Question

A perfectly 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 \(\left(q_{4}=20\right) .\) 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-rum supply schedule? b. What is the long-run equilibrium price \(\left(P^{2}\right)\) ? The total industry output \(\left(Q^{*}\right)\) ? The output of each firm \(\left(q^{*}\right) ?\) The number of firms? And the profits of each firm? c. The short-run total cost curve associated with each firm's long-rum equilibrium output is given by \\[ C=0.5 q^{2}-10 q+200 \\] Calculate the short-run average and marginal cost curves. At what output level does short-run average cost reach a minimum? d. Calculate the short-run supply curve for each firm and the industry short- run supply curve. e. 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 curve to recalculate the answers to g. What is the new long-run equilibrium for the industry?

Short answer

Question: In a perfectly competitive market, analyze the behavior of firms in the short run and long run, and how an increase in demand will impact the market. Answer: In a perfectly competitive market, firms produce at the lowest average cost in the long run, earning zero economic profit. When demand increases, price rises in the very short run when firms cannot adjust their output, and the number of firms remains the same. In the short run, firms can adjust their output, leading to a new equilibrium with a higher price and industry output. In the long run, new firms enter the market to re-establish the long-run equilibrium at the minimum average cost, with zero economic profit for each firm. This results in a higher industry output and number of firms in the market.

Step-by-step solution

a. Industry's Long-Run Supply Schedule

In a perfectly competitive market, the long-run supply schedule is horizontal at the minimum average cost, since each firm is efficient in the long run, producing at the lowest possible cost per unit. In this case, since the minimum average cost is $10 per unit, the industry's long-run supply schedule will be: \[P = \$10\]

b. Long-Run Equilibrium Price, Total Industry Output, Output of Each Firm, Number of Firms and Profits

In long-run equilibrium, the market price equals the minimum average cost, so: \[P^* = \$10\] To find the total industry output, we need to plug this equilibrium price into the market demand function: \[Q^* = 1500 - 50(\$10)\] \[Q^* = 1500 - 500\] \[Q^* = 1000\] To find the output of each firm, we need to use the q_min that minimizes long-run average cost, given as \(q_4 = 20\): \[q^* = 20\] To find the number of firms, we need to divide the total industry output by the output of each firm: \[N = \frac{Q^*}{q^*}\] \[N = \frac{1000}{20}\] \[N = 50\] In long-run equilibrium, perfectly competitive firms earn zero economic profit: \[\text{Profit} = 0\]

c. Short-Run Average and Marginal Cost Curves, Minimum Output Level

From the firm's short-run total cost curve, we can find the short-run average cost (AC) and marginal cost (MC) curves. AC: \[\text{AC} = \frac{\text{C}}{q}\] \[\text{AC} = \frac{0.5q^2 - 10q + 200}{q}\] MC: \[\text{MC} = \frac{d(\text{C})}{dq}\] \[\text{MC} = q - 10\] To find the output level where short-run average cost reaches a minimum, we need to take the derivative of the AC with respect to q, and set it equal to zero: \[ \frac{d(\text{AC})}{dq} = \frac{d(0.5q - 10 + \frac{200}{q})}{dq} = 0.5 - \frac{200}{q^2} = 0\] \[q = 20\]

d. Short-Run Supply Curve for Each Firm and the Industry

To find the firm's short-run supply curve, we need to equate MC with P: \[P = q - 10\] \[q = P + 10\] This is a firm's short-run supply curve. For the industry, multiply this by the number of firms (50) to get the industry short-run supply curve: \[Q = 50(P + 10)\]

e. Very Short-Run Outcomes with New Demand

When demand increases to: \(Q = 2000 - 50P\), in the very short run firms cannot change their output. We can solve for equilibrium price by equating the new demand function with the industry short-run supply curve: \[Q = 50(P + 10)\] \[2000 - 50P = 50(P + 10)\] Solving for P: \[P = \$16\] From the short-run supply curve, the output remains constant at \(q^* = 20\) for each firm in the very short run.

