to set up shop and cannot export in their first period. In equilibrium there is a free entry condition that determines the measure of entrants.

Aggregates

There is a government that collects export tariffs (calibrated to be 10%) and redistributes the proceeds lump sum to domestic consumers.

Equilibrium

Definition

An equilibrium is defined as allocations for consumers, intermediate good firms and final good firms in both countries as well as export decisions for intermediate good producers such that taking prices as given, the allocations are consistent with agent optimization, the free entry condition holds, market clearing and resource constraints hold.

Results

Two main calibrations:

1. Benchmark. Some features
   • Low fixed cost for becoming exporter relatively small (3.7% of the cost to enter as new firm)
   • Profits are back loaded: new exporters expect to run losses for at least the first 3 years
   • Iceberg transition matrix for exports is symmetric and the Markov process is very persistent.
   • Older exporters are typically more efficient (low iceberg cost). This provides an incentive to remain an exporter (don’t want to start over at high iceberg again)
2. Sunk cost model
   • Restricting the iceberg costs to be the same results in a sunk cost model
   • Entry cost vs continuation cost higher than in benchmark model
   • Profits are front loaded: exports pay a large cost up front, but expect to earn profits in their first period of exporting
   • Incentive to remain an exporter comes through a different channel: firms don’t want to pay large fixed cost again.

Experiments

The authors perform a number of experiments with different calibrations of the model.

Global Trade Liberalization: The first experiment is that tariffs are unexpectedly reduced to zero globally. The key features of the response to the new steady state are:

• Trade grows slowly and smoothly to a higher SS
• Consumption and output rise sharply and immediately, but follow a hump shaped path (overshoot new SS in early periods following policy change)
• Entry rate declines

They did the same experiment in the sunk cost model. Differing results were:

• Transition is much faster (after 3 periods trade growth 55% percent of long run value in benchmark model, but 90% in sunk cost model)
• Smaller welfare gain from policy change. Driven by stronger over shooting in consumption in benchmark model, which in turn is a driven by the fact that in sunk cost model new exporters export too large a share of aggregate exports. In the benchmark model new traders are less efficient, so they make up a smaller share of total trade

Also did a version where fixed costs are zero and iceberg costs constant (no export decision, no cost model). Differences in response to regime change are:

• Smaller welfare gain (relative to benchmark model) because consumption grows smoothly and doesn’t overshoot.
• This happens because The model without trade costs experiences smaller changes on the extensive margin than the model with costs.

References

This paper aims to explain the low exchange rate pass-through for exporters. Exchange rate pass-through is the response in export prices to movement in the exchange rate.

Model

The theoretical model is not the focus of the authors’ analysis in this paper. They write down a fairly complicated model, derive some of the equilibrium conditions, then use the implications of the equilibrium conditions as testable predictions they take to the data.

I will attempt to summarize only the portions of the model that are necessary for understanding the testable predictions.

The model is made up of two main components:

1. An oligopolistic competition model of variable markups following Atkeson and Burstein (2008). There are a continuum of sectors, each with a finite number of firms. Good are combined at the sector level using a CES technology. Then sector outputs are combined using another CES technology before being consumed by a representative household in each country. Household’s have preferences over the final consumption good and the labor they supply to firms.
2. A model of the firm’s choice to import intermediate inputs at a fixed cost as in Halpern, Koren, and Szeidl (2011). Firms use a CRS Cobb-Douglass production function to combine a continuum of inputs. Each input is the CES aggregation of a domestic and foreign variety of the input. Foreign varieties have a multiplicative productivity advantage in the CES aggregator. Firms pay a firm specific fixed cost for each input variety they choose to import. Inputs are by sorted total productivity factor (combination of effect at the CES and Cobb Douglass levels), which together with the fixed costs makes the import policy have the form of a cutoff rule.

The oligopolistic competition portion of the model generates the fact that firms set prices at a constant markup over marginal cost. Furthermore, the markup is fully characterized by production function parameters and the firm’s market share at the sector-destination level.

The intermediate input importing part of the model results in firms with larger total input costs or smaller fixed cost of importing have a larger import intensity. (defined as fraction of total variable costs spend on importing)

Finally, the main theoretical result of the paper is that in any general equilibrium in this framework, the first-order approximation of the elasticity of destination-specific firm prices to the exchange rate (e.g. exchange-rate pass through) is affine in the importing intensity and market share.

Data

The main data used in the paper is from the National Bank of Belgium. It consists of a comprehensive panel of Belgian trade flows by firm at the product (CN 8 digit level). It includes exports by destination and imports by source country. This is combined with firm-level characteristics from the Belgian Business Registry. Data is measured annually between 2000 and 2008.

The authors run a number of regressions. In each of them, the dependent variable is the log change in a firm’s export price of a good to a country. This is computed as the difference in the log of export value over export quantity.

A key variable from the theory is the sector-destination-time market share of each firm. This is computed in the data as the total value exported by a firm to a destination divided by the total value exported from to that destination from the sector.

The final key variable in the theory is the import intensity of the firm. This is defined as the total value of all non-euro zone imports over total variable costs.

The authors also use data on the exchange rate and change in marginal costs.

Stylized Facts

The data reveal a number of stylized facts about Belgian importers and exporters:

• Import intensive firms (firms whose import intensity is above the median level of 4.2%) operate at a much larger scale than non import intensive firms.
• Import intensity is skewed towards zero, but has wide support and high dispersion.

Results

The main empirical findings are summarized by the results of regressing the log change in export price on

• Change in exchange rate
• Lagged import intensity
• Lagged market share
• Interactions between change in exchange rate and both import intensity and market share
• Firm, destination fixed effects

Using variants of this specification, they document the following results:

• Including only the change in exchange rate and dummies, they find that exchange rate pass through (change in prices in response to change in exchange rate) is roughly 80%.
• Adding the interaction between the change in exchange rate and import intensity reveals that a 10 percent higher import intensity leads to a 6 percent lower pass-through. This is consistent with the model where pass- through is decreasing in import intensity.
• Controlling for changes in marginal costs causes the coefficient on the interaction of the exchange rate and import intensity to be cut in ½, but remain statistically significant at the 1% level. The coefficient on changes in marginal cost is almost twice as large as the import intensity term. This suggests that marginal cost is an important channel through which import intensity affects pass-through, but there is significant residual that operates through other channels. The theory indicates that the other channel is the markup channel.
• Regressing change in prices on changes in the exchange rate and interactions between import intensity and market share shows that both interaction terms have significant coefficients at the 1% level. Under the results of this regression, a small non-importer will have 96% pass through. A non-importing large firm with 75% market share will have pass-through of 73%. If additionally this large firm has an import intensity of 38%, the pas-through drops to 55%. This shows that variation in import intensity and market share explain a vast range of the variation in pass-through across firms.

Model

The model in this paper is fairly simple. The interesting analysis in this paper has to do with the information structure

Setup

Agents in the model either live in the mainland or on one of a continuum of islands

Each island:

• Is populated by a continuum of workers and a continuum of monopolistic firms.
• Within each island all firms are identical, as are workers
• Firms:
  • Hire workers through competitive labor markets
  • Use labor in a linear production technology: product of labor supply island specific TFP
    • The log of TFP in each period is the sum of an aggregate and an idiosyncratic shock
    • All shocks in the model are mean zero Gaussians, unless otherwise specified.
• Workers
  • Supply labor to firms
• Preferences for firms and workers are not considered here, as the economy will have a representative agent on the mainland.

So let’s turn to the mainland:

• Inhabited by a continuum of identical consumers
• Each consumer is tied to a single firm and a single worker on every island
  • This assumption lets the economy admit a representative agent
• The representative household maximizes the present discounted value of lifetime utility
• Utility is time separable and utility in each period is the difference between utility from consumption and disutility from labor supply:
  • The aggregate consumption good is valued with a CRRA utility function over the output of a nested CES structure:
    • Inner most level: goods produced by all firms on an island are aggregated into an island final good.
      • This aggregator is subject to markup or cost-push shocks (i.e. the exponent in the aggregator is a random variable)
    • The island final goods are then aggregated into the final consumption good.
  • Labor is aggregated by integrating another CRRA function of labor supply over each island and each firm
    • This sum is subtracted from the utility of consumption
• The representative agent faces a budget constraint where consumption expenditures and risk-less bond purchases must be less than or equal to
  • aggregate firm profits
  • the return on bond holdings
  • Aggregate labor income tax and a lump sum transfer used to balance the budget.
• The total wedge due to monopolistic power is a function of the island specific tax rate and the island specific markup or cost-push shock.

Information

A crucial aspect of this model is the information setup. The authors run two experiments:

1. Fix monopoly markup and let agents receive noisy signals about TFP
2. Fix TFP and let agents see noisy signals about markups

The signal structure mirrors that of Morris and Shin. Each period each island receives two signals:

1. A public signal seen by all agents. This signal is the sum of the actual fundamental and a mean zero Gaussian noise term
2. A private signal with the same structure, but seen only by the island.

Similar to the result from Morris and Shin, the best response in each island is a production rule that is affine in productivity, monopolist markup, and expected aggregate output.

The coefficient on the expected aggregate output determines the degree of strategic complementarity, or how much firms like to behave like each other. Unlike earlier work on beauty-contest models, this parameter is micro founded as a function of underlying preference and technology parameters.

Extra notes:

• Bergemann and Morris (2013) show that equilibrium results from any information structure with Gaussian signals can be replicated by the structure here.
• Thus, within the family of Gaussian signals, the structure considered in the paper is without loss of generality.

Experiments

Definitions:

Σ ≡ Var(logY−logY)

σ ≡ Var(logỹ_it−logỸ_t)

$$\Delta \equiv \frac{E[Y]}{Y/Y^}$$

$$\Lambda \equiv \Sigma + \frac{1}{1 -\alpha} \sigma$$

In the experiments the authors do, they consider how welfare changes with the precision of the signals.

Welfare is decomposed into the product of two functions, each taking one argument.

The first function is a strictly concave function of Δ

The second function is strictly decreasing in the sum of volatility of the aggregate output gap and the cross-sectional dispersion of local output gaps (Λ).

Let’s consider the main results for each experiment

Signals about TFP

In experiment with TFP shocks, the log of TFP of each island is modeled as the sum of an aggregate productivity shock common across all islands and an island specific idiosyncratic shock.

This paper develops a model with menu costs for adjusting prices and imperfect information about idiosyncratic productivity shocks. They conduct monetary policy experiments and conclude that the distribution of firm level uncertainty is important for the propagation of monetary shocks.

Model

In this model time is continuous. There is a representative consumer, a continuum of monopolistically competitive firms, and a monetary authority.

In the baseline model, the monetary authority keeps the money supply fixed at its initial level.

The representative consumer has log preferences over consumption and money holdings, with linear disutility from labor. The household discounts the future at a constant rate r. Consumption is the CES aggregation of firm specific goods, each multiplied by a firm specific quality shock. The lifetime budget constraint of the consumer says that the value of consumption expenditures plus the cost of holding money, less labor earnings and firm profits is no greater than the initial money supply.

There are a continuum of firms who each product and sell their product in a monopolistically competitive market. Each firm has access to a linear technology that produces one unit of output for each unit of labor, divided by the stochastic quality of the good (meaning a higher quality good requires more labor to produce).

This is the same quality that appears in the CES aggregator for consumption. The log of the quality shock follows a jump-diffusion process, without drift. This means that da(z) = sigma1 dW + sigma2 u dQ, where W is a Weiner process and u Q is a poisson process with standard normal innovations. It is assumed that sigma1 is much smaller than sigma2, such that when the poisson process jumps there is a large shock to the quality of the firm’s product.

Firms do not observe the quality directly and cannot learn about it using wage costs. The only information they receive about their quality is a noisy signal and the information about when the poisson process jumps. The evolution of the signal follows ds = a dt + gamma dZ, where Z is another independent Weiner process.

Firm prices are subject nominal rigidities in which firms can only change their price if they pay a fixed menu cost θ.

The flow profit to a firm is given by the demand for the good, multiplied by the price less the wage bill.

Firms use the discount rate of the household (because households own firms) and seek to maximize the expected discounted profit streams by choosing a sequence of prices and stopping times for price adjustments. When firms adjust prices, it is optimal to set them to a constant markup over marginal costs. Each period the firm does not adjust its price, there is a gap between the optimal markup in a frictionless economy (i.e. without both price and information frictions) and the markup being used by the firm. The equilibrium flow of profits can be expressed as minus a constant times this markup gap.

Because firms cannot directly observe the quality of their goods, they do not know the true value of the markup gap each period. Instead, they use the signals about the quality shock to form a signal extraction problem. A main contribution of the paper is the extension of the Kalman-Bucy filter to the environment where the hidden state follows a jump-diffusion process. One key output of the system of filtering equations is that innovations in the estimate of the markup gap are more volatile when there is high uncertainty about the estimate – e.g. when the variance of the estimate is high. An implication of this is that when uncertainty is high, firms place higher weight on their signals than they do on the current estimate when updating beliefs. In this scenario, the learning rate is higher, but also noisier.

The optimal stopping time for firms is characterized by an inaction region – as long as the filtered estimate for the markup gap is within certain bounds, the firm does not update prices. Once the price touches one of the borders, it immediately adjusts prices to set the estimated markup gap to zero.

Results

Here are some key results about the steady state equilibrium of this economy:

• The filtering equation for updating uncertainty (the variance of the state estimate) produces cycles of uncertainty for the firm. There is a deterministic component that causes uncertainty to fall to it’s minimum level (the volatility of the diffusion component of quality shocks) and a stochastic component that causes it to rise whenever Poisson process jumps. The time series for a simulated path has a saw-tooth pattern where sharp increases are followed by gradual declines in uncertainty.
• In times when uncertainty is high, the innovations to the filtered estimate of the markup gap are more volatile. This generates cycles in firm markup behavior where in times of high uncertainty firms choose to update prices more frequently. These cycles in uncertainty and pricing behavior are idiosyncratic because they are driven by idiosyncratic evolution of Brownian Motion and Poisson processes.
• The idiosyncratic cycles allow the authors to look at how the distribution of uncertainty relates to the response to monetary shocks. To do this they perform an experiment where the monetary authority does a one-time un-anticipated increase in the money supply by a known amount. When the monetary shock occurs, estimates of the markup gap are updated immediately. But, a firm’s price will only get update when its estimate leaves its inaction region. As long as some firms’ estimates remain in the inaction region, aggregate output will be different from its post-shock steady state value. As a summary statistic, they compute the total output effect, which is the integral over all time of the output gap relative to the steady state. They find that in the economy with information frictions, the output effect is larger up to 7 times larger and has a half life up to 5.3 times longer half life than in the perfect information economy. In this sense the authors claim that the distribution of firm level uncertainty is an important consideration when trying to evaluate the effects of monetary policy actions.

Empirical paper that tries to quantify the effect of import competition on the composition of the balance sheet of individuals between 2000 and 2007.

The main finding is that consumers in regions with higher exposure to import competition had debt growth of 20 percentage points more than consumers in regions with low import competition exposure.

Data

• New york FRB consumer credit panel: 5% of all US individuals with a credit record and social security number. Data is measured at a quarterly frequency. The important features for this study are a breakdown of household debt into mortgages, home equity credit, auto loans, credit card debt, and other loans. There is also limited demographic data such as the age, credit score, and zip code of participants.
• Census and IRS data to compute regional demographics
• House prices from CoreLogic
• Employment data from BLS
• Home Mortgage Disclosure Act data. the consumer credit panel doesn’t show what people did with rising debt. The authors here adopt the methodology explained in the paper Don presented last week
• PSID: track household income and debt over time
• Building Permits survey BPS: capture change in mortgages due to new house purchases
• Peter Schott’s database of shipping costs as a percentage of sale price at the product level

Strategy

Results

The authors of this paper use firm level data on Portuguese manufacturing firms and documents two new novel facts about the joint evolution of firm performance and prices:

1. Within product categories, firms with longer tenure in export markets tend to export larger quantities at similar prices
2. Older or more experienced exporters tend to import more expensive inputs

The authors then write down a model that can match these facts

Model

Time is discrete, there are N countries. Each country is populated by a unit mass of infinitely lived consumers. There re two sectors in each economy: a final goods sector and an intermediate inputs sector. The final goods sector is populated by a mass of monopolistically competitive firms that each supply a differentiated variety of varying quality. The intermediate inputs sector is perfectly competitive and operates with a CRS production technology.

Consumers have log preferences over a CES aggregation of final goods. Within the CES aggregator final good has a multiplicative term representing the firm chosen quality of the good as well as a multiplicative preference shock term. This is idiosyncratic across firms.

TODO: DESCRIBE WHAT THE STRUCTURE IS.

intermediate sector

Perfect competition leads to price being equal to marginal cost

final goods

Static problem: choose good quality and quantity based on expected profits in each economy.

