Household’s Problem
The problem of the representative household is given by max
"∞ X
Et
{Ct+τ ,ht+τ ,`t+τ ,Bt+τ }
# τ
β U (Ct+τ , `t+τ )
τ =0
subject to ht+τ + `t+τ 6 1 Bt+τ + Pt+τ Ct+τ 6 Rt+τ −1 Bt+τ −1 + Wt+τ ht+τ + Ωt+τ Using the fact that the utility function is strictly increasing in consumption and leisure (nonsatiation), the two constraints are necessarily binding, such that the problem rewrites max
Et
"∞ X
{Ct+τ ,ht+τ ,Bt+τ }
# τ
β U (Ct+τ , 1 − ht+τ )
τ =0
subject to Bt+τ + Pt+τ Ct+τ = Rt+τ −1 Bt+τ −1 + Wt+τ ht+τ + Ωt+τ This program can be solved either using dynamic programming arguments or the dynamic Lagrangian approach. We will use the second and form the Lagrangian L = Et
X ∞
β τ U (Ct+τ , 1 − ht+τ ) + Λt+τ Rt+τ −1 Bt+τ −1 + Wt+τ ht+τ + Ωt+τ − Pt+τ Ct+τ − Bt+τ
τ =0
In fact one can focus on choosing Ct , ht and Bt , because the problem of the household does not change from one period to the next such that the decisions are the same in each and every period. But then, the actual problem that needs to be solved for each variable can be greatly simplified. Consumption decision (Ct ): Let us define L c , the component of the Lagrangian that only features Ct . This component is simply obtained by setting τ = 0 and focusing on the terms involving Ct only.
L = Et U (Ct , 1 − ht ) − Λt Pt Ct c
Note that, in the beginning of period t, the household observes all the shocks that can drive the behavior of prices. This implies that the household does not have to formulate a forecast for the price Pt or the shadow value Λt . Likewise, given that the household is deciding both Ct and ht , 1
these are perfectly known to her. Hence, all variables involved in L c are known to the household and the expectation term is actually useless in the expression of L c which simplifies to L c = U (Ct , 1 − ht ) − Λt Pt Ct The first order conditions associated to the consumption decision is then given by ∂U (Ct , `t ) = Λt Pt ∂Ct
(1)
Just like in the case of consumption, let us define L h , the
Labor supply decision (ht ):
component of the Lagrangian that only features ht
L = Et U (Ct , 1 − ht ) + Λt Wt ht h
As for consumption, in the beginning of period t, the household observes all the shocks that can drive the behavior of wages. The household therefore does not have to forecast its value, nor that of Λt . Hence, all variables involved in L h are known to the household and the expression of L h simplifies to L h = U (Ct , 1 − ht ) + Λt Wt ht The first order conditions associated to the labor supply decision is then given by ∂U (Ct , `t ) = Λt Wt ∂`t Asset decision (Bt ):
(2)
As for consumption and labor, let us define L B , the component of the
Lagrangian that only features Bt . In this case, Bt appears in period t (at the time the household decides on bonds, τ = 0) and in period t + 1 (at the time she receives the bond income, τ = 1) "
L
B
#
= Et βΛt+1 Rt Bt − Λt Bt
(3)
Given that the household observes the shadow price, Λt , the interest rate, Rt , and decides on Bt , these variables are not random variables from the point of view of the household. They can therefore be pulled out of the expectation term. This is however not the case for Λt+1 that will only be known in period t + 1. The expectation terms shall then be maintained for this variable, such that L B rewrites L B = βEt [Λt+1 ]Rt Bt − Λt Bt The first order conditions associated to the bond accumulation decision is then given by Λt = βRt Et [Λt+1 ]
Transversality Condition: The transversality condition is given by lim β τ Et [Λt+τ Bt+τ ] = 0
τ →∞
2
(4)
Restating the system: Using Equation (1) to eliminate Λt , the system reduces to ∂U (Ct , 1 − ht ) Wt ∂U (Ct , 1 − ht ) = ∂`t ∂Ct Pt ∂U (Ct , 1 − ht ) Pt ∂U (Ct+1 , 1 − ht+1 ) = βRt Et ∂Ct Pt+1 ∂Ct+1 ∂U (Ct+τ , 1 − ht+τ ) Bt+τ lim β τ Et =0 τ →∞ ∂Ct+τ Pt+τ
(5) (6) (7)
Equation (5) determines the labor supply behavior of the household. By allocating a marginal unit of hours in productive activities, the household reduces her leisure time and therefore suffers a utility loss of
∂U (Ct ,1−ht ) . ∂`t
However, by supplying her labor in the productive sector she earns
extra real wage, Wt /Pt , which enables her to buy additional consumption good which increases her utility by
∂U (Ct ,1−ht ) Wt /Pt . ∂Ct
The household supplies labor until the gains and costs are
equalized. Equation (6) characterizes the consumption/saving behavior of the household. Assume the household is given an extra unit of good today. She can consume it immediately and therefore enjoy an extra utility of
∂U (Ct ,1−ht ) . ∂Ct
But she can also decide to postpone consumption and
invest this extra unit of good into bonds and earn the risk free return Rt . This will enable her to ,1−ht+1 ) purchase additional consumption good and therefore enjoy a total utility gain of Rt ∂U (Ct+1 . ∂Ct+1
This is obtained for a given value of the shocks tomorrow, and obviously shocks are drawn from a distribution, such that the gains the household can expect is given by Rt Et
h
∂U (Ct+1 ,1−ht+1 ) ∂Ct+1
i
.
Since this gain will be earned tomorrow, it has to be expressed in current period units, which is achieved through discounting. In other words, should the left hand side of the equation be greater (lower) than the right hand side, the household will consume (invest) the extra unit of good. Optimal behavior is achieved when no arbitrage is left. Note that this behavior exactly corresponds to the consumption smoothing behavior. The last condition, Equation (7), states that in the infinite of time, it is not beneficial to further accumulate wealth. One way to intuitively understand this condition is to think about an agent with finite life. Assume our agent is totally selfish and aware that she will die in the next period. In this case, her best interest is to totally eat any remaining wealth, such that in the next period (when she is dead) nothing is left unused and wasted. Hence, the utility gain of accumulation should be nil. Since our agent never dies, this will only occur in the limit. Viewed from today, this transversality condition states that the discounted expected utility from wealth accumulation is nil in the infinite of time. This actually gives us a terminal condition for the problem of the agent. Note that using equation (5) and (6), we obtain ∂U (Ct , 1 − ht ) Pt ∂U (Ct+1 , 1 − ht+1 ) Pt+1 = βRt Et ∂`t Wt ∂`t+1 Wt+1
To understand this relationship, let us again consider that the household is given one extra unit of the good. Since, the wage income is expressed in terms of good, Pt /Wt corresponds to 3
the transformation rate that converts units of the good in units of time. In other words, by applying Pt /Wt , the household can instantaneously turn good into leisure time, which yields her extra utility
∂U (Ct ,1−ht ) . ∂`t
However, she can invest this extra unit of good and get the yield
Rt . This will yields further extra discounted income βRt , which she will be able to turn into leisure applying next period transformation rate Pt+1 /W t + 1. She will then be able to enjoy more leisure time and get extra utility
∂U (Ct+1 ,`t+1 ) Pt+1 ∂`t+1 Wt+1 .
Hence tomorrow’s expected gain is
given by the right hand side of the preceding equation. At the end of the day, this intertemporal trade-off is akin to the consumption/saving tradeoff. In other words, the household is willing to smooth her leisure consumption profile (and hence her labor supply) the same way she was willing to smooth her consumption profile.
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