labor leisure model, quasilinear preferences

Question

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?

Answer 1

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.

Answer 2

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 $t$.