Question About Non-Discriminating Monopolist - Mathematics for Economists by Simon and Blume Chapter 17 Exercise 7

I am working on Mathematics for Economists by Simon and Blume Exercise 17.7. I know there is an Answers Pamphlet. However, the solution to this question does not make any sense to me. It seems that they put a solution to a wrong question there. But anyway, I tried that exercise myself, but I am not totally sure whether my answer is correct or not. I would really appreciate it if someone could help me check! Thanks in advance!

Here is the exercise:

A monopolist producing a single output has two types of customers. If it produces $$Q_1$$ units for customers of type 1, then these customers are willing to pay a price of $$50 - 5Q_1$$ dollars per unit. If it produces $$Q_2$$ units for customers of type 2, then these customers are willing to pay a price of $$100 - 10Q_2$$ dollars per unit. The monopolist's cost of manufacturing $$Q$$ units of output is $$90 + 20Q$$ dollars. Compute the demand function for the market as a whole, without price discrimination. Compute the firm's profit maximizing output for this situation.

This is how I did it:

Without price discrimination, we have $$p_1 = p_2$$. Then $$$$50 - 5Q_1 = 100 - 10Q_2.$$$$ Plug in $$p = 100 - 10Q_2$$, we have the market demand function: $$$$p = -\frac{10}{3}Q + \frac{200}{3}.$$$$ The firm's profit function is $$$$F(Q) = (-\frac{10}{3}Q + \frac{200}{3})Q - (90 + 20Q).$$$$ F.O.C.: $$$$\frac{\partial F}{\partial Q} = -\frac{20}{3}Q + \frac{140}{3} = 0 \Rightarrow Q = 7.$$$$ S.O.C.: $$$$\frac{\partial^2 F}{\partial Q^2} = -\frac{20}{3} < 0.$$$$ Thus, $$Q = 7$$ is the profit maximizing production. The profit then is $$F = \frac{220}{3}$$.

Basically, I am not sure whether it is correct to just set $$p_1 = p_2$$ to get the market demand function. I cannot figure out how intuitively that will make sense.

I don't know what is in the answer pamphlet.

One thing to be wary of is that you make the assumption that both types of consumers are served, $$Q_1,Q_2>0$$. You could still consider the case that the price is set so high ($$p>50$$) that type $$1$$'s do not purchase any units. Seems like maximal profit under that condition occurs at $$p=60$$, $$Q_2 = 4$$, neting $$\Pi = (60 - 20) \cdot (0+4) - 90 = 70,$$ so your solution is still the global maximum.

Your S.O.C. only checks whether your solution is a local maximum; here the profit function was not globally concave, it had a kink, thus your local maximum is not necessarily global.

• Thank you so much! I just want to confirm my understanding is correct: The reason that there is a kink is because it is possible for the monopolist to set the price so high so that customers of type 1 do not consume at all. Is that what you mean? Commented Jul 23, 2023 at 16:15
• Yes, that is what I mean. Commented Jul 23, 2023 at 19:26

Price discrimination implies that the monopolist is selling different units of output at different prices. In the above case, since there is no price discrimination the monopolist is selling different units of output at the same price.