<p>I`d propose you to follow these steps:</p>

<ol>
<li>Set up the minimization cost problem (<em>i.e.</em> for a given output quantity <span class="math-container">$y$</span> minimize costs):
<span class="math-container">\begin{align}
\min_{H,L,K}&amp;  \quad sH + wL + rK \tag{1} \label{1}\\
\text{such that} &amp;\quad  \min\{H,L\} + \min\{H, K\}\geq y \tag{2} \label{2}
\end{align}</span></li>
<li><p>In principle you have 3 cases, depending on price of factors <span class="math-container">$(s,w,r)$</span>.</p>

<p><strong>1)</strong> When <span class="math-container">$r&gt;s+w$</span>, then buying a unit of <span class="math-container">$L$</span> and one of <span class="math-container">$H$</span> is going to cost you <em>strictly less</em> than buying a unit ok <span class="math-container">$K$</span>. Consequently the monopolist opts for producing only using <span class="math-container">$H$</span> and <span class="math-container">$L$</span> (<span class="math-container">$K=0$</span>). Thus <span class="math-container">$H = L = y$</span> </p></li>
</ol>

<p><strong>2)</strong> The second case happens for <span class="math-container">$s&gt;r+w$</span> which is, anyway, <strong>ruled out</strong> by the assupmption <span class="math-container">$w&gt;s$</span> (if this was not the case, then producing only using <span class="math-container">$H$</span> and <span class="math-container">$K$</span> would have been optimal, <em>i.e</em> <span class="math-container">$H = K = y$</span> and <span class="math-container">$L = 0$</span>). </p>

<p><strong>3)</strong> Finally, when none of the above occurs, you have <span class="math-container">$L = H$</span> and <span class="math-container">$K = H$</span> because of the fixed proportions production function. Then the constraint \eqref{2} becomes <span class="math-container">$H + H \geq y$</span> and the inequality sign become equality. The equality sign comes from the fact that producing more than <span class="math-container">$y$</span> (the LHS of \ref{2} being strictly larger than the RHS) is just overshooting and inflates production costs for a <em>given</em> <span class="math-container">$y$</span>. <span class="math-container">$L = H$</span> and <span class="math-container">$K = H$</span> comes from the <em>Leontief production</em> set-up of the problem: inside the <span class="math-container">$\min\{...,...\}$</span> operator is never convenient to exploit more one of the two factors (because by decreaing the most used factor one still gets the same amount of output but a lower cost). Therefore, each of the two <span class="math-container">$\min\{...,...\}$</span> can just be seen as one of its inner arguments, namely, for convenience, <span class="math-container">$H$</span>. So you finally get <span class="math-container">$H+H = y$</span> and, from there: <span class="math-container">$H = y/2$</span> and <span class="math-container">$L = K = y/2$</span></p>

<ol start="3">
<li>Substitute the optimal values <span class="math-container">${\big(H^*\small{(s,w,r)},L^*\small{(s,w,r)},K^*\small{(s,w,r)}\big)}$</span> in \eqref{1} so to get an expression only in <span class="math-container">$(s,w,r)$</span>. This expression is the cost function <span class="math-container">$C(s,w,r,y)$</span></li>
<li>Now solve the monopolist's pricing problem for given factor prices:
<span class="math-container">\begin{equation}
\max_p \quad p\cdot D(p) - C\big(s,w,r,D(p)\big)
\end{equation}</span>
and find the profit maximizing price, <span class="math-container">$p^*$</span> for output level <span class="math-container">$y^* = D(p^*)$</span></li>
</ol>

<p>I hope this helps, comment below if you need any more hint.</p>

CC BY-SA 4.0
2019-10-28T04:11:23.453Z
I understand how to solve the problem but could you explain a little more about how you arrived at point 2? Thank you so much...
user24692
CC BY-SA 4.0
2019-10-28T08:22:40.203Z
I`ve just edit my answer, hopes it clarifies enough.
GabMac
CC BY-SA 4.0
2019-10-27T22:00:39.957Z
2019-10-28T22:13:56.123Z
2019-10-28T22:13:56.123Z
GabMac
<p>@GabMac, I don't think L=H,  K=H will always hold given this function. I think  L=H,   K=H holds because of the inequality r &lt; w + s. If r > w + s, the producer will always produce in H,L only and K=0 even when the production function remains fixed proportions. The answer is correct but I think only the reasoning for modifying the production function is wrong. Thus, L=H,K=H because of the given inequality r &lt; w + s. </p>

CC BY-SA 4.0
2019-10-28T22:14:34.013Z
You're right, I`ve edited my answer accordingly
GabMac
CC BY-SA 4.0
2019-10-28T19:11:40.327Z
2019-10-28T19:11:40.327Z
ghghghghghg
<p>I was doing my homework and got really confused about how to approach the optimal levels of inputs when there are three variables. My current understanding is that the problem is to solve the optimization problem</p>

<p><span class="math-container">$$\min_{H, L, K}\;{sH + wL +rK}$$</span> 
subject to
<span class="math-container">$$q = \min\{H,L\} + \min\{H,K\}$$</span></p>

<p>However, I don't know how to determine the inputs of H, L, and K. Below is the problem:</p>

<blockquote>
  <p>A monopolist can hire high skill labor <span class="math-container">$H$</span>, low skill labor <span class="math-container">$L$</span>, and robots <span class="math-container">$K$</span>. The per-unit price of these inputs are <span class="math-container">$s$</span>, <span class="math-container">$w$</span>, and <span class="math-container">$r$</span>. High skill labor is more expensive: <span class="math-container">$s &gt; w$</span>. The production function is <span class="math-container">$f(H, L, K) = \min\{H, L\} + \min\{H, K\}$</span>. The demand curve that the monopolist faces is <span class="math-container">$D(p) = A-p$</span>, where <span class="math-container">$A &gt; w + r + s$</span>. Suppose that robots are cheap: <span class="math-container">$r &lt; w + s$</span>.</p>
</blockquote>