Entry problem: given result of static problem, firms choose each period which markets to enter. If they enter they pay a fixed cost per market.

For each market the firm serves, they observe the equilibrium price that cleared the market for their good. They invert this price to uncover what the demand shock was in that period. They use this demand shock as a noisy signal though which they update their beliefs about the distribution of demand shocks they face.

eqm

They show that this model can generate dynamics that match the two empirical facts from before.

Counterfactual

They also do a very brief counterfactual exercise where they consider the welfare implications of imposing a minimum quality standard on all exported goods. They find that this limit decreases welfare. The mechanism is as follows:

• quality standard raises productivity threshold for entry into export market
• The decline in number of exporters reduces competition in each market, raising prices
• Another effect is less foreign firms allows for more low efficiency domestic firms to enter

There is also an impact on the intensive margin:

• firms close to the productivity threshold (small or young) will face a binding quality constraint
• in order to export these firms must comply and in order to do so they need to pick which quality intermediate inputs.
• higher quality inputs requires more labor
• This leads in a reallocation of resources among exporters toward young and small firms.

TODO FILL THIS IN!!!

An economist, a physicist, and two computer scientists walk into a bar…

This is a computational paper that describes and algorithm featuring an adaptive sparse grid and discuss implementation details on a sophisticated HPC cluster.

Model

They provide examples of their algorithm and computation using a standard international real business cycle model. This is not the interesting part of the paper, so I will not focus on it here.

Computation

Adaptive Sparse Grids

The first main contribution of this paper is to introduce economists to a specialized flavor of function approximation.

The authors use familiar linear Splines; but do so on a sparse, adaptive, and hierarchical grid.

By sparse I mean that the n-dimensional grid is not composed of the tensor product, or Cartesian product, of all univariate grids. This helps alleviate the curse of dimensionality.

By adaptive I mean that the knot vector for the grid in each dimension will change as the solution algorithm for the economic problem proceeds. This helps preserve accuracy of function approximation routines when the grid is sparse.

By hierarchical I mean that the grid for a particular level of refinement is a strict subset of the grid for all higher levels of refinement. This helps reduce computation costs as the basis functions for higher order terms only require a few additional function evaluations instead of a full basis matrix.

Algorithm

The second main contribution of the paper is an algorithm that leverages this special interpolation scheme to solve high dimensional dynamic stochastic models.

The main steps of the iterative portion of the algorithm are:

• Start with the coarsest refinement level l = 1 for all dimensions. Call the grid G. Choose a maximum refinement level
• Also Initialize G_old and G_new to be the empty set
• While G and G_old are not the same (NOTE: at end of this loop explain the while goes until we are at Lmax or until we don’t add refinement points)
  • For each grid point in G \ G_old
    • Solve the system of non-linear equations characterizing the optimal controls at that grid point (note you will need to interpolate over the current guess of the policy function) to obtain a new guess for the policy rule at that point
    • If the distance between the old and new guess for the policy at the grid point is greater than some threshold, add the neighboring points at the next refinement level to Gnew
  • Set G_old = G, G = G_old ∪ G_new, G_new = ∅ and l +  = 1
• Calculate the error for this iteration as the sup norm over the implied policy rule from the current iteration and the previous iteration

Notice that this algorithm will start with very few points across the entire domain and iteratively add points only in regions where the policy rule has dramatic changes. This will naturally cause the grid to adapt and add more grid points in areas of the state space that feature non-linearities or discontinuities.

Results

This algorithm was implemented by coders who know their stuff! They build on cutting edge software and run their code on state of the art super computers in Switzerland.

The results are impressive.

On just one node of the super-computer, they were able to achieve a 30x speedup for their code by utilizing a GPU and multi-threaded parallelism.

They were then able to scale that code from a single node to up to 2048 nodes to achieve a speedup on the order of 10,000x.

They compare their algorithm to a non-adaptive sparse grid and find that the log 10 average Euler errors from their algorithm are smaller on average than the sparse grid case, but run time is up 2 to 3 orders of magnitude smaller.

This paper provides an overview of various solution concepts that belong to the families of approximate dynamic programming and reinforcement learning.

I will discuss an alternative representation of a dynamic programming that is often used in this literature and provide an overview of some solution methods that are common with this representation.

Basics: Q-functions

In economics, the standard representation of a dynamic programming problem is to define a recursive function V that maps from a state space into the real line. Today I will call V the V function. The V function describes the value of being in a particular state.

Consider an alternative function Q that maps from the state-action space, into the real line. I will call this the Q-function. The Q function describes the value of being in a particular state and choosing a particular action.

As we do with the V-function, we can express the Q-function recursively using Bellman’s principle of optimality. The recursive form of the Q-function is

Q(s, x) = E_{s′ ∼ f(s, x,  ⋅ )}[r(s,x,s′)+βmax_x′Q(s′,x′)]

We can make some basic comparisons between Q and V by studying their recursive representations:

• Parameteric representations (i.e. splines or other interpolands) of Q are more expensive than representations of V because Q is defined over the state-action space instead of just the state space.
• Computing the maximization on the right hand side of Q is easier than the max on the RHS of V. This is because the expectation is outside the max in Q, but inside the max for V.

The Bellman operator associated with the Q function is a contraction mapping under the same conditions that Bellman operator associated V is a contraction mapping. When Q’s operator is a contraction mapping, the operators’ unique fixed point Q is the optimal Q function.

The optimal policy can be easily computed from Q: $h^* (s) = \underset{x}{\text{argmax}} Q^* (s, x)$

The optimal V-function can be obtained from Q: $V^ = \underset{x}{\max} Q^* (s, x)$.

Computing Q

We see that if we can find Q, we can get the value and policy functions we care about as economists. I want to briefly describe a few flavors of algorithms that can be used to compute Q

Q-learning

Q-learning is classified as an online, model-free algorithm. This means you can update your guess for Q whenever new data becomes available. Economists would classify this as a simulation algorithm.

Here’s the basics:

• Start with any initial guess for the Q function
• Then on each iteration k:
  • Given the state action pair on the ith iteration ((s_i, x_i)), the observation of the period i return r_i, and the state iteration i + 1 s_i + 1…
  • … update your guess for Q using:
    Q_i + 1(s_i, x_i) = (1 − α_i)Q_i(s_i, x_i) + α_i(r_i+βmax_x′Q_i(s_i + 1,u′))
    

Notice that the right hand side is a convex combination of the current guess of the Q function and the right hand side of the Q-function operator. This is very similar to standard value function iteration, but has the main advantage of not having any expectations.

This can be applied to any model for which you have a transition function that defines the state today as a function of the state and action yesterday and a random innovation from a distribution you can sample. Many economic models fit into this framework.

When the action space is continuous, you need do a modest alteration of the basic algorithm to ensure asymptotic convergence to Q.

Approximate Q-iteration

One strength of this literature is the number of different ways of reducing the dimensionality of a dynamic programming problem.

Approximate Q-iteration is one example of this type of algorithm. Here instead of operating on the Q-function directly, we instead operate on a parameterized approximation of the Q-function.

This can be done using linear dependence on coefficients (splines, Chebyshev polynomials) or non-linear parametric approximations (neural nets).

We’ll focus on the linearly parametrized version here. We will form an approximation $\hat{Q}(s, x) = \sum{l=1}^n \phi_l(s, x) \thetal = \Phi\theta$.

We and talk about a variant of the Q-learning we previously discussed that uses gradient information to update the coefficient vector.

Here’s the basic idea:

• Start with any guess of coefficients θ
• Then on each iteration k:
  • Given the state action pair on the ith iteration ((s_i, x_i)), the observation of the period i return r_i, and the state iteration i + 1 s_i + 1…
  • … update your guess for θ using:
    $$\theta{i+1} = \theta_i + \alpha_i \left[ri + \beta \max{x’} \hat{Q}i(s{i+1}, x’) - \hat{Q}_i(s_i, x_i)\right] \frac{\partial}{\partial \theta_i} \hat{Q}_i(s_i, x_i)$$
    
  • In the linear parameterization world we are in this becomes
    θ_i + 1 = θ_i + α_i[r_i+βmax_x′(Φ(s_i + 1,x′)θ_i)−Φ(s_i,x_i)θ_i]Φ(s_i, x_i)
    

As this is a Q-learning algorithm, the same computational benefits arise here, but now we additionally have the benefit of a potentially vastly reduced computational problem depending on the state-action space the choice of basis functions.

Model

Mechanics

Two agents, infinite horizon, discrete time, and each agent has a common discount factor δ.

Each period agent 1 can choose to enter or exit.

Player 2 has a countably finite set of actions, represented as the set 𝒜, which is equal to the natural numbers. Each period an i.i.d subset of A ⊂ 𝒜 is drawn. This is both the state of the economy in each period and agent 2’s feasible actions set for the period. Each action is a ∈ 𝒜 is drawn with the same probability p each period.

Each period t is broken into two stages. In stage 1, agent 1 chooses to enter or exit. Upon exit, both players get 0 flow payoffs, the game moves to period t + 1.

If agent 1 chooses to enter, he pays a constant fixed cost k while agent 2 gets a benefit π. Then the iid state A_t ⊂ 𝒜 is drawn and stage 2 begins.

In stage 2 player 2 chooses his action a ∈ A_t. This choice brings a deterministic cost c(a). With probability q the action succeeds and agent 1 gains a deterministic payoff b(a). With probability 1 − q the action fails and agent 1 gets 0.

Technology

In the set 𝒜 there are N productive actions (numbered 0 to N − 1).

The function c(a) takes on a constant value c for all productive actions, and 0 for all non-productive actions.

The function b(a) is positive (though not constant) for all productive actions, and 0 for all other actions.

There is no mechanism for agents to share utility.

Information

All agents know perfectly the parameters δ (discount rate), p (probability at which a particular action is drawn from 𝒜 each period), q (probability a productive action is successful), and N the total number of productive actions.

Both agents perfectly observe the state each period (though the time t state is realized after agent 1 makes his time t decision). Both players also see perfectly the action a taken by player 2 and the payoff paid to agent 1.

Information is asymmetric in that only agent 2 sees the cost of his actions – hence only agent 2 knows if the selected action is productive.

The realizations of the function b(a) are drawn from a known distribution B with compact support and held fixed throughout the game. Agents do not know these realizations ex ante, but share the same prior about the distribution from which they were drawn.

Agent 1 has an improper uniform prior over which actions are productive. Specifically

$$\forall A \subset \mathcal{A}, \forall a \in A, \quad \text{prob} \left{a \in \mathcal{N} | \text{card} A \cap N = n\right} = \frac{n}{ \text{card} A}$$

Parameters are such that agent 1 doesn’t exit in period 1¹.

Equilibrium concept

Let d_t ∈ {S, E} be agent 1’s decision to stay or exit at time 1. If d_t = E, then A_t = a_t = b̃(a_t)∅. There are three types of history:

1. ℋ¹ are histories of the form h¹ = {d₁,A₁,a₁,b̃(a₁),…,d_t − 1,A_t − 1,a_t − 1b̃(a_t − 1)} and correspond to agent 1’s information set at his decision node in period t
2. ℋ^2|1 histories are h¹ ⊔ {d_t, A_t} are agent 1’s information at agent 2’s decision node in period t.
3. ℋ² histories are h^2|1 ⊔ 𝒩 are agent 2’s information set at his decision node in period t.

A pure strategy for player 1 is s₁ : ℋ¹ → {S, E}. A pure strategy for player 2 is s₂ : ℋ² → 𝒜 such that forall histories h_t² ∈ ℋ², s₂(h_t²) ∈ A_t.

The equilibrium concept is a Pareto efficient perfect Bayesian equilibrium in pure strategies. The strategy profiles form the equilibrium.

Equilibrium Analysis

Complete information benchmark

Consider a benchmark model of full information where agent 1 also knows agent 2’s cost function. In all Pareto efficient equilibria of this game, player 1 never chooses to exit.

Because the payoff to agent 1 is bounded above, value functions for player 1 and player 2 are bounded. Denote the highest possible value function for agent 2 in the perfect information game as V̄₂. This will be used in later analysis.

Asymmetric information model

In the asymmetric information model, agent 1 learns which actions are productive. In the early stages of the game, monitoring is imperfect (agent 1 doesn’t know for sure if he got zero payoffs because agent 2 chose an unproductive action, or if a productive action failed.) As agent 1 learns, the game transitions to perfect monitoring.

An interesting feature of the asymmetric information model is that along this path to perfect monitoring, inefficient punishment (exit by player 1) is rational and on the equilibrium path.

Consider an equilibrium (s₁, s₂) A history h^2|1 ∈ ℋ^2|1 is called a revelation stage if there is non-zero probability that a productive action that has not been taken before will be taken. A history h_t + 1¹ ∈ ℋ¹ is a confirmation stage for action a ∈ A_t iff a_t = a player 1’s payoff is positive (called a confirmation stage because agent 1 now perfectly knows that action a is productive.). In this case we say a was confirmed in this confirmation stage.

A routine is a pair of strategies that starting from a particular history h_t + 1¹ include only confirmed or unproductive actions in the continuation game.

Optimal inefficiency

A main result of the paper is that along an equilibrium equilibrium, agent 1’s strategy can include inefficient exit.

To see why consider a history h_t^2|1 where N′ < N productive actions have been confirmed.

Then denote by $\underbar{V}_2^{N’}$ the value agent 2 gets by selecting a non-productive action in every period along paths where agent 1 never exits.

Then, if at this history, $\delta (\bar{V} - \underbar{V}_2^{N’}) < c$, exit must occur with positive probability on the continuation path.

This inequality means that the cost of choosing a productive action today, is greater than the discounted difference between maximal value in the full information game and the value of always choosing unproductive actions. In other words, given that agent 1 will never exit, agent 2 has no hope to make up losses in value induced by the cost today of choosing a productive action.

To make up for this, agent 1 then sometimes must choose to exit, which prevents agent 2 from gaining the flow utility, and lowers the value of the strategy of never choosing productive actions.

In this sense inefficient exit is used as a vehicle to motivate agent 2 to reveal information.

This paper extends the Meltiz framework to allow for optimal tariff policies.

Model

This model extends a two country Melitz framework in two ways:

1. Allows for heterogeneity in variable and fixed costs. This will be a policy instrument where governments can set firm specific tariffs
2. The representative consumer in each country has a constant elasticity of substitution across all varieties procued within a country, but that elasticity can differ across countries.

Preferences

In each country there is a representative consumer.

The consumer has CES preferences over all varieties in both countries, but the CES parameter might differ across countries.

Technology

Household inelastically supply a fixed amount of aggregate labor to their own country. Firms pay households a country specific wage for each unit of labor.

There are a continuum of firms in each country, indexed by their productivity.

Each firm takes in labor and produces consumption good. Labor demand for each firm is affine in the quantity produced. The slope and intercept coefficients can be arbitrary functions of productivity.

Markets Firms sell their goods in a monopolistically competitive market.

Taxes

We will consider a cost structure where each firm can potentially face both an import tariff in the consumer’s country and an export subsidy in the firm’s country. Note that a negative tariff acts like an import subsidy and a negative export subsidy acts like an export tax.

All tax revenues are rebated lump sum to the consumer in the country where the revenues are collected.

Equilibrium

We will consider a decentralized equilibrium in which

• consumers maximize their utility subject to their budget constraint
• firms choose their output to maximize their profits taking their residual demand curves a given
• firms optimally choose to enter domestic and export markets, based on productivity and costs
• The government’s budget is balanced in each country (all tax revenues are rebated to HH)

Solution

I felt the solution technique was interesting.

The authors consider a special case of the model just described where the government in country H is strategic and the the government in country F is passive.

This means that the H government chooses tariffs and subsidies to maximize home consumer’s utility, whereas all taxes and subsidies in F are zero.

The home government’s problem is written as a planning problem where the planner chooses taxes, subsidies, prices, and quantities.

Planning problems

To solve this model the authors use a three step procedure:

1. Solve the problem of minimizing labor cost, subject to producing Q_HH units of aggregate consumption for the home consumer and Q_HF units for the foreign consumer.
   • This problem is infinite dimensional (a continuum of firms) and non-smooth (kinks because of fixed costs)
   • However, the objective and constraint are additively separable in the controls (quantities for each firm type) so the problem can be solved firm by firm and market by market using a Lagrangian approach.
   • This breaks down the difficult original problem one dimensional subproblems and an conditions on Lagrange multipliers such that the aggregate production constraint holds
   • This problem has a unique solution that is characterized in closed form
2. Solve the problem of minimizing the cost of importing one unit of aggregate goods from country F, subject to aggregate imports being Q_FH.
   • The control here is import quantities for each foreign firm.
   • The problem faces the same issues as before, but is also additively separable in the control, so it can be solved using a similar Lagrangian approach
   • The solution to this problem is also unique and characterized in closed form
3. Solve the problem of optimally choosing Q_HH, Q_HF, and Q_FH to maximize the home consumer’s utility, subject to the constrains on aggregate quantities implied by aggregating the results of the previous two micro problems
   • The solution is again unique and characterized in closed form
   • The key output of this problem is that the optimal behavior can be characterized only by manipulating the terms of trade. This will be important for implementing this solution using taxes in the next section.

