<p>You want to model the decision problem of firm <span class="math-container">$M$</span>. In case they produce a total quantity of <span class="math-container">$q_M$</span>, they can split this between factories <span class="math-container">$1$</span> and <span class="math-container">$2$</span>. Firm <span class="math-container">$M$</span> can minimize the cost of producing <span class="math-container">$q_M$</span> units by solving the problem
<span class="math-container">$$
\min_{q_1,q_2} C_1(q_1) + C_2(q_2) = \min_{q_1,q_2} 20q_1 + q_1^2 + 20q_2 + q_2^2
$$</span>
subject to the constraint
<span class="math-container">$$
q_M = q_1 + q_2.
$$</span>
(And the usual non-negativity constraints, but those will not matter here.)</p>
<p>The solution to the above problem yields <span class="math-container">$C_M(q_M)$</span>. If the solution is such that <span class="math-container">$q_1,q_2$</span> are both positive, then no factory should be shuttered.</p>

CC BY-SA 4.0
2021-06-10T12:38:11.220Z
Thanks for your response!

I get $ C_m(q_m) = 20q_m + (1/2)q_m^2$, which yields a lower cost than when producing at only 1 factory . I suppose it makes sense - since $20q_m$ is linear so the difference between producing at one factory only and two factories will simply be in how the production is allocated across the polynomial term i.e if $q=10$, $5^2 + 5^2 = 50 = (1/2) 10^2$
Microman
CC BY-SA 4.0
2021-06-10T09:21:05.577Z
2021-06-10T09:21:05.577Z
Giskard
<p>Suppose I have 3 firms in a market. They have an identical, convex cost function, <span class="math-container">$C(q) = 20q + q^2$</span> = <span class="math-container">$C_1 = C_2 = C_3$</span>, and each firm produces in their own factory.</p>
<p>Market demand is linear, <span class="math-container">$P=200-Q$</span> , where <span class="math-container">$Q = q_1 + q_2 + q_3$</span></p>
<p>Now, suppose that Firm 2 and Firm 3 merge, but costs don't change. The new Firm (call it Firm <span class="math-container">$M$</span>), has the same cost function as above.</p>
<p>How would Firm <span class="math-container">$M$</span> decide whether to produce at 1 factory entirely and close the inactive one down, or split production between two factories? <strong>Specifically, which choice would lead to lower costs?</strong></p>
<p>Intuitively, since the cost function is convex,  it would be beneficial to split production between two factories. I don't know how to identify Firm M's cost function when it produces in both factories.</p>

<microeconomics>
Finding the cost-minimising production allocation in a Cournot merger with symmetric costs
Microman
# Finding the cost-minimising production allocation in a Cournot merger with symmetric costs

(Asked by: username_0 on 2021-06-10)

Suppose I have 3 firms in a market. They have an identical, convex cost function, $C(q) = 20q + q^2$ = $C\_1 = C\_2 = C\_3$, and each firm produces in their own factory.

Market demand is linear, $P=200-Q$ , where $Q = q\_1 + q\_2 + q\_3$

Now, suppose that Firm 2 and Firm 3 merge, but costs don't change. The new Firm (call it Firm $M$), has the same cost function as above.

How would Firm $M$ decide whether to produce at 1 factory entirely and close the inactive one down, or split production between two factories? **Specifically, which choice would lead to lower costs?**

Intuitively, since the cost function is convex, it would be beneficial to split production between two factories. I don't know how to identify Firm M's cost function when it produces in both factories.

**Tags:** microeconomics


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## 1 answers

### Answer by username_1 on 2021-06-10. Score: 1 | Accepted answer

You want to model the decision problem of firm $M$. In case they produce a total quantity of $q\_M$, they can split this between factories $1$ and $2$. Firm $M$ can minimize the cost of producing $q\_M$ units by solving the problem
$$
\min\_{q\_1,q\_2} C\_1(q\_1) + C\_2(q\_2) = \min\_{q\_1,q\_2} 20q\_1 + q\_1^2 + 20q\_2 + q\_2^2
$$
subject to the constraint
$$
q\_M = q\_1 + q\_2.
$$
(And the usual non-negativity constraints, but those will not matter here.)

The solution to the above problem yields $C\_M(q\_M)$. If the solution is such that $q\_1,q\_2$ are both positive, then no factory should be shuttered.

> **Comments:**
> **username_0**: Thanks for your response!  I get $ C_m(q_m) = 20q_m + (1/2)q_m^2$, which yields a lower cost than when producing at only 1 factory . I suppose it makes sense - since $20q_m$ is linear so the difference between producing at one factory only and two factories will simply be in how the production is allocated across the polynomial term i.e if $q=10$, $5^2 + 5^2 = 50 = (1/2) 10^2$  

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