Baley:2016wr (Firm uncertainty cycles and the propagation of nominal shocks)

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.