A multinational firm produces microchips at a home facility and at a foreign subsidiary according to the respective cost functions:
CH = 120QH and CF = 50QF + .5QF2
The firm sells chips in the home market and the foreign market where the inverse demand curves are
PH = 300 − DH and PF = 250 − .5DF,
Respectively: Here D denotes the quantity sold in each market, and Q denotes the quantity produced in each facility. Chips can be costlessly shipped between markets so that DH need not equal QH (nor DF equal QF). However, total production must match total sales: QH + QF = DH + DF.
a. Create a spreadsheet (based on the accompanying example) to model the firm’s worldwide operations. Use the spreadsheet optimizer to find the firm’s profit-maximizing outputs, sales quantities, and prices.
(Be sure to include the constraint that cell F9—extra output—must equal zero. That is, total sales must exactly equal total output.) Are chips shipped overseas? At the optimal solution, confirm that MRH = MRF = MCH = MCF.
b. Answer the questions in part a under an “antidumping” constraint; that is, the company must charge the same price in both markets. (Include the additional constraint that cell F12, the price gap, must equal zero.)
SOLUTION
a. Using the spreadsheet optimizer, we determine the multinational firm’s optimal plan: Maximize total profit in cell E20 by changing outputs and sales in cells C10, D10, C8 and D8, subject to cell F9 being greater or equal to zero. The solution is: QH = 150, QF = 70, DH = 90, DF = 130, and π = $19,000. At this solution, we see that MRH = MRF = MCH = MCF = 120.
b. If the firm must charge a uniform worldwide price, we reoptimize the spreadsheet after adding the constraint that the price difference in cell C12 must be equal to zero. The new solution is: QH = 150, QF = 70, DH = 106.67, DF = 113.33, P = 193.33, and π = $18,583.