Question

7. Consider a firm that produces sodium polyacrylate at a single plant then sells it to two different markets where they hold local monopolies. Below are the inverse demand equations for both markets A and B: PA= 34 −4qA; PB= 14 −qB

(a) Find the firm’s marginal revenue functions for both markets.

(b) Find an equation for qB in terms of qA that gives the optimal distribution of output across the two markets.

(c) Assume the firm manufactures the good in a central location so total costs are based on total output. If the firm’s marginal cost for total production is given by M C = 0.2(qA+ qB), what is the profit-maximizing level of output in both markets?

(d) What prices will the firm charge in each market? Based on your response, which market likely has more elastic demand?

Answer 1

To determine the marginal revenue functions, we derive the MR as the derivative of total revenue. For optimal output distribution, we equate the MR and solve for qB in terms of qA. Substituting into the MC function gives us the profit-maximizing output levels, from which we can determine the prices for each market.

Step-by-step explanation:

To find the marginal revenue functions for both markets A and B with inverse demand equations PA= 34 −4qA and PB= 14 −qB, we first need to derive the total revenue (TR) for each market. Total Revenue is found by multiplying price times quantity (TR=PxQ). For market A, TRA=PA*qA=(34-4qA)qA=34qA-4qA^2. The marginal revenue is the derivative of total revenue with respect to quantity, which for market A is MRA=d(TRA)/dqA=34-8qA. Similarly, for market B, TRB=PB*qB=(14-qB)qB=14qB-qB^2 and MRB=d(TRB)/dqB=14-2qB.

In order to find the equation for qB in terms of qA that gives the optimal distribution of output across the two markets, we set the marginal revenues equal to each other since the marginal cost is the same for both. So MRA=MRB leads to 34-8qA=14-2qB. Rearranging terms, we get qB=10-4qA.

Given the firm’s marginal cost for total production is MC=0.2(qA+qB), we can substitute qB from our previous equation into this to find the total output that maximizes profit. Setting MR equal to MC, for market A, we have 34-8qA=0.2(qA+(10-4qA)), and for market B, 14-2qB=0.2(qA+qB). Solving these equations gives us the profit-maximizing level of output qA and qB.

Once we have the profit-maximizing quantities, we can substitute them back into the original inverse demand equations to determine the prices charged in each market. The market with a steeper demand curve (more significant decrease in price for each additional unit sold) will likely have the more elastic demand.