Implementation

With the solution to the decentralized planner’s problem in hand, the authors then describe how to implement this equilibrium using a variety of possible tariff and subsidy schemes.

I will focus on the equilibrium with a fully flexible tax system that allows the home government to choose tariffs and subsidies for each producer/consumer pair.

The results can be summarized as follows:

• The ratio of home to home subsidy and home to home tariffs is constant for all productivity levels.
• The home to foreign subsidy is constant for all productivity levels
• The foreign to home tariff discriminates across productivity levels.
  • For all foreign producers above a certain productivity level, per unit tariffs are flat.
  • For all foreign producers below this level who still choose to export in this equilibrium tariffs are lower, but strictly increasing in the export profitability of the firm.

The set of firms who face the discriminatory tariffs are the most interesting part of this equilibrium.

The firms in this set are have export profits that are exactly equal to the fixed costs of production in the foreign to home market.

In the decentralized equilibrium, the planner found it optimal to adjust the quantity demanded from these firms relative firms for which the constraint was slack, to ensure that binding firms are willing to produce and export strictly positive amounts.

The way to implement his feature of the first-best equilibrium using taxes, is to have tariffs scale one for one with the profitability of the firms whose export profit constraint binds.

Conclusion

The two extensions of the Melitz model (heterogenous trade costs and market specific CES parameters) generated one main departure from the Melitz equilibrium: an optimal one-sided tariff and subsidy policy that discriminates against low productivity foreign exporters.

Intro

This paper takes an empirical look at a variant of the Melitz model from last week.

Model

Assumptions:

1. Domestic and foreign product markets are monopolistically competitive and segmented. In Melitz we got segmentation from CRS production.
2. Marginal costs do not respond to output shocks: domestic shocks do not impact optimal export levels
3. Firms are heterogeneous in marginal production costs and foreign demand functions (so profits differ)
4. Future exchange rate, production costs, and foreign demand shifters are unkown, but Markov
5. Firms must pay an up-front cost to enter export market in addition to per period fixed operating costs

For a firm already in the export market, export profits are characterized by log-linear (Cobb Douglass in levels) function of firm-specific characteristics, the real exchange rate, and a random disturbance.

The disturbance term is modeled as the sum of m independent AR(1) processes, which means it is equivalent to an ARMA(m, m-1) process.

The actual profit for a firm is this profit just described less fixed costs. For a firm that is already in the export market, these are a constant fixed cost. For forms that are just deciding to export, this is the same constant fixed cost plus an additional sunk cost for becoming and exporter. The sunk cost makes this model dynamic, because the return to becoming an exporter today includes the option value of being able to continue exporting without having to pay the sunk cost again.

Data

The data used in the estimation is on Columbian firms from 1981-1991 in three industries: leather products, knitted fabrics, and basic chemicals. For each industry, the data set includes plant and year specific information on total costs, domestic sales revenue, and realized export revenue. The data also includes plant-specific information such as location, size, business type.

Recall that the model was written in terms of firm profits, but that the data set includes information on revenues. The authors use the firm’s equilibrium pricing rule (constand markup over marginal cost) to create a link between revenues and profits.

The authors also use data on the real effective exchange rate over the same time horizon.

The following provides a map between model and data:

• The plant-specific demand elasticities are identified by plant-specific ratios of total revenues to total variable costs and by the fraction of revenues that come from exporting
• The revenue function parameters are identified by variation in export revenues among incumbent exporters.
• Sunk entry costs are identified by differences in exporting frequencies across plants that have comparable expected profit streams, but differ in terms of whether they exported in the previous period
• Finally, given profit streams and sunk costs, the frequency of exit among firms with positive gross profit streams identifies fixed costs

Estimation

The parameters of the model are estimated using a type II Tobit likelihood function, generalized to handle features of this model.

Features requiring generalizations are

• Serially correlated disturbances
• endogenous initial export conditions
• incidental parameter (demand elasticities).

The likelihood is not globally concave, so point estimates might be difficult to obtain. So, the authors employ a Metropolis Hastings algorithm and characterize the posterior distribution of the model’s parameters.

Priors

As they employ MCMC methods, the authors need to specify prior distributions for all model parameters. The authors assume that all parameters are independent, thus the prior is the product of univariate priors for each parameter. They choose very diffuse priors for all model parameters except foreign demand elasticities, where they ensure that prior and posterior have mean bounded above by unity.

Results

Some basic results in line with conventional thinking:

• Firms with large domestic revenues typically stand to profit more from exporting
• The elasticity of profits with respect to the real exchange rate is very high
• There is evidence for strong serial correlation in profit shocks
• Larger producers face lower average sunk cost to entering export market.

Option value

What I think is the most novel finding

The probability of remaining in the export market roughly 85% on average, with slight variation across industries.

Because the authors estimated a dynamic model, they are able to measure the option value of entering in one period and not having to re-pay sunk costs in the next.

They find that the option value is the largest component of export value for most producers

This means at least two things:

1. Modeling implication: Considering the dynamic components of the export entry decision are crucial to get Melitz flavor models to match the data.
2. Policy implications: Changes in option values (perhaps due to changes in expectations about future market conditions) can induce large changes in exporter decisions, even when short run profits are unchanged.

The second finding suggests that policies that wish to induce changes in export market participation should aim to adjust the expectation about future market conditions (e.g. expected changes in real interest rates) or directly subsidize the export fixed costs (sunk and/or fixed).

Overview

A primarily empirical paper that studies how allowing exporting firms to learn about demand for their product impacts firm dynamics in a Melitz style trade model.

Data and stylized facts

Let’s start with the data.

The data set used in this paper describes all imports by US buyers from Colombian exporters from 1992-2009. The source is the U.S. Census Bureau’s Longitudinal Foreign Trade Transactions Database (LFTTD).

Using this data, the authors report the following stylized facts:

• About 80% of Colombian exporter/US importer matches are 1-1, meaning the Colombian firm has no other US buyers and the US firm only buys from a single Colombian firm.
• About 70% of matches dissolve after one period. There are several additional features behind this fact:
  • If a match does survive the into a second period, there is a significantly higher chance it will survive into the third (about 50%). This trend continues as the duration of the match increases.
  • The survival probability of a new match increases with first year sales volume.
  • These facts suggest that there are many Colombian firms who merely “test the US waters” and export a small volume, find the conditions unfavorable, and exit. This might suggest these firms are trying to learn about foreign demand.

Model

Now let’s turn to the theoretical model. The model is set in continuous time and will feature Columbian firms who sell domestically and to US importers.

Firm pricing problem

Each Colombian firm competes monopolistically and seeks to maximize total profits. The details of the pricing problem solved by the firms are not given, but rather the authors jump to the Melitz result: prices are set to a constant markup over marginal costs. In this setting the profits of a firm can be written as a multiplicative function of three components:

1. A macroeconomic state that is common to all firms
2. Firm productivity
3. A time-varying match specific demand shifter

Relationships

At each instant in time the seller maintains relationships with an endogenous number of buyers. These relationships are the product of a search process that will be outlined shortly. The matches dissolve exogenously at a constant rate δ or if the seller no longer finds it profitable to pay a constant fixed cost F to maintain the relationship.

If the match is kept alive, it may be hit by one of three events:

1. With a constant hazard λ^b the buyer places another order
2. With a hazard q_xx′^X the macro state will jump to a new level
3. With a hazard q_yy′^Y the match specific demand shifter will jump to a new level

As the shocks to macro and idiosyncratic states are Markov, the stopping time characterizing when one event occurs is distributed exponentially. This allows the authors to write the firms expected profit recursively and obtain expectations in closed form.

Learning

Sellers conduct search in domestic and foreign markets. When searching in a particular market m, sellers understand that a constant fraction θ^m of potential buys will form a match.

Each seller draws θ^h and θ^f before meeting with any clients and the draws remain fixed over time. All draws of θ come independently from a beta distribution.

Sellers are assumed to have met with many domestic firms and know θ^h. They will; however, have to learn about θ^f by meeting with many firms.

For a given level of θ^f, the probability that a random sample of n potential foreign buyers will result in a matches is distributed binomally.

The beta and binomial distributions form a conjugate pair, so after n and a are realized, the posterior for θ^f is known in closed form.

Search

The final component of the model is the search problem. Each seller will continuously choose a market specific hazard rate s^m with which they meet with a potential buyer. At the moment of the meeting, the firm pays a flow cost of c(s^m, a), where a is still the number of successful previous matches.

The search problem for a firm with a specific productivity and matching efficiency can be written recursively with states being a, n and the macro state x. The expectations in this recursive formulation are again over the stopping time of seller events, which means they can also be written in closed form.

The optimal search hazard rate for each firm in both markets can be characterized in closed form.

Empirical Model

To implement this model empirically, the authors impose the following additional structure:

• A specific form for the search cost function
• Distributions for all exogenous states (seller productivity, and macro/micro states) are all assumed.
• The data is uninformative regarding the rate of time discount for sellers and the demand elasticity, so values for these parameters are directly assumed.

The distribution of the states is a set of independent Ehrenfest diffusion processes. This process is special because it converges to the well known Ornstein-Uhlenbeck process as the spacing between grid points in the state space converges to zero

Estimation

Estimation is broken into 2 stages:

Estimating the jump process for macro states

They proxy x in each market as real expenditures on manufacturing goods in that country. Their distributional assumption for the macro states allow them summarize the distribution in terms of a jump size and hazard rate.

Indirect inference

The rest of the estimation happens via indirect inference on the closed form results form the theoretical model.

The parameters are estimated to minimize distance between moments generated by the model and corresponding moments in the data. The data features that are targeted include

• Distribution of home and foreign sales
• Distribution of clients across exporters
• Sales per client, conditioned on the number of clients
• An autoregression of log domestic sales
• Transition probabilities in the number of clients
• Exporter exit hazard by duration in foreign market
• Match death probabilities and match sales

Basic results

There are many parameters, so I won’t enumerate how well the model does at targeting each parameter. The overall results are that the estimates match the signs of all empirical moments and the magnitude of most.

Some interpretation of the coefficients is helpful:

• Mature (5+ years) matches fail with a probability of 40%, but the estimated exogenous separation rate is only 27%. This suggests that the idiosyncratic demand shift shocks impact long-run match survival.
• On average, the success rate of matches is 0.18. However, the standard deviation of this estimate is 0.176, suggesting that there is significant room for learning.
• Network effects are very strong. This is summarized by the search cost falling sharply as the number of successful matches rises.
• Finally, the estimation also suggests that the network effects are strongest for sellers getting their first successful match (suggests that search costs are concave in the number of matches).

Experiments

The authors then do a few experiments to further probe the model.

No-learning model

They first consider a no learning model where firms are assumed to know both θ^h and θ^f with certainty at the onset of the model.

The main impact of this model restriction is:

• The exit of inexperienced exports is lower. This is driven both by having low θ^f sellers not try to learn about foreign demand and by no learning based exit.
• To overcome this and still match seller exit rates, exogenous separation probability doubles from 0.26 to 0.52.

This paper considers the relation between of firm size heterogeneity and income dispersion. Using two large panel data sets, they document a number of empirical facts about the link between these forms of heterogeneity, then build a model combining non-homothetic preferences for consumers and a Metliz’ style production environment that can match the facts.

Data

Kilts Nielsen consumer panel and retail scanner data..

• weekly upc code level data from households and firms
• contains prices and quantities
• also demographic info about households: discrete income binning, location
• Metadata about stores: brands they sell, location

Empirical finding

Main empirical finding is that wealthier households spend a higher share of consumption expenditure on goods from large firms. The relationship is monotonic across the income distribution. Finding is robust across level of aggregation (product level, product module level, aggregate spending, etc.) and time frame (6 month windows or across entire sample)

Model

TODO

Equilibrium

TODO

counter factuals

SEE NOTES IN PDF

This paper builds on the approximate linear programming work of Farias and van Roy that I presented a few weeks ago and applies a version of that technique to a dynamic oligopoly model.

The actual model is not novel to this research, so I will spend most of my time talking about the algorithm.

Model

• The model is set in discrete time and multiple firms compete in a single good market over an infinite horizon.
• Firms are identified by a firm specific state x that takes on an integer value between 0 and some upper limit xbar
• The aggregate state s is a histogram of the number of firms at each individual state (a vector of xbar+1 integers)
• The maximum number of incumbent firms is fixed at N. Each period there are N − sum(s) possible entrants. Entrants do not produce or earn profits in the first period.
• Incumbents choose an investment level that determines the probability of remaining in the same state, or moving up or down one step to a neighboring state in the next period.
• Each period the following events occur in this order:
  1. Incumbent firms draw a random sell-off cost and decide if they want to exit. If they stay, they make investment decisions.
  2. Each potential entrant draws a random entry cost and makes entry decision
  3. Incumbent firms compete in spot market and receive prices
  4. Exiting firms exit and receive sell off values
  5. Shocks are realized, each firm that stays transitions to a new state.

The equilibrium concept studied in this paper is a symmetric Markov perfect equilibrium. In this context a MPE is an investment/exit strategy for incumbents and an entry strategy for potential entrants such that:

• given that all other incumbents follow the exit/investment strategy, each incumbent does not want to deviate from that strategy
• For all states with a positive number of entrants, the cutoff entry value is equal to the expected discounted value of profits of entering the industry

Computation

Naive algorithm

To understand the contribution of this paper, it is helpful to have a basic understanding of a naive “brute force” algorithm.

The naive algorithm presented here is iterating on a best response operator and proceeds as follows:

• Choose some initial investment, exit, and entry policy
• Repeat the following until the policies are close enough:
  • Taking that N-1 players use the current guess for the policy, compute a best response for one agent. To do this you can apply standard dynamic programming algorithms
  • Compute some notion of a norm between the best response and the current guess for the policy
  • Set the current guess equal to the best response and continue if needed

This algorithm is robust, but has one major drawback: the curse of dimensionality. For in a model with 20 incumbents and 20 individual states, there are over a thousand billion states that must be iterated over when computing the best response.

Approximate dynamic programming algorithm

The authors of this paper make 4 modifications to the naive algorithm:

1. They form the linear program version of the dynamic programming problem to solve for the best response
2. They apply approximate linear programming approach of de Farias and van Roy to reduce the number of variables the solver must find
3. They use constraint sampling to only enforce a subset of the constraints of the linear program
4. They choose basis functions that are especially well suited to their problem

Results

The authors do a few numerical experiments. The experiments all revolve around solving for the MPE exactly or approximately and then computing implied long run aggregates, such as average producer and consumer surplus, average share of ith largest firm, and average investment.

Exercise 1: small model, exact comparison

For a relatively small version of the model (number of incumbents low), they are able to apply the naive algorithm and compute the true MPE. They also solve the model using their proposed algorithm and show that the aggregates are always within 8% of the true values, typically within 2%.

Runtime for their algorithm is on the order of minutes in each case. With a max of 3 incumbents, runtime for the naive algorithm is on the order of seconds. When the max number of incumbents is 5, the naive algorithm takes a few hours.

Exercise 2: large model, compare OE

They also run a similar experiment for a much larger model where they compare a current state of the art algorithm to their proposed algorithm.

They find that the aggregates are always within 13% of one another, but often within 5%.

As the state of the art is also an approximation, this doesn’t say much more than that both algorithms appear to approximate a similar thing.

They comment that the runtime for a version of the model with 20 incumbents is a couple of hours. Unfortunately they don’t elaborate on the difference in runtime between the algorithms.

Intro

Goal: Characterize the main conceptual composition of risk and shock elasticities and provide an example of their interpretation in a macro model.

Outline:

• General continuous time framework
  • Stochastic process for state
  • Growth processes and cash flows
  • Perturbations
  • Elasticities
• Simple example
  • Do the BL model. Can talk about choosing α_d = e_i for i = 1, 2, 3.

Mathematical Framework

I will briefly explain the mathematical framework used in this and related papers.

State

State is Markov diffusion: dX_t = μ(X_t)dt + σ(X_t)dW_t (W is standard BM, μ and σ may be non-linear.)

Functionals

Additive functionals: parameterized by state dependent drift β(X) and volatility α(X) coefficients integrated over time (A_t = ∫₀^tβ(X_u)du + ∫₀^tα(X_u) ⋅ dW_u)

Multiplicative functionals are also characterized by (β, α) and are the exponential of an underlying additive functional (M = exp (A))

Growth processes of economic interest, e.g. cash flows and consumption, will be modeled as multiplicative functionals.

