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ECON 5110 Class Notes
Learning
1 Introduction
This section relies heavily on the material in George Evans and Seppo Honkapohja’s book Learning and Expectations in Macroeconomics. Expectations of future economic variables play an important role in macroeconomic theory. Examples include the permanent-income lifecycle consumption hypothesis, monetary policy and asset pricing models. The evolution of expectations in macroeconomics can be classified as follows:
- Naive expectations. Under this mechanism, expectations of a future variable yt+1 are given by yet+1 = yt.
- Adaptive expectations. An example of adaptive expectations (AE) is
yet+1 = yet + λ(yt − yte )
where the parameter λ governs how current expectations adjust to the previous period’s forecasting errors. AE were commonly used in Keynesian models that dominated the macro landscape in the 1960s and 1970s. For example, the expectations-augmented Phillips curve often employed AE. In terms of policy, AE imply that policymakers can continually adjust policy instruments (such as government spending or the money supply) to manipulate macro aggregates.
- Rational expectations. The rational expectations (RE) revolution in macroeconomics began in the mid 1970s with the research of Robert Lucas and Thomas Sargent. It has dominated macroeconomic theory ever since. RE assumes economic agents are very sophisticated. They form expectations of future variables according to yet+1 = E(yt+1|Ωt) where E is the mathematical expectation operator and Ωt is the information set containing all informa- tion dated at time t and earlier. RE assumes agents know the structure of the economy and all relevant parameter values. In terms of policy, RE imply that policymakers are no longer able systematically manipulate macro aggregates — agents understand policymakers’ incentives to do so and adjust their behavior accordingly.
- Learning. Learning in macroeconomics is a reaction to the strong assumptions made with RE. In particular, it seems unreasonable to assume that economic agents know the relevant parameter values with certainty when even the best econometricians must themselves estimate the parameters. Learning generally assumes that, while agents are able to figure out the reduced-form equations governing the economy, they must continually update their estimates of the parameters. An interesting question is whether, through the learning process, agents can grope their way toward the rational expectations equilibrium (REE). Learning is also a useful tool to choose between multiple REE — only those that are stable under learning would be expected to be observed.
2 Learning Techniques
2.1 The Setup
Begin by considering a structural macroeconomic model (similar to the one discussed in the previous set of lecture notes):
yt = a + b 1 E t∗− 1 yt + b 2 E t∗− 1 yt+1 + cxt (1) xt = ρxt− 1 + et
where E t∗− 1 is some arbitrary expectations mechanism and E t∗− 1 et = 0. The REE solution takes the form
yt = φ¯ 0 + ¯φ 1 xt− 1 + ¯ηt (2)
where ¯ηt = cet. Assume now that agents do not know (¯φ 0 , ¯φ 1 ), but are able to figure the structure of equation (2). Agents instead specify a perceived law of motion (PLM)
yt = φ 0 + φ 1 xt− 1 + ηt (3)
where (φ 0 , φ 1 ) are the agents estimates of (φ¯ 0 , ¯φ 1 ). The actual law of motion (ALM) is found by substituting the forecasts for yt and yt+1 from the PLM into the structural model (1):
yt = a + b 1 [φ 0 + φ 1 xt− 1 ] + b 2 [φ 0 + φ 1 ρxt− 1 ] + cxt,
Substituting (7) into (6) gives
φt = φt− 1 + t−^1 R− t 1 zt− 1 ((T (φt− 1 )^0 − φ^0 t− 1 )zt− 1 + ηt) (8) Rt = Rt− 1 + t−^1 (zt− 1 z^0 t− 1 − Rt− 1 ), (9)
which is a recursive stochastic system. Showing convergence of this recursive least squares system is complicated (see chapter 6 of Evans and Honkapohja) and by no means obvious. Under learning, economic variables depend on agents’ econometric forecasts of a system, which in turn depends on their forecasts. This type of learning environment can lead to either divergence from or convergence to REE. Fortunately, the concept of expectational stability (E-stability) can be used to establish convergence (or lack thereof).
