# Elasticity calculation

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.

The elasticity, which we denote by $$\eta$$, is $$\eta = \frac{\partial h}{\partial w} \frac{w}{h}.$$
$$\frac{\partial h}{\partial w} = \frac{I A}{w^2(1+A)},$$
$$\eta = \frac{I A}{w^2(1+A)} \frac{w}{h} = \frac{I A}{w(1+A) h} = \frac{IA}{w-IA},$$
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 $$\frac{1-h}{h} = \frac{w A + IA}{w-IA} = \frac{w A }{w-IA} + \eta.$$