Perturbations

We will perturb the multiplicative functionals in a specific way that allow us to capture the marginal value of exposure risk in a particular direction.

Similar in spirit to a risk premium, which measures the average value of risk exposure (not marginal or isolated to a specific direction.)

How to construct Perturbations

NOTE: only talk about this if I have cruised through the above. I want to be at this point after 2-3 minutes.

We construct perturbations of a multiplicative functional M as the product of M and another multiplicative functional with precisely chosen drift and volatility coefficients. The perturbation coefficients are functions of a scalar parameter r. When r = 0 the perturbation is equal to 1 (i.e. perturbed cash flow is just the cash flow).

Often the drift is chosen so the perturbation itself is a martingale, or the perturbed cash flow is a martingale. This simplifies computation.

The volatility coefficient can be varied to isolate the impact of innovations to a particular dimension of the BM. An example of this will be shown later.

Risk elasticities

A risk elasticity for a cash flow M is defined as the derivative of the log expectation of the perturbed cash flow evaluated at r = 0.

The direct representation of the risk elasticity is additive over a particular time horizon t. To consider contributions that are localized in time, we build an integral representation.

This integral representation expresses the risk elasticity from a given initial state x over a time horizon t as the expectation (conditional on x), of the cash flow in time t times the integral of a function.

The function that is integrated over is called a shock elasticity function and captures the instantaneous contribution of the perturbation to the risk elasticity (i.e. shock elasticities are the building blocks for risk elasticities).

Example

I will now highlight how these theoretical building blocks can be used within a macro model.

Model

The state is 2 dimensional: x₁ represents consumption growth, and x₂ represents stochastic volatility.

The Brownian motion will be three dimensional, where dimensions correspond to independent shocks directly to consumption itself, consumption growth x₁, and stochastic volatility x₂

Multiplicative functionals take a specific form:

• The drift is affine in deviations of x₁ and x₂ from their mean values. The intercept and coefficients on deviations distinguish multiplicative functionals (β(x) = β̄₀ + β̄_t ⋅ (x₁ − ι₁) + β̄₂(x_t − ι₂))
• The volatility is the square root of x₂ times a constant vector $(\alpha(x) = \sqrt{x_2} \bar{\alpha}$).

Consumption growth is modeled directly as a multiplicative functional.

Preferences

Agents have either CRRA preferences or Epstein-Zin preferences over streams of consumption.

Part of the solution to the agent’s problem is a stochastic discount factor used to price cash flows in the economy.

The cash flow of interest will be the growth of the consumption process (C_t/C₀).

Elasticities

We consider two types of fundamental elasticities, as well as a third that is a function of the former two. These types are

1. Exposure elasticities, where the multiplicative function is consumption
2. Value elasticities, where the MF is the price of consumption (consumption times SDF)

We will also consider a price elasticity, which is the difference between the exposure elasticity and the value elasticity. The price elasticities are the functions that characterize the marginal value of exposure risk in a particular direction that we mentioned earlier.

We will allow the volatility coefficient of the perturbation to be each of the vectors [100]′, [010]′, or [001]′. This allows us to construct price elasticities in response to shocks to consumption, consumption growth, and stochastic volatility. This is the sense in which price elasticities characterize the exposure risk in a particular direction.

Results

The exposure elasticities for the two sets of preferences are the same. Remember the exposure elasticity is the elasticity for the cash flow, which is the same underlying consumption process in both models.

The Shock exposure elasticities (local increments) to each of the three dimensions of W are reported.

• Shock to consumption has permanent effect on consumption and the shock elasticity is constant over time
• The shock exposure elasticity to a positive shock in the growth rate increases over time as the impact of a higher growth rate compounds
• The volatility shock exposure elasticity declines over time as the volatility process mean reverts and the impact of the initial shock wears off.

Given the form of agent’s SDF, the shock value elasticities are scaled versions of the shock exposure elasticities.

References

This paper uses Compustat data to document that the aggregate capital share of income and average capital share of income have diverged over the last 20 years. Specifically, the average capital share has fallen while the aggregate share has risen.

They then write down a model that uses changes in firm level volatility of productivity to generate similar dynamics in general equilibrium.

The model is interesting in its own right. It features a 2 sided limited commitment contracting problem. The agent has all bargaining power ex ante, but after the principal faces idiosyncratic shocks he ends up with capturing almost all aggregate rents ex post.

This is related to a new literature on carefully modeling large firms in order to understand aggregates.

Model

The model is set in continuous time.

Unit measure of ex ante identical firms. Each firm has an idiosyncratic time-varying productivity. Firms are owned by risk neutral investors (or shareholders) and operated by skilled managers. Firms rent physical capital and employ unskilled labor to produce using a decreasing returns to scale production technology.

Firm productivity evolves as a geometric brownian motion that is subject to a negative poisson shock. If productivity reaches a strictly positive minimum value (determined in equilibrium), or if poisson process jumps, the firm exits immediately. There is a competitive fringe of potential entrants. When an investor creates a new firm they pay a fixed cost P. Each period firms enter until the expected value of entering is equal to P (free entry). After entering the initial productivity level is drawn from a Pareto distribution.

When a firm is formed, but before initial productivity is realized, the investor is matched with a risk adverse manager and the investor offers a long term contract. The contract is a sequence of payments to be made to the manager at each instant. The manager has convex preferences over this sequence of payments and discounts the future at a constant rate. The manager can choose to accept or reject the proposed contract. Upon rejection the manager is immediately matched with a new potential entrant.

Both parties are free to terminate the contract at any time. If the manager walks away from the firm, he will receive some outside option that is determined in equilibrium. The investor will choose to continue operations as long there is a positive net present value. The investor faces an optimal stopping time problem that characterizes when the firm is abandoned. The abandonment problem is to maximize the present discounted value of profits less managerial payments from the current period until the random stopping time, subject to the constraint that the manager doesn’t walk away. The constraint sets the NPV of managerial payments plus post-stopping value equal to the outside option. This constraint is always binding, so WLOG the authors restrict attention to contracts that gives the manager a fixed payment until the firm exits.

Equilibrium

The authors use this model to describe how to match the facts about the capital share of income. The main mechanism is that an increase in the volatility parameter in the productivity process will do two main things:

1. raise the real option value of waiting to abandon for firms with low productivity. This results in a decrease in the lower productivity threshold at which firms exit
2. Increase the number of firms with very high productivity.

In other words, higher volatility in innovations to productivity creates fat tails in the distribution of productivity across firms.

The average capital share of income is computed as the integral of profits less managerial payments divided by profits over all firms.

The aggregate capital share of income is computed as the integral of profits less managerial payments over all firms divided by the integral of profits over all firms.

The fat right tail will put more mass on the extremely productive firms who are earning profits far above the managerial payments. This drives up the aggregate capital share of income.

On the other hand, the additional mass of low productivity firms will bring the average capital share of income down.

Model

This is two-type agent, continuous time model of financial intermediaries. He and Krishnamurthy (2013)

Agents are either households or specialists.

Distinction will be made clear once we understand the market structure

Market structure

• One risk-free short run bond in zero net supply
  • All agents can freely buy this asset
  • Specialists can also short
• One risky asset whose dividends evolve as GBM
  • Only specialists can purchase this asset, however they can do so in behalf of households via an intermediary mechanism to be explained shortly
  • Net supply normalized to 1

Technology (how markets work)

Each period between t and t + dt is split into 5 mini-periods:

1. At t each specialist is randomly matched with a household to form and intermediary
2. Specialists allocate all w_t of their wealth to buy equity in intermediary. Households allocate a part, H_t, of their wealth to purchase equity
3. Specialists take w_t + H_t and allocates all of it between the risk free bond and risky. There are not restrictions in buying or shorting either asset
4. Returns are realized and distributed according to equity shares. Agents consume out of net wealth
5. At t + dt the match is broken and new one is formed

Note about choice of matching structure:

• Alternative would be standard Walrasian market.
  • Did this in 2012 paper.
  • Found with this market structure the specialists charge fees that rise in financial crisis – this is counterfactual.

Agents

• Households:
  • An overlapping generation of agents.
  • At each time t a unit mass of time t agents is born with wealth w_t^h (evenly distributed based on end of period wealth of previous generation – means they don’t need to track the wealth distribution)
  • They live between period t and t + dt, during which time they receive labor income that is constant fraction of risky asset dividends. (NOTE: without this it is possible to arrive in state where HH sector vanishes)
  • Authors assume a fraction λ can only invest in risk-free asset. This generates the + side of the zero net supply condition, making it possible for other agents to take a levered position in risky asset. HH cannot short the bond
  • Choose period t consumption and asset positions to maximize convex combination of log of consumption and expectation of log of continuation wealth (utility and bequest motive both log form)
• Unit mass of identical specialists. Each one:
  • Infinitely lived.
  • Operates a single intermediary (represent decision makers of bank, hedge fund, extc.)
  • Chooses sequence of consumption and portfolio shares in risky asset (acting as intermediary) to maximize the expected present discounted value of a CRRA utility function of consumption subject to…
  • Budget constraint: dw_t =  − c_tdt + w_tr_tdt + w_t(dR_t(α_t^I)−r_tdt), where α_t^I is intermediary share in risky asset and dR_t(α_t^I) is the associated return

Equilibrium outcomes

In this section we discuss the key outcomes in the paper.

Friction

Model contains one key financial friction that drives most results:

• Households are willing to invest no more than a constant fraction m of specialist wealth as equity in the intermediary: H_t ≤ mw_t for some constant m > 0
• Constrains intermediary’s ability to raise outside equity financing.
• Interpretation:
  • Managers usually have significant wealth tied in their own funds (aligns incentives)
• Equilibrium effect:
  • Effectively creates a boundary x^c such that if specialist wealth relative to value (price) of risky asset x = w/P falls below x_c, they are constrained in how much equity they can raise.
    • In constrained region we have H_t = mw_t
    • If unconstrained H_t < mx_t and value of m doesn’t impact decisions
  • Adds leverage effect such that when specialist is constrained he can’t raise enough capital in intermediary via equity, therefore must take levered position by shorting risk free asset and holding a very large position in risky asset (see figure 2)

The main asset pricing impact of this friction can be seen in how the risk premium (return on expected risky asset less risk free return) changes as a function of specialist wealth. Before getting there we need to understand three details

Equity premium

Risk aversion

First, note that specialist CRRA parameter γ is calibrated to be greater than 1 (i.e. specialists are more risk adverse than households). This causes all households who can to invest all their wealth in risky equity

Intuitive reasoning:

• Recall market clearing: zero net supply risk free asset and positive net supply risky asset ⇒ intermediary always holds more than 100% of wealth in risky asset
• Household is less risk adverse, so they would like to hold more risky asset than specialist.
• To hold more than specialist, who already holds more than 100%, household would have to be able to short bond, but they can’t. So they get as close as possible by spending all their wealth to purchase risky equity.

Risky asset position vs. specialist wealth

Second, the relationship between intermediary position in risky asset and specialist wealth is very asymmetric.

Let αⁱ = risky asset holdings/total assets. Then the relationship beween α^I and w_t can be decomposed in two parts:

• In unconstrained region (specialist wealth relatively high), α^I nearly independent of w_t
• In constrained region; strong, non-linear inverse relationship between α^I and w_t

This is driven by the leverage effect outlined above: a binding constraint forces specialists to short bond - ratcheting up αⁱ.

Risk premium vs. specialist wealth

Finally, the equilibrium risk premium is increasing in αⁱ.

Combining these points we see how the model delivers a state dependent risk premium. This result is different from similar results obtained in the literature on two fronts:

1. This model has “standard” CRRA utility. Others modify the utility function to introduce state dependence (Campbell & Cochrane (1999), Barberis, Huang, and Santos (2001))
2. The relationship between capital (wealth) and the risk premium is very asymmetric: might provide important window for studying crises

Dynamics

Wealth distribution is mean reverting. To see it notice what happens in the two tails:

• Specialist wealth very low -> risk premium high -> specialist wealth expected to increase -> wealth distribution mean reverts
• • Household wealth very low -> risk premium low -> aggregate household return similar to aggregate specialist return -> extra labor income for HH pushes their wealth up -> wealth distribution mean reverts
• That the equity premium is very high when intermediary leverage is high means that specialist wealth exhibits strong mean aversion.
• When household wealth is relatively low, so is the risk premium. Thus, as household’s save their labor income, their wealth is expected to grow, causing the wealth distribution to mean revert from the other side

References

The goal of this paper is to test the degree of risk sharing across regions and industries in the US.

The rough outline is to write down a simple model they use to motivate an empirical specification, then test if that specification holds in the data.

Model

There are H households that each live in one of R regions and work within one of I industries. Each period, households receive an endowment of the consumption good and are able to trade in complete state contingent markets. Households have time-separable CRRA preferences that are subject to multiplicative preference shocks.

The authors formulate the social planner’s problem for the entire economy. The first order condition for each household equates the Lagrange multiplier on the resource constraint to the Pareto weight times marginal utility of consumption.

They then take logs, sum across all households, and take first differences to obtain a key equation that says household log consumption growth is equal to aggregate log consumption growth plus the difference in deviations of individual preference shocks from the average preference shock times a scale factor.

There are two key things to understand about this relationship:

1. There is perfect within period risk sharing across households as their consumption is independent of their endowment.
2. There is also intertemporal consumption smoothing because the innovation to permanent income is the same across households because it is linked to aggregate consumption growth. When solving the planner’s problem it is difficult to write down the permanent income process, but that’s how it works.

They also consider market structures where there is perfect risk sharing only within a region or within an industry, but not across these groups. They solve for allocations in the within-group risk sharing economy using a social planner problem, just like before. The only difference is that the new planners only consider the agents in one group.

The optimality conditions are identical and that important equation relates individual consumption growth to group-wise aggregate consumption growth and preference shocks is the same. If you then aggregate these within-group conditions across the groups you arrive at the same condition we got at the end of the economy-wide planner’s problem, with the addition of one extra term. This term is a risk adjusted difference in group and economy Lagrange multipliers.

In order to derive the equation they use as the main specification for the empirical exercises, they need to remove these unobservable Lagrange multipliers. To do this they return one more time to optimality conditions for the different planners and manipulate them to write the difference in Lagrange multipliers as the difference in group and economy level consumption growth.

Empirics

The final equation they end up taking to the data is that individual log consumption growth is equal to the sum of aggregate log consumption growth, the difference between regional and aggregate consumption growth, the difference between industry and aggregate consumption growth, controls for changes in preferences, and individual income growth.

They use data from the PSID to run regressions on this model. The theory suggests that if there is complete risk sharing across regions and industries, that the coefficients on those terms should be zero. They find that these coefficients are significantly different from zero, suggesting that a large fraction of wealth is incompletely shared across regions and industries.

This paper presents a stochastic simulation method for solving dynamic economic models.

The ideas in this paper lean on a literature sometimes known as approximate dynamic programming and can enable us to solve models with many state variables and non-convexities in objectives and constraints.

My intent is to summarize the core theoretical ideas behind the algorithm.

Main idea: Post decision states

Notation: classic Bellman equation

Consider a stationary economic model where at time t the state is summarized by a vector s_t of endogenous state variables and a vector x_t of exogenous state variables.

The optimization problem of an agent is often summarized by a Bellman equation of the form

V(s_t, x_t) = max_ctu(c_t) + βE[V(s_t + 1, x_t + 1)]

subject to

s_t + 1 = f(s_t, x_t, c_t), c_t ∈ Γ(s_t, x_t), x_t + 1 = g(x_t, ϵ_t + 1)

Post-decision state value function

Note that that the transition function for the endogenous state takes s_t, x_t, and c_t and emits s_t + 1. We can think of s_t + 1 being chosen at the end of period t, meaning after the controls c_t have been decided.

In the standard (or pre-decision) Bellman equation, we have that the state at time t is the endogenous state defined at the end of period t − 1 and the exogenous state realized at the start of period t.

We will now consider a different representation of the state vector that couples the endogenous state defined at the end of period t − 1 with the exogenous state realized at the start of period t − 1. That is we will consider s_t and x_t − 1, which is known as the post-decision state at time t − 1.

Let V^x(s_t, x_t − 1) be the value of having post-decision state s_t, x_t − 1 in period t − 1. This is the maximum expected, discounted utility an agent can achieve after controls have been selected in period t − 1.

Because c_t − 1 is not chosen until after x_t − 1 is realized, we know that V^x(s_t, x_t − 1) is equal to the expectation of the maximum expected, discounted utility the agent will receive after x_t arrives. That is, we can write

V^x(s_t, x_t − 1) = E[V(s_t, x_t)|s_t, x_t − 1],

where V(s_t, x_t) is the pre-decision Bellman equation.

