Question

Suppose a firm wants to maximize output at the total cost of producing some fixed output Q' in two factories with cost functions C1=C(Q1) and C2=C(Q2). These two cost functions might be the same or different. Write this out as a calculus constrained-maximization problem and then solve for the first-order conditions for "output maximization."

Answer 1

In the case of multi-plant firm equilibrium is achieved at the point where

MC₁ = MC₂ = MR

MC₁ = Marginal Cost of First Factory

MC₂ = Marginal Cost of Second Factory

MR = Marginal Revenue

Inverse market demand: P = f(Q₁ + Q₂)

And cost structure of the factories are

C₁ = C(Q₁)

C₂ = C(Q₂)

The firm aims at the allocation of its production between Factory 1 and Factory 2 so as to maximise its profit

Profit = Total Revenue – Cost 1 – Cost 2

The first order condition for maximum profit requires

?Profit/?Q₁ = 0

And

?Profit/?Q₂ = 0

?Profit/?Q₁ = ?(Total Revenue)/?Q₁ – ?C₁/?Q₁

?(Total Revenue)/?Q₁ = ?C₁/?Q₁

MR₁ = MC₁

?Profit/?Q₂ = ?(Total Revenue)/?Q₂ – ?C₂/?Q₂

?(Total Revenue)/?Q₂ = ?C₂/?Q₂

MR₂ = MC₂

But MR₁ = MR₂ = MR

So we have

MR = MC₁ = MC₂