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

CC BY-SA 4.0
2024-05-01T13:30:14.637Z
I see nothing wrong in your approach. Maybe the question was wrong...?
VARulle
CC BY-SA 4.0
2024-05-01T13:48:42.763Z
Could be. But seems unlikely. This test is from a prestigious institute in my country.
Kanooz
CC BY-SA 4.0
2024-05-02T23:12:05.847Z
Even the Indian Statistical Institute can make errors...
VARulle
<macroeconomics><optimization><self-study>
Irrelevance of Heterogeneous Agent Modelling
Kanooz
# Irrelevance of Heterogeneous Agent Modelling

(Asked by: username_0 on 2024-05-01)

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.

**Tags:** macroeconomics, optimization, self-study

> **Comments:**
> **username_1**: I see nothing wrong in your approach. Maybe the question was wrong...?  
> **username_0**: Could be. But seems unlikely. This test is from a prestigious institute in my country.  
> **username_1**: Even the Indian Statistical Institute can make errors...  

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## 0 answers

