Question

Suppose the market for tennis shoes has one dominant firm and five fringe firms. The market demand is \(Q=400-2 P .\) The dominant firm has a constant marginal cost of \(20 .\) The fringe firms each have a marginal cost of \(\mathrm{MC}=20+5 q\) a. Verify that the total supply curve for the five fringe firms is \(Q_{f}=P-20\) b. Find the dominant firm's demand curve. c. Find the profit-maximizing quantity produced and price charged by the dominant firm, and the quantity produced and price charged by each of the fringe firms. d. Suppose there are 10 fringe firms instead of five. How does this change your results? e. Suppose there continue to be five fringe firms but that each manages to reduce its marginal cost to \(\mathrm{MC}=20+2 q\). How does this change your results?

Short answer

For a variety of setups, the dominant firm produced 360 units and priced them at $20 per unit, while the fringe firms would not produce anything.

Step-by-step solution

Verify the total supply curve for the five fringe firms

The marginal cost curve of each fringe firm is given as \(MC = 20 + 5q\), where \(q\) is the quantity supplied by each firm. Supply equals marginal cost for firms in perfect competition, so the supply equation for each firm is \(q = P/5 - 4\). Now, for five such firms, the total quantity supplied by the fringe firms would be \(Q_f = 5q\). This yields \(Q_f = 5(P/5 - 4) = P - 20\) as required.

Find the dominant firm's demand curve

The dominant firm's demand is the difference between the market demand and the fringe firms' supply. Thus, subtracting the total quantity supplied by the fringe firms from the market demand, we have \(Q_D = Q - Q_f = 400 - 2P - (P - 20) = 420 - 3P\). This is the demand curve for the dominant firm.

Find the profit-maximizing quantity and price

For the dominant firm, profit maximization occurs where MC equals the price derived from the demand curve. The MC is constant at 20. From the demand curve of the dominant firm which is \(Q_D = 420 - 3P\), setting \(P = MC = 20\) yields \(Q_D = 420 - 3*20 = 360\) which is the quantity produced by the dominant firm. The fringe firms collectively supply \(Q_f = P - 20 = 20 - 20 = 0\). This indicates that the firms are producing at their minimum supply point to avoid a loss.

Changes with 10 fringe firms

If the number of fringe firms increases to 10, the total supply curve becomes \(Q_f = 10q = 10(P/5 - 4) = 2P - 40\). Substituting this into the market demand curve to get the new demand curve of the dominant firm, we get \(Q_D = 400 - 2P - (2P - 40) = 440 - 4P\). MC still equals price, therefore, setting \(P = 20\) yields \(Q_D = 440 - 4*20 = 360\), same as before. Now, the fringe firms supply \(Q_f = 2P - 40 = 2*20 - 40 = 0\), indicating that they still produce at their minimum supply point.

Reduced marginal cost of the fringe firms

If the marginal cost of fringe firms reduces to \(MC = 20 + 2q\), then their supply curve becomes \(q = P/2 - 10\). This results in a total supply curve of \(Q_f = 5(P/2 - 10) = 2.5P - 50\). The demand curve of the dominant firm then shifts to \(Q_D = 400 - 2P - (2.5P - 50) = 450 - 4.5P\). MC still equals price, therefore, setting \(P = 20\) yields \(Q_D = 450 - 4.5*20 = 360\), same as before. The fringe firms now supply \(Q_f = 2.5P - 50 = 2.5*20 - 50 = 0\), indicating that they still produce at their minimum supply point.

Key concepts

Marginal Cost

Understanding marginal cost is crucial in analyzing how firms make production decisions. Marginal cost is the additional cost incurred when producing one more unit of a good. For a firm, a constant marginal cost means that each additional unit produced costs the same to make. In our scenario, the dominant firm has a marginal cost of 20. This means for every extra pair of tennis shoes produced, the cost remains steady at 20 dollars.

Each of the fringe firms has a different situation. Their marginal cost equation is given by \(MC = 20 + 5q\), where \(q\) represents the quantity supplied by each firm. This implies that the cost of production for these firms increases with the production quantity. Understanding this helps us to see why fringe firms may not produce large volumes independently, as it becomes increasingly expensive for them.

• The constant marginal cost for the dominant firm allows it to predict costs accurately and assist in setting competitive prices.
• For fringe firms, increasing marginal costs restrict them from scaling production without facing significant cost hikes.

Market Demand

Market demand is the total quantity of a product consumers are willing to purchase at various price levels in a theoretical market. In this framework, the market demand for tennis shoes is expressed with the equation \(Q = 400 - 2P\), where \(Q\) stands for quantity demanded and \(P\) is the price.

This inverse relationship means as the price of the tennis shoes increases, the overall quantity demanded decreases, which is a typical economic pattern. The dominant firm's strategy revolves around gauging this market demand to decide its output level and pricing strategy.

In a market dominated by one firm and several small fringe firms, the dominant firm can influence market dynamics significantly. By assessing the market demand, this firm can decide on a production level that maximizes its profits by aligning production costs and consumer demand effectively. The dominant firm's decision-making relies heavily on understanding the total market demand to carve out its slice efficiently.

• Market demand helps predict consumer behavior and informs production and pricing.
• The dominant firm's understanding of demand allows them to control market shares by adjusting their output.

Fringe Firms Supply

In a market where several smaller companies operate alongside a dominant firm, these smaller companies are referred to as fringe firms. Each fringe firm's supply is linked to its marginal cost structure, where the total supply impacts the market as a whole.

Originally, with five fringe firms each having a supply equation of \(q = P/5 - 4\), their collective supply equation becomes \(Q_f = P - 20\). This equation illustrates how the fringe firms' supply contributes to the market, acting externally to the dominant firm's production.

When changes occur, such as having 10 fringe firms, this equation shifts to \(Q_f = 2P - 40\), revealing how increased competition affects the supply curve. However, the presence of fringe firms doesn't always guarantee a change in output when their ability to produce at minimal costs is limited, as reducing their marginal cost to \(MC = 20 + 2q\) again reveals when leading to a new supply equation \(Q_f = 2.5P - 50\).

• Fringe firms' supply is more sensitive to changes in marginal cost and number of firms than the dominant firm.
• Increasing fringe firm presence shifts the overall market supply, but their capacity to undercut the dominant firm depends on their internal costs.