Question

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?

Short answer

#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.

Step-by-step solution

a. Finding optimal output and prices in separated markets

To find the optimal output and prices in separated markets, we need to set marginal revenue equal to marginal cost in each market. Market 1: The demand function is given by \(Q_1 = 55 - P_1\). First, we need to find the inverse demand function by solving it for \(P_1\). We get: \(P_1 = 55-Q_1\) The total revenue is: \(TR_1 = P_1 * Q_1 = (55-Q_1)*Q_1\) To find the marginal revenue, differentiate the total revenue with respect to \(Q_1\): \(MR_1 = \frac{d(TR_1)}{dQ_1}= 55-2Q_1\) Since the marginal cost is \(5\), we equate marginal revenue to marginal cost to find the optimal output: \(55 - 2Q_1 = 5\) Solving for \(Q_1\), we find \(Q_1 = 25\). By substituting the output into the inverse demand function, we can find the optimal price: \(P_1 = 55 -25 = 30\) Market 2: The demand function is given by \(Q_2 = 70 - 2P_2\). Again, we need to find the inverse demand function. Solving it for \(P_2\), we get: \(P_2 = 35-Q_2\) The total revenue is: \(TR_2 = P_2 * Q_2 = (35-Q_2)*Q_2\) We find marginal revenue by differentiating the total revenue with respect to \(Q_2\): \(MR_2 = \frac{d(TR_2)}{dQ_2}= 35-2Q_2\) We set the marginal revenue equal to the marginal cost: \(35 - 2Q_2 = 5\) Solving for \(Q_2\), we find \(Q_2 = 15\). By substituting the output into the inverse demand function, we can find the optimal price: \(P_2 = 35 -15 = 20\) We can now calculate the total profits in separated markets. Market 1 profits: \(\Pi_1 = (P_1 - MC)*Q_1 = (30 - 5) *25 = 625\) Market 2 profits: \(\Pi_2 = (P_2 - MC)*Q_2 = (20 - 5) *15 = 225\) Total profits: \(\Pi = \Pi_1 + \Pi_2 = 625 + 225 = 850\) In this situation, the firm should produce 25 units in market 1 and 15 units in market 2, with prices of \(30\) and \(20\), respectively, to achieve total profits of \(850\). #Phase 2: Determining optimal output and prices considering transportation costs and other scenarios# We will do this in the next answer.

Key concepts

Monopoly Pricing

Monopoly pricing is a strategy utilized by a monopolist to maximize profits by setting prices above marginal costs. In a monopoly, the firm is the sole provider of a particular good or service, allowing it to control the market price. The monopolist determines the price that will maximize profits by considering the demand curve and equating marginal revenue with marginal costs. Unlike in perfect competition, where prices equal marginal costs, a monopoly can charge a higher price due to a lack of competition.

When considering separate markets, a monopolist can use monopoly pricing to set different prices in each market. This is based on the understanding of the demand elasticity in each market. If a firm can perfectly separate these markets and prevent arbitrage, where consumers buy in one market and sell in another, it can set the optimal output and prices to maximize profits in each segment independently. The pricing decision is crucial, as it impacts both consumer surplus and the firm's total revenue.

Marginal Revenue

Marginal revenue (MR) is the additional revenue a firm earns by selling one more unit of output. For a monopolist, it is crucial to understand how marginal revenue behaves, as it affects the decision about how much to produce and what price to charge.

To calculate marginal revenue in a given market, differentiate the total revenue (TR) with respect to quantity (Q). For instance, when the demand function is known, such as \(Q_1 = 55 - P_1\), the inverse demand function \(P_1 = 55 - Q_1\) helps calculate the total revenue \(TR_1 = P_1 \times Q_1 = (55-Q_1)\times Q_1\). From this, derive the marginal revenue as \(MR_1 = \frac{d(TR_1)}{dQ_1}= 55-2Q_1\).

A monopolist will continue to produce additional units as long as marginal revenue exceeds marginal cost. Once MR equals marginal cost, producing beyond that point would reduce the firm's profits. Understanding the concept of marginal revenue is essential for optimizing production and is a fundamental principle in determining monopoly pricing.

Total Revenue

Total Revenue (TR) refers to the total income a firm receives from selling a product or service. In the monopoly context, total revenue is calculated by multiplying the price (P) by the quantity sold (Q). Understanding TR is essential for a monopolist since it provides insights into how changes in price or quantity affect overall profitability.

In separate market scenarios, the monopolist calculates total revenue for each market individually. For example, if market demand is expressed as \(Q_1 = 55 - P_1\), the inverse function \(P_1 = 55 - Q_1\) allows for the total revenue calculation \(TR_1 = P_1 \times Q_1 = (55-Q_1)\times Q_1\).

• For Market 1: \(TR_1 = (55-Q_1)\times Q_1\)
• For Market 2: \(TR_2 = (35-Q_2)\times Q_2\)

Total revenue helps determine the optimal output and price level by illustrating where profits are maximized, usually when marginal revenue equals marginal cost in each respective market. To maximize profit, the monopolist analyzes changes in total revenue relative to output levels.

Two-Part Tariff

A two-part tariff pricing strategy involves charging a fixed fee in addition to a per-unit price. This method can be beneficial for a monopolist in maximizing profits while allowing different groups of consumers to participate in the market. The fixed fee can vary by consumer group, helping to capture more consumer surplus than single pricing models.

In the context of monopoly markets, a two-part tariff can allow a firm to integrate marginal pricing across different markets while customizing entry fees. By charging equal marginal prices in both markets, the monopolist adheres to cost minimization principles. Meanwhile, variable entry fees enable the company to manage demand variations and collect surplus through the fixed portion. This method serves as a tool for price discrimination while ensuring consumption across both markets remains profitable. By adopting such a pricing strategy, firms can achieve higher profits, as it efficiently segments the market without losing out on consumer participation.