Análisis de la entrevista inicial (EI)

The sequence of events is as follows. At the beginning of period t, the capital stock K_t is inherited from the last period, given decisions from the last period. Then, total

175

factor productivity is revealed. With this knowledge, …rms choose factor employment and households choose consumption (and thereby savings).

c

w

r

K

A

Figure 8.1.1 Timing of events

Hence, only “at the end of the day”does one really know how much one has produced.

This implies that wages and interest payments are also only known with certainty at the end of the period.

The state of the economy in t is completely described by Kt and the realization of At: All variables of the model are contingent on the state.

8.1.3 Firms

As a consequence of this timing of events, …rms do not bear any risk and they pay the marginal product of labour and capital to workers and capital owners at the end of the period,

w_t = p_t@Y_t

@Lt

; (8.1.3)

r_t= p_t@Y_t

@K_t = p_tA_t L_t K_t

1

: (8.1.4)

All risk is therefore born by households through labour and capital income.

In what follows, the price will be set to unity, pt 1. All other prices will therefore be real prices in units of the consumption good.

8.1.4 Intertemporal utility maximization

This is the …rst time in this book that we encounter a maximization problem with un-certainty. The presentation will therefore be relatively detailed in order to stress crucial features which are new due to the uncertainty.

General approach

We consider an agent that lives for two periods and works and consumes in a world as just described. Agents consume in both periods and choose consumption such that they maximize expected utility. In all generality concerning the uncertainty, she maximizes

max E_tfu (c^t) + u (c_t+1)g ; (8.1.5)

where is the subjective discount factor that measures the agent’s impatience to con-sume. The expectations operator has an index t, similar to E0 in (7.4.3), to indicate that expectations are based on the knowledge concerning random variables which is available in period t. We will see in an instant whether the expectations operator Et is put at a meaningful place in this objective function. Placing it in front of both instantaneous consumption from ct and from ct+1 is the most general way of handling it. We will also have to specify later what the control variable of the household is.

Imagine the agent chooses consumption for the …rst and the second period. When consumption is chosen and given wage income wt; savings adjust such that the budget constraint

w_t = c_t+ s_t (8.1.6)

for the …rst period holds. Note that this constraint always holds, despite the uncertainty concerning the wage. It holds in realizations, not in expected terms. In the second period, the household receives interest payments on savings made in the …rst period and uses savings plus interests for consumption,

(1 + r_t+1) s_t= c_t+1: (8.1.7)

One way of solving this problem (for an alternative, see the exercise) is to insert consumption levels from these two constraints into the objective function (8.1.5). This gives

maxst

E_tfu (w^t s_t) + u ((1 + r_t+1) s_t)g :

This nicely shows that the household in‡uences consumption in both periods by choosing savings st in the …rst period. In fact, the only control variable the household can choose is st:

Let us now take into consideration, as drawn in the above …gure, that consumption takes place at the end of the period after revelation of productivity At in that period.

Hence, the consumption level in the …rst period is determined by savings only and is thereby certain. Note that even if consumption ct (or savings) was chosen before reve-lation of total factor productivity, households would want to consume a di¤erent level of consumption ct after At is known. The …rst choice would therefore be irrelevant and we can therefore focus on consumption choice after revelation of uncertainty right away.

The consumption level in the second period is de…nitely uncertain, however, as the next period interest rate rt+1 depends on the realization of At+1 which is unknown in t when decisions about savings st are made. The objective function can therefore be rewritten as

maxst

u (w_t s_t) + E_tu ((1 + r_t+1) s_t) :

For illustration purposes, let us now assume a discrete random variable Atwith a …nite number n of possible realizations. Then, this maximization problem can be written as

maxst

u (w_t s_t) + ⁿ_{i=1 i}u ((1 + r_i;t+1) s_t)

where i is the probability that the interest rate ri;t+1is in state i in t+1: This is the same probability as the probability that the underlying source of uncertainty At is in state i:

The …rst-order condition then reads

u⁰(w_t s_t) = ⁿ_{i=1 i}u⁰((1 + r_i;t+1) s_t) (1 + r_i;t+1) = E_tu⁰((1 + r_t+1) s_t) (1 + r_t+1) (8.1.8) Marginal utility of consumption today must equal expected marginal utility of consump-tion tomorrow corrected by interest and time preference rate. Optimal behaviour in an uncertain world therefore means ex ante optimal behaviour, i.e. before random events are revealed. Ex post, i.e. after resolution of uncertainty, behaviour is suboptimal when compared to the case where the realization is known in advance: Marginal utility in t will (with high probability) not equal marginal utility (corrected by interest and time prefer-ence rate) in t + 1. This re‡ects a simple fact of life: “If I had known before what would happen, I would have behaved di¤erently.”Ex ante, behaviour is optimal, ex post, prob-ably not. Clearly, if there was only one realization for At, i.e. ₁ = 1 and i = 08 i > 1;

we would have the deterministic …rst-order condition we had in exercise 1 in ch. 2.

The …rst-order condition also shows that closed-form solutions are possible if marginal utility is of a multiplicative type. As savings st are known, the only quantity which is uncertain is the interest rate rt+1: If the instantaneous utility function u (:) allows us to separate the interest rate from savings, i.e. the argument (1 + ri;t+1) s_tin u⁰((1 + r_i;t+1) s_t) in (8.1.8), an explicit expression for st and thereby consumption can be computed. This will be shown in the following example and in exercise 6.

An example - Cobb-Douglas preferences

Now assume the household maximizes a Cobb-Douglas utility function as in (2.2.1). In contrast to the deterministic setup in (2.2.1), however, expectations about consumption levels need to be formed. Preferences are therefore captured by

E_tf ln c^t+ (1 ) ln c_t+1g : (8.1.9) When we express consumption by savings, we can express the maximization problem by max_stE_tf ln (w^t s_t) + (1 ) ln ((1 + r_t+1) s_t)g which is identical to

maxst

ln (wt st) + (1 ) [ln st+ Etln (1 + rt+1)] : (8.1.10) The …rst-order condition with respect to savings reads

w_t s_t = 1

s_t (8.1.11)

and the optimal consumption and saving levels are given by the closed-form solution c_t= w_t; c_t+1 = (1 ) (1 + r_t+1) w_t; s_t= (1 ) w_t; (8.1.12)

just as in the deterministic case (2.4.7).

Thus, despite the setup with uncertainty, one can compute a closed-form solution of the same structure as in the deterministic solutions (2.2.4) and (2.2.5). What is peculiar here about the solutions and also about the …rst-order condition is the fact that the ex-pectations operator is no longer visible. One could get the impression that households do not form expectations when computing optimal consumption paths. The expectations operator “got lost” in the …rst-order condition only because of the logarithm, i.e. the Cobb-Douglas nature of preferences. Nevertheless, there is still uncertainty for an indi-vidual being in t : consumption in t + 1 is unknown as it is a function of the interest-rate in t + 1.

Exercise6shows that closed-form solutions are possible also for the CRRA case beyond Cobb-Douglas.