<p>For some intuition, rewrite the problem as the consumption problem <span class="math-container">$\max_{(c, l)\in\mathbb{R}_{+}^{2}} (cl)^{0.5}$</span> subject to the constraint <span class="math-container">$c/w + l \leq 1$</span>.
The optimal level of <span class="math-container">$l$</span> is then given by <span class="math-container">$l = 1/2$</span> as per the usual solution for Cobb-Douglas preferences.
The intuition for why <span class="math-container">$l$</span> does not depend on <span class="math-container">$w$</span> in this example is the same intuition for Cobb-Douglas consumers, who spend a fixed fraction of their income on each good.</p>

CC BY-SA 4.0
2019-11-13T13:37:38.093Z
2019-11-13T13:37:38.093Z
induction_is_a_laddah
<p>Consider the model <span class="math-container">$\max U = \ln c + \ln l$</span> subject to <span class="math-container">$c=Lw$</span> and <span class="math-container">$1=l+L$</span>, where <span class="math-container">$c$</span> denotes consumption, <span class="math-container">$l$</span> leisure, <span class="math-container">$L$</span> the labour and <span class="math-container">$w$</span> the wage rate.</p>

<p>The optimal choices of <span class="math-container">$L$</span> and <span class="math-container">$l$</span> can be easily computed to be 1/2. I don't understand the intuition why the labour supply decision is independent of the wage rate in this case.
Is it possible to argue in terms of substitution effect and income effect of a wage rate change?</p>

<microeconomics>
Why is the labour supply independent of the wage in this case?
Alex Wang
# Why is the labour supply independent of the wage in this case?

(Asked by: username_0 on 2019-11-13)

Consider the model $\max U = \ln c + \ln l$ subject to $c=Lw$ and $1=l+L$, where $c$ denotes consumption, $l$ leisure, $L$ the labour and $w$ the wage rate.

The optimal choices of $L$ and $l$ can be easily computed to be 1/2. I don't understand the intuition why the labour supply decision is independent of the wage rate in this case.
Is it possible to argue in terms of substitution effect and income effect of a wage rate change?

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2019-11-13. Score: 3

For some intuition, rewrite the problem as the consumption problem $\max\_{(c, l)\in\mathbb{R}\_{+}^{2}} (cl)^{0.5}$ subject to the constraint $c/w + l \leq 1$.
The optimal level of $l$ is then given by $l = 1/2$ as per the usual solution for Cobb-Douglas preferences.
The intuition for why $l$ does not depend on $w$ in this example is the same intuition for Cobb-Douglas consumers, who spend a fixed fraction of their income on each good.


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