# labor leisure model, quasilinear preferences

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?

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.

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 $$t$$.