Incentives in Information Acquisition

AppendixBthoroughly discusses my approach for solving the rational inattention problem of the firms. Here I discuss the incentives of firms in acquiring information.

In addition to the strategic incentives discussed in the static model, the specification of a firm’s problem in Equation (6) shows how their information set becomes the source of a new dynamic trade-off. At every period, firms understand that the information they choose to see will not only inform them about their contemporaneous optimal price, but also about their future payoffs. While the dynamic incentives of a rationally inattentive agent is the main focus of Afrouzi and Yang (2016), the main objective here is to understand the effects of the strategic trade-off that imperfectly competitive firms face in allocating their attention.

To better understand the separate roles that these strategic incentives play in the their price- setting behavior of firms, in this paper I shut down the dynamic incentives of firms completely by assuming that they do not endogenize the continuation value of information in their maximization

problem. This does not eliminate the necessity of tracking firms’ signals over time, as they still rely on their previous signals in forming their beliefs about the fundamental and the prices of their competitors. However, the lack of dynamic incentives implies that firms ignore these continuation values in choosing their information.

To solve the firms’ problem, I derive the second order approximation of firms’ losses from sub-optimal pricing following the rational inattention literature, and assume that they minimize the expected net present value of these losses subject to the attention constraint.35 At any time, given a realization of Pj, k,tand Qt, a firm’s profit loss from charging a price Pj,k,t is given by

L(Pj,k,t,Pj, k,t,Wt) ⌘ max_x P(x,Pj, k,t,Wt) P(Pj,k,t,Pj, k,t,Wt) = (pj,k,t (1 a)wt a_{K 1}1

Â

l6=k

pj,l,t)2.

Here, small letters denote percentage deviations from the steady state, and a is the degree of industry level strategic complementarity which is now directly linked to the micro-foundations of the model. The following Proposition derives the form of the signals that firms choose to see in the equilibrium.

Proposition 5. Given a strategy profile for all other firms in the economy, a particular firm prefers to see only one signal at any given time. Moreover, the optimal signal of firm j,k at time t is

Sj,k,t = (1 a)qt+apj, k,t(Stj, k) +ej,k,t

where qt is the nominal aggregate demand, pj, k,t is the average price of j,k’s competitors, and ej,k,t is the rational inattention error of the firm.

The closed form for the optimal signal in this case shows how firms incorporate the mistakes of their competitors into their information sets. To see this given the strategy of others, decompose

35_{Notice that profit maximization is equivalent to minimizing these losses over time. The second order approxi-}

mation reduces the state space of the problem from a whole distributions to its covariance matrix by implying that the distribution is a multivariate Normal. Moreover, since optimal signals under Gaussian fundamentals and quadratic objectives are also Gaussian, it allows us to only focus on Gaussian signals without loss of generality.

their average price at time t to its projection on the history of fundamentals and the part that is orthogonal to them

pj, k,t(Stj, k) = pj, k,t(Sj, k,t)|q

| {z }

projection on realizations of all qt t’s

+ vj, k,t

| {z }

orthgonal to all realizations of qt t’s .

This is analogous to the decomposition that I did in the static model. It separates the average prices of others to a part that is linearly projected on current and past realizations of the fundamental, and a part that is orthogonal to it, denoted by vj, k,t. Similar to before, we call these the mistakes of a firm’s competitors in pricing. Notice that the finiteness of the number of competitors immediately implies that var(vj, k,t)6= 0. Given this decomposition, the optimal signal of the firm is

Sj,k,t =

predictive of industry price changes

z }| {

(1 a)qt+apj, k,t(Sj, k,t)|q

| {z }

predictive of qt t’s

+avj, k,t+ej,k,t.

This decomposition of the signal illustrates the main departure of this paper from models that assume a measure of firms. Since var(vj, k,t)6= 0, the signal of a firm co-varies more with the price changes of its competitors than with the fundamentals of the economy.36 When there is a measure of firms, however, the termavj, k,tdisappears and these two covariances converge to one another. Intuitively, going back to the result in Proposition 3, this implies that when a is large enough, and there are a finite number of firms in industries, firms are more informed about their own industry prices than the fundamentals of the economy. In the next subsection, I show how for largea’s, it is these expectations that mainly drive the inflation in the economy.

Moreover, given the joint stochastic process of these signals, the best pricing response of a firm

36_{This in itself does not mean that the signal is more predictive of a firm’s competitors’ prices than the aggregate}

economy since predictive power of a signal also depends on the volatility of the variable that is being predicted, and industry prices are more volatile than the aggregate economy. However, as we showed in Proposition3, oncea gets large enough, this difference is large enough so that firms end up with more information about their own industry price changes than the fundamental.

reduces to a Kalman filtering problem, which then implies that in a stationary equilibrium pj,k,t(Stj,k,t) =

•

Â

t=0dtSj,k,t t,

where (dt)•t=0 is a summable sequence.37 Intuitively, these dt’s represent the confidence of the agent on how informative each element of her information set is about the optimal price that she would like to charge at time t. If a firms did not make mistakes, then the only signal that would matter for it at time t would be Sj,k,t = (1 a)qt+apj, k,t(St_{j, k}), so thatd0=1,dt=0,8t 1. However, making mistakes over time reduces the informativeness of a firm’s signals and it finds it optimal to put some weight on their previous signals in setting their prices. Therefore, the more uninformative the signals, the more persistent the response of firm’s prices would be to a shock over time.

Given the result in this Proposition5, solving the model reduces to finding the following fixed point: a symmetric stationary equilibrium is a stationary joint stochastic process for signals of firms, and a pricing strategy (d_t⇤₎•

t=0, such that for any firm whose competitors set their prices according to this sequence, the firm finds it optimal to use (d⇤

t)•t=0 for setting its prices. Appendix

Blays out the computational algorithm that I use to solve for the joint stochastic process of signals and pricing strategies.