<p>Assuming that the firm is a price-taker in the input market, we can determine the cost function associated with plant 1 and 2 by solving the cost minimisation problem in the routine way, and we get:</p>
<ul>
<li>For <span class="math-container">$f_1(l_1, k_1) = \min(l_1, k_1)$</span>, cost function is <span class="math-container">$c_1(w,r,q_1) = (w+r)q_1$</span></li>
<li>For <span class="math-container">$f_2(l_2, k_2) = l_2^{\frac{1}{2}}k_2^\frac{1}{2}$</span>, cost function is <span class="math-container">$c_2(w,r,q_2) = (2\sqrt{wr})q_2$</span></li>
</ul>
<p>Clearly, Marginal cost of firm 2 is always less than or equal to marginal cost of firm 1 because <span class="math-container">$2\sqrt{wr} \leq w+r$</span> for all <span class="math-container">$w &gt; 0,r &gt; 0$</span>. Therefore, one of the cost minimising choice for firm 2 is to continue to use its own plant and not use firm 1's plant in any capacity no matter what the input prices are. In other words, the overall cost function is <span class="math-container">$c(w,r,q) = (2\sqrt{wr})q$</span>.</p>

CC BY-SA 4.0
2023-03-28T13:28:47.080Z
2023-03-28T13:28:47.080Z
Amit
<p>I think, firm 2 after the acquisition of firm 1, it will have two different production plants. In order to find optimal quantity and price, because it has become monopoly, we should find marginal cost of these two plants and equate it with the marginal revenue.</p>
<p>Firm 1' marginal cost:</p>
<p><span class="math-container">$f(K,L)=min(K,L) $</span></p>
<p><span class="math-container">$K=L=Q$</span></p>
<p><span class="math-container">$TC1=Q\times w+Q\times r=Q\times(w+r)$</span></p>
<p><span class="math-container">$MC1=w+r$</span></p>
<p>Firm 2' marginal cost:</p>
<p><span class="math-container">$Q=f(K,L)=L^{0.5}\times K^{0.5} \rightarrow L=Q\times \sqrt{r/w}$</span></p>
<p><span class="math-container">$K=Q\times\sqrt{w/r}$</span></p>
<p><span class="math-container">$TC2= 2\times Q\sqrt {w\times r}$</span></p>
<p><span class="math-container">$MC2=2\times \sqrt {w\times r}$</span></p>
<p>Because of <span class="math-container">$ 2\times \sqrt {w\times r}&lt;w+r$</span> production facility of first firm should be used until <span class="math-container">$MC1=MR=(P\times Q)(dTR/dQ) $</span> And after exceeding this the remaining production should be held by second facility until <span class="math-container">$MC2=MR$</span></p>
<p>Thanks @Amit for the correction. I forgot to derive demand functions.</p>

CC BY-SA 4.0
2023-03-28T11:58:45.303Z
2023-03-28T17:11:01.980Z
2023-03-28T17:11:01.980Z
Tunay Sabri Yüksel
<p>Let's say firm 1 has a production function <span class="math-container">$\;=min(L1,K1)$</span>
And firm 2 has a production function <span class="math-container">$\;= \sqrt{L}\times \sqrt{K}$</span>
Total labour supply <span class="math-container">$\;=L1+L2\;$</span> Total capital <span class="math-container">$\; =K1+K2\;$</span>
And firm 2 takes over firm 1.
How to derive the new cost function? Should it be equivalent to adding cost function of both firms after accounting for cost minimization?
Since firm 1 has constant returns to scale I'm having difficulty deriving it.</p>

<microeconomics>
How to derive a cost function when a firm takes over another firm to form a monopoly
Anjali
# How to derive a cost function when a firm takes over another firm to form a monopoly

(Asked by: username_0 on 2023-03-28)

Let's say firm 1 has a production function $\;=min(L1,K1)$
And firm 2 has a production function $\;= \sqrt{L}\times \sqrt{K}$
Total labour supply $\;=L1+L2\;$ Total capital $\; =K1+K2\;$
And firm 2 takes over firm 1.
How to derive the new cost function? Should it be equivalent to adding cost function of both firms after accounting for cost minimization?
Since firm 1 has constant returns to scale I'm having difficulty deriving it.

**Tags:** microeconomics

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## 2 answers

### Answer by username_1 on 2023-03-28. Score: 3

Assuming that the firm is a price-taker in the input market, we can determine the cost function associated with plant 1 and 2 by solving the cost minimisation problem in the routine way, and we get:

* For $f\_1(l\_1, k\_1) = \min(l\_1, k\_1)$, cost function is $c\_1(w,r,q\_1) = (w+r)q\_1$
* For $f\_2(l\_2, k\_2) = l\_2^{\frac{1}{2}}k\_2^\frac{1}{2}$, cost function is $c\_2(w,r,q\_2) = (2\sqrt{wr})q\_2$

Clearly, Marginal cost of firm 2 is always less than or equal to marginal cost of firm 1 because $2\sqrt{wr} \leq w+r$ for all $w > 0,r > 0$. Therefore, one of the cost minimising choice for firm 2 is to continue to use its own plant and not use firm 1's plant in any capacity no matter what the input prices are. In other words, the overall cost function is $c(w,r,q) = (2\sqrt{wr})q$.


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### Answer by username_2 on 2023-03-28. Score: 1

I think, firm 2 after the acquisition of firm 1, it will have two different production plants. In order to find optimal quantity and price, because it has become monopoly, we should find marginal cost of these two plants and equate it with the marginal revenue.

Firm 1' marginal cost:

$f(K,L)=min(K,L) $

$K=L=Q$

$TC1=Q\times w+Q\times r=Q\times(w+r)$

$MC1=w+r$

Firm 2' marginal cost:

$Q=f(K,L)=L^{0.5}\times K^{0.5} \rightarrow L=Q\times \sqrt{r/w}$

$K=Q\times\sqrt{w/r}$

$TC2= 2\times Q\sqrt {w\times r}$

$MC2=2\times \sqrt {w\times r}$

Because of $ 2\times \sqrt {w\times r} production facility of first firm should be used until $MC1=MR=(P\times Q)(dTR/dQ) $ And after exceeding this the remaining production should be held by second facility until $MC2=MR$

Thanks @Amit for the correction. I forgot to derive demand functions.


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