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)) = 4000 – 3Q(1)Q(2) + 2Q(1)2 + 3Q(2)2 and management has approached you for advice.

a. 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)?

b. 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? Provide evidence to support your conclusion.

Answer 1

We are given the cost function C(Q(1),Q(2)) = 4000 – 3Q(1)Q(2) + 2Q(1)^2 + 3Q(2)^2.

a. We need to check if there are cost complementarities in this case, implying that, if the marginal cost of producing a good declines when production of other good increases. Note that we use the coefficient of combined output term Q1Q2 to measure cost complementarities which is in this case, given by 'a' = -3. When a < 0, an increase in Q2 would reduce the marginal cost of producing Q1. Here a is -3 so it is less than 0. Hence the cost of producing Q1 falls as Q2 is increased.

b. To check if it would be better to produce the two products separately than together, we need to find if there are economies of scope. For this we must have f > aQ1Q2 where f is fixed cost.  Given that Q(1) = 2 and Q(2) = 3, we have 4000 > 3*2*3 or 4000 > 18 which is true. Hence, there are economies of scope existing. It is therefore true that  it would be better to produce the two products separately than together.