<p>Once again self-study question:</p>
<p>Given a household's utility function be:</p>
<p><span class="math-container">$$U = C^\alpha (1-L)^\beta$$</span></p>
<p>where, <span class="math-container">$L$</span> is labor supplied. Total labor time is normalized to <span class="math-container">$1$</span>. <span class="math-container">$C$</span> is read household consumption, which is taken to be equal to its wage income, i.e., real wage into labor.</p>
<ol>
<li>Find FOC for utility maximization and derive household's labor supply (<span class="math-container">$L$</span>) for a given real wage, <span class="math-container">$w$</span>. Does <span class="math-container">$L$</span> depend on <span class="math-container">$w$</span>? Give explanation.</li>
</ol>
<p>My attempt: I simply substituted <span class="math-container">$C = wL$</span> in the utility function and differentiated (after taking log - to make it simple) w.r.t <span class="math-container">$L$</span> to obtain the following:</p>
<p><span class="math-container">$$L^* = \alpha / (\alpha + \beta)$$</span></p>
<p>I am unable to properly explain why is this independent of <span class="math-container">$w$</span>. On the contrary it feels counter intuitive because real wage should be equal to MRS which is doesn't seem to be constant.</p>
<ol start="2">
<li>Assuming for the economy production function is <span class="math-container">$Y= AL^\theta$</span>. And aggregate demand of economy is given by <span class="math-container">$C+I$</span> where <span class="math-container">$I$</span> is exogenously given investment level. Find equilibrium output for this economy for some given <span class="math-container">$I$</span>. What role does <span class="math-container">$\theta$</span> play? Provide and intuitive answer.</li>
</ol>
<p>My attempt: shouldn't answer simply be <span class="math-container">$A(L^*)^\theta$</span>?</p>
<hr />
<p>EDIT: Revisiting this question, I made some progress. For part (1), as suggested in comments, I can use the fact that elasticity of substitution for this utility function is <span class="math-container">$1$</span>. Therefore, the ratio of consumption to leisure increases by same amount as <span class="math-container">$w$</span>. Since consumption is directly proportional to <span class="math-container">$w$</span>, leisure consumption remains same.</p>
<p>For part (2), introducing another variable <span class="math-container">$n$</span>: number of households employed at equilibrium. So:</p>
<p><span class="math-container">$$w = Y_L|_{L = nL^*} =  \theta A (nL^*)^{\theta - 1}$$</span>
Now I think a confusion arose because of same choice of variable symbols in the question, denoting <span class="math-container">$c$</span> by aggregate consumption (somewhat opposite of convention though):</p>
<p><span class="math-container">$$c = nwL^* = \theta A (nL^*)^\theta$$</span></p>
<p><span class="math-container">$$Y = A(nL^*)^\theta$$</span></p>
<p>If the above is correct, <span class="math-container">$\theta$</span> acts as the <span class="math-container">$1-s$</span>, where <span class="math-container">$s$</span> is aggregate saving rate.</p>
<p>Since <span class="math-container">$I$</span> is exogenous, <span class="math-container">$n$</span> will adjust to bring equilibrium. I still don't know what 'intuitive' explanation can be given here.</p>

CC BY-SA 4.0
2022-09-11T06:26:04.143Z
I don't think it is independent of w. You could use a monotonic transformation instead f(u) = ln(u), which makes your calculation easier.
Fiona Jiang
CC BY-SA 4.0
2022-03-17T05:32:09.730Z
Have you ever encountered Cobb Douglas type utility functions before, do you know if they have any sort of special property?
Giskard
CC BY-SA 4.0
2022-03-17T05:57:11.283Z
@Giskard: Constant elasticity of substitution?
Dayne
CC BY-SA 4.0
2022-03-17T06:01:50.197Z
Thanks for the pointer. I think I found the answer on wiki: "...*when the elasticity of substitution is exactly one (as in the Cobb–Douglas case), expenditure on $c_{2}$ relative to $c_{1}$ is independent of the relative prices*". How about second part of the question? Is my answer correct?
Dayne
<macroeconomics><self-study>
Finding Labour Supply in Cobb Douglas type Utility Function
Dayne
# Finding Labour Supply in Cobb Douglas type Utility Function

(Asked by: username_0 on 2022-03-17)

Once again self-study question:

Given a household's utility function be:

$$U = C^\alpha (1-L)^\beta$$

where, $L$ is labor supplied. Total labor time is normalized to $1$. $C$ is read household consumption, which is taken to be equal to its wage income, i.e., real wage into labor.

1. Find FOC for utility maximization and derive household's labor supply ($L$) for a given real wage, $w$. Does $L$ depend on $w$? Give explanation.

My attempt: I simply substituted $C = wL$ in the utility function and differentiated (after taking log - to make it simple) w.r.t $L$ to obtain the following:

$$L^\* = \alpha / (\alpha + \beta)$$

I am unable to properly explain why is this independent of $w$. On the contrary it feels counter intuitive because real wage should be equal to MRS which is doesn't seem to be constant.

2. Assuming for the economy production function is $Y= AL^\theta$. And aggregate demand of economy is given by $C+I$ where $I$ is exogenously given investment level. Find equilibrium output for this economy for some given $I$. What role does $\theta$ play? Provide and intuitive answer.

My attempt: shouldn't answer simply be $A(L^\*)^\theta$?

---

EDIT: Revisiting this question, I made some progress. For part (1), as suggested in comments, I can use the fact that elasticity of substitution for this utility function is $1$. Therefore, the ratio of consumption to leisure increases by same amount as $w$. Since consumption is directly proportional to $w$, leisure consumption remains same.

For part (2), introducing another variable $n$: number of households employed at equilibrium. So:

$$w = Y\_L|\_{L = nL^\*} = \theta A (nL^\*)^{\theta - 1}$$
Now I think a confusion arose because of same choice of variable symbols in the question, denoting $c$ by aggregate consumption (somewhat opposite of convention though):

$$c = nwL^\* = \theta A (nL^\*)^\theta$$

$$Y = A(nL^\*)^\theta$$

If the above is correct, $\theta$ acts as the $1-s$, where $s$ is aggregate saving rate.

Since $I$ is exogenous, $n$ will adjust to bring equilibrium. I still don't know what 'intuitive' explanation can be given here.

**Tags:** macroeconomics, self-study
> **Comments:**
> **username_2**: Have you ever encountered Cobb Douglas type utility functions before, do you know if they have any sort of special property?  
> **username_0**: @Giskard: Constant elasticity of substitution?  
> **username_0**: Thanks for the pointer. I think I found the answer on wiki: "...*when the elasticity of substitution is exactly one (as in the Cobb–Douglas case), expenditure on $c_{2}$ relative to $c_{1}$ is independent of the relative prices*". How about second part of the question? Is my answer correct?  
> **username_1**: I don't think it is independent of w. You could use a monotonic transformation instead f(u) = ln(u), which makes your calculation easier.  

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## 0 answers

