EH_ch1_4

Learning of a steady state is formally identical to that in Section 4.2, equations (4.1) and (4.2) with qt replaced by nt . The argument given in Section 3.4 of Chapter 3 can again be applied here to show that a steady state n = F(n ) is locally stable under learning if F (n ) < 1. It follows that nL and nH are locally stable, while nU is unstable. In this model it is therefore possible for the economy to become stuck in a low-activity, low-welfare steady state.

4.6.2Adaptive Learning and Economic Policy

The possibility of the economy becoming trapped in a low-level steady state raises the issue of whether economic policy can be effective in dislodging it from such inefficient equilibria. In the context of a positive production externality, it is natural to consider the effects of a proportional subsidy ρ to the price of output financed by a lump-sum tax T = ρpt f (nt , nt+1). Here pt is now the producer price, so that the price paid by the consumer is (1 − ρ)pt . If ρ < 0, this is interpreted as an ad valorem tax which is redistributed as lump-sum transfer payments. Allowing for this subsidy changes the equilibrium equation to

(1 − ρ)V (nt )f (nt , nt )/f1(nt , nt )

(4.13)

= Et f (nt+1, nt+1)U (f (nt+1, nt+1)).

For our earlier specification of production functions and utility functions, and assuming point expectations, this can be written

nt = F(net+1, ρ),

where

F(n, ρ) = (α/(1 − ρ))1/(1+ε)

× Anα max(I , λ n/(1 + aλ n)) β (1−σ )/(1+ε).

For the case 0 < σ < 1, in which F(n, ρ) is an increasing function of n, an increase in ρ rotates F(net+1, ρ) counterclockwise. For sufficiently large values of ρ, the graph will be like Fa in Figure 4.5, with a single (high-activity) interior steady state. For sufficiently low (possibly negative) values of ρ, the graph will be like Fb, with a single (low-activity) interior steady state. A range of intermediate values of ρ will give three interior steady states, as with the graph F.

Changing the value of ρ can have dramatic effects under adaptive learning. Consider slow, gradual changes in ρ such that the learning dynamics are fast relative to changes in ρ. Starting from nL in Figure 4.5, as ρ is increased the

Figure 4.6.

level of economic activity approximately shifts slowly up along the 45-degree line, tracking the increasing value of nL. However, at a sufficiently high value of ρ, the low steady state disappears and adaptive learning drives economic activity to naH . Thus, in addition to the usual effects, changes in policy can in certain circumstances lead to large discrete changes by shifting the economy between distinct equilibria.

Furthermore, changes in the subsidy rate ρ exhibit irreversibilities over certain ranges. Starting from the high level of ρ associated with the graph Fa and steady state naH , if the level of ρ is slowly reduced to the original level associated with graph F, the level of economic activity will continue to track the high-activity equilibrium nH .

The approximate relationship between ρ and n is as shown in Figure 4.6. There are two branches to the relationship, a high-activity branch AB and a lowactivity branch CD. Over intermediate ranges of ρ, the level of n depends via the learning dynamics on the history of the variables. Values of ρ outside this intermediate range can be used to shift the two equilibrium branches.

4.6.3Sunspot Equilibria

The model of increasing returns can be used to illustrate the phenomenon of “sunspot equilibria” which have been much used recently in macroeconomics to model endogenous fluctuations.6 These rational expectations solutions can be viewed as a modern formulation of a long tradition in economics which emphasizes the possibility of endogenous fluctuations in market economies. The

6The initial investigations were done in Shell (1977), Azariadis (1981), Azariadis and Guesnerie (1982), and Cass and Shell (1983). See Chiappori and Guesnerie (1991) and Guesnerie and Woodford (1992) for recent surveys.

central idea is that in some models it is possible for the expectations of firms or households to depend on an extraneous random variable in a way which is entirely rational. Shifts in the variable (often called a “sunspot”) would then trigger self-fulfilling shifts in expectations, prices, and quantities, creating or amplifying economic fluctuations.7 We introduce such equilibria in the context of the current model and then take up then issue of whether such equilibria can emerge under adaptive learning.

