# Price discriminating monopolist question

A price discriminating monopolist sells in two markets. Inverse demand in market 1 is given by: $$P_1(Q_1) = 80 - (1/2)Q_1$$ and inverse demand in market 2 is given by: $$P_2(Q_2) = 100 - Q_2$$ The monopolist's cost function is $C(Q) = (Q_1 + Q_2)^2$
a. Formulate the monopoly's function as a function of $Q_1$ and $Q_2$.
b. Calculate the monopoly's profit maximizing quantity sold in markets 1 and 2 and the corresponding prices.

Now suppose the government forbids price discrimination, so the monopolist can only set a single price for the two markets.

c. Compute the monopoly price and quantity.
d. How much is sold in market 1 and market 2, respectively?
e. Was the government intervention beneficial for social welfare or not?

a. This one is straightforward. $$= Q_1(80 - Q_1/2) + Q_2(100 - Q_2) - (Q_1 + Q_2)^2$$ b. This one I believed to be straightforward - set MR = MC. Thus, $$MR_1 = 80 - Q_1$$ $$MR_2 = 100 - 2Q_2$$ $$MC_1 = 2Q_1$$ $$MC_2 = 2Q_2$$ which gives $$(Q_1, Q_2, P_1, P_2) = (\frac{80}{3}, 25, \frac{200}{3}, 75)$$ c. This is where I start to get a bit messed up. Setting $P_1 = P_2$ gives a total demand function $$Q_1 + Q_2 = Q = 260 - 3P$$ I then solve for price, and get $$MR = \frac{260}{3} - \frac{2Q}{3}$$ and $$MC = 2Q$$ Thus, I'm left with $$(P,Q) = (75.8333, 32.5)$$

However, I feel as though I must have made a mistake. The profit when the monopolist is able to price discriminate actually ends up being lower than the profit when limited to one price. This surely can't be the case - where is my thinking going wrong here? Is my methodology off?

• Wouldn't that make sense that the monopolist's profit is higher when he can charge different customers the maximum price for that customer? Oct 2, 2017 at 1:58
• @CWill that certainly does make sense - I actually mistyped, because it in fact turns out that the profit is lower when he's able to price discriminate. Thus, I'm afraid my methods might be incorrect. Oct 2, 2017 at 2:06
• Why do you say $MC_i=2Q_i$? Oct 2, 2017 at 6:43
• @Ubiquitous $$C(Q_1, Q_2) = (Q_1 + Q_2)^2$$ $$Q_1 + Q_2 = Q_i$$ $$C(Q_i) = (Q_i)^2$$ $$MC_i = 2Q_i$$ Or do you mean for $i \in 1,2$? This I was unsure about. Oct 2, 2017 at 13:53
• I meant $i\in1,2$. Marginal cost for good $1$ is calculated as $\partial C/\partial Q_1=2(Q_1+Q_2)$. So it is only true that $MC_1=2Q_1$ if $Q_2=0$. But you are looking for a solution in which $Q_1$ and $Q_2$ are both different from zero, so you have used the wrong marginal cost function. Oct 2, 2017 at 14:41

I think if the government prohibits price discrimination, then the sum of MR in each market should equal to the MC: $$2(Q_1+Q_2) = 180 - Q_1 - 2Q_2$$ Also the price in each market should be the same: $$P_1 = P_2$$. Therefore: $$80-Q_1/2 = 100-Q_2$$. Aligning these two equations we get $$Q_1 = 20; Q_2 = 30; P_1=P_2 = 70.$$ Hope this helps!