<p>In order to determine the labor supply function, we need to solve the following optimization problem:</p>
<p><span class="math-container">\begin{eqnarray*} \max_{l, L, x} &amp; \ x^{0.5}+l^{0.5} \\ \text{s.t. } &amp;  px \leq w(T-l) + m \\ \text{and} &amp; \ L+l=T, \\ &amp; l\geq 0, L\geq 0, x\geq 0\end{eqnarray*}</span></p>
<p>We can re-write the above problem as an optimization problem in two variables:
<span class="math-container">\begin{eqnarray*} \max_{l, x} &amp; \ x^{0.5}+l^{0.5} \\ \text{s.t. } &amp;  px +wl \leq wT + m \\ \text{and} &amp; \ 0\leq l \leq T, x\geq 0\end{eqnarray*}</span></p>
<p>Solving the problem we get the following as demand for leisure <span class="math-container">$l$</span>:
<span class="math-container">\begin{eqnarray*} l^d(p,w,m,T) = \min\left(\dfrac{p(wT+m)}{w(w+p)},T\right)\end{eqnarray*}</span>
and therefore, labor supply is given by:
<span class="math-container">\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*}</span></p>

CC BY-SA 4.0
2023-04-28T06:20:11.960Z
2023-04-28T06:20:11.960Z
Amit
<p>I want to know if I'm taking the right steps to derive a labor equation from a utility function. Suppose <span class="math-container">$U(x,L)=x^{0.5}+l^{0.5}$</span> where <span class="math-container">$L$</span> is labor, <span class="math-container">$x$</span> is our one good of interest, and <span class="math-container">$l$</span> is leisure. We also have a budget constraint <span class="math-container">$px+wL=m+wT$</span>. We begin with taking our partials and get our optimality condition:
<span class="math-container">\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}</span>
Next I plug this solution into our budget constraint. Here, <span class="math-container">$m$</span> is our non-labor income, <span class="math-container">$w$</span> our wage, <span class="math-container">$T$</span> our time endowment, and <span class="math-container">$p$</span> is the price of good <span class="math-container">$x$</span>. Since I'm interested in the labor supply equation <span class="math-container">$L^{\ast}=T-l$</span>, I solve for <span class="math-container">$l$</span>.
<span class="math-container">\begin{align*}
p\left(\frac{w^2l}{p^2}\right)+wL=m+wT.
  \end{align*}</span>
When we solve for <span class="math-container">$l$</span>, we get
<span class="math-container">\begin{align*}
l=\frac{mp+pwT-pwL}{w^2}
  \end{align*}</span>
Does this mean that the labor function in this instance is:
<span class="math-container">\begin{align*}
L^{\ast}=T-\frac{mp+pwT-pwL}{w^2}
  \end{align*}</span>
I fear that I've made an error at some point.</p>

CC BY-SA 4.0
2023-04-28T06:28:28.640Z
Budget constraint is incorrect.
Amit
<mathematical-economics><labor-economics>
Derivation of Labor Function
Jared Greathouse
# Derivation of Labor Function

(Asked by: username_0 on 2023-02-13)

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.

**Tags:** mathematical-economics, labor-economics
> **Comments:**
> **username_1**: Budget constraint is incorrect.  

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## 1 answers

### Answer by username_1 on 2023-04-28. Score: 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\*}


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