# 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? – CWill Oct 2 '17 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. – mizichael Oct 2 '17 at 2:06
• Why do you say $MC_i=2Q_i$? – Ubiquitous Oct 2 '17 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. – mizichael Oct 2 '17 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. – Ubiquitous Oct 2 '17 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!