<p>In the utility function you have labor supply will respond to <span class="math-container">$w$</span> and <span class="math-container">$t$</span>.</p>
<p>If we maximize utility we get:</p>
<p><span class="math-container">$$u'=0 \\ (1-t)w = p_l'dl  \\ \frac{(1-t)w}{p_l'}=dl$$</span></p>
<p>so labor supply changes negatively with tax <span class="math-container">$t$</span> and positively with wage <span class="math-container">$w$</span>. You would also expect this to happen since <span class="math-container">$t$</span> is distortionary as the total wage depends on <span class="math-container">$l$</span> and there is no offset and utility is decreasing <span class="math-container">$t$</span>. Utility increases in <span class="math-container">$w$</span> and it depends on choice of <span class="math-container">$l$</span> so you would expect positive effect of <span class="math-container">$w$</span> on <span class="math-container">$l$</span>.</p>
<p>You can offset change in <span class="math-container">$t$</span> with change in <span class="math-container">$w$</span> so that <span class="math-container">$dl=0$</span> but this holds for most of utility functions and budget constraints so its not special.</p>

CC BY-SA 4.0
2023-07-20T16:13:54.103Z
2023-07-20T16:13:54.103Z
1muflon1
<p>I interpret your (quasilinear) utility function as <span class="math-container">$u = \text{income from working} - \text{disutility from working}$</span></p>
<p>where the salary <span class="math-container">$w$</span> is taxed at a rate <span class="math-container">$t \in (0,1)$</span>.</p>
<p>This would give a univariate problem with respect to labor supply <span class="math-container">$l$</span>.</p>
<p>Suppose <span class="math-container">$p(l)$</span> is increasing and convex (as many costs/bads are assumed to be). Then <span class="math-container">$p’(l)$</span> is increasing.</p>
<p>Note the first term is linear and the second term is concave (being the negative of the convex function <span class="math-container">$p(l)$</span>), yielding that <span class="math-container">$u$</span> is concave.</p>
<p>We would then maximize utility by setting <span class="math-container">$\frac{du}{dl} = 0$</span></p>
<p><span class="math-container">$\implies (1-t) w = p’(l)$</span></p>
<p>The left hand side (and hence <span class="math-container">$p’(l^\star)$</span>) is increasing in <span class="math-container">$w$</span> and decreasing in <span class="math-container">$t$</span>.</p>
<p>Therefore, by monotonicity of <span class="math-container">$p’(l)$</span>, <span class="math-container">$l^\star$</span> is increasing in <span class="math-container">$w$</span> and decreasing in <span class="math-container">$t$</span>.</p>

CC BY-SA 4.0
2023-07-20T16:25:30.643Z
2023-07-20T16:31:17.517Z
2023-07-20T16:31:17.517Z
Nicolas Torres
<p>There is a quasilinear utility function <span class="math-container">$u= (1-t)wl - p(l)$</span>, where <span class="math-container">$l$</span> is labor supply. I don't quite understand what happens, if the budget changes (due to <span class="math-container">$w$</span> or <span class="math-container">$t$</span>) since it is quasilinear. Does it mean the labor supply will not change since the ratio stays the same and only the IC?</p>

<microeconomics><utility><labor-economics><preferences>
labor leisure model, quasilinear preferences
tk420
# labor leisure model, quasilinear preferences

(Asked by: username_0 on 2023-07-11)

There is a quasilinear utility function $u= (1-t)wl - p(l)$, where $l$ is labor supply. I don't quite understand what happens, if the budget changes (due to $w$ or $t$) since it is quasilinear. Does it mean the labor supply will not change since the ratio stays the same and only the IC?

**Tags:** microeconomics, utility, labor-economics, preferences

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## 2 answers

### Answer by username_1 on 2023-07-20. Score: 0

In the utility function you have labor supply will respond to $w$ and $t$.

If we maximize utility we get:

$$u'=0 \\ (1-t)w = p\_l'dl \\ \frac{(1-t)w}{p\_l'}=dl$$

so labor supply changes negatively with tax $t$ and positively with wage $w$. You would also expect this to happen since $t$ is distortionary as the total wage depends on $l$ and there is no offset and utility is decreasing $t$. Utility increases in $w$ and it depends on choice of $l$ so you would expect positive effect of $w$ on $l$.

You can offset change in $t$ with change in $w$ so that $dl=0$ but this holds for most of utility functions and budget constraints so its not special.


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### Answer by username_2 on 2023-07-20. Score: 0

I interpret your (quasilinear) utility function as $u = \text{income from working} - \text{disutility from working}$

where the salary $w$ is taxed at a rate $t \in (0,1)$.

This would give a univariate problem with respect to labor supply $l$.

Suppose $p(l)$ is increasing and convex (as many costs/bads are assumed to be). Then $p’(l)$ is increasing.

Note the first term is linear and the second term is concave (being the negative of the convex function $p(l)$), yielding that $u$ is concave.

We would then maximize utility by setting $\frac{du}{dl} = 0$

$\implies (1-t) w = p’(l)$

The left hand side (and hence $p’(l^\star)$) is increasing in $w$ and decreasing in $t$.

Therefore, by monotonicity of $p’(l)$, $l^\star$ is increasing in $w$ and decreasing in $t$.


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