Suppose a monopoly can produce any level of output it wishes at a constant marginal (and average) cost of \(\$ 5\) per unit. Assume the monopoly sells its goods in two different markets separated by some distance. The demand curve in the first market is given by $$a,=55-p_{n}$$ and the demand curve in the second market is given by $$\mathrm{Q}_{2}=70-2 \mathrm{P}_{2}$$ a If the monopolist can maintain the separation between the two markets, what level of output should be produced in each market, and what price will prevail in each market? What are total profits in this situation? b. How would your answer change if it only cost demanders \(\$ 5\) to transport goods between the two markets? What would be the monopolist's new profit level in this situation? c. How would your answer change if transportation costs were zero and the firm was forced to follow a single-price policy? d. Suppose the firm could adopt a linear two-part tariff under which marginal prices must be equal in the two markets but lump-sum entry fees might vary. What pricing policy should the firm follow?

#tag_title# b. Considering transportation costs #tag_content# Now, we will consider a transportation cost of $5 per unit between markets. For simplicity, we will assume that this cost is incurred when selling products in market 2, and can be added to the marginal cost. The new marginal cost in market 2 is $10 (original marginal cost of $5 plus the transportation cost of $5). We will now set the marginal revenue equal to the new marginal cost in market 2: \(35 - 2Q_2 = 10\) Solving for \(Q_2\), we find \(Q_2 = 12.5\). By substituting the output into the inverse demand function, we can find the optimal price: \(P_2 = 35 - 12.5 = 22.5\) Market 1 output, price, and profits remain the same as in the previous situation: \(Q_1 = 25\), \(P_1 = 30\), and \(\Pi_1 = 625\). Market 2 profits: \(\Pi_2 = (P_2 - MC - T)*Q_2 = (22.5 - 5 - 5) * 12.5 = 156.25\) Total profits: \(\Pi = \Pi_1 + \Pi_2 = 625 + 156.25 = 781.25\) With the $5 transportation cost, the firm should produce 25 units in market 1 and 12.5 units in market 2, with prices of \(30\) and \(22.5\), respectively, to achieve total profits of \(781.25\). #tag_title# c. Single-price policy without transportation costs #tag_content# Under a single-price policy and no transportation costs, the firm needs to set the same price in both markets. To determine the optimal output and price for each market, we consider the total demand function aggregating both markets: The total demand function: \(Q_T = Q_1 + Q_2 = (55 - P) + (70 - 2P) = 125 - 3P\). The inverse total demand function: \(P = \frac{125 - Q_T}{3}\). The total revenue function: \(TR = P * Q_T = \left(\frac{125 - Q_T}{3}\right)*Q_T\). We find the marginal revenue by differentiating the total revenue with respect to \(Q_T\): \(MR_T = \frac{d(TR)}{dQ_T} = \frac{125}{3} - 2Q_T\) Set the marginal revenue equal to the marginal cost: \(\frac{125}{3} - 2Q_T = 5\) Solving for \(Q_T\), we find \(Q_T = 35\). By substituting the total output into the inverse total demand function, we find the single price: \(P = \frac{125 - 35}{3} = 30\) Now, we distribute this output among the two markets according to their demand functions: Market 1 demand function: \(Q_1 = 55 - P = 55 - 30 = 25\) Market 2 demand function: \(Q_2 = 35 - Q_T + Q_1 = 35 - 35 + 25 = 25\) Market 1 profits: \(\Pi_1 = (P - MC) * Q_1 = (30 - 5) * 25 = 625\) Market 2 profits: \(\Pi_2 = (P - MC) * Q_2 = (30 - 5) * 25 = 625\) Total profits: \(\Pi = \Pi_1 + \Pi_2 = 625 + 625 = 1,250\) Under a single-price policy and no transportation costs, the firm should produce 25 units in each market and set a price of \(30\) in both markets, resulting in total profits of \(1,250\). #tag_title# d. Linear two-part tariff #tag_content# A linear two-part tariff consists of two components: a fixed fee (F) and a per-unit price (P). The marginal prices must be equal in both markets. The firm needs to determine the optimal fixed fee and per-unit price to maximize profits. It is optimal for the firm to set the per-unit price equal to the marginal cost, which is $5, as this maximizes output and consumer surplus and then capture that surplus through the fixed fee. Market 1 quantity based on the new price: \(Q_1 = 55 - P = 55 - 5 = 50\) Market 2 quantity based on the new price: \(Q_2 = 70 - 2P = 70 - 10 = 60\) Calculate the consumer surplus in each market: Market 1 consumer surplus: \(\frac{1}{2} * (30 - 5) * 50 = 625\) Market 2 consumer surplus: \(\frac{1}{2} * (20 - 5) * 60 = 450\) The firm can set the fixed fee equal to the consumer surplus in each market: Optimal fixed fee for market 1: \(F_1 = 625\) Optimal fixed fee for market 2: \(F_2 = 450\) Total profits: \(\Pi = \Pi_1 + \Pi_2 = 625 + 450 = 1,075\) Under a linear two-part tariff policy, the firm should set a per-unit price of $5 in both markets and charge a fixed fee of $625 in market 1 and $450 in market 2, resulting in total profits of $1,075.