<p><strong>Short Run Cost Minimisation problem</strong>:</p>
<p><span class="math-container">\begin{eqnarray*} \min_{l\geq 0} &amp; \ wl +rk \\ \text{s.t. } &amp; y\leq \min(\sqrt{l},\sqrt{k}) \end{eqnarray*}</span>
where <span class="math-container">$y\geq 0$</span>, <span class="math-container">$k\geq 0$</span>, <span class="math-container">$w&gt;0$</span>, <span class="math-container">$r&gt;0$</span> are given.</p>
<p>This problem has no solution if <span class="math-container">$y&gt;\sqrt{k}$</span> because the set of <span class="math-container">$l$</span> satisfying the constraint is an empty set. Solving this problem for <span class="math-container">$y\leq \sqrt{k}$</span>, we get the following short-run conditional labor demand function:
<span class="math-container">\begin{eqnarray*} l^{sc}(w,r,k,y)=y^2\end{eqnarray*}</span></p>
<p>Therefore, short-run cost function is
<span class="math-container">\begin{eqnarray*} c^{s}(w,r,k,y)=\begin{cases}wy^2+rk &amp; \text{if } y\leq \sqrt{k}\\ \infty &amp; \text{if }y&gt;\sqrt{k}\end{cases} \end{eqnarray*}</span></p>
<p>where cost at <span class="math-container">$y&gt;\sqrt{k}$</span> is <span class="math-container">$\infty$</span> suggests that it is impossible to reach an output of <span class="math-container">$y$</span> irrespective of how much labor is employed when capital is sufficiently low.</p>
<p><strong>Short Run Profit Maximisation problem</strong>:</p>
<p><span class="math-container">\begin{eqnarray*} \max_{y\geq 0} &amp; \ py - c^s(w,r,k,y) \end{eqnarray*}</span>
where <span class="math-container">$p\geq 0$</span>, <span class="math-container">$k\geq 0$</span>, <span class="math-container">$w&gt;0$</span>, <span class="math-container">$r&gt;0$</span> are given.</p>
<p>Solving this problem, we get the short-run supply as</p>
<p><span class="math-container">\begin{eqnarray*} y^{ss}(p,k,w,r) =\min\left(\dfrac{p}{2w},\sqrt{k}\right)\end{eqnarray*}</span></p>
<p>Substituting this into the short-run conditional labor demand obtained earlier will provide us with the short-run labor demand as follows:
<span class="math-container">\begin{eqnarray*} l^{sd}(p,k,w,r) =\min\left(\left(\dfrac{p}{2w}\right)^2,k\right)\end{eqnarray*}</span></p>
<p><strong>Long Run Cost Minimisation problem</strong>:
<span class="math-container">\begin{eqnarray*} \min_{l\geq 0,k\geq 0} &amp; \ wl +rk \\ \text{s.t. } &amp; y\leq \min(\sqrt{l},\sqrt{k}) \end{eqnarray*}</span>
where <span class="math-container">$y\geq 0$</span>, <span class="math-container">$w&gt;0$</span>, <span class="math-container">$r&gt;0$</span> are given.</p>
<p>Solving this problem we get the following long-run conditional labor and capital demand functions:
<span class="math-container">\begin{eqnarray*} (l^{c},k^{c})(w,r,y)=(y^2,y^2)\end{eqnarray*}</span></p>
<p>Therefore, long-run cost function is
<span class="math-container">\begin{eqnarray*} c(w,r,y)=wy^2+ry^2=(w+r)y^2  \end{eqnarray*}</span></p>
<p><strong>Long Run Profit Maximisation problem</strong>:</p>
<p><span class="math-container">\begin{eqnarray*} \max_{y\geq 0} &amp; \ py - c(w,r,y) \end{eqnarray*}</span>
where <span class="math-container">$p\geq 0$</span>, <span class="math-container">$w&gt;0$</span>, <span class="math-container">$r&gt;0$</span> are given.</p>
<p>Solving this problem, we get the long-run supply as</p>
<p><span class="math-container">\begin{eqnarray*} y^s(p,w,r) =\dfrac{p}{2(w+r)}\end{eqnarray*}</span></p>
<p>Substituting this into the long-run conditional labor demand obtained earlier will provide us with the long-run labor and capital demand as follows:
<span class="math-container">\begin{eqnarray*} (l^d,k^d)(p,w,r) =\left(\left(\dfrac{p}{2(w+r)}\right)^2,\left(\dfrac{p}{2(w+r)}\right)^2\right)\end{eqnarray*}</span></p>

