Dynamic Consumption-Labor Framework
We build upon the dynamic consumption-savings framework by adding a labor/leisure decision
for each period. In other words, we are making the labor income endogenous in our dynamic
framework, allowing us to study consumption/savings decisions and labor/leisure decisions si-
multaneously.
1.1 Preferences
Preferences will be represented by the utility function:
V (c1, l1,c2, l2) = u(c1, l1) + βu(c2, l2) (1)
where V (·) is associated with well-behaved indi�erence curves, u(·) is a `sub-utility’ function, and ct and lt are consumption and leisure for t = 1,2.
→ We typically assume that u(·) takes the same function form each period. → Note that since V (·) is de�ned over 4 dimensions, we cannot graph it. Nonetheless, you can think of the associated indi�erence curves as simultaneously de�ned over the (c, l) dimensions
for t = 1,2, the (c1,c2) dimensions we considered in the previous chapter, and an additional
(l1, l2) that we have not yet considered. The indi�erence curves in each of those 4 dimensions
give us 4 MRS’s that are relevant for optimal choice.
1.2 Budget Constraints
Recall the budget constraint for the dynamic consumption-savings model in nominal terms:
Ptct + At = Yt + At−1(1 + i) (2)
for t = 1,2. The budget constraint here makes the additional speci�cation income Yt depends
on endogenous labor supply so that:
Yt = (1− τt)Wt(1− lt) (3)
where as in the static consumption-labor model, τt is a tax on wage income, Wt is the nominal
wage rate, and labor supply is given by time not spent on leisure so that nt = 1− lt. → Using equation (3) into equation (2) we have our period 1 and 2 budget constraints:
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Period 1: P1c1 + A1 = (1− τ1)W1(1− l1) + (1 + i)A0
Period 2: P2c2 + A2 = (1− τ2)W2(1− l2) + (1 + i)A1
From the above period budget constraints, we can derive the lifetime budget constraint by
isolating A1 in the period-1 budget constraint, and using that into the period-2 budget constraint.
Assuming the initial and terminal conditions A0 = A2 = 0:
P2c2 = (1− τ2)W2(1− l2) + (1 + i) ((1− τ1)W1(1− l1)−P1c1)
P2c2 1 + i
= (1− τ2)W2(1− l2)
1 + i + (1− τ1)W1(1− l1)−P1c1
⇒ P1c1 + P2c2 1 + i
= (1− τ1)W1(1− l1) + (1− τ2)W2(1− l2)
1 + i (4)
→ This lifetime budget constraint has the same interpretation as all of the others we have seen thus far. It equates the present discounted value of lifetime expenditures to the present
discounted value of lifetime resources.
→ While this lifetime budget constraint is in nominal terms, we can transform it into real terms either by applying the Fisher equation directly to equation (4) or applying the Fisher
equation to each of the period-t budget constraints.
→ There the relative slopes (and intercepts) of the lifetime budget constraint in the (c, l) dimensions for t = 1,2, the (c1,c2) dimension, and an additional (l1, l2) dimension. Each of
these slopes give the e�ective price ratio that is relevant for optimal choice.
1.3 Optimal Choice
The household chooses the optimal (c1∗,c2∗, l1∗, l2∗) bundle which maximizes their utility sub- ject to the lifetime budget constraint.
→ c1, c2, l1, and l2 are the endogenous variables → P1, P2, W1, W2, τ1, τ2 and i are the exogenous variables
As with the previous models, the optimal choice will be such that the Marginal Rate of Substitu-
tion between each choice variable is equal to the relative prices of those choice variables.
⇒There will be two Intratemporal Optimality Conditions that characterize the consumption- leisure choice for each period t = 1,2 and one Intertemporal Optimality condition that
characterizes the consumption-savings choice across periods.
⇒ The three optimality conditions together with the LBC fully characterize the solution to the model, (c1∗,c2∗, l1∗, l2∗).
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1.3.1 Lifetime Lagrangian Formulation
max c1,c2,l1,l2
u(c1, l1) + βu(c2, l2)
subject to: P1c1 + P2c2 1 + i
− (1− τ1)W1(1− l1)− (1− τ2)W2(1− l2)
1 + i = 0
⇒L = u(c1, l1) + βu(c2, l2) + λ ( P1c1 +
P2c2 1 + i
− (1− τ1)W1(1− l1)− (1− τ2)W2(1− l2)
1 + i
) Taking FOCs:
∂L ∂c1
= 0 −→ ∂u
∂c1 + λP1 = 0 −→
∂u
∂c1
1
P1 = −λ (5)
∂L ∂c2
= 0 −→ ∂u
∂c2 β + λ
( P2 1 + i
) = 0 −→
∂u
∂c2
β(1 + i)
P2 = −λ (6)
∂L ∂l1
= 0 −→ ∂u
∂l1 + λ(1− τ1)W1 = 0 −→
∂u
∂l1
1
(1− τ1)W1 = −λ (7)
∂L ∂l2
= 0 −→ ∂u
∂c2 β + λ
( (1− τ2)W2
1 + i
) = 0 −→
∂u
∂l2
β(1 + i)
(1− τ2)W2 = −λ (8)
The next step is to combine the FOCs to derive the optimality conditions, but given that there
are now four FOCs, you might be wondering how to combine them. The key is to keep in mind
what dimensions of optimization are relevant for each optimality condition.
→ We want (i) one intertemporal optimality condition for consumption-savings across peri- ods, and (ii) two intratemporal optimality conditions for consumption-leisure in each period.
→ Starting with (i), we want to use Equations (5) and (6):
∂u
∂c1
1
P1 =
∂u
∂c2
β(1 + i)
P2∂u∂c1 / ∂u∂c2 = β(1 + i)P1P2 (9)
⇒ Equation (9) is the intertemporal optimality condition in nominal terms. Note that if you apply the Fisher Equation to this, you would obtain it in real terms.
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→ Next for (ii), we want to use equation (5) and (7) for t = 1, and equations (6) and (8) for t = 2. Starting with t = 1:
∂u
∂c1
1
P1 = ∂u
∂l1
1
(1− τ1)W1
∂u
∂l1 / ∂u
∂c1 =
(1− τ1)W1 P1
(10)
⇒ Equation (10) is the intratemporal optimality condition for consumption-leisure in nomi- nal terms for period one.
→ And for (ii) in t = 2:
∂u
∂c2
1
P2 = ∂u
∂l2
1
(1− τ2)W2
∂u
∂l2 / ∂u
∂c2 =
(1− τ2)W2 P2
(11)
⇒ Equation (11) is the intratemporal optimality condition for consumption-leisure in nomi- nal terms for period two.
→ Note that we have not talked about an intertemporal optimality condition for leisure across periods. You should convince yourself that if the two intratemporal optimality conditions for
consumption leisure hold, there is implicitly a condition that holds relating the MRS for leisure
across periods to the relative price of leisure across periods.
