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A monopolist produces output with constant marginal cost equal to 1. There are two of consumers that are potentially in the market for the good. Consumer A has inverse demand function $$p_A(x) = 7 − x,$$ and consumer B has inverse demand function $$p_B(x) = 5 − x.$$

Suppose the monopolist cannot identify which consumer is which. What is the best pair of purchase options $(x_A, R_A)$ and $(x_B, R_B)$ the monopolist should offer? (Hint: You’ll need to show that $x_B = 2.$) What is the firm’s profit?

My attempt:

The question doesn't make clear what $R_A, R_B$ are. But I guess the question is asking about 2nd degree price discrimination. So type A is high-demand consumer and B is low-demand consumer. $MR_B=5-2q_B=1$, then $x_B^*=2$ and at this point, the price is 3. So the first bundle designed for low-demand consumer is (2 units, whole bundle price is 2x3=6). I still need to design the other bundle for high-demand consumer.

If high-demand consumer pretends to be low-demand and buys the above bundle, so he or she buys 2 units and pays $6. His or her WTP is 12. So consumer surplus is 12-6=6. Then how should I do next?

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