Assuming rational expectations, equation (4.12) or (4.13) can be written as

where W(nt ) denotes the left-hand-side function and G(nt+1) denotes the right- hand-side function following Et . In the case (4.13) in which the production subsidy is present, ρ is implicit in W. Earlier, when studying steady-state learning, we were able to use the notation F = W−1 ◦ G, but we must now drop the assumption of point expectations since in a sunspot equilibrium the economy will be undergoing random shifts.

The definition of a sunspot equilibrium involves the idea that economic agents condition their expectations on some (random) variable st which otherwise does not have any influence on the economy. Though different types of sunspot solutions have been considered in the literature, we will focus here on REE that take the form of a finite Markov chain.

We simplify even further by assuming that the extraneous random variable st {1, 2} is a two-state Markov chain with a constant transition matrix = (πij ), 0 < πij < 1, i, j = 1, 2. Here πij denotes the probability that st+1 = j given that st = i. A two-state Markov chain is specified by probabilities π11 and π22 since π12 = 1 − π11 and π21 = 1 − π22. A (two-state) stationary sunspot

where F (y) = G(W−1(y)). Note that since the transformation y = W(n) is 1–1, the qualitative features of Figure 4.5 are preserved for the map F . In particular, there are three interior steady states yL, yU , yH .

7This is one way of modeling the dependence of expectations on “animal spirits,” as emphasized, for example, in the title of Howitt and McAfee (1992).

Figure 4.7.

An SSE (n1, n2) can equivalently be represented as an SSE (y1 , y2 ), i.e., a process yt = y1 if st = 1 and yt = y2 if st = 2 which satisfies equation (4.15). This is equivalent to the equations

y1 = π11F (y1 ) + (1 − π11)F (y2 ), y2 = (1 − π22)F (y1 ) + π22F (y2 ).

Note that geometrically an SSE exists if there exist y1 , y2 in the open interval formed by F (y1 ) and F (y2 ). (The transition probabilities must be chosen to correspond to y1 , y2 .) In particular, it is possible to construct sunspot solutions near a pair of two interior steady states. This is illustrated in Figure 4.7 for an arbitrary mapping F (y) which has two steady states. In a sunspot equilibrium of this kind the economy alternates stochastically between yt = y1 and yt = y2 with transition probabilities πij .

4.6.4Learning Sunspot Equilibria

Is it possible that agents in the economy can be led to such an equilibrium if they use a simple learning rule to make their forecasts? We will systematically examine this issue in Chapter 12.8 Here we briefly take up an example of sunspot equilibria and learning based on the model of increasing social returns.

As shown above, this model can have multiple interior steady states and sunspot equilibria which are near a pair of distinct steady states. For the analysis

8The seminal work of Woodford (1990) demonstrated that indeed learning can converge to sunspot solutions in the basic OG model.

of learning we return to the formulation in terms of labor supply, nt . A two-state Markov chain SSE is a process nt = n1 if st = 1 and nt = n2 if st = 2 which satisfies equation (4.14), i.e., the equations

n1 = W−1 π11G(n1) + (1 − π11)G(n2) ,

n2 = W−1 (1 − π22)G(n1) + π22G(n2) .

If agents try to learn the two values G(n1) and G(n2), a natural estimator for them is the following. Divide a sample of data G(n1), G(n2), . . . , G(nt ) into two groups in accordance with the realization of the sunspot process st . That is, for periods t in which st = 1 one puts the data point G(nt ) into the first group, and for periods in which st = 2 the data point is put in the second group. Then form the averages of the data points separately for each group, taking into account the number of observations in the group. The average value in each group is the estimate for G(ni ), i = 1, 2, respectively. That is, G(ni ) is estimated, for i = 1, 2, by

where Ni (t − 1) denotes the set of data points in periods j ≤ t − 1 for which sj = i and #Ni (t − 1) denotes the number of these points in periods 1, . . . , t − 1. The estimates for a group are then updated using the new data point, provided the realization of st corresponds to that group. If it does not, then the estimate is not changed. Finally, given these estimates, the economy evolves according to

Here Et G(nt+1) = πi1φ1,t + πi2φ2,t when st = i, because we are assuming that the transition probabilities πij are known. If they are not known, they can also be estimated.

