# What is the economic interpretation of this utility function?

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?

• Can I claim that there is no income effect in this case? – Chenli Zhou Nov 27 '19 at 17:59

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.

• Adding further clarification, since it is an important concept in macro, $\frac{1}{\rho}$ also represents the intertemporal elasticity of substitution. – lunar_props Nov 27 '19 at 17:26
• 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? – Chenli Zhou Nov 27 '19 at 17:52
• @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 Nov 27 '19 at 18:05