if you have a quasi linear utility function, for example $U(l,c)=c-l^{1+γ}/(1+ γ)$ the supply function of l is inelastic right? But can we extend this argument and say that any quasi linear function is going to give you an inelastic function of the non lineal good? Forgot to mention the production function. But this is the function $y(l)=A(l)^{1-a}$. A is the productivity So if you take the efficiency condition you get that $l=(A(1-a))^{1/(a+γ)}$ Thank you.
Thank you.
We solve the utility maximization problem of the individual whose utility function is $u(c, l) = c - \frac{l^{1+\gamma}}{1+\gamma}$ to get the supply function. The problem can be written as: \begin{eqnarray*} \max_{c,l} & & c - \frac{l^{1+\gamma}}{1+\gamma} \\ \text{s.t.} && c \leq wl\end{eqnarray*} In this problem, we are assuming that the only source of income of the consumer is his wage income. When we solve the problem we get the labor supply function as: \begin{eqnarray*} l(w) = w^{1/\gamma}\end{eqnarray*} The elasticity of labor supply curve is this case is constant and equal to $\frac{1}{\gamma}$. Supply will be elastic if $0 < \gamma < 1$ and inelastic if $\gamma > 1$.
The supply function will be perfectly inelastic if it does not depend on the wage. Will this happen with quasilinear? If you substitute out $c=w(1-l)$ you will find the wage is in the labor supply function. You need a first order condition where the wage drops out, like in $u=cl$ . However if you have some exogenous wealth source in addition to your work, think $c=w(1-l)+\pi$, then this will no longer be the case.
The linear good in quasilinear absorbs all the wealth effects, but with labor/leisure, the other good is a source of the wealth so it behaves differently than with a typical consumer problem with exogenous wealth.
So if $c=wl$ then a sufficient condition for perfectly inelastic labor supply would be homogeneity of degree 1 in consumption.