It follows that we can write

V(s_t, x_t) = max_ct + βV^x(s_t + 1, x_t).

These equations can be manipulated to produce the recursive form of the post-decision state Bellman equation:

V^x(s_t, x_t − 1) = E[max_ctu(c_t) + βV^x(s_t + 1, x_t)|s_t, x_t − 1].

Notice that the expectation is outside the max operator, meaning that the maximization problem is deterministic.

Algorithm

Now that we have the post-decision state Bellman equation, the algorithm is fairly straightforward. I will present the algorithm from the paper in the context of Markov exogenous processes, but I believe it is incorrectly specified. I’ll discuss how I’d change it later.

1. Setup
   • Discretize endogenous state space
   • Choose a simulation length T
   • Choose initial endogenous and exogenous states
   • Construct an initial guess for the value function at the discritized endogenous and exogenous states.
2. Iterations
   • Construct a time series of Exogenous states for t=1, 2, …, T
   • For time t = 1, 2, …, T perform the following 3 steps:
     1. Choose controls c_t to maximize the term inside the expectation on the RHS of V(s_t, xt − 1). To do this we need to using the value function from the previous iteration for the future value function
     2. Compute the expectation implicitly by updating the guess of the value function using a convex combination of the previous iteration’s value function and the value computed above
     3. Using the chosen controls and realization of exogenous state, apply the endogenous transition equation to iterate the endogenous state forward one period
3. Convergence:
   • Check a convergence criterion that compares the discretized value function across multiple iterations.
   • If converged, return the discretized value function and run a regression on the time series to obtain a policy function from the time series of controls

Comments on the algorithm

Here are a few comments about the algorithm:

• Because the expectation operator is outside the max operator, we don’t have to spend time computing expectations when solving the optimization problem in each period of the simulation. This speeds up computation quite a bit.
• Expectations are computed implicitly when we update our guess for the post-decision state value function

My compliant I think it is incorrect to re-generate a time series of exogenous states on each iteration. Doing so will not allow the algorithm to ever converge as the updated value function in iteration n is dependent on the randomness from the exogenous simulation in period n.

Using the same simulated time series for exogenous states in every iteration (as in other simulation algorithms in the literature) will allow this algorithm to converge; subject to a particular exogenous path. To ensure that the solution is accurate for the underlying data generating process and not just the simulation you use, make sure that the length of the time series is large.

This paper presents and analyzes a stylized model of poverty traps in developing economies.

Baseline model

In the baseline model there are a finite number of agents.

Agent’s are characterized by a constant level of innate ability α and an evolving stock of capital k.

Each period, agents choose which of two DRS production technologies they wish to operate in that period. Switching technologies is costless. Both technologies have innate ability multiplied by physical capital raised to a power less than one. The more productive technology has a higher exponent, but requires payment of a fixed cost to operate.

Agent’s have CRRA preferences over consumption.

Agents choose consumption and technology each period to maximize the expected discounted value of lifetime utility from consumption, subject to a few constraints:

• Consumption can be no more than capital stock plus production output
• The capital stock is always non-negative
• Next period capital stock is the unconsumed portion of resources multiplied by an asset shock less depreciation. When this shock is less than unity, some capital is destroyed. The shocks are drawn iid from a known distribution.

Model Solution

The dynamics induced by agents’ optimal behavior can be understood by considering two interacting effects.

First, the choice of production technology is governed by a cutoff rule for capital as a function of innate ability. If an agent’s capital stock is above this threshold, they will attempt to accumulate capital so that they can employ the high productivity technology. Otherwise the agent will only pursue the low technology and accumulate the relatively smaller stock of capital needed to operate that technology optimally.

Second, the state space can be partitioned into three regions along the innate ability dimension

1. The unskilled worker region, where regardless of the amount of capital agents always find it optimal to use the low productivity technology. For each innate ability level, there is a unique optimal capital stock to target. Poverty is defined as having capital stock equal to this level.
2. A high skill region where for all levels of capital agents prefer to operate the high productivity technology.
3. An intermediate region where depending on the current capital stock and sequence of asset shocks, agents may choose to operate either technology.

These regions define two forms of poverty trap:

1. Low skill agents who will always be in poverty
2. Middle skill agents who are vulnerable to being pushed into poverty if they receive sufficiently unfavorable asset shocks.

Policy

The authors use this framework to analyze a few competing forms of government intervention. They analyze the effectiveness of each policy using a simulation experiment. In each experiment they randomly initialize 300 agents to be 25% in the low and high skill region and 50% in the intermediate region. The capital stock is initialized by independent draws from uniform distribution on [0, 10]. They they simulate the model forward under a given policy for 50 periods and track the distribution of agents.

As a baseline, we first consider autarky, or no government transfer program. The results of the simulation are a clear increase in the poverty level relative to the initial conditions. At the start, about 60% of the population chose to operate the high productivity technology. At then end of the simulation this has dropped to 40%.

The first policy considered is a progressive policy that targets the poorest agents in the population. All agents below the poverty level are given a transfer that brings them exactly to the poverty line. If there are insufficient government funds, each agent is given a share of total government resources proportional to their distance from the poverty line. Agents do not anticipate the transfers. The results of simulating in this environment are qualitatively identical to autarky.

The second policy targets the middle skill agents near the cutoff rule for switching between production technologies. Specifically, if an agent starts the period in the high technology region and recipes a poor enough asset shock to move to the low region – the government provides a transfer that brings the agent exactly back to the cutoff level. Again the transfer is unanticipated. The simulation results here are quite different. Aggregate output rises by 10% and poverty falls from 55% to 25% – meaning 75% of agents operate the high productivity technology.

The difference between these two policies brings up an ethical issue regarding which subset of agents government policies should target. The authors provide some discussion, but leave it as an open ended question.

The final experiment also targets the middle skill agents near the cutoff, but this time the transfers are anticipated. The anticipation brings about two competing moral hazard forces:

1. The positive force is that when agents know they will receive a transfer if their investment is subject to poor shocks, they choose to invest more.
2. The negative force is that agents would like to remain as close to the cutoff as possible so they can get the transfer more often.

The results of the simulation under these conditions falls between the two previous examples. The authors don’t report numbers, so I can’t be precise. However, from the figures the main takeaway is that agents in the middle skill region who are close to the production cutoff are now much less likely to end up operating the low productivity technology in the long run.

Outline

The authors of this paper do 3 main things:

1. Describe a particular class of Dynamic economic games
2. Describe two algorithms for computing Markov perfect equilibria of those games
3. Give two detailed, non-trivial examples of how to apply the algorithms to games in this class

We won’t have time to touch on the examples, but my goal today is to describe the class of models and explain the key components of the algorithms.

Dynamic Directional Games

The class of dynamic games considered in this paper have the following characteristics

• We consider a dynamic stochastic game G with n players and T periods. T might be ∞.
• Payoffs for each player are given by time separable von-Neumann Morgenstern utility functions.
  • Time t payoffs depend on the state s in time t as well as the vector of actions a_t chosen by all players
• The state for time t + 1 is Markov, conditional on time t state and actions
• We restrict the state space to be finite
• Players perfectly observe the current state as well as past actions
• For each player the discount factor, set of feasible actions given the state, and the Markov transition probabilities are common knowledge
• We will consider Markov perfect equilibria over behavior strategies

The authors further refine this class of games by introducing the concept of directionality in one or more state variables. Specifically, they partition the state vector into two components: D and X. The set D contains the elements of S that can be represented by a directed acyclic graph. X contains all other states.

What does this mean? The set D can be further divided into subgroups and ordered such that once the state moves from subgroup i to subgroup j > i, there is 0 probability of ever returning to subgroup i.

Algorithm 1: State Recursion Algorithm

How does directionality help? It allows them to define the first main algorithm in the paper: the state recursion algorithm.

Suppose that the set D has been divided into M groups, ordered such that the model begins in group 1 and terminates in group M. Then the state recursion algorithm roughly proceeds as follows:

• Starting with subgroup M, iterating down to subgroup 1
  • For each state d ∈ D_M
    • Compute all MPE of the subgame starting from s = d × x, ∀x ∈ X, taking as given optimal strategies already computed in subgroups j > i
    • If multiple such equilibria exist, use some deterministic equilibrium selection rule to choose a unique MPE
• When the recursion terminates, the selected equilibria from each subgame are joined to form the MPE of the original game.

To understand how the State Recursion Algorithm works, consider two simple examples:

1. Let the only directional state variable be time. Then SRA is equivalent to familiar backwards time recursion:
   • start in the terminal period, solve the model for each state in that period
   • Step backwards once in time and for each state in time T − 1, compute the equilibrium, taking as given the optimal cap T strategy.
2. A stylized version of Rubenstein’s 2 bargaining over a pie problem.
   • Suppose that 2 agents are bargaining over how to split a single perfectly divisible pie.
   • Further assume that the size of the pie evolves according to a 4 state Markov chain, where states are ordered in decreasing size of the pie (i.e. state 1 is the full pie) with an upper triangular transition matrix. Specifically, suppose that from state 1, there is non-zero probability of remaining in state 1, or moving to either state 2 or state 3. From state 2, you can remain in state 2 or move to state 4. From state 3 you can stay or move to 4. 4 is absorbing.
   • The state is partitioned into 3 subgroups [(1,), (2,3), (4,)]
   • After moving from 1 to 2 or 3, there is zero probability of ever moving back to 1. Similarly, once the state goes to 4, it has zero probability of ever returning to 1, 2, or 3

From the second example we can see how SRA is a generalization of backwards time recursion: the state can include variables other than time and have a stochastic law of motion between groups.

The SRA algorithm makes 2 assumptions:

1. That we know how to solve for an MPE starting from each element in D
2. That we have a deterministic equilibrium selection rule in mind

The output of the algorithm is a single MPE of the original game.

Algorithm 2: Recursive Lexicographical Search

The second algorithm is called “Recursive Lexicographical Search”. This algorithm relaxes the equilibrium selection rule assumption and in return finds all the MPE of the original game.

The core concept behind this algorithm is that you have an outer loop over all possible equilibrium selection rules, and use the SLA algorithm to compute a particular MPE associated with that ESR.

A key feature of their description of the algorithm is an extremely efficient and clearly explained algorithm for iterating over only the feasible ESRs. The output is that you can compute all the MPE of the original game in linear time (i.e. computational time increases linearly with the number of equilibria). Naively iterating over the possible ESRs would require checking K^N candidates, where K is the maximal number of MPE for any particular stage-game and N is the total number of stage-games. In one example this reduces the number of candidates from 1.7445e26 to 1.

Model

Firm Problem

The model has a representative firm that uses capital and labor within CRS Cobb Douglas production technology, subject to TFP shock.

Capital evolution is standard: k_t + 1 = (1 − δ)k_t + i_t

Firms also raise resources through debt and equity financing:

• Debt financing comes through the issuance of a non-state contingent one period bond.
  • Debt financing has a tax advantage relative to equity (R_t = 1 + r_t(1 − τ))
• Equity financing comes through dividend payments (or receipts) from shareholders
  • Total cost to firms of dividend payouts is the payment amount plus a quadratic term in deviation of payout from target (steady state).

Timing in each period is in three stages:

1. Firms choose investment, labor, dividend payments, and issue new bonds.
   • Because all these payments happen at the start of the period firms raise within period working capital through an intra-period loan.
   • This loan is repaid at the end of each period and bears no interest
   • The value of the loan is equal to production, making budget constraint hold
2. Production happens
3. The working capital loan is paid off

Firms are allowed to default on debt, so in equilibrium there is an enforcement constraint. Suppose that at the moment the debt is contracted, with probability 1 − ξ_t the liquidation value of k_t + 1 is zero, otherwise it is k_t + 1. ξ_t is AR(1) in logs and is called a financial shock. Firms and lenders will negotiate to arrive at an enforcement constraint:

$$\xit \left(x{t+1} - \frac{b_{t+1}}{1+r_t} \right) \ge l_t$$

The constraint is tighter with more intra-or inter-period debt and looser with a higher capital stock.

In summary, the firm’s financing problem has a few features. Debt financing is preferred to equity financing because of the tax benefit and lack of additional deviation from steady state cost. However, the amount of debt financing available is constrained in equilibrium.

Household Problem

A representative household has time separable preferences over consumption and labor. The household chooses sequences of labor supply, bond holdings, and equity shares to maximize the expected present discounted value of period utility.

Each period equity payouts are shares times the sum of dividend payments plus the price of equity p_t. When choosing shares for next period, they pay s_t + 1p_t.

The household also pays lump sum taxes that finance the tax benefit firm’s receive by issuing debt.

Equilibrium

Some key equilibrium results:

• If the tax benefit τ is positive, then the enforcement constraint binds in a deterministic steady state. Can be read directly off the FOC for firm debt next period: if τ > 0 then the multiplier on the enforcement constraint is also positive.
  • With uncertainty the constraint might be slack if τ is not large enough. The authors consider τ big enough that the constraint always binds.
• When the tax benefit and the equity costs are zero the constraint is always slack (even in stochastic environment) and changes in the capital liquidation value shock do not impact labor or investment decisions.
  • So, to get the main result that the financial matters in equilibrium we need debt financing to be better than equity financing on both of these dimensions

Empirical results

The consider the response to the model under a variety of stochastic environments and asses how closely the model simulation matches the data on GDP and hours worked:

• TFP shocks only: dosn’t match the data well.
• Financial shocks only: does much better at maching the data.
  • To see why we need to understand that the marginal condition for the firm choosing labor demand is MKL = w × Labor wedge.
  • This labor wedge is increasing in both the multiplier on the enforcement constraint and dividend costs.
  • As financial shocks go up, the constraint tightens, increasing the multiplier – altering labor demand even more.

Conclusion

They also do many other experiments and consider an extended model.

References

A mostly empirical paper that examines the inputs used by Chinese firms to produce exported goods.

DVAR

The empirical analysis in this paper is centered around a variable named DVA, which stands for domestic value added in exports.

To derive DVA, we start with total revenue. The authors break total revenue into the sum of

• profits
• labor costs
• capital costs
• materials from domestic sources
• materials from foreign sources

Because intermediate good producers can get their inputs from China or foreign sources, the domestic and foreign materials sources are decomposed into Chinese and non-Chinese components.

DVA is then defined as the sum of

• profits
• labor costs
• capital costs
• Chinese component of materials from domestic sources
• Chinese component of materials from foreign sources

The authors restrict their analysis to processing exporters, which means exporters who do not sell any of their final good in domestic markets. With this restriction the value of total exports is equal to total revenue. In this case DVA can be written as total exports less cost of imported materials plus and adjustment to account for foreign content materials from domestic materials.

This is the form of DVA that is used throughout the paper. The important takeaway is that DAV is increasing in total exports and decreasing in cost of imported materials.

Data

The authors use Chinese customs data that includes all exporting firms from 2000-2007. They apply three criterion to narrow down this universe to the final data set used in their analyses:

1. Look at only processing firms
2. Look at firms that operate in a single industry (difficult to decompose share of imports and exports within a firm across industries)
3. Look at firms that aren’t “too extreme” in their importing and exporting behavior (an extreme exporter is a firm that imports strictly more than it exports – selling the additional imported goods to other domestic producers)

In all analyses they normalize DVA by total exports for each firm. The resulting variable is named DVAR.

Main findings

I’ll talk about 3 main findings:

1. Aggregate DVAR increased from near 45% to 55% from 2000 to 2007. This finding is also robust across industries, where 15 reported industries showed a similar evolution
2. The increase in DVAR is driven by firms actively substituting imported materials for domestic materials, not by rising domestic production costs (e.g. labor and capital costs). This has policy implications. They do some regressions that show that changes in import tariffs for domestic, non-processing firms – firms who sell input goods to the firms in our sample – and rising FDI liberalization made significant contributions to rising DVAR.
3. Restricting the analysis to processing exporters is not a bad estimate of total behavior. Recent work has used input-output tables to document facts about aggregate DVAR movement. The numbers reported here (using transactions level data, but only for processing firms) accounts for almost all the aggregate change.

<!– My notes

• Model the economy as families of individuals. There is perfect risk sharing within a family.
• Individuals within a family can be employed or gain income income through home production
• Families also own firms and gain income from firm profits – because firms are owned by families, firms and families discount the future at the same rate
• Individuals (workers) choose employment status to maximize the present discount value of working.
  • If they choose to be employed they earn a state dependent wage, if not they earn a constant referred to as home production.
  • Workers have a productivity level $z$ –>

    Model

    The model is a search and matching model in the flavor of DPM, with a few modifications.

