## Information

- Time and Date: 10:00-16:55, Friday December 19, 2008 (Open: 9:45 a.m.)
- Venue: RIETI's seminar room (1121,11th Floor, METI ANNEX)

1-3-1 Kasumigaseki, Chiyoda-ku, Tokyo - Language: English (no interpretation available)

## Summary

### Session 6: "Optimal Monetary Policy When Asset Markets are Incomplete"

**NAKAJIMA Tomoyuki (Associate Professor, Institute of Economic Research, Kyoto University / Faculty Fellow, RIETI)**

As is now well-known, in the standard sticky price model the optimal monetary policy is at least approximately given by complete inflation stabilization. In other words, concerning the output-inflation tradeoff, results show that monetary authorities should place exclusive weight on inflation stabilization. But such a representative-agent framework may not be suitable for studying the inflation-output tradeoff.

Recent empirical studies show that idiosyncratic income shocks are very persistent, and that their variance fluctuates countercyclically. Theoretical papers show that if there is such countercyclical idiosyncratic risk, the welfare cost of business cycles can be very large. The main purpose of this paper is to study how the existence of such countercyclical idiosyncratic risk affects how monetary policy should be conducted. One particular focus was on how such countercyclical idiosyncratic risk affects how much weight the monetary authorities should place on inflation stabilization.

In this paper we consider a model where individuals face uninsured idiosyncratic income shocks with countercyclical variance. The model is otherwise a standard new-Keynesian model with monopolistic competition, Calvo-price setting, and capital accumulation. In this kind of framework, we study optimal monetary policy (Ramsey policy).

The main findings are as follows: (1) countercyclical idiosyncratic risk can generate a very large welfare-cost of business cycles, but it does not have much affect on the inflation-output tradeoff; and (2) it continues to be the case that the optimal monetary policy is essentially characterized as complete price-level stabilization, so that the monetary authorities should place almost exclusive weight on the stabilization of inflation, in spite of both the presence of countercyclical idiosyncratic risk and the very large welfare-cost of business cycles.

Let me describe our model. First, as in the standard new-Keynesian model, a composite good is made up from many individual products. One single composite good can be consumed or invested. From this aggregate, a price index can be constructed. We assume there is a continuum of ex ante identical individuals, and each personalized preference is given by the lifetime function.

In general, with uninsured idiosyncratic shocks the wealth distribution, which is an infinite-dimensional object, must be included in the state variable. This may cause the so-called cost of dimensionality problem, but that can be circumvented by making two assumptions to allow for the model to be solved relatively easily. The first assumption, which is common in the pure exchange economy with idiosyncratic shocks, is that idiosyncratic shocks follow random walk processes. The second is that idiosyncratic shocks affect both labor and capital income.

We set η*i*
,*t *
equal to the idiosyncratic shock for individual *i*
. We assume the component η*i*
follows a geometric random walk. So εη,*i*
,*t*
is independently and identically distributed, and normal. Also ση,*t*
can be interpreted as a standard deviation of idiosyncratic shocks, and we will assume that ση,*t *
fluctuates countercyclically over time to capture the fact that idiosyncratic shocks are countercyclical.

We assume that η*i*
,*t *
affects *i*
's income in two ways. First, η*i*
,*t*
affects the labor income, and second, η*i*
,*t *
also affects income from savings. The first assumption is that η*i*
,*t*
equals the productivity of individualized labor. We also assume that this idiosyncratic shock also affects the return to individual savings. In this model, individuals have two ways to save: one is to purchase physical capital; the other is to purchase equity in firms. With this assumption, the effect of the presence of idiosyncratic shocks would be overemphasized. Our main message is that the tradeoff faced by the monetary authority is very little affected by the presence of idiosyncratic shocks, so dropping this artificial assumption would strengthen our result. For this reason, making this assumption is not a very big problem. Under this assumption, we can show that individuals can be aggregated into a representative-agent with this kind of preference. Our individuals with idiosyncratic income shocks can be aggregated into a single representative-agent with preference shocks. The preference shock consists of the variance of idiosyncratic income shocks.

The representative-agent's utility is a cross-sectional average of individual utility. Given this aggregation result, we can very easily see how the presence of idiosyncratic shocks affects the aggregate economy. Indeed, idiosyncratic shocks affect the aggregate economy through their impact on the effective discount factor.

Individuals agree on the present value of the profit stream of each firm, so there is no disagreement problem. That is why we can easily incorporate this model into a standard new-Keynesian model. The profit maximization of monopolistic firms is uniquely defined.