CC BY-SA 4.0
2024-04-08T15:23:17.903Z
2024-04-08T15:23:17.903Z
Amit
<p>I'll solve the Cost-Minimization and Profit-Maximization problems for the Leontief Production Function in the Long Run</p>
<p>Solution: Here we are given the production as, <span class="math-container">$$y=f(l,k)=min\{l^{0.5},k^{0.5}\}$$</span>
i.e., we are dealing with a DRS Technology</p>
<p><span class="math-container">$$y=f(l,k)=min\{l^{0.5},{k}^{0.5}\}$$</span></p>
<p>Therefore, the Cost Minimization problem will be,
<span class="math-container">$$min_{l,k}\hspace{3mm}(wl+r{k})$$</span>
<span class="math-container">$$st. l \ge 0,y \le f(l,{k})=min\{l^{0.5},{ k}^{0.5}\}$$</span></p>
<p>The equilibrium will be when,
<span class="math-container">$$y=l^{\frac{1}{2}}={k}^\frac{1}{2}$$</span>
<span class="math-container">$$\Rightarrow y^2=l^\star={k}^\star$$</span></p>
<p>Now we will get the cost function as, <span class="math-container">$c(y;w,r)=wl+r{k}=(w+r)y^2$</span></p>
<p>Assuming that the firm is a price-taker, The Long Run Profit Maximization problem will now become,</p>
<p><span class="math-container">$$max_y \hspace{4 mm} p(y)-(w+r)y^2$$</span>
<span class="math-container">$$st. y\ge 0$$</span></p>
<p>Solving the above problem, we will get the Long-Run Supply Function of the Firm as ,
<span class="math-container">$$y^s(p,w,r)=\begin{cases}\frac{p}{2(w+r)}, &amp; \text {if $p &gt; 0$} \\
0, &amp;\text{otherwise}\end{cases}$$</span></p>

CC BY-SA 4.0
2024-04-08T06:59:04.920Z
2024-04-08T17:08:08.287Z
2024-04-08T17:08:08.287Z
SGP
<p>I am solving for the Leontief production function in the form of y= min {L^1/2 , K^1/2} , in the short run (fixed K) and in the long run respectively. I am asked to solve the production in terms of w, r the prices of labour and capital, and the sales price p.</p>
<p>I understand that for the Leontief function of the form:</p>
<p>y = min {z/a , z2/b} the optimal solution of inputs would be solved by letting z/a=z2/b then solving for z, but in this scenario, how would one express their answer in terms of w,r, and p?</p>

CC BY-SA 4.0
2024-04-08T05:37:39.823Z
What do you mean by ``solving the production function''. Are you looking at the profit maximisation problem or the cost minimisation problem. Also, could you maybe show what you have done so far in terms of trying to solve the question.
tdm
<microeconomics><leontief>
How to solve a Leontief Production function?
coco
# How to solve a Leontief Production function?

(Asked by: username_0 on 2024-04-06)

I am solving for the Leontief production function in the form of y= min {L^1/2 , K^1/2} , in the short run (fixed K) and in the long run respectively. I am asked to solve the production in terms of w, r the prices of labour and capital, and the sales price p.

I understand that for the Leontief function of the form:

y = min {z/a , z2/b} the optimal solution of inputs would be solved by letting z/a=z2/b then solving for z, but in this scenario, how would one express their answer in terms of w,r, and p?

**Tags:** microeconomics, leontief
> **Comments:**
> **username_1**: What do you mean by ``solving the production function''. Are you looking at the profit maximisation problem or the cost minimisation problem. Also, could you maybe show what you have done so far in terms of trying to solve the question.  

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## 2 answers

### Answer by username_2 on 2024-04-08. Score: 2 | Accepted answer

**Short Run Cost Minimisation problem**:

\begin{eqnarray\*} \min\_{l\geq 0} & \ wl +rk \\ \text{s.t. } & y\leq \min(\sqrt{l},\sqrt{k}) \end{eqnarray\*}
where $y\geq 0$, $k\geq 0$, $w>0$, $r>0$ are given.

