<p>The optimization problem we want to solve is</p>
<p><span class="math-container">$\max_{c,l} U(c,l) = c^{\frac{1}{3}} l^{\frac{2}{3}}$</span></p>
<p>subject to <span class="math-container">$c = w (4000-l) + I$</span></p>
<p>Here <span class="math-container">$w$</span> is the real wage (wage divided by the consumption good´s price) to make our lives easier, <span class="math-container">$I$</span> is the exogenous income (the stipend).</p>
<p>To save ourselves a few algebra steps with the Lagrangian, we know the condition for optimality is <span class="math-container">$MRS = \frac{p_c}{p_l}$</span> (relative prices):</p>
<p><span class="math-container">$\frac{\frac{\partial U}{\partial c}}{\frac{\partial U}{\partial l}} = \frac{p_c}{p_l} \implies \frac{\frac{1}{3} c^{-\frac{2}{3}} l^{\frac{2}{3}}}{\frac{2}{3} c^{\frac{1}{3}} l^{-\frac{1}{3}}} = \frac{1}{w} \implies \frac{l}{2c} = \frac{1}{w} \implies \frac{2c}{l} = w \implies c = \frac{wl}{2}$</span></p>
<p>Substituting this expression for <span class="math-container">$c$</span> in our budget constraint,</p>
<p><span class="math-container">$\frac{wl}{2} = w (4000-l) + I \implies l = 8000 -2l + 2I \implies 3l = 8000 + 2I \implies l^\star = \frac{8000 + 2I}{3} \implies l^\star = \frac{2}{3} (4000+I)$</span></p>
<p>Now we substitute back our marshiallian demand <span class="math-container">$l^\star$</span> back into our expression for <span class="math-container">$c$</span>,</p>
<p><span class="math-container">$c^\star = \frac{w}{3} (4000+I)$</span></p>
<p>Note the agent can't consume more than <span class="math-container">$4000$</span> hours of leisure, so we need to pay attention to the case where our expression for <span class="math-container">$l^\star \geq 4000$</span>.</p>
<p>We have that</p>
<p><span class="math-container">$l^\star \geq 4000 \iff \frac{2}{3} (4000 + I) \geq 4000 \iff 4000 + I \geq 6000 \iff I \geq 2000$</span>.</p>
<p>So in reality,</p>
<p><span class="math-container">$l^\star = \frac{2}{3}(4000 + I), I &lt; 2000$</span></p>
<p><span class="math-container">$l^\star = 4000, I \geq 2000$</span>.</p>
<p>Note in the latter case consumption would equal the stipend <span class="math-container">$c^\star = I$</span>.</p>
<p>Now we analyze both cases.</p>
<ol>
<li><span class="math-container">$w = 7.5, I = 20000$</span></li>
</ol>
<p>Since <span class="math-container">$I = 20000 \geq 2000$</span>, the agent wouldn't work and spend exactly their stipend on the consumption good:</p>
<p><span class="math-container">$l^\star = 4000, c^\star = 20000$</span></p>
<ol start="2">
<li><span class="math-container">$w = 15, I = 10000$</span></li>
</ol>
<p>Since <span class="math-container">$I = 10000 \geq 2000$</span>, the agent wouldn't work either:</p>
<p><span class="math-container">$l^\star = 4000, c^\star = 10000$</span></p>
<p>Normally we would compute the values of <span class="math-container">$U(c^\star,l^\star)$</span> and compare, but in this case it is very clear that the agent would be better off under plan #1.</p>
<p>This happens because the agent wouldn't work, spend all their time on leisure, and consume with exactly their stipend in either case but in #1 the stipend is higher and the agent would be able to consume more.</p>

CC BY-SA 4.0
2023-03-04T14:58:43.343Z
2023-03-04T14:58:43.343Z
Nicolas Torres
<p>I was wondering if anyone could help me check my work for the following question, and if I am wrong, help me correct my mistakes?</p>
<p>Question:
<a href="https://i.sstatic.net/SrELu.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/SrELu.png" alt="question" /></a></p>
<p>Work:</p>
<p><a href="https://i.sstatic.net/n5Spb.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/n5Spb.png" alt="1" /></a></p>
<p><a href="https://i.sstatic.net/b8cdx.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/b8cdx.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-03-01T22:12:31.487Z
How do you get to the equations you then solve? Do you do any utility maximisation?
BrsG
CC BY-SA 4.0
2023-03-02T09:45:02.143Z
These equations make no sense. The first one interprets $C$ as income (or as quantity of the consumption good, if its price is normalized to \$1) if you work the full 4000 hours. The second one interprets $C$ as hours worked. Moreover, "solving" these equations by substituting the first in the second gives you a negative $L$. So where do you get these 3200 hours worked from?
VARulle
<microeconomics><utility><preferences><consumption>
Help with checking work for preferences over consumption and leisure question
josephjones1472
# Help with checking work for preferences over consumption and leisure question

