Suppose that BMW can produce any quantity of cars at a constant marginal cost equal to \(\$ 20,000\) and a fixed cost of \(\$ 10\) billion. You are asked to advise the CEO as to what prices and quantities BMW should set for sales in Europe and in the United States, The demand for BMWs in each market is given by $$Q_{E}=4,000,000-100 P_{E}$$ and $$Q_{u}=1,000,000-20 P_{\mathrm{U}}$$ where the subscript \(E\) denotes Europe, the subscript \(u\) denotes the United States. Assume that BMW can restrict U.S. sales to authorized BMW dealers only. a. What quantity of BMWs should the firm sell in each market, and what should the price be in each market? What should the total profit be? b. If \(\mathrm{BMW}\) were forced to charge the same price in each market, what would be the quantity sold in each market, the equilibrium price, and the company's profit?

Price discrimination is a strategy where a company charges different prices for the same product to different groups of consumers, often based on their willingness to pay, geographical location, or other distinguishing characteristics. The goal is to capture consumer surplus, which is the difference between what consumers are willing to pay and what they actually pay, thus increasing a company's revenue and profits.

In the context of the exercise, BMW has the opportunity to implement price discrimination by setting different prices for its cars in Europe and the United States. The demand functions provided (Q_E = 4000000 - 100P_E for Europe and Q_U = 1000000 - 20P_U for the United States) allow BMW to determine the optimal price in each market to maximize profit based on differing demand elasticities. As long as BMW can prevent arbitrage, i.e., customers buying in one market and selling in another, it can successfully practice price discrimination and enhance its profits.

Profit maximization is the process by which a company determines the price and output level that delivers the highest profit. The fundamental condition for profit maximization is that the company produces up to the point where marginal cost (MC) equals marginal revenue (MR). Marginal cost is the increase in total cost when one more unit is produced, and marginal revenue is the additional income from selling one more unit of a product.

In this case, BMW's marginal cost is constant at \(20,000. The profit-maximizing strategy requires BMW to set prices in both markets where the resulting quantity demanded will equal the quantities at which the marginal cost is \)20,000. The provided exercise walks through the calculation to determine the optimal quantity and price for BMW cars in each market to achieve maximum profit.

In economics, the demand function is a mathematical representation that defines the quantity of a good or service that consumers are willing and able to purchase at various prices, assuming all other factors remain constant (ceteris paribus). The demand function typically has a negative slope, indicating an inverse relationship between price and quantity demanded.

The exercise presents two demand functions for BMW in different markets: Q_E = 4000000 - 100P_E for Europe and Q_U = 1000000 - 20P_U for the United States. These equations suggest that as the price increases, the quantity demanded decreases. Solving these functions given the marginal cost allows BMW to determine the price to charge for each car sold in both markets to align with consumer demand and maximize profits.

Market equilibrium occurs when the quantity demanded by consumers equals the quantity supplied by producers, resulting in a stable market price. At this point, the economic forces of supply and demand are balanced, and there is no tendency for the price to change, assuming other factors are held constant.

When BMW considers charging the same price in both Europe and the United States, it is attempting to find a market equilibrium that will apply across both markets. However, because the demand functions are different for each market, this single pricing strategy might not result in the profit-maximizing equilibrium found in each market separately. In part b of the exercise, we investigate the outcome of a single price across both markets and calculate the new quantity sold, the equilibrium price, and the resulting company profit.