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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?

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  • $\begingroup$ Can I claim that there is no income effect in this case? $\endgroup$
    – Alex Wang
    Commented Nov 27, 2019 at 17:59

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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.

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    $\begingroup$ Adding further clarification, since it is an important concept in macro, $\frac{1}{\rho}$ also represents the intertemporal elasticity of substitution. $\endgroup$
    – lunar_props
    Commented Nov 27, 2019 at 17:26
  • $\begingroup$ 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? $\endgroup$
    – Alex Wang
    Commented Nov 27, 2019 at 17:52
  • $\begingroup$ @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 $\endgroup$
    – 1muflon1 ♦
    Commented Nov 27, 2019 at 18:05

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