Question

3. Suppose that Perfect Labs produces autoclaves (laboratory equipment) at a constant marginal cost equal to $20,000 and a fixed cost of $10 billion. Perfect Labs sells its autoclaves in Canada and in Germany. You are trying to determine the best pricing strategy for Perfect Labs and you know that the demands in each country are the following:

Q_G =4,000,000 - 100P_G and Q_C =1,000,000 - 20P_C

where subscript G denotes Germany and subscript C denotes Canada.

(a) What is the quantity of autoclaves that Perfect Labs will sell in each market and at what price(s)? What is the total profit?

(b) If Canada and Germany sign a trade agreement which forces Perfect Labs to charge the same price in both markets, how many units will be sold in each market and at what price? What is the firm’s total profit in that case?

(c) Now suppose that Perfect Labs considers charging a two-part tariff in each coun- try. What is the profit-maximizing fixed fee and per unit price in each country? What is the total profit for Perfect Labs in this case?

(d) If Perfect Labs were able to employ perfect price discrimination, what would be its total profit? Explain.

(e) If Perfect Labs could choose, which of the three pricing strategies (a)-(c) would it choose? Why?

(f) If consumers were allowed to vote for one of those three pricing strategies (a)-(c), which one would they vote for? Explain.

Answer 1

So, here the demand for autoclaves in Canada and in Germany are given below.

=> QG = 4,000,000 – 100*PG, => PG = 4,000,000/100 – QG/100, => PG = 40,000 – QG/100.

=> PG = 40,000 – 0.01*QG, the demand in Canada.

=> QC = 1,000,000 – 20*PC, => PC = 1,000,000/20 – QC/20, => PC = 50,000 – QC/20.

=> PC = 50,000 – 0.05*QC, the demand in Germany. Now, the cost function is given by, TC = 10B + 20,000*Q, where “Q=QC + QG”and “10B = 10 billion.

So, here the “Perfect Labs” will charge different price in Canada and in Germany such that the overall profit will be maximum.

So, here the optimum condition is given by, “MRC = MRG = MC”, where “MRC” be the marginal revenue of “autoclaves” in Canada and “MRG” be the marginal revenue “autoclaves” in Germany.

=> PG = 40,000 – 0.01*QG, => MRG = 40,000 – 0.02*QG = MC = 20,000.

=> 40,000 – 0.02*QG = 20,000, => 0.02*QG = 20,000, => QG = 1,000,000.

So, the optimum price of autoclaves is “PG = 40,000 – 0.01*QG”.

=> PG = 40,000 – 0.01*1,000,000, => PG = 30,000, => PG = 30,000.

Similarly, “MRC = MC”, => PC = 50,000 – 0.05*QC, => MRC = 50,000 – 0.1*QC = MC = 20,000.

=> 50,000 – 0.1*QC = 20,000, => 0.1*QC = 30,000, => QC = 300,000.

So, the optimum price of autoclaves is “PC = 50,000 – 0.05*QC”.

=> PC = 50,000 – 0.05*300,000, => PC = 35,000, => PC = 35,000.

=> So, the total profit is given by, ? = PC*QC + PG*QG – TC.

=> ? = 35,000*300,000 + 30,000*1,000,000 – 10B – 20,000*Q, where Q=QC+QG=1,300,000.

=> ? = 35,000*300,000 + 30,000*1,000,000 – 10B – 20,000*1,300,000.

=> ? = 10,500,000,000 + 30,000,000,000 – 10B – 26,000,000,000.

=> ? = 10.5B + 30B – 10B – 26B, => ? = 4.5B.

b).

Now, if Canada and Germany sign a trade agreement which forces Perfect Labs to charge the same price in both the markets. So, here Perfect Lab will determine the optimum price on the basis of the aggregate demand and the aggregate MR curve.

=> MRC = 50,000 – 0.1*QC be the MR of autoclaves in Canada and MRG = 40,000 – 0.02*QG, be the MR of autoclaves in Germany.

=> MRC = 50,000 – 0.1*QC, => MRC/0.1 = 50,000/0.1 – QC,

=> 10*MRC = 500,000 – QC, => QC = 500,000 – 10*MRC.

Now, MRG = 40,000 – 0.02*QG, => MRG/0.02 = 40,000/0.02 – QG.

=> 50*MRG = 2,000,000 – QG, => QG = 2,000,000 – 50*MRG.

=> the aggregate MR is the horizontal summation of the individual one.

=> Q = QC + QG = 500,000 – 10*MRC + 2,000,000 – 50*MRG = 2,500,000 – 60*MR.

=> Q = 2,500,000 – 60*MR, => MR = 41,666.67 - 0.0167*Q .

=> the optimum quantity production will be determined by the following condition.

=> MR = MC, => 41,666.67 - 0.0167*Q = 20,000, => Q = 1,297,405.39.

Now, the aggregate demand curve is given by.

=> Q = QC + QG = 1,000,000 – 20*PC + 4,000,000 – 100*PG = 5,000,000 – 120*P.

=> 120*P = 5,000,000 – Q = 5,000,000 – 1,297,405.39 = 3,702,594.61.

=> P = 3,702,594.61/120 = 30,854.96.

=> the total profit is given by, ? = P*Q – TC = 30,854.96*1,297,405.39 – 10B – 20,000*Q.

=> ? = 40.03B – 10B – 20,000*1,297,405.39, => ? = 40.03B – 10B – 25.95B.

=> ? = 40.03B – 10B – 25.95B. = 4.08B, => ? = 4.08B.

c).

As we know that the fix fee is the “consumer surplus" that area under the demand curve ans above the per unit fee. The demand for autoclave in Canada and Germany are given by, PC = 50,000 – 0.05*QC and PG = 40,000 – 0.01*QG respectively.

So, under two part tariff the per unit price is exactly equal to “MC”, => So, the total quantity demanded in Canada is given by.

=> PC = 50,000 – 0.05*QC = 20,000, => QC = 600,000.

=> the optimum fixed fee in Canada is “FC = 0.5*(50,000 – 20,000)*600,000 = 9B.

Similarly, by PG = MC, => 40,000 – 0.01*QG = 20,000, => QG = 2,000,000.

=> the optimum fixed fee in Germany is “FG = 0.5*(40,000 – 20,000)*2,000,000 = 20B.

So, here the per unit fee is same for both country that is “MC=20,000”, but “fixed fee” is differ. So, “fixed fee” of Canada is “FC = 9B” and of Germany is “FG = 20B”.

=> total profit is given by, ? = FC + FG + 20,000*Q – 10B – 20,000*Q = 9B + 20B – 10B = 19B.

=> ? = 19B.

d).

Now, the Perfect Labs were able to employ perfect price discrimination then also the profit will be the same as the “two part tariff” case. Since here the total revenue will be the sum of “fixed fee” and the “total per unit fee” and “Q” will be determined by the condition “P=MC, => here also the profit will be “19B”, => 19 billion.