Question

A perfcctly competitive industry has a large number of potential entrants. Each firm has an identical cost structure such that long-run average cost is minimized at an output of 20 units \(\left(q_{i}=20\right) .\) The minimum average cost is \(\$ 10\) per unit. Total market demand is given by \\[Q=1,500-50 P.\\] a. What is the industry's long-run supply schedule? b. What is the long-run equilibrium price \(\left(P^{*}\right)\) ? The total industry output \(\left(Q^{*}\right)\) ? The output of each firm \(\left(q^{*}\right)\) ? The number of firms? The profits of each firm? c. The short-run total cost function associated with each firm's long-run equilibrium output is given by \\[C(q)=0.5 q^{2}-10 q+200.\\] Calculate the short-run average and marginal cost function. At what output level does shortrun average cost reach a minimum? d. Calculate the short-run supply function for each firm and the industry short-run supply function. c. Suppose now that the market demand function shifts upward to \(Q=2,000-50 P\). Using this new demand curve, answer part (b) for the very short run when firms cannot change their outputs. f. In the short run, use the industry short-run supply function to recalculate the answers to (b). g. What is the new long-run equilibrium for the industry?

Short answer

In long-run equilibrium, firms produce 20 units each at $10, leading to 50 firms. Short-run supply with new demand stabilizes at $20, output stays 1000 units. Long-run readjusts back at $10 with 1500 output and 75 firms.

Step-by-step solution

Understanding the Industry's Long-Run Supply Schedule

In perfect competition, the industry's long-run supply schedule is determined by the minimum points on the firms' long-run average cost curves. Since each firm's minimum average cost is $10 per unit at an output of 20, the long-run supply schedule is a horizontal line at a price of $10.

Calculating Long-Run Equilibrium Price, Output, and Firm Details

The long-run equilibrium price is where market demand equals market supply, at the price where all firms cover their costs. Here, it is $10. Substituting into the demand equation, \(Q = 1,500 - 50 \times 10 = 1,000\). Each firm produces 20 units (q). The number of firms is total output divided by each firm's output, \(1,000 / 20 = 50\). Since price equals average cost, profits are zero.

Finding Short-Run Cost Functions and Minimum Point

Short-run average cost is \( SAC(q) = \frac{0.5q^2 - 10q + 200}{q} \). Short-run marginal cost is \( SMC(q) = q - 10 \). To find the minimum SAC, set its derivative equal to zero and solve: \( \frac{d(SAC)}{dq} = 0\), leading to \(q = 10\).

Deriving Short-Run Supply Function

Each firm's short-run supply function equals its marginal cost above the shutdown point, here when price equals ten (or when SMC equals SAC). So, the individual supply function is \( S(q) = q - 10 \) for \( q \geq 10 \). The aggregate short-run supply function for 50 firms is \( Q_s = 50(q - 10) = 50q - 500 \) for \( q \geq 10 \).

Adjusting for Very Short Run with Increased Demand

With the new demand curve \( Q = 2,000 - 50P \) and can't change output, use previous outputs (1000 at $10) to adjust price. Solve \(2,000 = 50P + 1,000\), resulting in price \( P = 20\). Output capacity remains unchanged; firms can't adjust quickly.

Re-Evaluating Industry Short-Run with Increased Demand

Substitute \(P = 20\) in the short-run supply equation \(Q_s = 50(q - 10)\). Solve for \(q\) with new \(P\) in \(Q_s = 750 \) matches \(2,000 - 50 \cdot P = 1,000 \) gives \( q = 20\), keeping firm output the same in short run.

Assessing New Long-Run Equilibrium After Demand Shift

In the long run, firms adjust until they once again produce where price equals average cost, now requiring \(10\). With demand \(Q = 2,000 - 50P\) set to \(P = 10\), so total output returns to equilibrium with \(P = 10, Q = 1500\). The number of firms increases to \(75\) as \(1500/20 \).

Key concepts

Long-Run Supply

In a perfectly competitive market, the long-run supply schedule is a crucial element that determines how the industry behaves over time. In our scenario, each firm in the industry has an identical cost structure, with the long-run average cost being minimized at an output of 20 units, and the minimum average cost is $10 per unit.
This means that the industry's long-run supply curve becomes a horizontal line at the price of $10.

Here's why this happens: In the long run, the price must equal the minimum average cost for firms to stay in business without making losses or supernormal profits. If the price were higher than $10, new firms would enter, increasing the supply until the price falls back to $10. Conversely, if the price were lower, firms wouldn't cover their costs, prompting some to exit the market. Thus, the industry's long-run supply is perfectly elastic at the price where the average cost is minimized, ensuring firms earn zero economic profit.

Short-Run Cost Functions

Short-run cost functions are vital for understanding how firms operate when they can't fully adjust all their resources. In this case, each firm's short-run total cost function is given as \( C(q) = 0.5 q^2 - 10 q + 200 \).
To analyze the costs, we derive the short-run average cost (SAC) and short-run marginal cost (SMC) from this function.

• The short-run average cost is expressed as \( SAC(q) = \frac{0.5q^2 - 10q + 200}{q} \), which represents the average cost per unit of output.
• The short-run marginal cost is found by taking the derivative of the total cost function with respect to quantity, \( SMC(q) = q - 10 \). This tells us how much additional cost is incurred for producing one more unit of output.

To find where the SAC reaches a minimum, set its derivative equal to zero. This calculation reveals a output level of 10 units, where the costs are minimized in the short run.

Market Demand

Market demand is the total quantity of goods that consumers are willing and able to purchase at various price levels over a specific period. In our example, the market demand is initially presented as \( Q = 1,500 - 50 P \).
This equation shows a linear relationship between price and the quantity demanded, meaning that an increase in price leads to a decrease in demand and vice versa.

After a shift in the demand curve due to external factors, it becomes \( Q = 2,000 - 50 P \), signifying that consumers are now willing to buy more at the same prices. The shift can be prompted by changes in consumer preferences, income, or prices of related goods, demonstrating how market demand can fluctuate and influence the market equilibrium in both the short run and long run.

Equilibrium Price

Equilibrium price in a perfectly competitive market is the price at which the quantity supplied equals the quantity demanded.
It's determined where the market supply and demand curves intersect. In our exercise, initially, the long-run equilibrium price is set at \(10, where the market is stable and firms cover their costs.
With the initial demand equation \( Q = 1,500 - 50P \), entering this price gives a total market output of 1,000 units, with each firm producing 20 units.

However, when demand increases due to external changes, as reflected in the new curve \( Q = 2,000 - 50P \), the price needs adjusting. In the very short run, where firms can't change output, the price may briefly rise until capacities can adjust, but in the long run, it will return to \)10 as firms enter to meet full demand at the minimum average cost, ensuring a balance where there are neither profits nor losses among firms.