# Irrelevance of Heterogeneous Agent Modelling

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]

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.

• I see nothing wrong in your approach. Maybe the question was wrong...? Commented May 1 at 13:30
• Could be. But seems unlikely. This test is from a prestigious institute in my country. Commented May 1 at 13:48
• Even the Indian Statistical Institute can make errors... Commented May 2 at 23:12