    Consumers

    There are a continuum of identical consumers that have the following characteristics:

    • They face both idiosyncratic and aggregate risk (both productivity shocks that are AR(1) in logs)
    • Have a constant hazard probability ϕ of dying each period
    • Are organized into families that pool all idiosyncratic risk and exhibit perfect idiosyncratic risk sharing within the family

    The family problem:

    • Maximize the expected, discounted lifetime utility from consumption, where utility is CRRA in consumption and separable over time
    • Subject to a budget constraint that equates per period consumption and one-period ahead savings to security holding payoffs plus total income, plus profits from their family owns

    The individual employment problem:

    • Given aggregate and idiosyncratic productivity levels, the individual chooses wether or not to work in order to maximize the present value of income
      • If the individual chooses to be employed, they earn the market wage. Otherwise, they provide a constant household production to the family
      • Human capital (idiosyncratic risk) follows and AR(1) in logs, but with different processes depending on employment status. Specifically, human capital is, on average, increasing for employed workers and decreasing for unemployed workers.
      • This makes the employment decision dynamic as it not only impacts current period income, but also future productivity

    Firms

    Firms pay a vacancy cost to form a match with individuals, then produce according to individual and aggregate productivity when matched, and redistribute divided to the family of the worker.

    Matches persist between periods, but can be destroyed by an exogenous shock with constant probability, or by either individuals or firms if the match is no longer profitable (possible because wages are because home production is constant while labor income is a function of stochastic productivity)

    Matching, Bargaining, and Surplus

    The authors assume a standard Cobb-Douglass style aggregate matching technology

    Wages are renegotiated each period and are set by a generalized Nash bargaining protocol

    The dynamics of the match surplus drives incentives for firms and individuals. Using the value functions of individuals and firms it can be written as the sum of two components:

    1. A standard term that is the discounted sum of the difference of production from employed and unemployed workers
    2. A new second term that captures the benefit from human capital accumulation for employed workers over unemployed individuals

    Key Results

    The working paper only solves the model without aggregate shocks. There is an experiment where aggregate productivity unexpectedly drops 1% below its steady state level and then deterministically returns to the steady state.

    The main findings are:

    • Employment drops, and this drop is much more persistent than the drop in productivity
      • Most of this drop can be attributed to a drop in the market tightness (vacancies/unemployed)
    • The concavity of preferences amplifies the response of employment: the model with CRRA parameter set to 5 has an employment response 6 times larger than the model with linear preferences
    • On the job human capital accumulation makes the response to employment larger and more persistent

    Background

    When a Markov decision process (MDP) is formulated as a dynamic programming problem, the reinforcement learning literature proposes are two classic classes of algorithms to solve them. Let’s briefly review these types of algorithms and point out strengths and weaknesses of each.

    1: Actor only methods

    We can think of an actor as a fictitious character that operates on a policy rule.

    When I talk about the performance of an actor, I mean the value of following a policy.

    These methods are often implemented by estimating the gradient of the performance of an actor using simulation.

    There are two main issues with these “policy iteration”-esque algorithms:

    1. Gradient estimators can have high variance
    2. As the policy changes, a new gradient is estimated independently. This means there is no sense of learning from past data.

    2: Critic only methods

    We can think of a critic operating on either a Q or V value function.

    These methods rely exclusively on value function approximation and try to learn the solution to Bellman’s equation.

    The main issues with critic only methods are:

    1. They are indirect in that they do not try to optimize directly over the policy space
    2. Even with an accurate approximation of the value function, results that guarantee the near-optimality of the corresponding policy are difficult to guarantee.

    Main idea

    This paper suggests two actor-critic hybrid methods that aim to maintain the best features of each algorithm, while overcoming the shortcomings mentioned above.

    The main idea behind the algorithms is that the critic uses a linearly parameterized approximation scheme and simulation to learn a value function. The actor then uses the learned information to update parameters on the policy function in a direction of performance improvement.

    Aside to tie back to econ: This feels like modified policy iteration or Howard’s improvement algorithm, but it is different in a few ways:

    1. There is a learning element to these algorithms, which means we don’t have to compute expectations explicitly.
    2. We will be learning Q functions, which describe the value of being in a state and taking any feasible action (instead of the V function that describes the value of being in a state and choosing the optimal action).

    Algorithms

    The presentation is very technical and relies on assumptions that aren’t necessarily applicable to the models we write down, so I won’t the paper exactly as it was written. Instead, I will sketch the algorithm and explain the key insight the authors have that makes the algorithm tractable.

    Setup

    We will represent the critic using three variables:

    • A coefficient vector of length m that describe a linear parametrization of Q in terms of basis functions.
    • A scalar α that represents the average value of following the actor’s policy
    • A vector of length m that represents Sutton’s eligibility trace. This vector is used to form a bridge between fixed point methods and Monte Carlo methods

    The actor is represented by a suitable parametric representation of the policy function.

    Algo

    In order to understand the algorithm, I need to provide two related definitions. A temporal difference is the difference between the current approximation of a variable and a realization of that variable. In other words it is the error in our approximation for a particular sample.

    We say we update parameters or approximations using a temporal difference if the new approximation is the sum of the current approximation and a scaled temporal difference.

    The algorithm roughly proceeds as follows:

    • Initialize the actor and critic
    • Perform one step updates of the actor and critic as follows:
      • Because we learn Q, we enter a time period with a particular state and action in hand
      • The actor will dictate a new action and we need to simulate a new state, potentially given that action
      • The critic’s parameters are updated according to:
        • Average value of policy: temporal difference update (using flow implied by state and action)
        • Coefficient vector for Q: temporal difference update, scaled by eligibility trace
        • Eligibility trace:
      • The actor’s parameter vector is updated using a gradient approach that resembles Newton’s method. It takes into account updates to the actor

    The key insight the authors have that make this algorithm tractable is the following:

    Actors have to update a small number of parameters compared to the number of states. So the critic doesn’t need to form an approximation over the entire domain of Q, but rather a special projection of the Q onto the space spanned by the actor’s parameter vector.

    Intro

    An early paper that introduced a now common framework for thinking about firm entry, export decision, and exit in trade models.

    Model

    I’ll describe the model in two stages:

    1. A closed economy version of the model with a single country
    2. An open economy version with many similar countries

    Closed economy

    Consumers

    There is a representative consumer with CES a utility function over a continuum of goods. The elasticity of substitution between any two goods is σ > 1.

    We can apply Dixit Stiglitz type analysis to model this consumer’s choice as choosing only the aggregate, with an associated price. The demand an expenditure on individual varieties is then a function of the aggregated consumption good, aggregate price, and variety prices.

    Firms

    There is a continuum of firms, who each produce a single variety demanded by the consumer. Production is a function of only labor, which is inelastically supplied at an aggregate level of L.

    Technology is characterized by a cost function with constant marginal costs and a fixed overhead cost. This results in labor being an affine function in quantity: l = f + q/φ. Here φ characterizes the productivity of the firm. Higher productivity is interpreted as producing an identical variety at a lower marginal cost. This allows us to distinguish firms not by the variety they produce, but by their productivity.

    Optimal pricing is the standard constant markup over marginal cost.

    Firm revenue and profits can also be expressed in terms of the aggregate expenditure and price from the consumer problem.

    Results are that more productive firms are have higher output and revenues, charge a lower price, and earn more profits than less productive firms.

    Firm entry and exit

    There is an unbounded pool of ex ante identical prospective entrants into the industry. Each firm must pay a fixed entry cost (labor units) to draw a productivity from a distribution g. This productivity remains constant throughout the life of the firm.

    After receiving its productivity draw a firm can either exit immediately or begin producing. Once a firm decides to produce, there is an exogenous probability δ that the firm will be forced to exit.

    Because there are no changes in productivity, the firm’s entry decision is static and characterized by a cutoff threshold φ. At φ a firm is indifferent between entering or not. All firms who draw φ > φ choose to enter; others exit. These firms all earn non-negative profits in equilibrium.

    To close the economy, we also must consider a free entry condition. This condition says that the average value of entering net of entry costs must be zero.

    Equilibrium

    There exists a unique equilibrium in this model in which aggregate variables are constant across time. This equilibrium can be understood by a single figure:

    Consider φ on the horizontal axis and firm profits on the vertical axis. Then we put two lines in this space:

    1. The “zero profit cutoff”, which characterizes average profits as a function of the cutoff productivity. This will be a downward sloping line
    2. The free entry condition: expresses average profits as an increasing function of cutoff productivity

    Melitz proves that these lines always intersect exactly once, and the intersection is at the point (φ, π̄).

    Open economy

    Now consider a world where there are n additional countries, identical to the one we have been studying, with which our country can trade. In order to trade internationally, each country must pay a fixed cost to enter the export market and a variable trade cost (iceberg cost) per unit of export. These export costs are symmetric.

    Equilibrium

    Equilibrium in this setup is very similar to before. The exogenous environment for individual firms remains unchanged. Symmetric export costs imply that a firm will either choose to export to all foreign countries, or no foreign countries.

    Again, because a firm’s productivity is unchanged after the initial draw, the entry and export decisions are static and each is characterized by a cutoff. The entry cutoff is the same as before. The export cutoff φ_x?φ is characterized by the productivity that sets profits from foreign markets to zero.

    Impact of trade

    We will consider one experiment. Suppose a country initially starts in autarky and then is opened to trade.

    When this happens, the zero profit condition curve (downward sloping one) will shift up and the entry cutoff and average firm profits will both increase. Some firms who did enter in autarky will be forced to exit. Furthermore, only firms with productivity above the export cutoff will enter the export market.

    This will have a number of effects on equilibrium aggregates:

    • The number of domestic firms decreases (cutoff higher)
    • Domestic consumers have access to foreign varieties.
      • Typically this leads consumers to have access to more total varieties. Thus, consumer welfare rises as does aggregate firm profits because of the higher average productivity.
      • When trade costs are high, opening the economy might lead to a net decrease in the varieties available to consumers (reducing welfare). However, Melitz shows that even in this case the welfare gain from higher average productivity dominates this welfare loss.

    Why does trade cause less productive firms to exit?

    You might think it is because domestic firms now face more competition from foreign ones. This heightened competition would lower prices, making it unprofitable for low productivity firms to remain in the market. Because of the assumption of monopolistic competition and CES consumer preferences, optimal prices do not respond to the number of competing varieties.

    What actually does happen is entirely carried about in the labor market. The short description is that more productive firms now have access to more consumers, which makes them want to produce more, for which they need to hire more labor, which drives labor demand up, this increases the real wage, which makes it so less productive firms can no longer compete in the labor market.

    This paper looks at the class of equations used to represent and solve heterogenous agent models in continuous time and presents a solution approach that is efficient and tractable.

    Model

    A few models were presented in the paper. Here we’ll take the simplest one: a continuous time version of the Huggett economy.

    In this economy there are a continuum of individuals that are heterogeneous in their wealth and income.

    Individuals value streams of consumption using CRRA preferences and a constant discount factor ρ.

    Income is exogenous and is given in units of the consumption good. Wealth is accumulates via a risk free bond. Wealth evolves deterministically via

    
    da_t = (z_t+r_ta_t−c_t)dt
    

    Households are subject to a fixed, exogenous borrowing constraint.

    Income evolves stochastically and follows a two state Poisson process.

    The only price in the economy is the interest rate. It will be determined by a zero net supply condition on the bond.

    Because agents are heterogeneous, a state variable in this economy is the joint distribution of agents across wealth and income.

    Theoretical contributions

    The authors make various theoretical contributions. We’ll review those before moving on to the described solution approach.

    1. Agents in this economy are never borrowing constrained on the interior of the wealth state space. This means that the distribution is smooth everywhere, except possibly at the constraint. Very not true in discrete time. (multiple mass points in discrete economy). Also very easy to implement as they the only place the borrowing constraint enters the problem is in boundary conditions for the system of PDEs.
    2. In the stationary equilibrium (constant r), agents hit the constraint at a rate the authors characterize analytically
    3. An analytic characterization of the shape of both tails of the wealth distribution. Difficult to do in discrete time
    4. Equilibrium interest rate has a tight link to the number of constraint agents
    5. And an existence proof

    Solution method

    The economy we discussed above (and a class of other economies) can be boiled down to a coupled system of two PDEs:

    1. An HJB equation that describes the optimization behavior of agents
    2. A Kolmogorov Forward (or Fokker Plank) equation that describes the evolution of the distribution of agents

    Here’s the main steps of the algorithm they describe to compute the stationary equilibrium of the model:

    1. Given an interest rate, solve the HJB equation using a finite difference method (details below). Output is a savings rate function for each type
    2. Given the savings rules, solve the Kolmogorov Forward equation for the distribution
    3. Given the distribution, aggregate the savings rule and check the equilibrium (market clearing) condition(s).
    4. If the equilibrium condition(s) are not met, adjust r and iterate until convergence

    This algorithm is very standard, but there are a few key features here that you may not find in similar discrete time algorithms:

    • The distribution is characterized by a well known PDE – it is often very straightforward to solve
    • There is an established literature for solving HJB equations. The authors are able to leverage theoretical results (Barles and Souganidis (1991)) that guarantee the convergence of their finite difference scheme.
    • Borrowing constraint is completely handled via a boundary condition – something that already required for solving the PDEs, not something additional you need to build the algorithm around.

    Intro

    This introduces heterogenous banks into a general equilibrium trade model in the flavor of Melitz. There are also empirical exercises that show that the model can generate some features of German banking data.

    Model

    We will describe the model in two stages:

    1. Closed economy, where we will characterize economic setting and the optimization problem of each agent.
    2. Open economy, where combine 2 closed economies and allow them to trade

    Closed economy

    The economy consists of the following agents:

    • A continuum of capitalists of measure K. Capitalists can choose to become bankers or depositors.
      • A banker has two roles (the optimization problem will be described later):
        1. Channel capital from depositors to firms and
        2. Monitor the firms they lend to. Bankers earn a return R on all capital they lend to firms (their own, plus that collected from depositors)
      • A depositor lends their capital endowment to banks and earn a return R^D <  = R on their investment. R^D is an equilibrium object. Preferences dictate that they then consume all that they have
    • A continuum of potential entrepreneurs that run firms.
      • Firms are perfectly competitive and operate a production technology that exhibits constant returns to scale in capital and labor.
      • They rent capital and labor and seek to maximize revenues less input costs
      • Firms have a random probability of being productive. If they are productive, they produce and those who supplied capital and labor are paid their rents. If they are not productive, output is zero and those who supplied capital and labor get nothing.
    • A continuum of workers of measure L that inelastically supply labor to the firms and consume all labor income with linear utility.

    So; the depositor, worker, and firm problems are all trivial. Bankers are the interesting agents in this model, and we will describe their problem now.

    The model consists of two time periods

    In the first time period the following happens:

    • Capitalists draw a monitoring cost from a uniform distribution and then choose to become either a banker or depositors.
    • Depositors then lend all their capital to banks
    • Bankers collect capital from depositors and then make two choices: how much to lend to which firms and whether or not to monitor those firms:
      • The objective of bankers is to maximize the returns they earn, net of payouts to depositors and monitoring costs.
      • As stated above, parameters are such that the banker always chooses to monitor all firms. To credibly commit to monitoring, the banker invests his own capital into the firms he lends to.
      • Profit maximization boils down to maximizing the number of firms the banker lends to. This is a finite number because the banker has finite personal capital.

    Then in the second period

    • Firm production is realized
    • Rents are paid to banks and workers
    • Banks pay creditors at a rate R^d (all depositors earn the same rate), keeping the rest
    • Workers, depositors, and bankers then consume everything they are left with (note all consumption must come out of produced good – i.e. do not value capital or labor for consumption)

    Equilibrium

    In equilibrium there is free entry to become a banker. This boils down to having a cutoff in monitoring costs that says all capitalists with monitoring cost above the cutoff become depositors while all others become bankers.

    The second equilibrium condition is that all capital in the economy is used in production.

    These two conditions admit a semi-analytical solution for the cutoff monitoring cost and rate of return paid to depositors R^d.

    Open economy

    In the open economy there are two countries that typically differ in endowment of capital and labor as well as distribution over monitoring costs. Workers, firms, and depositors only operate within their home country – bankers are allowed to interact across the border.

    Bankers can choose to raise deposits at home and abroad as well as invest in domestic or foreign firms.

    International bankers face an variable “iceberg” costs where their return is a fraction less than one of what a banker from the foreign country would earn. Also payout costs to foreign depositors are a fraction higher than similar costs for domestic depositors.

    Operating internationally also has associated fixed costs that come in 3 varieties:

    1. There is one fixed cost associated with becoming an international lender
    2. Another (higher) fixed cost for bankers who with to borrow across the border
    3. A third (highest) fixed cost associated with foreign direct investment where a banker establishes a foreign affiliate and then interacts through the affiliate and faces no variable costs.