2.3 Expectational Stability
Before presenting the conditions necessary for E-stability, first note that the REE solution (φ¯ 0 , φ¯ 1 ) is a fixed point of the mapping φ = T (φ). We will show this explicitly in an example below. We say the REE is E-stable if the REE is locally asymptotically stable under the differential equation
d dτ
φ 0 φ 1
= T
φ 0 φ 1
φ 0 φ 1
where τ denotes artificial time. In other words, an REE is E-stable if small deviations from an REE under a perceived law of motion and a given learning rule, gradually return back to the REE. Using the framework above, we would look for the conditions under which the equations
dφ 0 dτ =^ a^ +^ φ^0 (b^1 +^ b^2 )^ −^ φ^0 =^ a^ +^ φ^0 (b^1 +^ b^2 −^ 1) dφ 1 dτ =^ φ^1 (b^1 +^ ρb^2 ) +^ cρ^ −^ φ^1 =^ φ^1 (b^1 +^ ρb^2 −^ 1) +^ cρ
generate stability in a neighborhood of the REE. Assuming that 0 ≤ ρ ≤ 1 , a sufficient condition for E-stability is b 1 + b 2 < 1.
3 Economic Applications
3.1 Cobweb Model
Consider a competitive market for a single good. The demand for the good is given by
dt = α 0 − α 1 pt + νdt
and since there is a production lag, supply depends on expected price
st = β 0 + β 1 E t∗− 1 pt + νst
where νdt and νst are mutually uncorrelated, mean-zero white-noise shocks. Assuming markets clear (i.e., dt = st), then we have the reduced-form equation
pt = a + bE∗ t− 1 pt + ηt
where a = (α 0 − β 0 )/α 1 , b = −β 1 /α 1 < 0 , and ηt is mean-zero white noise.
3.1.1 Naive Expectations
Under naive expectations (E t∗− 1 pt = pt− 1 ), we have
pt = a + bpt− 1 + ηt. (11)
There are two cases:
- Irregular case. If the supply curve is steeper than the demand curve (i.e., 0 > b > − 1 ), then equation (11) is a stationary stochastic process, the equilibrium is indeterminate and the fixed-point p = (1−b)−^1 a is a "sink". (Note: This appears to be inconsistent with the results in the previous set of notes, but notice that equation (11) is written in its backward-looking, as opposed to forward-looking, form).
- Regular case. Conversely, if the demand curve is steeper than the supply curve (i.e., b < − 1 ), the unique, fundamental equilibrium is pt = (1 − b)−^1 a + ˜ηt, a noisy steady-state.
3.2 Lucas Aggregate Supply Model
Lucas’ aggregate supply function is
qt = q + π(pt − E t∗− 1 pt) + t (15)
where qt is aggregate output, pt is the price level, π, q > 0 and t is mean-zero white noise. Aggregate demand is derived from the quantity equation
mt + vt = pt + qt (16)
where vt is a velocity shock and mt is the money supply, which is white noise around a constant mean m
mt = m + μt. (17)
All variables are measured in logarithms. Some simple algebra produces the reduced form
pt = a + bE∗ t− 1 pt + ηt (18)
where a = m 1 +^ − π q, b = (^) 1 +π π and ηt = (^) 1 +^1 π (μt + νt − t).
Since equation (18) is in the same form as the Cobweb equation, it has the same condition for stability under learning b < 1 =⇒ π < (1 + π).
This condition is satisfied so that the REE from the Lucas supply model is always stable under learning.
3.3 Ramsey Model
3.3.1 Framework
Consider a discrete-time version of the Ramsey growth model, which abstracts from population growth, technology shocks and depreciation. Labor supply (Nt) is normalized to one. The representative agent maximizes E∗ t^ P∞ i=0 βt+i(1 − σ)−^1 Ct^1 +−iσ
subject to Ct + Kt+1 = wt + (1 + rt)Kt.
Firms, given the CRS production function f (Kt) = Ktα , maximize profits given by
f (Kt) − rtKt − wt.
This produces the standard Euler equations
rt = f 0 (Kt) wt = f (Kt) − Ktf 0 (Kt).
Plugging these into the consumer’s problem (and assuming perfect foresight) gives
Ct+1 = Ct[β(1 + α(Kt + Ktα − Ct)α−^1 )]^1 /σ Kt+1 = Kt + Ktα − Ct.
The Ramsey model has a unique equilibrium, involving a saddle path that converges to a non-stochastic steady state ( C,¯ K¯). In other words, for a given K 0 , there is a unique choice of C 0 that will put the economy on a convergent path to the steady state. All other choices for C 0 will lead to divergent paths that violate some non-negativity constraint or transversality condition.
3.3.2 Learning
Now let’s introduce some uncertainty and learning. Given the knife-edge nature of the equilibrium, it is an open question as to whether the economy will converge to the rational expectations equilibrium when agents start with non-rational expectations and use some sort of adaptive learning. Begin by linearizing the system
ˆct = a 1 E∗ t ˆct+1 + a 2 E t∗ ˆkt+1 (19) ˆkt+1 = b 1 cˆt + b 2 ˆkt. (20)
