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 \(\left(q_{i}=20\right) .\) The minimum average cost is \(\$ 10\) per unit. Total market demand is given by \\[ Q=D(P)=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 short-run 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 \(Q=D(P)=2,000-50 P\). 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

f. Recalculate the answers to (b) using the industry short-run supply function. #tag_title#New Long-run Equilibrium Price#tag_content#The new long-run equilibrium price \(P^{*} = \$25\). #tag_title#New Total Industry Output#tag_content#We can find the new total industry output (\(Q^{*}\)) by equating market demand with market supply at the equilibrium price (\(P^{*}\)): \(Q^{*} = 2,000 - 50P^{*}\) Since \(P^{*} = \$25\), \(Q^{*} = 2,000 - 50(25)\) \(Q^{*} = 1,750\) #tag_title#New Output of Each Firm#tag_content#Assuming that each firm still produces 20 units in the short run, the output of each firm (\(q^{*}\)) remains at 20. #tag_title#New Number of Firms and Profits#tag_content#Let there be n firms in the industry. The total industry output is equal to the sum of each firm's output: \(Q^{*} = n*q^{*}\) Using the new value of \(Q^{*}\) and the value of \(q^{*}\) from above: \(1,750 = n * 20\) \(n = 87.5\) However, since there cannot be a fractional number of firms, the number of firms must be either 87 or 88. In the long-run equilibrium, profits of each firm should be zero in a perfectly competitive market. However, in the short-run equilibrium, firms may earn positive profits due to excess demand that resulted from the increased market demand. g. Find the new long-run equilibrium for the industry. In the long run, the firms in the industry will adjust their outputs in response to the increased market demand to bring the market back to long-run equilibrium. This will happen when the market price returns to the minimum average cost level, which remains at \(P^{*} = \$10\). To find the new total industry output \(Q^{*}\), we again equate the shifted market demand to market supply at the equilibrium price: \(Q^{*} = 2,000 - 50P^{*}\) Since \(P^{*} = \$10\), \(Q^{*} = 2,000 - 50(10)\) \(Q^{*} = 1,500\) The new long-run equilibrium has a price of \(P^{*} = \$10\) and a total industry output of \(Q^{*} = 1,500\). The output of each firm (\(q^{*}\)) remains at 20 units, and the number of firms adjusts to 75 (since \(1,500 = n * 20\)). This reallocation of firms will bring the market back to the long-run equilibrium with zero profits for each firm in the perfectly competitive market.

Step-by-step solution

Long-run Supply Schedule

The industry's long-run supply is P = $10. b. What is the long-run equilibrium price \(P^{*}\)? The total industry output \(Q^{*}\)? The output of each firm \(q^{*}\)? The number of firms? The profits of each firm?

Long-run Equilibrium Price

Since the long-run supply schedule is a horizontal line at the level of minimum average cost, the long-run equilibrium price \(P^{*} = \$10\).

Total Industry Output

To find the total industry output (\(Q^{*}\)), we equate market demand with market supply at the equilibrium price (\(P^{*}\)): \(Q^{*} = 1,500 - 50P^{*}\) Since \(P^{*} = \$10\), \(Q^{*} = 1,500 - 50(10)\) \(Q^{*} = 1,000\)

Output of Each Firm

Each firm produces 20 units to minimize long-run average cost. Hence, the output of each firm \(q^{*} = 20\).

Number of Firms and Profits

Let there be n firms in the industry. The total industry output is equal to the sum of each firm's output: \(Q^{*} = n*q^{*}\) Using the values of \(Q^{*}\) and \(q^{*}\) from above: \(1,000 = n * 20\) \(n = 50\) So, there are 50 firms in the industry. As a perfectly competitive market has no long-run profit, the profit of each firm in long-run equilibrium is 0. c. Calculate the short-run average and marginal cost functions. At what output level does short-run average cost reach a minimum?