(Asked by: username_0 on 2023-03-01)

I was wondering if anyone could help me check my work for the following question, and if I am wrong, help me correct my mistakes?

Question:
[![question](https://i.sstatic.net/SrELu.png)](https://i.sstatic.net/SrELu.png)

Work:

[![1](https://i.sstatic.net/n5Spb.png)](https://i.sstatic.net/n5Spb.png)

[![enter image description here](https://i.sstatic.net/b8cdx.png)](https://i.sstatic.net/b8cdx.png)

**Tags:** microeconomics, utility, preferences, consumption
> **Comments:**
> **username_1**: How do you get to the equations you then solve? Do you do any utility maximisation?  
> **username_2**: These equations make no sense. The first one interprets $C$ as income (or as quantity of the consumption good, if its price is normalized to \$1) if you work the full 4000 hours. The second one interprets $C$ as hours worked. Moreover, "solving" these equations by substituting the first in the second gives you a negative $L$. So where do you get these 3200 hours worked from?  

---

## 1 answers

### Answer by username_3 on 2023-03-04. Score: 2

The optimization problem we want to solve is

$\max\_{c,l} U(c,l) = c^{\frac{1}{3}} l^{\frac{2}{3}}$

subject to $c = w (4000-l) + I$

Here $w$ is the real wage (wage divided by the consumption good´s price) to make our lives easier, $I$ is the exogenous income (the stipend).

To save ourselves a few algebra steps with the Lagrangian, we know the condition for optimality is $MRS = \frac{p\_c}{p\_l}$ (relative prices):

$\frac{\frac{\partial U}{\partial c}}{\frac{\partial U}{\partial l}} = \frac{p\_c}{p\_l} \implies \frac{\frac{1}{3} c^{-\frac{2}{3}} l^{\frac{2}{3}}}{\frac{2}{3} c^{\frac{1}{3}} l^{-\frac{1}{3}}} = \frac{1}{w} \implies \frac{l}{2c} = \frac{1}{w} \implies \frac{2c}{l} = w \implies c = \frac{wl}{2}$

Substituting this expression for $c$ in our budget constraint,

$\frac{wl}{2} = w (4000-l) + I \implies l = 8000 -2l + 2I \implies 3l = 8000 + 2I \implies l^\star = \frac{8000 + 2I}{3} \implies l^\star = \frac{2}{3} (4000+I)$

Now we substitute back our marshiallian demand $l^\star$ back into our expression for $c$,

$c^\star = \frac{w}{3} (4000+I)$

Note the agent can't consume more than $4000$ hours of leisure, so we need to pay attention to the case where our expression for $l^\star \geq 4000$.

We have that

$l^\star \geq 4000 \iff \frac{2}{3} (4000 + I) \geq 4000 \iff 4000 + I \geq 6000 \iff I \geq 2000$.

So in reality,

$l^\star = \frac{2}{3}(4000 + I), I < 2000$

$l^\star = 4000, I \geq 2000$.

Note in the latter case consumption would equal the stipend $c^\star = I$.

Now we analyze both cases.

1. $w = 7.5, I = 20000$

Since $I = 20000 \geq 2000$, the agent wouldn't work and spend exactly their stipend on the consumption good:

$l^\star = 4000, c^\star = 20000$

2. $w = 15, I = 10000$

Since $I = 10000 \geq 2000$, the agent wouldn't work either:

$l^\star = 4000, c^\star = 10000$

Normally we would compute the values of $U(c^\star,l^\star)$ and compare, but in this case it is very clear that the agent would be better off under plan #1.

This happens because the agent wouldn't work, spend all their time on leisure, and consume with exactly their stipend in either case but in #1 the stipend is higher and the agent would be able to consume more.


---