<p>Essentially, we're asked to find the optimal levels (yields lowest costs) of input in terms of q for H,L,K</p>

<p>I initially thought since <span class="math-container">$r&lt;w+s$</span>, and <span class="math-container">$w&lt;s$</span>, we can deduce that <span class="math-container">$r&lt;s+s$</span>, and for this reason, the production function would give either <span class="math-container">$q = H+K$</span>, <span class="math-container">$q = L+H$</span>, or <span class="math-container">$q = L+K$</span> (making <span class="math-container">$H+H$</span> the only impossible combination). However, I'm super unsure if this is the right thought process, and I don't know how to proceed from here. </p>

<p>Any help is greatly appreciated. </p>

<microeconomics><mathematical-economics><input-output><cost-functions>
Optimal Production Input in Relation to Cost Minimization Problem
user24692
# Optimal Production Input in Relation to Cost Minimization Problem

(Asked by: username_0 on 2019-10-27)

I was doing my homework and got really confused about how to approach the optimal levels of inputs when there are three variables. My current understanding is that the problem is to solve the optimization problem

$$\min\_{H, L, K}\;{sH + wL +rK}$$
subject to
$$q = \min\{H,L\} + \min\{H,K\}$$

However, I don't know how to determine the inputs of H, L, and K. Below is the problem:

> A monopolist can hire high skill labor $H$, low skill labor $L$, and robots $K$. The per-unit price of these inputs are $s$, $w$, and $r$. High skill labor is more expensive: $s > w$. The production function is $f(H, L, K) = \min\{H, L\} + \min\{H, K\}$. The demand curve that the monopolist faces is $D(p) = A-p$, where $A > w + r + s$. Suppose that robots are cheap: $r < w + s$.

Essentially, we're asked to find the optimal levels (yields lowest costs) of input in terms of q for H,L,K

I initially thought since $r<w+s$, and $w<s$, we can deduce that $r<s+s$, and for this reason, the production function would give either $q = H+K$, $q = L+H$, or $q = L+K$ (making $H+H$ the only impossible combination). However, I'm super unsure if this is the right thought process, and I don't know how to proceed from here.

Any help is greatly appreciated.

**Tags:** microeconomics, mathematical-economics, input-output, cost-functions


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## 2 answers

### Answer by username_1 on 2019-10-27. Score: 1

I`d propose you to follow these steps:

1. Set up the minimization cost problem (*i.e.* for a given output quantity $y$ minimize costs):
   \begin{align}
   \min\_{H,L,K}& \quad sH + wL + rK \tag{1} \label{1}\\
   \text{such that} &\quad \min\{H,L\} + \min\{H, K\}\geq y \tag{2} \label{2}
   \end{align}
2. In principle you have 3 cases, depending on price of factors $(s,w,r)$.

   **1)** When $r>s+w$, then buying a unit of $L$ and one of $H$ is going to cost you *strictly less* than buying a unit ok $K$. Consequently the monopolist opts for producing only using $H$ and $L$ ($K=0$). Thus $H = L = y$

**2)** The second case happens for $s>r+w$ which is, anyway, **ruled out** by the assupmption $w>s$ (if this was not the case, then producing only using $H$ and $K$ would have been optimal, *i.e* $H = K = y$ and $L = 0$).

**3)** Finally, when none of the above occurs, you have $L = H$ and $K = H$ because of the fixed proportions production function. Then the constraint \eqref{2} becomes $H + H \geq y$ and the inequality sign become equality. The equality sign comes from the fact that producing more than $y$ (the LHS of \ref{2} being strictly larger than the RHS) is just overshooting and inflates production costs for a *given* $y$. $L = H$ and $K = H$ comes from the *Leontief production* set-up of the problem: inside the $\min\{...,...\}$ operator is never convenient to exploit more one of the two factors (because by decreaing the most used factor one still gets the same amount of output but a lower cost). Therefore, each of the two $\min\{...,...\}$ can just be seen as one of its inner arguments, namely, for convenience, $H$. So you finally get $H+H = y$ and, from there: $H = y/2$ and $L = K = y/2$

3. Substitute the optimal values ${\big(H^\*\small{(s,w,r)},L^\*\small{(s,w,r)},K^\*\small{(s,w,r)}\big)}$ in \eqref{1} so to get an expression only in $(s,w,r)$. This expression is the cost function $C(s,w,r,y)$
4. Now solve the monopolist's pricing problem for given factor prices:
   \begin{equation}
   \max\_p \quad p\cdot D(p) - C\big(s,w,r,D(p)\big)
   \end{equation}
   and find the profit maximizing price, $p^\*$ for output level $y^\* = D(p^\*)$

I hope this helps, comment below if you need any more hint.

> **Comments:**
> **username_0**: I understand how to solve the problem but could you explain a little more about how you arrived at point 2? Thank you so much...  
> **username_1**: I`ve just edit my answer, hopes it clarifies enough.  

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### Answer by username_2 on 2019-10-28. Score: 0

@username_1, I don't think L=H, K=H will always hold given this function. I think L=H, K=H holds because of the inequality r < w + s. If r > w + s, the producer will always produce in H,L only and K=0 even when the production function remains fixed proportions. The answer is correct but I think only the reasoning for modifying the production function is wrong. Thus, L=H,K=H because of the given inequality r < w + s.

> **Comments:**
> **username_1**: You're right, I`ve edited my answer accordingly  

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