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I have a utility function depending on consumption and leisure, $u(c, 1-h) = log(c) + Alog(1-h)$ subject to $c = wh + I$, where $w$ is wage, $I$ is income, $h$ is hours worked. I set up the optimization problem and I have derived an expression where $h = \frac{w-AI}{w+Aw}$, and I need to calculate elasticity of h with respect to w, the wage elasticity of labor supply. I computed this using derivative of h with respect to w and got an expression for elasticity = $\frac{I(A+A^2)}{(w+Aw)^2}$. But the correct answer is $\frac{1-h}{h}$ and I dont understand how to get that.

So I am now unsure how to proceed, I think I got the correct answer but intuitively, $\frac{1-h}{h}$ makes more sense.

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The elasticity, which we denote by $\eta$, is $$\eta = \frac{\partial h}{\partial w} \frac{w}{h}.$$

Since

$$\frac{\partial h}{\partial w} = \frac{I A}{w^2(1+A)},$$

we have

$$\eta = \frac{I A}{w^2(1+A)} \frac{w}{h} = \frac{I A}{w(1+A) h} = \frac{IA}{w-IA},$$

where I used optimal level of $h$ to obtain the last equality. Note, however, that his elasticity is not equal to $(1-h)/h$ as $$\frac{1-h}{h} = \frac{w A + IA}{w-IA} = \frac{w A }{w-IA} + \eta. $$

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