We will study systematically the convergence of learning based on this type of learning rule in Chapter 12. Here we note the basic result for the above example: a sunspot equilibrium near a pair of distinct steady states is locally stable under learning if it is near two distinct steady states both of which, as steady states, are stable under learning. Applying this result to the model of increasing social returns, we get the result that a sunspot equilibrium (sufficiently) near the steady states (nL, nH ) is locally stable under learning. Such a pair (n1, n2) is shown for the graph F in Figure 4.5. In contrast, a sunspot is not stable if it is in the neighborhood of (nU , nH ), because the “middle” steady state nU is not stable under learning.

Figure 4.8.

We illustrate learning of sunspot equilibria using a simulation. Assume the production and utility functions given in the previous section. We set the following parameter values: σ = 0.1, ε = 0.25, a = 0.025, α = 0.9, β = 1, = 40, λ = 0.5, I = 14.1935, A = 0.0792, π11 = 0.98, π22 = 0.995. Initially, the subsidy level is set at zero. For these parameter settings the economy has three interior steady states and therefore there exist nearby SSEs. Agents use a recursive version of the above learning rule which conditions expectations on the observed sunspot state. The initial estimates of employment levels for the two states were 1 and 2.5.9

Figure 4.8 up to period 2000 illustrates how the economy learns the sunspot equilibrium at n1 = 1.020 and n2 = 2.263. At time t = 2000, an unanticipated regime change was made to occur: the subsidy level was raised to ρ = 0.05. For

9We remark that in the simulation we actually used a constant-gain learning rule with gain parameter value equal to 0.5. Constant gain can give convergence in this model since there is no intrinsic noise, and it allows for faster convergence to a new equilibrium following a regime switch. See Evans and Honkapohja (1993b) for further details on the simulations.

Finally, although nonlinear models are usually developed in a nonstochastic framework, it is perfectly possible to develop nonlinear models with stochastic productivity, taste, or policy shocks. Such models have REE solutions taking the form of noisy steady states or cycles or noisy SSEs. These solutions can also be investigated for stability under learning. Such models are discussed in Chapters 11 and 12.

4.8 Appendix

We here give results on adaptive learning for the Ramsey and Diamond models. Since the Ramsey and Diamond models are deterministic models, we give direct arguments for stability conditions.

4.8.1Learning in the Ramsey Model

We log-linearize the model (4.9) with the parametric form used in equation (4.8). (For details of this technique see Chapter 10.) We obtain

kt+1 = dcct + dkkt .

The REE for this model is given by the saddle path and locally takes the form

The gain parameter γt can be taken to be either a small fixed gain γ > 0 or a decreasing-gain sequence of the form γt = t−1.

Convergence can be verified numerically, e.g., using a fixed gain, and conver-

4.8.2Learning in the Diamond Model

We now consider learning in the Diamond model as specified in equations (4.10) and (4.11). We first linearize the equation (4.10) and rt = f (Kt ) at a steady state, which gives

Kt = ak Kt−1 + ar rte, rt = bk Kt ,

where ak = sw , ar = ws , and bk = f . Here w = f − Kf and note that ak > 0 and bk < 0. (The functions and the derivatives are evaluated at the steady state under consideration.) For convenience we take here the variables to be deviations from the steady state.

We examine the case of a small constant gain in the learning rule (4.11), so it is written in the form

rte+1 = rte + γ rt − rte .

The system is of second order. It is straightforward to show that for small enough values of the gain parameter γ , its eigenvalues are real and both have absolute value less than 1 if and only if ak < 1 and ak + ar bk < 1. These are, therefore, the conditions that determine local convergence under learning to the steady state in question.

In order to relate these stability conditions to the steady states shown by the perfect-foresight map in Figure 4.3, we compute the slope of the perfectforesight map Kt+1 = s(f (Kt+1))w(Kt ). Differentiating, we get the slope

It is easily verified that the steady state is stable under learning if and only if