EH_ch1_4

Chapter 3

Variations on a Theme

3.1 Introduction

In this chapter we discuss some extensions and variations of econometric learning. Several issues arise naturally. So far we have assumed representative agent learning, although diversity of expectations should surely be treated. One can also consider alternative adaptive learning schemes and the possibility that the agents do not know the true model. In this chapter we show how such issues can be readily addressed in the context of the basic cobweb and asset pricing models.

We also take up learning in nonstochastic frameworks and obtain the key conditions for local stability under adaptive learning of perfect-foresight steady states. Since some standard textbook models are often presented in a nonstochastic setting, we will occasionally draw on these results in later chapters. In this chapter we show how the technique can be applied directly to assess local stability of equilibria in simple nonstochastic coordination games.

3.2 Heterogeneous Expectations

For expository purposes we simplify the cobweb model of the previous chapter. Dropping the intercept and assuming a scalar stationary shock wt−1, the model

45

This equation and the updating rule (3.2) define a standard stochastic recursive algorithm as introduced in Section 2.7. As pointed out there, the analysis of convergence in these algorithms is carried out by deriving the associated ODE. Following the same steps as in Section 2.8 of Chapter 2, we obtain

where M = E(wt2).

Since Ri → M globally for i = 1, . . . , N, stability of this differential equation system is governed by stability of

This system can be written in the matrix form

The coefficient matrix of this linear system has one eigenvalue of α − 1 and N − 1 eigenvalues of −1 and is therefore globally asymptotically stable with

Appealing to the results on convergence of stochastic recursive algorithms as in the previous chapter, we have the striking result that the convergence condition under heterogeneous expectations is the same as that for the case of a representative agent. That is, provided α < 1, all agents learn asymptotically the rational expectations equilibrium.

We remark that the preceding argument implicitly makes the “representative agent” assumption of structural homogeneity. Thus equation (3.1) is assumed to arise from an economy in which the cost functions of individual suppliers are identical. In principle structural heterogeneity could be tackled by the same techniques.

3.3 Learning with Constant Gain

So far we have considered cases where there is asymptotic convergence to the rational expectations equilibrium. In models with stochastic shocks, this can only happen if the learning algorithm is of the “decreasing-gain” type, i.e., γt → 0. We now briefly analyze the case of a constant gain, i.e., γt = γ , a small positive constant.

For our example we pick the model

pt = α + βEt pt+1 + vt ,

where vt is iid with mean zero and Et pt+1 denotes the (rational or nonrational) expectation of the next period price. We here use Et pt+1 in place of pte+1 to emphasize that the expectations are formed at time t. This model was introduced in Chapter 1 as the “Cagan Model” and it was noted there that it could also be interpreted as the standard asset pricing model with risk neutrality.2 For both cases we have 0 < β < 1.

We focus on learning of the market fundamental rational expectations solution. This is a stochastic steady state pt = a¯ + vt , where a¯ = (1 − β)−1α. We first begin with the standard decreasing-gain learning rule which does converge to the REE. We assume that forecasts take the form Et pt+1 = at , where at is the estimated mean which is updated according to

at = at−1 + t−1(pt−1 − at−1).

Since pt−1 = α + βat−1 + vt−1, we have

at = at−1 + t−1(α + (β − 1)at−1 + vt−1).

The associated ODE takes the form

da = α + (β − 1)a. dτ

Applying the stochastic approximation results, it follows that at → a¯ provided β < 1. (Using the global convergence results of Chapter 6, it can in fact be shown that convergence occurs with probability 1.) The condition β < 1 can easily be seen to be the E-stability condition, so that we obtain the expected result that when this condition is satisfied there is convergence to the REE.

2For further discussion of this model see Section 8.6 of Chapter 8.

52 View of the Landscape

between a0 and a. Then at each time t one has f (at ) > at , implying at+1 > at . Since γt < 1, it follows that at+1 ≤ a. To show that at → a, suppose to the

Alternative timing assumption: Under equation (3.6) we instead have the implicit equation

at = at−1 + γt (f (at ) − at−1).

In general, there need not be a unique solution for at given at−1, though this will be assured if γt is sufficiently small and if f (a) is bounded. Assuming uniqueness and focusing on local stability near a, we can approximate this equation by

at − a = 1 − γt f (a) −1(1 − γt )(at−1 − a).

Under the fixed-gain assumption this leads to the stability condition that either f (a) < 1 or f (a) > 2/γ − 1 (these possibilities correspond to the cases γf (a) < 1 and γf (a) > 1). Under decreasing gain the condition is again simply f (a) < 1.

The results of this section can be summarized as follows. Fixed-gain learning can converge to perfect-foresight steady states in nonstochastic models and such rules have somewhat different stability conditions than the decreasing-gain rules which are standard for stochastic models. The stability conditions also depend on assumptions made concerning the timing of information. However, for the small gain case, i.e., decreasing gain or a sufficiently small constant gain, the condition for local stability under adaptive learning is not affected by the timing assumption and is simply f (a) < 1, generalizing our earlier results for linear models. This stability condition has a straightforward interpretation in terms of E-stability: For a PLM yt = a, the corresponding ALM is yt = f (a). The E-stability differential equation is then da/dτ = f (a) − a with stability condition f (a) < 1.

The methods of this section can also be easily applied equally to models of the form pt = f (Et−1pt ). For this setup, the expectation Et−1pt cannot depend on pt and the timing assumption appears clear. Supposing that Et−1pt = at−1 with at = at−1 + γt (pt − at−1) leads to the recursive algorithm

6Recall that a bounded monotonic sequence always has a limit.

at = at−1 + γt (f (at−1) − at−1) as with the preceding model under the standard timing assumption. It follows that a perfect-foresight steady state pt = p¯, where p¯ = f (p)¯ , is locally stable under decreasing gain if f (a) < 1 and under constant gain γt = γ provided 1 − 2/γ < f (a) < 1.

3.4.1Application: Coordination Problems

The general methods of this section can be applied to study adaptive learning in certain nonstochastic macroeconomic models formulated as games. In this section we consider coordination games involving strategic complementarities along the lines of Cooper and John (1988) and Cooper (1999). A large finite number of agents i = 1, . . . , I, each choose an action xi [0, 1]. We consider a representative agent model in which the payoff for agent i is given by U (xi , Y (x−i )), where x−i denotes the vector of actions by other agents and Y (x−i ) is an aggregate statistic. U (xi , Y (x−i )) is assumed to be twice continuously differentiable and strictly concave in the first argument. We focus on symmetric outcomes in which each agent chooses action x and we adopt the notation Y (x) for Y (x−i ). We assume that Y (x) is twice continuously differentiable and that Y (x) > 0 for all x [0, 1].

A symmetric Nash equilibrium is an action x which maximizes U (x, Y ) with respect to x, given Y , and where Y = Y (x). In general, there may be multiple equilibria. For this game, a necessary condition for this is strategic complementarity, defined by ∂2U/∂x∂Y > 0. Let φ(Y ) denote the best response of the representative agent to Y . In the case of strategic complementarity we obtain an increasing function φ(Y (x)) as in Figure 3.1.

Figure 3.1.