Consider An Industry In Which Firms Can Expect To Sell 1000

Consider An Industry In Which Firms Can Expect To Sell 1000 Units Ann

Consider an industry in which firms can expect to sell 1,000 units annually at a market price of P. Before firms enter, they do not know their production costs with certainty. Instead, they believe that unit costs can be $2, $4, $6, or $8 with equal probability. Annualized sunk production costs are 1,500 – firms cannot recover this expense should they choose to exit. What is the equilibrium price at which firms are indifferent about entering? What is the average profit of firms that are producing?

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The industry described involves firms deciding whether to produce based on their uncertain production costs and the prevailing market price. Analyzing such a setting requires understanding the incentives for entry and production, considering the distribution of costs, and calculating the equilibrium price at which firms are indifferent about entering. Additionally, determining the average profit of incumbent firms predicated on their production costs provides insight into their profitability under these conditions.

Initially, firms face uncertainty about their marginal costs, which are equally likely among four discrete values: $2, $4, $6, and $8. These costs influence their decision to produce, as firms will only produce if the market price P exceeds or equals their marginal cost. Since marginal costs are discrete, firms with costs of $2 or $4 will produce when P ≥ $2 or P ≥ $4, respectively, but firms with costs of $6 or $8 will only produce when P ≥ $6 or P ≥ $8.

The key question is: What market price P would make firms indifferent about entering or not? This occurs when firms with the highest costs—that is, those at the margin—are just willing to produce. Since there is a sunk cost of $1,500, firms will engage in production only if the expected profit exceeds or equals zero. The profit for a firm with a given cost c equals:

Profit = (P - c) × Q - Sunk costs

Here, Q (the quantity sold per firm) is 1,000 units. The firm's expected profit depends on the probability it has a certain cost and the resulting payoffs.

Given the equal probability distribution, the probability that a firm's cost is less than or equal to a particular P is based on the costs defined: $2, $4, $6, and $8. Firms with costs below or equal P will produce if P ≥ c. Therefore, at a market price P, the proportion of firms willing to produce are those with costs less than or equal to P.

To find the equilibrium price, consider the highest cost at which firms are willing to produce, which is P = $8. For the firms with costs of $2, $4, and $6, they will produce if P exceeds their costs. The firms with costs at or below P contribute to the total supply, which in turn affects the equilibrium price.

Firms with costs at $2 and $4 will produce if P ≥ $4, whereas those with costs at $6 and $8 will produce only if P ≥ $6 or P ≥ $8, respectively. The key is identifying the equilibrium point where the firms with the highest costs just cover their sunk costs through production, considering the expected market price P that clears the market in a setting with fixed costs.

Given the sunk costs and the costs structure, the equilibrium market price must at least cover the average marginal cost across the producing firms to ensure their profitability. Since the highest cost among firms that will produce in equilibrium is $8, the firms with costs at $8 will only produce if P ≥ $8, which would mean all firms with costs less than or equal to P would also produce.

But, considering the firms with costs of $2, $4, and $6, their costs are lower than P, making it profitable for them to produce at P ≥ $4, $6, and $8, respectively. The firms with costs significantly above the equilibrium price will choose not to produce, so the market will clear when firms with costs of $2 and $4 are producing, and the market price is driven toward the highest cost at which profit is still positive, typically near the highest cost's average.

To determine the precise equilibrium selling price, P*, the firms will be indifferent between entering and not entering when their expected profit equals zero. This occurs when the expected revenue from production just offsets the sunk cost of $1,500, considering the probability distribution of costs and corresponding production decisions.

Assuming an expected market price P, the expected profit per firm is:

Expected Profit = Probability of producing × (P - c) × Q - 1,500

Given the probabilities are equal (¼ for each cost), and firms produce only if P ≥ c, the average profit across all potential producers can be calculated, and the equilibrium price P* solved by setting the expected profit to zero, i.e.,

0 = Σ [Probability(c) × (P* - c) × 1,000] - 1,500,

where the sum is over all costs c for which P* ≥ c.

Calculations reveal that the firms with costs of $2 and $4 will produce at lower prices, whereas those with costs of $6 and $8 require higher prices. Given symmetry and the costs, the equilibrium market price P must be at least high enough to justify producing for the highest-cost firms, which is P ≈ $8. However, since the expected profit must be zero at equilibrium, we find that the market price equates the expected average profit across these cost scenarios, leading to an approximate equilibrium price of around $6, where firms with costs less than or equal to $6 are producing, balancing the costs and sunk expenses.

Based on this analysis, the equilibrium market price at which firms are indifferent about entering is approximately $6, as at this price, the expected profit per firm equals zero when considering the likelihood of incurring different costs and the sunk costs involved. Under this price, firms with costs of $2, $4, and $6 are producing, while those with $8 costs are not, since the marginal revenue does not cover their higher costs.

Furthermore, the average profit for firms producing at this equilibrium price can be estimated by considering the firms with costs at or below $6. Their profits per firm are: 

- For $2-cost firms: Profit = (6 - 2) × 1,000 - 1,500 = (4) × 1,000 - 1,500 = 4,000 - 1,500 = $2,500

- For $4-cost firms: Profit = (6 - 4) × 1,000 - 1,500 = (2) × 1,000 - 1,500 = 2,000 - 1,500 = $500

- For $6-cost firms: Profit = (6 - 6) × 1,000 - 1,500 = 0 - 1,500 = -$1,500 (not profitable, hence they may not produce)

Therefore, the average profit of firms producing is weighted by the probability they produce. Since only firms with costs of $2 and $4 are profitable at P=6, and their probabilities are equal (each ¼), the aggregate expected profit per firm among producing firms is computed considering only those profitable scenarios.

In conclusion, the equilibrium market price is approximately $6, where firms with costs $2 and $4 are producing, and the profit margins are substantial for the lower-cost firms, while the higher-cost firms ($6 and $8) are not producing. The average profit for firms that are producing at this equilibrium price is roughly $1,250, considering only the profitable firms weighted by their probabilities.

References

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