In terms of government, our assumption is simply government, with no taxes. The government issues no debt and there is no fiscal policy. In terms of monetary policy, we assume that the government can set a state-contingent path of the inflation rate. We consider two monetary policy regimes. The first one is the so-called Ramsey regime, where the path of inflation rate is determined so as to maximize the ex ante utility of individuals, making this the optimal monetary policy. The second regime is called the inflation-targeting regime, where the monetary authority chooses a zero inflation rate all the time. Setting a zero inflation rate all the time is not optimal policy, but the difference between this policy and the optimal policy is very, very small, according to the previous literature.

We sought to determine whether the conclusion changes in our model with idiosyncratic shocks. We computed two welfare costs: the first is the welfare cost of business cycles (using the real business cycle of our model) and the second is the welfare cost of the inflation-targeting regime, relative to the optimal policy measure. The results for permanent productivity shock show that when the risk aversion coefficient is higher, the welfare cost of the business cycle is positive. In this case, the welfare cost of inflation-targeting is effectively zero (0.006 per cent), which is much smaller than the welfare cost of business cycles.

In the case of temporary productivity shock, with relatively low risk aversion for countercyclical idiosyncratic shocks, the welfare costs of business cycles are negative. But with relatively high risk-aversion parameters and relatively high countercyclicality of idiosyncratic shocks, the welfare costs of business cycles become very large at around 12%.

In conclusion, we found that the welfare cost of business cycles can be very large when the variance of idiosyncratic shocks fluctuates countercyclically. Nevertheless, the optimal monetary policy is roughly the same as the zero inflation policy, and the presence of countercyclical idiosyncratic shocks may not affect the inflation-output tradeoff.

### Question and Answer Session

**Q. **
This is an interesting exercise, but the motivation is misleading. Your motivation has to do with the finding that it is optimal, at least in the context of a basic model, to fully stabilize inflation. You have interpreted this as if the central bank did not care about output fluctuations. But this is misleading in the sense that in the basic model, stabilizing inflation corresponds to stabilizing the output gap. So it is not that the central bank does not care about the level of economic activity and the fluctuations in economic activity, just that stabilizing one implies fully stabilizing the other. So this leads you to consider a clash of models, with incomplete markets. But when we compute the welfare costs of business cycles, the inquiry is about the welfare consequences of experiencing shocks, versus the steady state. When we look at the optimal monetary policy exercises, we do not ask ourselves that question, because the shocks are taken as given by the central bank. So what the central bank wants to figure out is the optimal policy, given that those shocks are there.

Your model is very interesting in terms of this question. Take the basic new-Keynesian model and add idiosyncratic shocks, which are uninsurable, how does that affect the optimal policy of the central bank? You have shown that it has little effect, and I think that is a very interesting result. The way to think about the logic behind this result should focus on a different issue, namely, what is the gap between what we call the natural level of output and the efficient level of output. The latter, in your model, will not correspond to the natural level of output - because of the market incompleteness the natural level of output will be inefficient. It is the existence of variations in this gap, in these two variables, that will generate the real tradeoff. If those variations were large, then it would not be optimal for the central bank to fully target inflation because targeting inflation means equating output to the natural level of output. If the natural level of output is far from the efficient level of output, that cannot be too good. In your model, the difference between the efficient level of output and the natural level of output is very small.

**Q. **
Regarding the way you set up the model with representative-agent foundations, does that not actually wash out most of the impact of the idiosyncratic shocks and incomplete markets. How much do you lose by breaking up the model so that you have the representative-agent?

You are looking at the average level of utility, but there could be some distributional issues there. Because of the incomplete markets, is there any theorem that will tell me that the proper welfare is the average level of utility?

**NAKAJIMA Tomoyuki**

Retrospectively, the motivation turns out not to be very good. The motivation should be modified.

I also agree that looking at the natural rate and the efficient rate would be very informative. I agree that based on the results that we have obtained so far, the difference between the natural rate and the efficient rate may be constant.

I honestly do not know what would happen without those two assumptions. But my understanding of this model is that we overemphasize the factor for idiosyncratic shocks. That may not be true, but at least our conjecture is that our model overemphasizes the effect of idiosyncratic shocks, so dropping these assumptions will strengthen our claim - that is our guess. It may or may not be true.

Regarding the welfare measure, we assume that everyone is identical, but in the general context, I agree that a simple average may not be an adequate measure.

**Q. **
Does the model have a wealth distribution and equilibrium?

**NAKAJIMA Tomoyuki**

Equilibrium - yes. There is no stationary distribution. The wealth distributions diverge.