    Equilibrium

    Equilibrium behavior in this model is similar in spirit to the equilibrium of Melitz: lower cost bankers will

    1. Take on higher fixed costs
    2. Have a larger throughput (lend to more firms)
    3. Earn higher profits

    The author proves a unique equilibrium exists.

    Simplified model

    The general version of the model is quite complicated. Each banker has 7 potential profit streams (lend domestic capital to foreign firms, lend foreign capital to foreign firms, lend foreign capital to domestic firms, those three repeated but via foreign affiliates, and pure domestic profits).

    So consider a simplified version where we assume that factor endowments are such that bankers in country 1 earn a higher return than bankers in country 2 AND bankers in country 1 are more efficient than those in country 2 (have a lower lower bound on the uniform distribution from which costs are drawn).

    In this case bankers in country 2 will operate only domestically, become international lenders, or operate via foreign direct investment.

    Bankers in country 1 will operate domestically, become international borrowers, or operate via FDI.

    Experiment

    Consider an experiment where we compare the equilibrium outcomes when countries must live in autarky to the open economy equilibrium.

    A few things will happen:

    • The returns for bankers in the two countries will move closer to one another (that is the country with low returns will have higher returns than in autarky). This is driven by capital flowing into the higher return country, but not all the way to set returns in the two countries equal because of the fixed costs.
    • The cutoff for entry into domestic markets has an asymmetric impact.
      • In the more productive country the cutoff rises (loosens) and more banks enter. These bankers are more efficient than the foreign ones and are able to displace less efficient bankers.
    • Banks that operate internationally out of either country become larger.
      • For the borrowers in the high return/productivity country, the larger balance sheet is driven by access to reduced funding costs (rate paid to depositors falls)
      • For lenders in the less productive country, this is driven by access to markets with higher returns.

    Empirics

    The author also conducts an empirical analysis using data on the foreign assets and liabilities of all (1998) German banks for a particular year (2005).

    The reported findings support the predictions of the model:

    • The probability that a bank has foreign assets/liabilities and net FDI increase with the size of domestic business and efficiency.
    • Larger banks also hold more foreign assets and liabilities.
    • Fixed costs are a key component to foreign operations of banks. Only larger banks can overcome the fixed costs.

    Outline

    This is paper is more empirical than theoretical.

    Theoretical contributions are to specify more flexible stochastic environment than in other long run risk models a la Bansal et al. (2007)

    Empirical contributions are

    • form a linear approximation of the stochastic environment to apply state space methods within an MCMC algorithm
    • Show how to incorporate data of various frequencies and accuracy in the inference algorithm

    Overview of this talk:

    • Background: consumption data is available at varying frequencies and varying degrees of of measurement error. The authors want to use all data available to identify innovations in consumption and dividend growth as well as asset returns.
    • They will build a simple representative agent exchange economy that includes short and long run components to consumption growth and stochastic volatility
    • Will use semi-linearized solutions to this model to define a non-linear state space system
    • Use a Gibbs sampler to characterize posterior of parameter distribution – characterizing consumption growth innovations
    • They compare results of this posterior to moments in the data

    Model

    Preferences

    Endowment economy with a representative agent with Epstein Zin preferences.

    The agent maximizes lifetime utility subject to the simple Budget constraint W_t + 1 = (W_t − C_t)R_{c, t + 1}, where R_{c, t + 1} is the return on all invested wealth.

    They also consider an extension with a time rate of preference shock (the flow utility C is shocked). The growth of this shock follows an AR(1) with innovations independent of all other processes in the model.

    Technology/Endowment

    The authors save on some notation by jumping to the equilibrium outcome of C_t = Y_t and they describe the growth process for C directly.

    • They decompose consumption growth g into a persistent component x and a transitory shock that has stochastic volatility
    • The persistent component follows and AR(1) with its own stochastic volatility process
    • They also model dividend streams that have levered exposures to (linear in) both persistent and transitory components of consumption growth and its own stochastic volatility process.

    All stochastic volatility processes are AR(1) in logs with Gaussian innovations, so that each stochastic volatility is log normal.

    Information

    The agent observes current wealth, consumption growth, dividend growth, and asset returns in every period.

    Solution

    The focus of their work is on a unique estimation procedure. To facilitate this, they want a closed form solution for the model.

    Non-Guassian dynamics for each volatility process prevents them from writing a closed form solution.

    They use a linear approximation to the Log normal process. This linear approximation is Gaussian.

    Key results from solution:

    • State variables are the level x as well as the stochastic volatility of innovations to both g and x
    • The log price consumption ratio (price of consumption good) and risk free rate are affine in state variables
    • State prices are reflected in innovations to the SDF (m_t + 1 − E[m_t + 1]), which in equilibrium are linear innovations to g, x, and the two associated volatilities.

    State space representation

    This closed form solution to the model is used to identify the coefficient matrices in a state space representation of the model. Characteristics of the state space form are

    • The model is quite large
      • 22 parameters
      • 30 states: most deal with x and the various innovation realizations
      • 3-6 measurement variables
    • The state space model is non-linear because the levels of stochastic volatility are nonlinear.
    • Measurement equation for consumption must be flexible in two ways:
      • Allow various frequency of observations (annual pre-1959 and monthly from 1959 to 2011)
      • Allow for the potentially larger measurement errors for the monthly frequency data.

    Bayesian Inference

    The authors use a Gibbs sampling scheme to draw from the posterior of the parameter vector of the non-linear state space system. They split the parameters into two blocks:

    1. stochastic volatility levels | all other parameters
    2. all other parameters | stochastic volatility levels

    In each step of the MCMC algorithm:

    • The stochastic volatility block is updated using a non-linar particle filter
    • Then, taking as given levels of stochastic volatility, the rest of the equations form a linear Guassian state space. Thus, the prediction error decomposition within a Kalman filtering framework is used to update the parameters in this block.

    A common result with particle filters is that their accuracy and stability degrade with the number of parameters. Isolating the non-linearity allows the authors to applying the Kalman filter to compute the exact conditional likelihood for the majority of the parameters. This substantially reduces the error inherent in large scale particle filters.

    Results

    They run the MCMC algorithm on three versions of the model:

    1. Without rate of time preference shocks and only using consumption and divided growth data (they drop the risk free rate and market returns from the measurement equation)
    2. Without rate of time preference shocks and using consumption and divided growth data as well as market returns and the risk free rate
    3. With rate of time preference shocks and using consumption and divided growth data as well as market returns and the risk free rate

    Key empirical results:

    Common across all three versions:

    • Strong evidence for a persistent component to consumption growth (robust to sample used – i.e. pre/post 1959 and all). Shown in high correlation coefficient in AR(1) for x
    • Strong evidence for all three independent forms of stochastic volatility

    Differences when including return data:

    • Auto correlation increases from 0.97 to 0.99 (captures part of equity premium)
    • Volatility of long run risk innovations increases (reflects long run info about growth uncertainty contained in prices)
    • Predictability of consumption growth drops to levels that are closer to data

    Differences when including time rate of preference shocks:

    • Much better able to capture movements in risk free rate.
      • In fact, they do a variance decomposition of market returns, the price divided ratio, and the risk free rate in terms of x, the growth rate of the preference shock, and the volatility of x
      • They find that the growth rate of the preference shock explains almost no variation in the market returns or the price dividend ratio, but explains between 40-90% of the variation in the risk free rate over the sample.
      • Driven by the fact that the rate of time preference shock directly moves the SDF and that movement in the SDF maps directly into movement in the risk free rate.

    References

    This paper documents facts about Brazilian manufacturing firms that switch their bundle of exported products over time. The author then builds a Melitz-style trade model that attempts to explain these facts.

    Empirics

    The author uses data product level data on Brazilian manufacturing firms to document several stylized facts.

    • 72% of continuing exporters alter their product mix every year (add new exported products or drop existing exported products)
    • 83% of all Brazilian exports come from these product switching firms
    • The proportion of exporters who do product switching falls with age in export market
    • The frequency with which exporters engage in product switching falls with age
    • The exit rate of product switching firms is lower than aggregate exit rate for all ages of exporter.
    • Conditional on an exporter adding new products, over ½ of exporter products are added products

    Model

    The model in this paper extends Melitz (2003) in two ways:

    1. Consumers have firm specific demand shocks in their consumption aggregator
    2. Firms of a particular brand produce a finite, discrete number of products

    Here’s the main points of the model:

    Consumers

    In each of N countries, there is a representative consumer that has log preferences over an aggregate consumption good.

    The aggregate good is a CES aggregated bundle of firm goods from all N countries. Each firm good is hit with a time-varying, firm-and-import-country-specific demand shock.

    The firm goods are a CES aggregation of a finite number of differentiated products produced by the firm.

    Consumers inelastically supply a country specific fixed labor each period and earn labor income from firms. They also own firms in their country and retain all profits.

    Consumers take prices and wages as given maximize the utility of consuming the aggregate good subject to total expenditures being equal to labor income plus firm profits.

    The output of the consumer problem is a demand function for each product from each firm in each country.

    This demand function is a function of the consumer’s income, the current demand shock for the firm and prices. Crucially, it can be inverted to give prices as a function of quantity and the demand shock.

    Firms

    Firm’s differ in their ability φ to produce all their products. This ability or productivity is constant over time and across products and is drawn from a Pareto distribution.

    The demand shock in the consumer’s CES aggregator is the sum of a constant firm specific component and a mean zero normally distributed iid shock drawn each period. The firm specific constant component is drawn from a normal distribution with known (to the firm) mean and variance. The prior beliefs of every firm are that the firm specific demand shock is drawn from its true distribution. Because signals and priors are normal, posterior beliefs are also normal and sufficient statistics are the mean and variance of the firm specific component.

    Firms choose the quantity of each product to be sold in each country, before seeing the value of the demand shock for their firm in a given period. Consumers’ see this quantity and use their optimality conditions to declare the price they are willing to pay for the good. The firm observes this prices, which can be manipulated to reveal the current level of the demand shock. This acts as a signal about the firm specific constant component of the demand shock. Firms are Bayesian and update their beliefs about this component each period after observing the demand shock.

    Firms face fixed per-period cost of selling into each country. These costs increase with the number of products produced.

    Conditional on entering into a specific market, each period firms choose the number of products to be sold into the market as well as a quantity of each of these goods to maximize the expected value of their one period profits. Firm’s operate in monopolistically competitive markets, so they understand how their quantity decision will impact the market clearing price. However, they do not know the current value of the demand shock, so they form expectations of the price using their current beliefs about the firm specific demand shock. The output of this problem is a quantity for each product as well as an expected profit for each profit.

    Firms continue to add products until the expected profits from adding more varieties (taking into account the fixed costs) are negative.

    Finally, firms face an entry and exit decision. A firm with productivity φ has as state variables the sufficient statistics for their beliefs regarding the firm specific demand shock. They then choose whether or not to enter each market using the expected profit functions we just discussed.

    Results

    Timoshenko looks at a symmetric, stationary equilibrium of this economy. Some properties of this equilibrium are are:

    • The market participation policy is given by a cutoff threshold for the current expected value of the firm specific demand shock in terms of productive and other sufficient statistics. This cutoff is decreasing in productivity and increasing in the precision (inverse of variance).
    • The quantity adjustment in response to seeing another signal about the demand shock is positive when the posterior mean is sufficiently high and negative when it is lower. The cutoff is a function of the prior mean and variance and the variance of the signal.
    • Profits scale with quantities, so many high signals expands quantities and profits, and causes firms to add new products. A sequence of low signals has the opposite effect: lower quantities and profits leading to dropping products.
    • Firms posterior precision increases deterministically, meaning asymptotically firms perfectly learn the firm specific component of the demand shocks. This generates age effects in quantity decisions and profits, which in turn generate age effects in adding and dropping products that match the data.
    • If trade costs are lowered, the quantity of all current products is expanded. This is not supported empirically. In the data, lower trade costs tend to have firms specialize more – meaning they increase qualities of their most profitable products and scale back on quantities of marginal products.

    Quantitative results

    A brief quantitative section is given. The main message is that the model generates more modest values of the number of exporters that engage in product switching, the age dependent survival rate of exporters, and the age dependence of product switching. This suggests that the learning mechanism in the model is significant and supported by the data, but not sufficient to fully explain product switching behavior of Brazilian firms.

    The authors of this paper use confidential Irish data to document 4 novel facts about the lifecycle of exporting firms and then combine two existing modeling pieces to builds a partial equilibrium model that can reconcile the reported facts.

    Data

    The authors use two confidential micro data sets from Ireland:

    1. The Irish census of industrial production
    2. Irish custom records

    They are able to link the datasets to build a panel dataset at the firm-product-destination market level.

    Empirics

    The main empirical exercise is to determine how one of log revenue, log quantity, or log price varies with both the firm-product duration in a particular market and the length of a firm-product-market export spell. The export spell is defined as the number of consecutive periods a firm exports a particular product to a particular market. Note that in the regressions this is a constant number for the entire spell, while the export tenure rises from 1 to the duration of the spell.

    The authors also control for destination market fixed effects and firm-product-year fixed effects.

    There are 4 key results from the estimation:

    1. Quantities grow dramatically in the first five years of successful export spells, defined as spells that last at least seven years. This growth is statistically significant up to a horizon of four years and is not driven purely by part-year effects in the first year (i.e. there is economically and statistically significant growth between years 2 and 4).
    2. Within successful export spells, there are no statistically or economically significant dynamics in prices.
    3. Higher initial quantities predict longer export spells: for spells lasting between one and four years, all pairwise comparisons of initial quantities are statistically different.
    4. Initial prices do not predict export spell length.

    The authors do a number of robustness checks and report that the results are qualitatively unchanged when the data is cut differently or other controls are used.

    Model

    I now turn to the model. The use of the model is not as interesting or enlightening as the components themselves, so I will focus my discussion on why they made the assumptions they did.

    The authors make a quick note that common supply-side tricks for generating revenue and size dynamics (productivity shocks, capital adjustment costs, capacity or financial constraints, market-specific cost shocks, etc.) have a difficult time generating the observed dynamics in quantity without introducing dynamics in prices. For this reason they choose to focus on demand-side features that can generate dynamics.

    They choose two of the more common demand side bells and whistles to include in their model:

    1. Learning about unobserved idiosyncratic shocks.
    2. Consumer capital: firms build up a consumer base that deprecates over time and add consumers by direct investment in marketing or other costly acquisition methods.

    After estimating the model with simulated method of moments (targeting moments about revenues and quantities over export spells), the authors show that the model can match all 4 of the key facts.

    They also show that both learning and consumer capital are necessary in their framework. To do this they remove one at a time, re-estimate the model, and show that the model generates price dynamics.

    The main contribution of this paper is to provide a model where technology adoption and innovation interact to determine the equilibrium characteristics of the economy.

    They end up using a model where adoption an the rate of innovation are both choice variables for the firm, but describe the main mechanisms and intuitions where only adoption is a control variable and innovation is stochastic. The qualitative results from both models are the same, so I to will use the simpler model with exogenous innovation.

    Model

    Continuous time, infinite horizon.

    A continuum of firms produce a homogenous good. Firms are indexed by their productivity and their innovation ability. The production technology is equal to productivity. Innovation ability can either be high or low. Whenever innovation ability is high, productivity grows at a constant rate. When ability is low, there is no productivity growth. Transitions between high and low ability states occur randomly at a fixed rate.

    Each period a firm has the option to adpot a new technology. When adoption takes place, innovation ability is reset to be low and productivity is drawn from the current distribution of productivity in the economy. Firms disount the future at a constant rate and choose only whether or not to adpot a new technology.

    Consumers All the firms in the economy are owned by a representative consumer who has log preferences over aggregate consumption and discounts the future at a constant rate. The consumer makes no decisions and just serves as a way to measure welfare.

    Results

    The authors describe a balanced growth path equilibrium. Dividing by the growth rate of the economy allows us to think of this in terms of a stationary equilibrium.

    I’ll describe the main results in question and answer form.

    Growth rates

    First, does the optimal adoption policy influence the long run growth rate of the economy?

    Answer:

    It depends on how the option value adoption impacts the incentive to innovate. The policy dictates that firms at the bottom of the productivity distribution will adopt a new techonlogy and get a higher productivity. Firms in the upper tail of the productivity distribution will not choose to adopt a new technology. The growth rate of the economy is intimately related to the constant rate of innovation for high ability firms.

    The key margin governing whether or not the adoption policy influences the long run growth rate of the economy is how high the option value of adopting is for medimum to high productivity firms.

    In calibrations of the model where the distribution of productivity is infinite or unbounded, the productivity difference between frontier firms and potential adopters is very large – giving almost no option value for high productivity firms. This means that the long run growth rate will not
    be impacted by adoption. However, if the support of productivity is bounded, then the option value is non-trivial, so adoption rates will determine growth rates.

