Derivation of Labor Function

Question

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: \begin{equation} \frac{\frac{1}{2 \sqrt{l}}}{\frac{1}{2 \sqrt{x}}}=\frac{p}{w} \Rightarrow x=\frac{w^2l}{p^2}. \end{equation} 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.

Answer 1

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

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

Answer 2

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*}