A number of recent papers (Milani, 2011, Eusepi and Preston, 2011) include an intercept in the learning rule, interpreted as capturing households’ uncertainty about the steady state. It is straightforward to augment the model of this paper with an intercept by changing the measurement vector (15) to ist = h
1 rt wts i0
, (see the Appendix for details). When this is done the restricted-perceptions equilibrium (which is unchanged) remains e-stable and the convergence properties of the model are very similar.
However the addition of an intercept has signi…cant e¤ects on the properties of the model - table 8 shows the moments for di¤erent gains. Now at a gain of 0:001 consump-tion and output are around 10% more volatile than at the RPE, in contrast to the model without an intercept in which there was almost no ampli…cation.
[TABLE 8 HERE]
To see why this happens, …rstly consider the steady state distribution of beliefs.
Figure 8 shows the distribution of beliefs about the intercept of the wage equation (the distributions of the other components of beliefs are broadly similar to those shown in
…gure 5). Note the high volatility of beliefs: this is a consequence, from (9), of the high volatility of the idiosyncratic shock.
[FIGURE 8 HERE]
Why should uncertainty about the intercept translate into high volatility? To answer this, modify the simple example of section 5.3 to include an intercept. Consumption is
then
ct= r
1 + r (1 + r) bt+ 1
1 (1 + r) 1 1+ 1
1 t(1 + r) 1yt (48) where the second term picks up the e¤ect of the intercept, a discounted forward sum of a constant. For the discount factor of the baseline calibration, 1 (1+r)1 1 100 which, if the persistence of income is 0:9 is around 10 times higher than the coe¢ cient on income.
So variations in 1 are greatly ampli…ed21. This happens to such an extent that for values of the gain greater than 0:005 , the economy becomes unstable so no values are reported.
Why does this lead to instability? Imagine a household learning a large positive value for the constant. This means their consumption increases dramatically, which will increase aggregate consumption. Other things equal, higher aggregate consumption means lower aggregate capital and lower aggregate labour supply, so the return to capital and the wage increase and in the next period which will increase 1 further; and hence consumption still further leading to instability.22 Such instability is more of a problem in a heterogeneous agent economy than in one with identical agents since the higher volatility of idiosyncratic shocks is translated into wider distributions of beliefs so a higher probability of a draw leading to instability. It also arises only if households are very forward looking - in models that take the "Euler equation learning" approach (for example Milani, 2011, which uses an intercept) it is not an issue.
While using a low gain, as in Eusepi and Preston (2011) avoids this problem, it seems informational implausible for two reasons. Firstly, a gain of 0:001 implies that households place around half as much weight on data two centuries old as they do on new information23. Secondly, as shown in section 5.1, with a gain of 0:001 the economy takes many thousands of periods to converge to the RPE - so starting the economy from the steady state distribution is equivalent to endowing households with the very knowledge they are supposed to be learning.
6 Discussion
This paper has taken a model in which agents have limited information, both about the structure of the economy and the variables relevant to their decisions. Despite this, the economy is shown to converge quickly to an equilibrium which is similar to the equilibrium with model-consistent expectations. Learning does not introduce strong dynamics at the
21This also explains the strong e¤ect of updating beliefs present in Eusepi and Preston (2011) but absent from the impulse response of this paper - a small change in the element of relating to the intercept has a massively ampli…ed e¤ect on consumption.
22Clearly this will only lead to instability of the aggregate economy if households are su¢ ciently large, however even with 20; 000 households the economy shows instability with gains much above 0:005 and computational constraints prevent more households being used.
23The weight on information periods old is (1 ) .
aggregate level, though more interesting things happen at a household level. Another way of putting this is, at least in the context of this model, assumptions about information are important for aggregates; assumptions about the ability of households to form model-consistent expectations less so.
One strength of the approach taken in this paper is that the informational assumptions are clear. Households’ information sets are constrained by the markets in which they trade and they use all the information at their disposal to make optimal decisions. They have no other knowledge either of the structure of the aggregate economy or of the values of aggregate variables.
On the one hand, as pointed out in Graham and Wright (2010) the assumption of market-consistent information is itself a strong one - households clearly have many other sources of information than factor prices - but adding such information (for example a noisy signal of output) would only further reduce the impact of learning. On the other hand, section 5.7 showed that if an intercept is added to the learning rule, learning can have a bigger e¤ect on the properties of the economy. However this comes at the cost of instability unless the gain is small, and although plausible, the addition of an intercept seems arbitrary. This is related to a point made by Grandmont (1998) on the speci…cation of perceived laws of motion. What variables should be included in them? What econometric speci…cations should be used? Such choices would be far more complicated if the model included features such as non-linearity, structural breaks or non-ergodic shocks.
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