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?
MC = 2Q1 + 2Q2 for both demand functions. Set MR1 = 2Q1 + 2Q2, and MR2 = 2Q1 + 2Q1 and then you have a system of 2 unknowns and 2 equations. The solution ends up being Q1 = 15, P1 = 72.5, Q2 = 17.5, P2 = 82.5. In this case Profits under PD > Profits Under single price. See you in 468 tomorrow.