Short-run Average and Marginal Cost Functions

The given short-run total cost function is: \(C(q) = 0.5q^{2} - 10q + 200\) To find the short-run average cost function (SAC), divide the total cost by the output q: \(SAC(q) = \frac{C(q)}{q} = 0.5q - 10 + \frac{200}{q}\) To find the short-run marginal cost function (SMC), take the derivative of the total cost function with respect to output q: \(SMC(q) = \frac{dC}{dq} = q - 10\) Now, to find the output level at which short-run average cost (SAC) reaches a minimum, we'll take the derivative of SAC with respect to output q and equate it to zero: \(\frac{d(SAC)}{dq} = 0.5 - \frac{200}{q^{2}} = 0\) Solving for q: \(q^{2} = \frac{200}{0.5}\) \(q = \sqrt{400} = 20\) The short-run average cost reaches a minimum at an output level of 20 units. d. Calculate the short-run supply function for each firm and the industry short-run supply function.

Short-run Supply Function for Each Firm

The short-run supply function for each firm is determined by equating the marginal cost with the output price: \(SMC(q) = P\) \(q - 10 = P\) \(q = P + 10\)

Industry Short-run Supply Function

To find the industry short-run supply function, multiply each firm's supply function by the number of firms (n = 50): \(Q = 50(P + 10)\) e. Suppose now that the market demand function shifts upward to \(Q=D(P)=2,000-50P\). Find the new long-run equilibrium for the industry. In the very short run, when firms cannot change their outputs, the industry short-run supply function remains unchanged. To find the new long-run equilibrium, we update the market demand function and solve for the new equilibrium price and total industry output: \(Q = 2,000 - 50P = 50(P + 10)\) Solving for P: \(2,000 - 50P = 50P + 500\) \(2,500 = 100P\) \(P = 25\) The new long-run industry equilibrium is at a price of \(P^{*} = \$25\). Please confirm if you want the information for parts f and g, since there was a typo in your question and no e provided.

Key concepts

Perfect Competition

In the world of microeconomic theory, perfect competition represents a market structure characterized by numerous small firms, all of which have no power to influence the price of their products. This is because the products are homogeneous, and consumers have perfect information about them. The presence of many firms assures that none can control the market price, leading to a situation where firms are 'price takers.' A key outcome of perfect competition is that, in the long-run, firms earn zero economic profit, which means they cover all their costs including a normal profit, but do not earn above-normal profits.

Each firm aims to maximize profit, but since the price is determined by market forces, their only choice is to manage production in a way that minimizes costs. A perfectly competitive industry is also characterized by free entry and exit of firms, meaning potential competitors can join or leave the market with ease, affecting the overall market supply.

Long-Run Equilibrium

Long-run equilibrium in perfect competition occurs when firms have adjusted their output and resources in a way that they only earn normal profits, meaning there are neither economic profits nor losses. At this point, each firm in the industry operates at its most efficient scale, producing the quantity of goods where average total costs are minimized.

With the given cost structure where firms minimize long-run average cost at an output of 20 units, and the price is constant at $10 per unit, firms will not have the incentive to expand or reduce their size. This balance ensures long-run stability in the market, where the number of firms adjusts based on the overall demand so that supply equals demand at the equilibrium price.

Cost Structure

Understanding the cost structure is essential in microeconomics, especially in determining the production decisions of a firm. The cost structure includes all the costs a firm incurs, from fixed costs that do not change with the level of output, like rent for factory space, to variable costs that do fluctuate with production levels, like costs for raw materials.

In the context of the exercise, the long-run average cost is given and is minimized at a certain output level. This cost minimization point is crucial as it dictates the most efficient scale for firms in the industry. Firms aim to produce at this output level to stay competitive and survive in the long-run.

Market Demand

Market demand plays a significant role in the economics of perfect competition. It represents the total quantity of a good or service that consumers in the market are willing and able to purchase at various prices. Generally described by a downward sloping line on a graph, market demand decreases as prices increase and vice versa.

In the exercise, the market demand is expressed algebraically as a function of price, illustrating an inverse relationship between price and quantity demanded. Adjustments in market demand, such as a shift upwards, can temporarily affect market prices and outputs, but ultimately in the long-run, the market adjusts to restore the equilibrium.

Supply Schedule

A supply schedule lists the quantity of goods that producers are willing to supply at different price levels. In a perfectly competitive market, the supply schedule can be thought of as a sum of all the individual supply schedules of the firms in the market.

For a single firm in long-run equilibrium, as seen in the exercise, the supply function reflects a horizontal line at the level of the minimum average cost, since the firm produces its most efficient output indifferent to price changes. Industry supply, however, will shift as firms enter or exit the market based on profitability, which in turn is influenced by shifts in market demand and cost structures.