How to derive a cost function when a firm takes over another firm to form a monopoly

Question

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.

Answer 1

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 $c(w,r,q) = (2\sqrt{wr})q$.

Answer 2

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}<w+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.