<p>This looks like constant relative risk aversion (CRRA) utility . Usually CRRA is written like <span class="math-container">$U = \frac{C^{1-\rho}-1}{1-\rho}  $</span> (I omitted second part for brevity) in your case <span class="math-container">$a=1-\rho$</span>. <span class="math-container">$\rho$</span> is the relative risk aversion. By extension <span class="math-container">$a$</span> is the function of <span class="math-container">$\rho$</span> so as <span class="math-container">$a$</span> increases (due to smaller <span class="math-container">$\rho$</span>) the person should become less risk averse. </p>

CC BY-SA 4.0
2019-11-27T17:26:50.863Z
Adding further clarification, since it is an important concept in macro, $\frac{1}{\rho}$ also represents the intertemporal elasticity of substitution.
lunar_props
CC BY-SA 4.0
2019-11-27T17:52:00.500Z
Can I interpret alpha as the substitutability between consumption and leisure. The higher the alpha, the more inclined the consumer is to substitute consumption for leisure. Is that a plausible explanation?
Alex Wang
CC BY-SA 4.0
2019-11-27T18:05:27.230Z
@ChenliZhou  i think that in this case where you don’t have time periods that $1\rho$ would be an elasticity of substitution between C and L. substituting for rho the elasticity would be $1/(1-a)$ so as a increases the elasticity of substitution decreases. I would recommend trying to actually derive the elasticity of substitution and double checking because I did not do any calculations and I am relaying on intuition
1muflon1
CC BY-SA 4.0
2019-11-27T17:18:55.560Z
2019-11-27T17:32:33.910Z
2019-11-27T17:32:33.910Z
1muflon1
<p>I have a utility function <span class="math-container">$U(c,l)={{C^\alpha-1}\over {\alpha}}+{{l^\alpha-1}\over {\alpha}}$</span> where C denotes consumption, l denotes leisure. What is the economic interpretation of the term <span class="math-container">$\alpha$</span> (<span class="math-container">$\alpha&lt;1)$</span>? Does changes in <span class="math-container">$\alpha$</span> affect the economic interpretation of the utility function as a whole?</p>

<p>The full problem is to derive the optimal labour supply. I get the answer <span class="math-container">$L={{1}\over {1+[w(1-t)^{{\alpha}\over {\alpha-1}}]}}$</span> where t represents  the tax rate. I am trying to understand the effect of <span class="math-container">$\alpha$</span> on the labour supply as I increase the tax rate <span class="math-container">$t$</span>. How can I intuitively explain the role of <span class="math-container">$\alpha$</span> in this case?</p>

CC BY-SA 4.0
2019-11-27T17:59:22.443Z
Can I claim that there is no income effect in this case?
Alex Wang
<microeconomics>
What is the economic interpretation of this utility function?
Alex Wang
# What is the economic interpretation of this utility function?

(Asked by: username_0 on 2019-11-27)

I have a utility function $U(c,l)={{C^\alpha-1}\over {\alpha}}+{{l^\alpha-1}\over {\alpha}}$ where C denotes consumption, l denotes leisure. What is the economic interpretation of the term $\alpha$ ($\alpha<1)$? Does changes in $\alpha$ affect the economic interpretation of the utility function as a whole?

The full problem is to derive the optimal labour supply. I get the answer $L={{1}\over {1+[w(1-t)^{{\alpha}\over {\alpha-1}}]}}$ where t represents the tax rate. I am trying to understand the effect of $\alpha$ on the labour supply as I increase the tax rate $t$. How can I intuitively explain the role of $\alpha$ in this case?

**Tags:** microeconomics
> **Comments:**
> **username_0**: Can I claim that there is no income effect in this case?  

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## 1 answers

### Answer by username_1 on 2019-11-27. Score: 7

This looks like constant relative risk aversion (CRRA) utility . Usually CRRA is written like $U = \frac{C^{1-\rho}-1}{1-\rho} $ (I omitted second part for brevity) in your case $a=1-\rho$. $\rho$ is the relative risk aversion. By extension $a$ is the function of $\rho$ so as $a$ increases (due to smaller $\rho$) the person should become less risk averse.

> **Comments:**
> **username_2**: Adding further clarification, since it is an important concept in macro, $\frac{1}{\rho}$ also represents the intertemporal elasticity of substitution.  
> **username_0**: Can I interpret alpha as the substitutability between consumption and leisure. The higher the alpha, the more inclined the consumer is to substitute consumption for leisure. Is that a plausible explanation?  
> **username_1**: @ChenliZhou  i think that in this case where you don’t have time periods that $1\rho$ would be an elasticity of substitution between C and L. substituting for rho the elasticity would be $1/(1-a)$ so as a increases the elasticity of substitution decreases. I would recommend trying to actually derive the elasticity of substitution and double checking because I did not do any calculations and I am relaying on intuition  

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