You are the manager of a firm that produces products X and Y at zero cost. You k

Question

You are the manager of a firm that produces products X and Y at zero cost. You know that different types of consumers value your two products differently, but you are unable to identify these consumers individually at the time of the sale. In particular, you know there are three types of consumers (1,000 of each type) with the following valuations for the two products.

Consumer Type Product X Product Y
1 $60 $50
2 $50 $125
3 $25 $140

a. What are your firm’s profits if you charge $25 for product X and $50 for product Y?
b. What are your profits if you charge $60 for product X and $140 for product Y?
c. What are your profits if you charge $110 for a bundle containing one unit of product X and one unit of product Y?
d. What are your firm’s profits if you charge $175 for a bundle containing one unit of X and one unit of Y, but also sell the products individually at a price of $60 for product X and $140 for product Y?

Explanation / Answer

a. Zero cost implies that total revenue equals profit. Let's use V to denote profit. If we charge a price of $25 for X and $50 for Y, then consumers 1, 2, and 3 will buy product X because they all value product X at least as much as $25. So, the revenue from product X is: R(X) = P*Q R(X) = 25*3 R(X) = 75 If we charge $50 for product Y, then all three consumers will buy because their valuations are all above $50. R(Y) = P*Q R(Y) = 50*3 R(Y) = 150 Profit is the sum of these. V = R(X) + R(Y) V = 75 + 150 V = 125 There are 1000 of each consumer, so really V = $125,000. I like to do this at the end to keep the math simple. b. We do the same process. Now, only consumer 1 buys product X and only consumer 3 buys product Y because they are the only consumers with valuations as large as the new, higher prices. R(X) = P*Q R(X) = 60*1 R(X) = 60 R(Y) = P*Q R(Y) = 140*1 R(Y) = 140 Profit is the sum of these. V = R(X) + R(Y) V = 60 + 140 V=200 Again, the real V is $200,000 because there are 1000 of each type. c. Let's see how much each type values a bundle of 1 of each Bundle 1 $60 $50 110 2 $50 $125 175 3 $25 $140 165 So, all three types will purchase for $110 V(B) = 110*3 V(B) = 330 But, of course, this means V = $330,000 d. People buy the bundle if the consumer surplus of the bundle is greater than the consumer surplus of buying individual items and people buy individually if the consumer surplus of the bundle is less than the consumer surplus of buying individual items. Consumer surplus is the difference between what you are willing to pay and what you have to pay. CS(Bundle) = 175 - valuation Group CS(Bundle) CS(ind) 1 64 0 2 0 N/A 3 -10 0 Consumer 1 will buy the bundle. Consumer 2 will purchase the bundle because he values the bundle at $175 and it costs $175. Alternatively, consumer two would not value either X or Y at their individual per unit prices. Consumer 3 will purchase product Y for $140. So, the profit equals the sum of the three revenues. V = R(1) + R(2) + R(3) V = 175 + 175 + 140 V = 490 But of course V = $490,000

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