<p>The elasticity, which we denote by <span class="math-container">$\eta$</span>, is
<span class="math-container">$$\eta = \frac{\partial h}{\partial w} \frac{w}{h}.$$</span></p>
<p>Since</p>
<p><span class="math-container">$$\frac{\partial h}{\partial w} = \frac{I A}{w^2(1+A)},$$</span></p>
<p>we have</p>
<p><span class="math-container">$$\eta = \frac{I A}{w^2(1+A)} \frac{w}{h} = \frac{I A}{w(1+A) h} = \frac{IA}{w-IA},$$</span></p>
<p>where I used optimal level of <span class="math-container">$h$</span> to obtain the last equality. Note, however, that his elasticity is not equal to <span class="math-container">$(1-h)/h$</span> as
<span class="math-container">$$\frac{1-h}{h} = \frac{w A + IA}{w-IA} = \frac{w A }{w-IA} + \eta. $$</span></p>

CC BY-SA 4.0
2022-02-16T13:58:26.773Z
2022-02-16T13:58:26.773Z
Green.H
<p>I have a utility function depending on consumption and leisure, <span class="math-container">$u(c, 1-h) = log(c) + Alog(1-h)$</span> subject to <span class="math-container">$c = wh + I$</span>, where <span class="math-container">$w$</span> is wage, <span class="math-container">$I$</span> is income, <span class="math-container">$h$</span> is hours worked. I set up the optimization problem and I have derived an expression where <span class="math-container">$h = \frac{w-AI}{w+Aw}$</span>, 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 = <span class="math-container">$\frac{I(A+A^2)}{(w+Aw)^2}$</span>. But the correct answer is <span class="math-container">$\frac{1-h}{h}$</span> and I dont understand how to get that.</p>
<p>So I am now unsure how to proceed, I think I got the correct answer but intuitively, <span class="math-container">$\frac{1-h}{h}$</span> makes more sense.</p>

<microeconomics><elasticity>
Elasticity calculation
alohaspreader
# Elasticity calculation

(Asked by: username_0 on 2022-02-16)

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.

**Tags:** microeconomics, elasticity

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## 1 answers

### Answer by username_1 on 2022-02-16. Score: 5 | Accepted answer

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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