This is a question from a previous year PhD entrance exam. I have outlined how I have tried to tackle the problem as well:
N.B. 1 This exam is of 100 points and this particular problem is of 25 points.
Consider an economy with a continuum $[0,1]$ of agents. All agents derive utility from consumption and leisure with identical preference structure. However, they differ in their acquired education level $e$. Assume that the distribution of $e$ across agents is represented by $F(e)$. An agent with exogenous education level $e$ earns a total wage of $e\times l$ if she chooses to work for $l$ hours. The total numbers of available hours to each agent is $a ≥ l$. Therefore, $a − l$ represents leisure. The government runs a balanced budget program where a proportional tax at the rate $t ∈ (0, 1)$ is imposed (on income of each agent) and it is distributed to agents in a lump-sum way (say, the amount for each agent is Z). Utility of each agent is represented by $U = c^β(a − l)^{1-\beta} $ , where $c$ represents consumption and parameter $β ∈ (0, 1)$.
Answer the following questions.
- Write down the optimization problem of an agent. [$9$ points]
- Suppose there is no government. Therefore, there is no tax
or lump-sum redistribution.
- Show that the optimal choice of working hours ($l$) varies with exogenous level of education $e$. [$4$ points]
- Does individual utility (welfare) increase with the level of education $e$? [$4$ points]
- Suppose the government is present. Therefore, there are taxes and redistributive lump-sum payments. If $Z > 0$, will an increase in e result in higher optimum labor supply $l$? [$8$ points]
My trial answer:
An individual agent's optimization problem:
$$\begin{array}{ll} \text{maximize} & c^β(a − l)^{1-\beta} \\ \text{subject to}& c = (1-t)el + Z \\ &0 \le l \le a. \end{array}$$
For the second question, we will set $t=0$, $Z=0$. But for time being let them be.
Then $L = (c)^\beta (a-l)^{1-\beta} + \lambda \cdot [(1-t) \cdot el+Z-c] $
FOCs give $c^* =\beta (1-t) (ae+Z) $
and $l^* = a - \frac{(1-\beta)\cdot (ae+Z)}{e}$
Now set $t=0$, $Z=0$
$c^* =\beta (ae)$
$l^* = \beta a$
Clearly I have gone wrong somewhere. As $l^*()$ doesn't depend on $e$
I can't seem to figure out how to incorporate $Z$ more sophisticatedly. Like I thought, i can write $Z=\int_{0}^{1}l(i)e(i)di$ where $l(i)$ is the labor of the ith indivual and $e(i)$ is the education level. Then I am not able to incorporate $F(e)$ after this. I have tried using the property of a non-negative random variable $X$ (in our case $e$) that its expected value is the the integral of $1-F_X(x)$. I haven't reached anywhere worthwhile. Further, 9 points for writing three lines for a simple optimization problem seems excessively irresponsible. Kindly guide.
