Derivation of Labor Function

I want to know if I'm taking the right steps to derive a labor equation from a utility function. Suppose $$U(x,L)=x^{0.5}+l^{0.5}$$ where $$L$$ is labor, $$x$$ is our one good of interest, and $$l$$ is leisure. We also have a budget constraint $$px+wL=m+wT$$. We begin with taking our partials and get our optimality condition: $$$$\frac{\frac{1}{2 \sqrt{l}}}{\frac{1}{2 \sqrt{x}}}=\frac{p}{w} \Rightarrow x=\frac{w^2l}{p^2}.$$$$ Next I plug this solution into our budget constraint. Here, $$m$$ is our non-labor income, $$w$$ our wage, $$T$$ our time endowment, and $$p$$ is the price of good $$x$$. Since I'm interested in the labor supply equation $$L^{\ast}=T-l$$, I solve for $$l$$. \begin{align*} p\left(\frac{w^2l}{p^2}\right)+wL=m+wT. \end{align*} When we solve for $$l$$, we get \begin{align*} l=\frac{mp+pwT-pwL}{w^2} \end{align*} Does this mean that the labor function in this instance is: \begin{align*} L^{\ast}=T-\frac{mp+pwT-pwL}{w^2} \end{align*} I fear that I've made an error at some point.

• Budget constraint is incorrect.
– Amit
Commented Apr 28, 2023 at 6:28

It's in the right direction, just not complete. You have $$wl + pL = wmp + w^2pT$$

$$\implies w(T-L) + pL = wmp + w^2pT \\ \implies L(p-w) = wmp + w^2pt-wT \\ \implies L^{*}(w,m,p,T) = \frac{wmp + w^2pT-wT}{p-w}$$

• This is incorrect because budget constraint is incorrect. Another way to see that this is incorrect is because as $m$ (non-labor income) increases, the labor supply must not go up in this environment, but in your case labor supply is an increasing function of $m$.
– Amit
Commented Apr 29, 2023 at 5:11

In order to determine the labor supply function, we need to solve the following optimization problem:

$$\begin{eqnarray*} \max_{l, L, x} & \ x^{0.5}+l^{0.5} \\ \text{s.t. } & px \leq w(T-l) + m \\ \text{and} & \ L+l=T, \\ & l\geq 0, L\geq 0, x\geq 0\end{eqnarray*}$$

We can re-write the above problem as an optimization problem in two variables: $$\begin{eqnarray*} \max_{l, x} & \ x^{0.5}+l^{0.5} \\ \text{s.t. } & px +wl \leq wT + m \\ \text{and} & \ 0\leq l \leq T, x\geq 0\end{eqnarray*}$$

Solving the problem we get the following as demand for leisure $$l$$: $$\begin{eqnarray*} l^d(p,w,m,T) = \min\left(\dfrac{p(wT+m)}{w(w+p)},T\right)\end{eqnarray*}$$ and therefore, labor supply is given by: $$\begin{eqnarray*} L^s(p,w,m,T) = T-\min\left(\dfrac{p(wT+m)}{w(w+p)},T\right) = \max\left(\dfrac{Tw^2-pm}{w(w+p)},0\right)\end{eqnarray*}$$