This problem has no solution if $y>\sqrt{k}$ because the set of $l$ satisfying the constraint is an empty set. Solving this problem for $y\leq \sqrt{k}$, we get the following short-run conditional labor demand function:
\begin{eqnarray\*} l^{sc}(w,r,k,y)=y^2\end{eqnarray\*}

Therefore, short-run cost function is
\begin{eqnarray\*} c^{s}(w,r,k,y)=\begin{cases}wy^2+rk & \text{if } y\leq \sqrt{k}\\ \infty & \text{if }y>\sqrt{k}\end{cases} \end{eqnarray\*}

where cost at $y>\sqrt{k}$ is $\infty$ suggests that it is impossible to reach an output of $y$ irrespective of how much labor is employed when capital is sufficiently low.

**Short Run Profit Maximisation problem**:

\begin{eqnarray\*} \max\_{y\geq 0} & \ py - c^s(w,r,k,y) \end{eqnarray\*}
where $p\geq 0$, $k\geq 0$, $w>0$, $r>0$ are given.

Solving this problem, we get the short-run supply as

\begin{eqnarray\*} y^{ss}(p,k,w,r) =\min\left(\dfrac{p}{2w},\sqrt{k}\right)\end{eqnarray\*}

Substituting this into the short-run conditional labor demand obtained earlier will provide us with the short-run labor demand as follows:
\begin{eqnarray\*} l^{sd}(p,k,w,r) =\min\left(\left(\dfrac{p}{2w}\right)^2,k\right)\end{eqnarray\*}

**Long Run Cost Minimisation problem**:
\begin{eqnarray\*} \min\_{l\geq 0,k\geq 0} & \ wl +rk \\ \text{s.t. } & y\leq \min(\sqrt{l},\sqrt{k}) \end{eqnarray\*}
where $y\geq 0$, $w>0$, $r>0$ are given.

Solving this problem we get the following long-run conditional labor and capital demand functions:
\begin{eqnarray\*} (l^{c},k^{c})(w,r,y)=(y^2,y^2)\end{eqnarray\*}

Therefore, long-run cost function is
\begin{eqnarray\*} c(w,r,y)=wy^2+ry^2=(w+r)y^2 \end{eqnarray\*}

**Long Run Profit Maximisation problem**:

\begin{eqnarray\*} \max\_{y\geq 0} & \ py - c(w,r,y) \end{eqnarray\*}
where $p\geq 0$, $w>0$, $r>0$ are given.

Solving this problem, we get the long-run supply as

\begin{eqnarray\*} y^s(p,w,r) =\dfrac{p}{2(w+r)}\end{eqnarray\*}

Substituting this into the long-run conditional labor demand obtained earlier will provide us with the long-run labor and capital demand as follows:
\begin{eqnarray\*} (l^d,k^d)(p,w,r) =\left(\left(\dfrac{p}{2(w+r)}\right)^2,\left(\dfrac{p}{2(w+r)}\right)^2\right)\end{eqnarray\*}


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### Answer by username_3 on 2024-04-08. Score: 1

I'll solve the Cost-Minimization and Profit-Maximization problems for the Leontief Production Function in the Long Run

Solution: Here we are given the production as, $$y=f(l,k)=min\{l^{0.5},k^{0.5}\}$$
i.e., we are dealing with a DRS Technology

$$y=f(l,k)=min\{l^{0.5},{k}^{0.5}\}$$

Therefore, the Cost Minimization problem will be,
$$min\_{l,k}\hspace{3mm}(wl+r{k})$$
$$st. l \ge 0,y \le f(l,{k})=min\{l^{0.5},{ k}^{0.5}\}$$

The equilibrium will be when,
$$y=l^{\frac{1}{2}}={k}^\frac{1}{2}$$
$$\Rightarrow y^2=l^\star={k}^\star$$

Now we will get the cost function as, $c(y;w,r)=wl+r{k}=(w+r)y^2$

Assuming that the firm is a price-taker, The Long Run Profit Maximization problem will now become,

$$max\_y \hspace{4 mm} p(y)-(w+r)y^2$$
$$st. y\ge 0$$

Solving the above problem, we will get the Long-Run Supply Function of the Firm as ,
$$y^s(p,w,r)=\begin{cases}\frac{p}{2(w+r)}, & \text {if $p > 0$} \\
0, &\text{otherwise}\end{cases}$$


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