f. Short-Run Outcomes with New Demand

Use the industry short-run supply curve to determine the short-run equilibrium price and industry output with the new demand: \[Q = 50(P + 10)\] \[2000 - 50P = 50(P + 10)\] \[P = \$14\] Now, use this price to find the industry output: \[Q = 50(\$14 + 10)\] \[Q = 1200\]

g. New Long-Run Equilibrium for the Industry

In response to the higher demand, firms will enter the market until the long-run equilibrium is re-established where price equals minimum average cost ($10) and firms earn zero economic profit. We can find the new industry output with the new demand function: \[Q^* = 2000 - 50(\$10)\] \[Q^* = 1500\] To find the new number of firms in the market, divide the new industry output by the output of each firm: \[N = \frac{Q^*}{q^*}\] \[N = \frac{1500}{20}\] \[N = 75\]

Key concepts

Long-Run Supply Curve

In perfect competition, the long-run supply curve is a vital concept because it reflects the industry's production capabilities without constraints. In this scenario, firms produce where average costs are minimized to remain viable in the market. Perfect competition assumes many firms producing identical products with free market entry and exit. Over time, firms aim to operate at the lowest average cost. This leads to a perfectly elastic long-run supply curve in competitive markets at the minimum average cost price.
The minimum average cost is achieved at the lowest point on the long-run average cost curve. For this exercise, it is given as $10 per unit. Thus, the industry's long-run supply curve will be a horizontal line at this price level, indicating that any increase or decrease in demand will not affect the price.
Firms will efficiently supply any quantity of goods demanded by the market at this price. The curve ensures that resources are allocated optimally, promoting efficiency and sustainability in the industry.

Equilibrium Price

The equilibrium price in a perfectly competitive market is where the quantity supplied equals the quantity demanded. This price ensures that there are no surpluses or shortages in the market, promoting efficient resource allocation. In long-run equilibrium, the price is equal to the minimum average cost, as firms earn zero economic profit under these conditions.
Using the given market demand function in the exercise: \[Q = 1500 - 50P\] and setting the price at $10 (the minimum average cost), we find the long-run equilibrium quantity is 1000 units. This ensures that supply meets demand at this efficient price point. It reflects the market's stability and firms' balance between costs and revenue.
This equilibrium ensures every firm operates at optimal full efficiency, producing the necessary output to meet market demand while covering costs, with no economic profits.

Short-Run Costs

Short-run costs include all the expenses a firm incurs when producing goods within a limited time frame, where some inputs are fixed. These costs are essential for understanding how firms operate when immediate changes can't be made to capita or technology.
For this exercise, the short-run total cost is expressed by the equation: \[ C = 0.5q^2 - 10q + 200 \]From this, we can derive the short-run average cost (AC) and marginal cost (MC) curves as follows:

• The average cost (AC) is the total cost per unit of output: \[ AC = \frac{0.5q^2 - 10q + 200}{q} \]
• The marginal cost (MC) is the derivative of the total cost with respect to output (q), indicating the cost of producing one more unit:\[ MC = q - 10 \]

Short-run average cost reaches its minimum at the output level where marginal cost equals average cost. In our example, this occurs at an output level of 20 units, indicating the most efficient production scale in the short-run.

Market Demand Shift

Market demand shifts occur when factors such as consumer preferences, income changes, or external economic shifts modify the demand for goods. In this exercise, an upward shift in market demand reflects increased consumer willingness to buy at every price level.
With the new demand function: \[Q = 2000 - 50P\], the industry faces new challenges in both the short and long term. In the very short run, firms cannot change their outputs, leading to an upward pressure on price until production can adjust to meet the new demand level.
In the short run, firms operate along a supply curve that reflects existing capacities, leading to a new equilibrium price of \(14 and industry output of 1200 units. Over time, the market adapts, prompting entry of new firms or expansion of current production to meet the increase in demand, eventually returning to a long-run equilibrium where price and supply stabilize back at \)10 but with more firms and higher total output.