Optimal consumption in the neoclassical consumption model

Question

In some economy, individuals live for two periods (periods 1 and 2). An individual works in the first period and earns an income of $wL$, with $L$ representing the proportion of the day worked. In the second period, which can be thought of as retirement, the individual consumes the remainder of their savings. Lifetime utility is given by:

$U=log(c_1)+log(c_2)+log(1-L)$

The question is asking to solve for optimal consumption in each period, and the optimal work effort.

I started by writing the budget constraints for each period:

Period 1
$c_1+s=wL$, with $s$ being savings

Period 2
$c_2=s(1+r)$, with $r$ being the interest rate

Intertemporal budget constraint
$c_1+\frac{c_2}{(1+r)}=wL$

To maximise utility from consumption, I would normally just optimise for $c_1$ and $c_2$ subject to the intertemporal budget constraint. However, I don't know what to do with the $log(1-L)$ element in the original lifetime utility function. Any pointers would be appreciated, and sorry for the elementary nature of this question.

Answer 1

The problem we want to solve is:

\begin{eqnarray*} \displaystyle\max_{c_1, c_2, R} & \log c_1 + \log c_2 + \log R \\ \text{s.t.} & \ c_1 + \dfrac{c_2}{1+r} + wR = w \\ \text{and } & c_1 \geq 0, c_2\geq 0, 0 \leq R \leq 1 \end{eqnarray*}

Here $R$ is the leisure and is equal to $1-L$.

This is a standard Cobb-Douglas utility maximisation problem with linear constraint, and the solution to the problem is:

\begin{eqnarray*}(c_1^d, c_2^d, R^d) = \left(\frac{w}{3},\frac{(1+r)w}{3}, \frac{1}{3} \right) \end{eqnarray*}

Consequently, the labor supply is \begin{eqnarray*}L^s = 1-R^d = \frac{2}{3} \end{eqnarray*}

Answer 2

You need to optimize with respect to $L$, $c_1$, and $c_2$ subject to the intertemporal budget constraint. This will yield the optimal consumption and labor choices. You should end up with

1. an Euler equation that governs the optimal intertemporal consumption choice, stating that the marginal utility derived from consuming in the two periods must be equal (after discounting)
2. a labor FOC stating that the marginal rate of substitution between consumption and leisure is equal to the real wage