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 $$Q_{u}=1,000,000-20 P_{u}$$ 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?

Understanding the concept of marginal cost is essential in microeconomics, particularly when discussing production and pricing strategies. Marginal cost refers to the additional cost incurred to produce one more unit of a good or service. It's a crucial component for businesses to consider when seeking to maximize profits, as it influences the decision on how many items should be produced.

In the BMW example, we're told that the marginal cost is constant at \(\$ 20,000\) per car. This implies that for every additional car BMW produces, the cost rises by this amount. When companies like BMW are setting prices for their products, they must ensure that the price covers the marginal cost, otherwise, they will not be covering their production costs on additional units sold, leading to losses.

• The marginal cost is the additional cost of producing one more unit.
• For BMW, the marginal cost is constant, unlike many scenarios where it changes with the level of output.
• Setting prices above marginal cost is necessary for profit.

For profit maximization, it's essential that the marginal cost is less than the price at which the product is sold so that each additional unit contributes to covering fixed costs and adding to profit.

A fixed cost is a type of business expense that does not change with the level of goods or services produced by the firm. These costs are 'fixed' over a specified period, regardless of the production volume. Common examples include rent, salaries, and in the case of our BMW example, the initial \(\$ 10\) billion investment.

For BMW, this massive \(\$ 10\) billion cost is incurred regardless of the number of vehicles it manufactures or sells. When advising on pricing and sales strategies, it's critical to retrieve this sunk cost through the pricing of cars over time. The fixed cost does not change with the quantity produced, making it a key concept in the calculation of total cost and in turn, profit.

• Fixed costs do not fluctuate with production levels.
• BMW's huge \(\$ 10\) billion fixed cost must be factored into their pricing strategy to ensure profitability over time.

Although fixed costs do not impact the marginal cost, they are vital in determining the break-even price and the profit-maximizing quantity.

The demand function is a mathematical representation that shows the relationship between the quantity of a product that consumers are willing and able to buy and the product's price, along with other factors like consumer income and prices of related goods. In the BMW exercise, we're given separate demand functions for Europe and the United States, indicating how many cars are expected to be sold at different price points.

Using the provided demand functions, \(Q_{E}=4,000,000 - 100 P_{E}\) and \(Q_{U}=1,000,000 - 20 P_{U}\), we can derive how quantity demanded varies in each market in response to price changes. These equations are key to understanding consumer behavior and setting optimal prices for different markets—core components in microeconomics and pricing strategy.

• Demand functions reveal the sensitivity of demand in relation to price.
• BMW needs to use these functions to determine the appropriate pricing strategy for both Europe and the United States market to maximize their sales and profits.

For BMW, these functions are pivotal for price differentiation between markets and are used to calculate the profit-maximizing price and quantity in each market.

Profit maximization is the short-run or long-run process by which a firm determines the price and output level that returns the greatest profit. The condition for profit maximization is where marginal cost equals marginal revenue. That's when the cost of producing one additional unit equals the revenue it generates.

In the BMW scenario, we aim to find the price and quantity that maximize profits in both the European market (\(P_E, Q_E\)) and the U.S. market (\(P_U, Q_U\)). Profit is calculated by deducting total costs, which include fixed and variable costs, from total revenues. The profit is maximized when the addition of the total revenue equals the addition of the total cost; this can be calculated by differentiating the profit equation with respect to quantity and setting the derivative to zero.

• The sweet spot for profit maximization occurs where marginal cost equals marginal revenue.
• BMW needs to deduce the sales quantity for each market that achieves this balance to maximize their profits.

By manipulating the demand functions and including the cost structures, we can formulate an equation to express profit and subsequently find its maximum value by calculating derivatives, as seen in the exercise solution.