Question

You have been hired by a firm that produces two products, Q(1) and Q(2). As the economic consultant, the production function is: C(Q(1),Q(2)) = 5000 – 2Q(1)Q(2) + 4Q(1)3 + 5Q(2)2 and management has approached you for advice.

First, management is considering increasing production of Q(2) in response to gaining access to a new market. If the firm does increase the production of Q(2) and holds the production of Q(1) constant, what is the impact of this decision on the costs associated with the production of Q(1)?

Second, some in management believe it would be better to produce the two products separately than together. Given that current production is 2 million units for Q(1) and 3 million units for Q(2), would this, if production levels were held constant, be a wise course of action?

Answer 1

The given equation is C(Q(1),Q(2)) = 5000 – 2Q(1)Q(2) + 4Q(1)3 + 5Q(2)2 of a firm that produces two products, Q(1) and Q(2).

We are also given that the production is 2 million units for Q(1) and 3 million units for Q(2).

The profit maximising condition for a firm is when the marginal cost (MC) of the firm is equal to the marginal revenue (MR) of the firm. We will solve the give problem with this theorem.

Therefore, MR = MC would be profit maximising condition for the given firm and every firm would try to operate at the profit maximising level of output.

Now, lets look at the equation again. C(Q(1),Q(2)) = 5000 – 2Q(1)Q(2) + 4Q(1)3 + 5Q(2)2 This is the total cost faced by the firm. We need to calculate the marginal cost from this.

Therefore, which gives us MC(Q(1)) = -2Q(1) + 10

Similarly, which gives us MC(Q(2)) = -2Q(2) + 12

MC(Q(1)) shows us partial change is Q1 due to change occured in Q2. Which answers our first question that If the firm does increase the production of Q(2) and holds the production of Q(1) constant, there would be a change of -2Q(1) + 10 with every unit change of Q(2).

Now let us look at the profit maximising condition MR = MC

For Q(1) : MR = MC(Q(1))

MR = -2Q(1) + 10      ( Equation 1 )

Similarly, for Q(2) : MR = MC(Q(1))

MR = -2Q(2) + 12      ( Equation 2 )

Equating equation 1 and equation 2 we get

-2Q(2) + 12 = -2Q(1) + 10

Q(2) - 1 = Q(1) , This would be the optimal proft maximising quantity production for the firm.

We were given that current production is 2 million units for Q(1) and 3 million units for Q(2),

Substituting these in the above equation we get that LHS = RHS and hence satisfies the condition.

Therefore it is better to produce both products together at constant level of prodution of 2 million units for Q(1) and 3 million units for Q(2).