Question

It is relatively easy to extend the single-period model of labor supply presented in Chapter 16 to many periods. Here we look at a simple example. Suppose that an individual makes his or her labor supply and consumption decisions over two periods. \(^{14}\) Assume that this person begins period 1 with initial wealth \(W_{0}\) and that he or she has 1 unit of time to devote to work or leisure in each period. Therefore, the two-period budget constraint is given by \(W_{0}=c_{1}+c_{2}-w_{1}\left(1-h_{1}\right)-w_{2}\left(1-h_{2}\right),\) where the \(w^{\prime}\) s are the real wage rates prevailing in each period. Here we treat \(w_{2}\) as uncertain, so utility in period 2 will also be uncertain. If we assume utility is additive across the two periods, we have \(E\left[U\left(c_{1}, h_{1}, c_{2}, h_{2}\right)\right]=U\left(c_{1}, h_{1}\right)+E\left[U\left(c_{2}, h_{2}\right)\right]\) a. Show that the first-order conditions for utility maximization in period 1 are the same as those shown in Chapter \(16 ;\) in particular, show \(\operatorname{MRS}\left(c_{1} \text { for } h_{1}\right)=w_{1}\) Explain how changes in \(W_{0}\) will affect the actual choices of \(c_{1}\) and \(h_{1}\) b. Explain why the indirect utility function for the second period can be written as \(V\left(w_{2}, W^{*}\right),\) where \(W^{*}=W_{0}+\) \(w_{1}\left(1-h_{1}\right)-c_{1} .\) (Note that because \(w_{2}\) is a random variable, \(V\) is also random.) c. Use the envelope theorem to show that optimal choice of \(W^{*}\) requires that the Lagrange multipliers for the wealth constraint in the two periods obey the condition \(\lambda_{1}=E\left(\lambda_{2}\right)\) (where \(\lambda_{1}\) is the Lagrange multiplier for the original problem and \(\lambda_{2}\) is the implied Lagrange multiplier for the period 2 utility-maximization problem \() .\) That is, the expected marginal utility of wealth should be the same in the two periods. Explain this result intuitively. d. Although the comparative statics of this model will depend on the specific form of the utility function, discuss in general terms how a governmental policy that added \(k\) dollars to all period 2 wages might be expected to affect choices in both periods.

Short answer

Answer: The optimal choice condition for an individual's wealth in a two-period model requires that the expected marginal utility of wealth should be the same in both periods. This ensures that the individual maximizes their utility by allocating their wealth across the two periods to equalize their marginal utilities. A government policy that adds a constant amount to all period 2 wages will lead to higher expected wealth in the second period, affecting choices in both periods. The changes in labor supply and consumption will depend on the relative strengths of income and substitution effects in both periods. In general, the individual may choose to adjust their consumption and work more or less in both periods to smooth their consumption and leisure over the two periods.

Step-by-step solution

Set up the problem

We have the two-period budget constraint: $$W_0 = c_1 + c_2 - w_1(1-h_1) - w_2(1-h_2),$$ and the expected utility function: $$E[U(c_1, h_1, c_2, h_2)] = U(c_1, h_1) + E[U(c_2, h_2)].$$ We want to maximize the expected utility function subject to the budget constraint.

Introduce the Lagrange multiplier and set up the Lagrangian

Introducing the Lagrange multiplier \(\lambda_1\), the Lagrangian for the problem is: $$\mathcal{L} = U(c_1, h_1) + E[U(c_2, h_2)] + \lambda_1 \left(W_0 - c_1 - c_2 + w_1(1-h_1) + w_2(1-h_2)\right).$$

