Coke and Pepsi are two American soft drink companies operating in Russia. The ma
ID: 1195885 • Letter: C
Question
Coke and Pepsi are two American soft drink companies operating in Russia. The market demand curve for soft drinks in Russia is given by Q= 119-0.5P. Coke's short-run total and marginal costs are given by STC=3q^2 +48q+572 and SMC= 6q+48. Pepsi's short-run total and marginal costs are given by STC = 6q^2+18q+849 and SMC=12q+18. A) If Coke and Pepsi form a cartel to market soft drinks in Russia, calculate the cartel's profit- maximizing output and price. B) Calculate the profit-maximizing output produced by each firm. C) Calculate the profits earned by the cartel and each firm.
Explanation / Answer
When the firms form a cartel, this cartel acts like a monopolist. That is, the cartel doesn't take the price as given (as in a perfectly competitive market); rather it chooses a price-quantity combination (which depends on the demand curve and its cost function) such that the profit is maximized. There is, however, a 'twist' on this problem with respect to the monopolist's problem: in this case the cartel must also choose the quantity that each of the firms has to produce in order to maximize profits. So the problem here is to choose the quantities Q1 (quantity of Cokes) and Q2 (Pepsis) such that the following function is maximized: Max P(Q1+Q2)*(Q1+Q2) - C(Q1) - C(Q2) Q1, Q2 where C(Q1) and C(Q2) are the cost functions of each of the firms, and P(Q1+Q2) is the inverse demand function; which tells, given that the market quantity is Q1+Q2, what is the price consumers are willing to pay for this quantity. Clearly, this function represents the combined profits of both firms. We already know C(Q1) and C(Q2). We find P(Q) by isolating P in the demand function: Q = 119 - 0.5P 0.5P = 119 - Q P = 238 - 2Q So P(Q1+Q2) is: P = 238 - 2*(Q1+Q2) So the function we must maximize is: Max [238-2(Q1+Q2)]*(Q1+Q2) - 3Q1^2-48Q1-572 - 6Q2^2-18Q2-849 Q1, Q2 In order to solve this, we must the first order conditions: we must find the derivative of this function with respect to Q1 and equate it to 0, and then do the same derivating with respect to Q2. With this, we'll have two equations with two unknowns (Q1 and Q2). These will be the quantities that each firm must produce (the answer to question b). The sum of these quantities, and the price that comes from plugging this sum into the inverse demand function is the answer to question a. Finally, plugging these quantities into the above function will give you the cartel's profits; and calculating the income of each firm (quantity of each firm times market price) minus each firm's cost will give you the profits earned by each firm. This constitutes the answer to c. We have to choose Q1 and Q2 in order to maximize Max [238-2(Q1+Q2)]*(Q1+Q2) - 3Q1^2-48Q1-572 - 6Q2^2-18Q2-849 Q1,Q2 The derivative of this function with respect to Q1 is: -2(Q1+Q2) + [238-2(Q1+Q2)] - 6Q1 - 48 = 0 238 - 4(Q1+Q2) - 6Q1 - 48 = 0 238 - 10Q1 - 4Q2 -48 = 0 190 - 10Q1 - 4Q2 = 0 The derivative of the function with respect to Q2 is: -2(Q1+Q2) + [238-2(Q1+Q2)] - 12Q2 - 18 = 0 238 - 4(Q1+Q2) - 12Q2 - 18 = 0 238 - 4Q1 - 16Q2 - 18 = 0 220 - 4Q1 - 16Q2 = 0 Therefore, we have the system: 190 - 10Q1 - 4Q2 = 0 220 - 4Q1 - 16Q2 = 0 That is a system of 2 equations with 2 unknowns which can be easily solved. The solution is then: Q1 = 15 Q2 = 10 So Coke produces 15 units and Pepsi produces 10. The cartel as a whole produces 25 units. The price that is paid for these units is P = 238 - 2Q P = 238 - 2*25 P = 188 The profits of Coke are: Profits = Price*Quantity - Costs = 188*15 - 3*15^2 - 48*15 - 572 = 853 The profits of Pepsi are calculated in a similar fashion. Pepsi's profts are 251. The cartel's profits are 853+251=1104