    It’s more poignant in models where innovation rates are endogenous, but we kinda have a discrete version of that right now where firms can choose to adopt, which means they voluntarily leave the innovation state (and might get a draw that puts them back in it).

    Productivity distribution shape

    Next, how do adoption and innovation shape the stationary distribution of productivity?

    Along a balanced growth path, high innovative, high productivity firms will push the productivity frontier. This lengthens the right tail of the productivity distribution.

    Adoption, on the other hand, is done by firms in the low end of the productivity distribution and acts as a way to redistribute these firms back towards the center of the distribution. Thus, adoption causes the distribution of productivity to contract and become more peaked.

    The overall shape of stationary distribution of de-trended productivity will be determined by how these forces compete.

    <!— In calibrations where the support of productivity is infinite, there are a continuum of equilibria featuring fat-tails in the productivity distribution indexed by the thickness of the tail (for each shape paramater there is an equilibrium growth rate strictly greater than the innovation rate).

When the support of z is unbounded, but finite, there is a unique equilibrium where the growth rate of the economy is equal to the innovation rate. In this setting, the adoption policy –>

Outline

An empirical paper that examines the relationship between the real exchange rate and the relative price of non-traded vs traded goods.

Modes of thought

At the time of writing the paper, there were two main modes of thought for thinking about real exchange rate movements

• Traditional view: all movements in bilateral real exchange rate between two countries are due to fluctuations in the bilateral relative price of non-traded to traded goods.
• New Open Economy Macroeconomics view (the “new view”): models that segment traded goods, allowing for deviations in the law of one price and have nominal rigidities that support these deviations. This means that purely monetary shocks to nominal exchange rates have persistent impact on prices and thus the real exchange rate. There is no distinction between traded and non-traded goods.

An influential 1999 paper by Engel found evidence that we should abandon the traditional view and ignore the distinction between traded and non-traded goods.

This paper finds empirical evidence that suggests a substantial link between real exchange rates and relative prices of non-traded to traded goods.

Key equation

The real exchange rate can be written as the product of a nominal exchange rate and a ratio of price indices.

If we multiply and divide by a price index for only traded goods for each country, we can decompose this product into two pieces:

• Nominal exchange rate times relative price of traded goods – we call this the bilateral real exchange rate of traded goods
• Overall price index ratio divided by traded good price index ratio – this is a ratio of one country relative price indices. This is the price ratio of non-traded vs traded goods mentioned earlier.

Data

The authors use data on 50 countries: all the countries they could find quarterly real exchange rate and price data for over the period 1980 through 2005.

The list accounts for 83.5% of all world trade in that time span. The main country missing from the list is China, for which the authors only had access to yearly data.

Qualitative description of main findings

The authors report 8 main findings:

1. The correlation between real exchange rate and the price ratio is on average just above 0.5
2. The volatility of relative prices is about 2/3 the volatility of the real exchange rate
3. The relative price accounts for about 1/3 of the variance in the real exchange rate
4. The relationship between the price ratio and real exchange rates is strengthened as the trade intensity between countries increases
5. The relationship is stronger when the variance of the real exchange rate between two countries is low
6. The strength of the relationship does not depend systematically on whether the US is the pair of countries being studied
7. The relationship is not biased upwards when the authors account for high/low inflation pairings or rich/poor country pairings (in contrast to common beliefs at the time)
8. The anaomaly: the relationship between the price ratio and the real exchange rate between EU/NAFTA and EU/US pairs is extremely weak compared to pairing from any other trade blocs.

Conclusion

The authors document a string and robust statistical relationship between the real exchange rate and the relative price of non-traded to traded goods.

References

Section 1 (intro):

• Two channels:

1. productivity channel says move from low productivity to high productivity country
2. risk-sharing channel says move from low marginal utility (MU) to high MU country

Section 2 (lit review):

• Differ from Tretvol on three dimensions: (1) look at long run shocks, (2) Tretvoll considers standard BKK capital accumulation (TODO: what is used here) (3) parameter values are closer to asset pricing than RBC literature (RRA = 10, IES > 1 where tretvoll has RRA=100, IES < 1)

Section 3 (empirics):

• Used data on G7 countries to show three main results:
• Positive short run productivity news leads to net inflow of capital
• Positive long run productivity news leads to net outflow of captial
• Did estimation for US vs UK and then for US vs all other G6. Results were robust across rather large cross section of developed countries.

Section 4 (model):

Model features

• Complete markets (let’s them solve Pareto problem)

Two countries - GHH consumption/leasure aggregator – avoids issue brought up by Raffo (2008) with CES aggregators - Homogeneous EZ preferences - Cobb-Douglass consumption good aggregator with home bias

Two goods - Country specific production with Cobb douglass aggregation of country specific labor and capital - Country specific capital and labor stock, but frictionless trade in consumption and investment good

Exogenous process: - constant volatility - with long and short run shocks. - Productivity across countries is cointegrated

Intuition:

• EZ prefs provide two “risk” motivators over additive preferences

1. A preference for early (more common) or late resolution of uncertainty casues the variance of future Value function to impact current welfare (early resolution means higher variance -> lower welfare)

2. Extra term appears in SDF (relative to additive case) (future utility over its certainty equivalent) appears. Why does this matter? Creates a risk-sharing channel to impact flow of resources. How? Two things: first, with EZ prefs there is a 1-to-1 map between lifetime wealth and the value function; and second, SDF is used to price risk, so now there is a link between lifetime wealth and risk adjustments. In this model lifetime wealth changed by long run news (positive news about long run productivity => higher lifetime welath => higher value function => lower marginal value => planner moves resources toward other, high marginal value agent.)

Section 5

Matching moments:

• Volatility of output growth and consumption growth relatively close to data (same as BKK)
• Volatility of investment growth and labor growth too low (same as BKK)
• Equity premium generated by all three models (BKK with high/low RRA and EZ) is too low
• All models match data mometns on correlation between c, n, i, nx. the nx one is harder for the high RA additive model to match

Conclusion

• Paper provides empirical evidence for G7 countries that short-and-long-term productivity news have differing effects on capital flows (short  ↑ ⇒ captial in . long  ↑ ⇒ capital out)

• They start with a classic BKK model (two agents, two goods, CRRA utility, constant volatility, persistent shocks to productivity, frictionless trade across countries). Here there is one main channel that prompts resource flows: productivity. So good news about either long run or short run productivity growth will cause resources to flow into that country.

• They also present an almost identical version of the model, the only change being endowing agents with Epstein Zin recursive preferences instead of additive preferences. In addition to the productivity channel inherited from the BKK structure, the EZ preferences bring a risk-channel or motive for resource allocation. This motive can be seen by considering 3 related things:

1. Under EZ preferneces there is a 1-to-1 mapping between lifetime wealth and the value function.
2. In the planner’s version of this problem, the parteto weights are monotonic in the ratio of SDFs.
3. The SDF under EZ preferences has an additional term that involves future utility and its certainty equivalent. When agents have a preference for early resolution of unceratinty (common parameterization), this extra term causes the SDF to fall as the future value function increases.

Keeping those three in mind, consider the effect of positive news about productivity growth in the domestic country:

Raises lifetime domestic wealth => Raises lifetime domestic utility => Causes domesic SDF to fall, relative to the foreign SDF=> Causes domestic pareto weight to fall => planner to immdetiately move resources out of that country.

• They present some empirical results showing that their model does better than the standard BKK model at matching moments from international data.
• They still fall short on 5 main fronts. Their problems were:

1. investment is too smooth (not volatile enough)
2. correlation between domestic and foreign investment growth is negative (risk sharing result – planner moves resources to less productive country, part of that is used in that country’s investment)
3. Risk premium is too small
4. Exchange rate volatility is too small
5. consumption growth rates are more correlated than output growth rates across countries

• They then extend their baseline recursive model by allowing for a different home bias parameter for consumption and investment goods and by introducing the concept of vintage captial, or that captial becomes more productive over time. With these changes they at least partially resolve all 5 concerns above, without significant departures in other areas.

• They then do sensitivity analysis and conclude.

This paper was published in Operations Research and as such they use a different notation and jargon than economists. I’ll present some of their main results, but in a language and notation that is familiar to me.

Background

Consider a typical dynamic programming problem faced by an economic agent, which we summarize by the following Bellman equation:

$$V(x) = \underset{c \in \Gamma(x)}{\max} u(x, c) + \beta E V(x’)$$

Now define the natural operator associated with the Bellman, which we call T:

$$T v = \underset{c \in \Gamma(x)}{\max} u(x, c) + \beta E v(x’)$$

For ease of presentation we will directly assume that T is a contraction mapping and that v belongs to the set of bounded and continuous functions.

Let v be the optimal value function. Under our assumptions v is the unique fixed point of Bellman’s equation.

Mathematical programs

Under our assumptions, the operator T is monotonic.

This means that for any v such that v ≥ Tv we have v ≥ v.

Also ∀vstv ≤ Tv, v ≤ v.

Typically this property of our contraction mapping is used to show that value function iteration will converge for any initial guess of v. Today we will use it in a slightly different way.

Let c be any vector of all positive elements. Consider the mathematical program defined by

$$\begin{aligned}&\underset{V}{\min} c’v
s.t. & \quad v \ge T v \end{aligned}$$

By the monotonicity of T it is easy to see that any v satisfying the constraint must be at least as big at v.

Then notice that the objective is to minimize the inner product between a strictly positive vector c and the choice v.

These two facts mean that the unique solution to the programming problem is v.

Linear program

I told you that we would use linear programming, but notice that the constraint is nonlinear because T is nonlinear.

However, if we stare at the definition of T for long enough we will notice that if we consider the constraint v ≥ Tv state by state, (e.g. v(x) ≥ Tv(x)) we will notice that we can replace the single non-linear constraint per state with a system of linear constraints for each state.

This system is defined by enumerating all feasible actions for each state and writing down the right hand side of Bellman’s equation for that state and control. The program looks as follows:

$$\begin{aligned}&\underset{V}{\min} c’v
s.t. & \quad v(x) \ge u(x,c) + \beta \sum_{x’} P(x’| x,c)v(x’) \quad \forall x \in X \; \forall c \in \mathcal{A}x \end{aligned}$$

Curse of dimensionality

This linear program has an S dimensional state with an S * A dimensional constraint matrix. When S or A are large, this problem can quickly become subject to the curse of dimensionality.

The authors of this paper propose approximate linear programming as a way to resolve this issue.

Specifically they choose to represent v as the product of a basis matrix and a vector of coefficients r.

They then write down a linear program

$$\begin{aligned}&\underset{V}{\min} c’\Phi r
s.t. & \quad \Phi r(x) \ge u(x,c) + \beta \sum{x’} P(x’| x,c) \Phi r(x’) \quad \forall x \in X \; \forall c \in \mathcal{A}_x \end{aligned}$$

Notice now that we have swapped out a vector of length S, for a vector with length equal to the number of columns in the basis matrix. This is something that we have control over, thus we can choose the “size” of this problem. Thus the objective is smaller, but the number of constraints is exactly the same.

However, if we choose each column of Φ to have finite support (i.e. we use splines), most constraints become inactive and the large constraint matrix becomes sparse.

The remainder of the paper, and its main contribution, is to bound the error we are subject to by solving the approximate linear program instead of the exact linear program.

In 2013; Mian, Rao, Sufi used proprietary data on the US housing market (obtained from Core Logic) and personal consumption expenditures (obtained from master card) from 2006-2009 to estimate that the elasticity of consumption expenditures to changes in the housing share of household net worth.

This paper replicates the main results from Mian, Rao, and Sufi using data that is more easily accessed by economists in academia.

Data

The authors proxy the housing data from Core Logic, using housing data from Zillow. The data is freely downloadable from the Zillow website.

Instead of the mastercard data, the authors use data from the Kilts Neilsen scanner retail survey. This survey includes weekly price and quantity levels for sales at the bar code level for about 40,000 US stores from 2006-2009 (the panel is still ongoing and currently runs to 2014). KMV estimate that this subset is approximately 40% of aggregate US consumption on non-durables.

Replication

Using the Core logic and mastercard data to, MRS report an elasticity of consumption expenditures to changes in the housing share of household net worth between 0.33 and 0.36 (depending on the controls in the regression and regression technique – OLS vs IV2SLS).

Using the Zillow and Kilts Neilsen data, KVM report an elasticity between 0.24 and 0.36.

The similarity of these findings despite the very different data sets is encouraging.

New contributions

In addition to replicating the results from MRS, KMV have 3 main findings:

1. They show that the interaction between the fall in local house prices and the size of initial leverage is not statistically significant, after controlling for the direct impact of house price changes.
2. They separate the price and quantity components in the fall in consumption expenditure during the great recession. They construct a proxy measure of the quantity of household expenditure by aggregating quantity sold from all stores at the product level, and then multiplying by an average price for the product. When this is used as the dependent variable in the regression, elasticities are approximately 20% lower.
3. They use the Diary Survey of the Consumer Expenditure Survey to estimate the elasticity of total non-durable goods and survives to the counter part found in the Kilts Nielsen dataset. They obtain an elasticity between 0.7 and 0.9, meaning that their estimated consumption to household share of wealth elasticity should be lowered by approximately 20% when applied to all non-durable goods and services.

Intro

The authors did some empirical work with data on Columbian firms to arrive at two stlyized facts about new exporters (firms who recently began exporting):

1. On average, entrant exporters are smaller than long-time exporters
2. The probability of exiting the export market falls with tenure

The goal of the paper is to write a model that can match the data on these two dimensions.

Model

Partial equilibrium version of the model I presented last week. There will be three varaints:

1. A baseline model similar to Melitz
2. A model that perturbs the baseline by making foreign demand grow with tenure in export market. This is called the delayed demand model
3. A model that perturbs the delayed demand model by making the fixed entry cost stochastic, rather than constant

Baseline model

Domestic demand:

• Representative consumer supplies labor inelastically and aggregates differentiated varieties of the consumption good using a CES technology.
• The consumer maximizes the level aggregated consumption, subject to the budget constraint that the sum of price times variety is equal to total income

Foreign demand:

• The “rest of the world” is populated by another representative agent with the same preferences.
• Only difference from domestic consumer is in demand for varieties, which differs with the exchange rate (varieties have different prices in different countries)

Plants are indexed by the variety they produce and operate under monopolistic competition.

Plants operate a potentially DRS Cobg-Douglass production technology, with idiosyncratic TFP shocks.

Each period, each plant chooses domestic/foreign prices, domestic/foreign supply, input demand, and export status to maximize profits. Profits are the sum of revenues earned in both markets, less input costs.

The plant problem can be split into static and dynamic parts:

• Static problem:
  • Given export status, choose prices, production and factor demand
• Dynamic problem:
  • Sunk entry costs make the export entry decision dynamic
  • When a plant enters the export market he must pay a fixed cost f₀
  • An incumbent exporter that chooses to continue exporting pays fixed cost f₁ < f₀
  • The plant’s state variables are its productivity and the real exchange rate. Both are modeled as an AR(1) in logs with Gaussian innovations.
  • In this problem the firm makes a discrete decision regarding

Results

This model misses both empirical facts. In equilibrium, new exporters are too large relative to incumbents and probability of exit increases with time in export market.

The reason for this is that the discrete entry decision such that firms enter when either they have high productivity or face a favorable exchange rates. In short, the strongest, or largest, firms in the economy are the entrants. The size of these firms raises the initial export to sales ratio rise above the levels in the data.

Furthermore, the persistence of the process for both productivity and the exchange rate will cause profits to be front-loaded. As these state variables return to long run levels, firms choose to exit.

Another important point is that with front-loaded profits, the value function of the typical exporter is quite large. To match empirical moments on the entry rate, the entry cost must be very high.

Extension 1: delayed demand

The first extension to the baseline model is to make foreign demand grow slowly over time. This is done exogenously by causing foreign demand for each variety to rise with time in the export market.

This modeling assumption bakes in a smaller initial export to total sales ratio for entrant firms. This calibrated model matches the data on the export to sales ratio almost exactly, but still does very poorly at dealing addressing the inverted exit probability.

Extension 2: delayed demand and stochastic entry costs

To get the model to match the other dimension of the data, the authors perturb the delayed demand model to allow some “bad” firms to enter. The mechanism that makes this happen is stochastic entry costs: in most periods the entry cost is positive, but in rare periods it is zero.

In the periods where entry costs are zero, some less productive firms who would not normally choose to enter the export market become exporters. When fixed costs are high again, they choose to exit.

The model with both extensions in place are able to match both stylized facts from the data that teh baseline model fails to produce.

References

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1. In the first period agent 1 makes his decision based solely on his prior. If he chooses to exit, he learns nothing, and next period must make his decision with the same information. Because we look for pure strategy equilibria, he will make the same decision forever and the game will be trivial.↩