Find the first-order conditions

Differentiate the Lagrangian with respect to \(c_1\), \(h_1\), \(c_2\), and \(h_2\) and set the derivatives equal to zero. For period 1, we get: $$\frac{\partial \mathcal{L}}{\partial c_1} : \frac{\partial U(c_1, h_1)}{\partial c_1} - \lambda_1 = 0 \Rightarrow \lambda_1 = \frac{\partial U(c_1, h_1)}{\partial c_1}$$ $$\frac{\partial \mathcal{L}}{\partial h_1} : \frac{\partial U(c_1,h_1)}{\partial h_1} + \lambda_1 w_1 = 0 \Rightarrow \frac{\partial U(c_1, h_1)}{\partial h_1} = -\lambda_1 w_1 .$$ When we divide these equations: $$\operatorname{MRS}(c_1 \ \text{for} \ h_1) = \frac{\frac{\partial U(c_1, h_1)}{\partial h_1}}{\frac{\partial U(c_1, h_1)}{\partial c_1}} = w_1.$$ This result is the same as in the single-period model from Chapter 16.

Analyze the effect of changes in initial wealth

An increase in initial wealth \(W_0\) leads to an outward shift of the budget constraint. As the marginal rate of substitution (MRS) stays the same, the individual will choose higher consumption (\(c_1\)), and since the wage (\(w_1\)) is given, their labor supply will be affected depending on the relative strength of income and substitution effects. #b. Write the indirect utility function for the second period#

Define the wealth in the second period

Use the budget constraint to define wealth in period 2: $$W^* = W_0 + w_1(1-h_1) - c_1.$$

Write the indirect utility function

The indirect utility function for the second period can be written as: $$V(w_2, W^*) = \max_{c_2,h_2} \ U(c_2, h_2) \ \text{subject to} \ W^* = c_2 - w_2(1-h_2).$$ Since \(w_2\) is a random variable, the function \(V\) is also random. #c. Use the envelope theorem to derive the optimal choice condition#

Calculate the period 2 Lagrange multiplier

The period 2 Lagrangian can be written as: $$\mathcal{L}_2 = U(c_2, h_2) + \lambda_2 \left(W^* - c_2 + w_2(1-h_2)\right).$$ Differentiate with respect to \(c_2\) to find: $$\frac{\partial \mathcal{L}_2}{\partial c_2}: \frac{\partial U(c_2, h_2)}{\partial c_2} - \lambda_2 = 0 \Rightarrow \lambda_2 = \frac{\partial U(c_2, h_2)}{\partial c_2}.$$

Apply the envelope theorem

The envelope theorem states that for any small change in \(W^*\): $$\frac{dV}{dW^*} = \lambda_2.$$ Since only \(W_0\) and \(W^*\) are free variables in the overall problem, we also have: $$\frac{dE[V]}{dW^*} = E\left(\frac{dV}{dW^*}\right) = E[\lambda_2].$$ Now, for the optimal choice of \(W^*\) and since both \(\lambda_1\) and \(\lambda_2\) are multipliers associated with wealth constraints, we require: $$\lambda_1 = E(\lambda_2).$$ This condition implies that the expected marginal utility of wealth should be the same in the two periods. Intuitively, the individual maximizes their utility by allocating their wealth across the two periods to equalize their marginal utilities. #d. Discuss the effect of a government policy that adds k dollars to all period 2 wages#

Describe the policy and its impact

Under this policy, every individual's wage in period 2 will be increased by a constant amount, \(k\). This means that their period 2 budget constraint will shift outward, and the range of feasible consumption and leisure options in the second period will expand.

Discuss the effects on choices in both periods

While the specific changes in the individual's choices will depend on the functional form of their utility function, we can make some general observations. The increase in period 2 wages will lead to higher expected wealth in the second period. As a result, the individual may choose to consume more or work less in the first period to smooth their consumption and leisure over the two periods. The change in period 1 labor supply will depend on the relative strengths of income and substitution effects. If the substitution effect dominates, the worker may work more in period 1, anticipating higher future wages. If the income effect is stronger, the worker may choose to work less in period 1, due to greater expected wealth from the wage increase in period 2. In the second period, higher wages will again lead to changes in labor supply based on the relative strengths of the income and substitution effects. The individual may work more if the substitution effect dominates, attracted by the higher wage rate, or they may work less if the income effect dominates, as they have additional wealth from the wage increase.