Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 14: Problem 6

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 \(\left(q_{t}=20\right) .\) 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 (P*)? The total industry output ((?*)? The out put of each firm \((

Short Answer

Expert verified
#summary#: This exercise analyzes the long-run and short-run equilibria in a perfectly competitive industry. The key outcomes are: - The long-run supply schedule in the industry is perfectly elastic at a price of $10. - The long-run equilibrium has 50 firms, each producing 20 units of output, and the total industry output is 1,000 units. - In the short run, the average cost function is \(AC(q) = 0.5q - 10 + \frac{200}{q}\) and the marginal cost function is \(MC(q) = q - 10\). - The industry short-run supply curve is \(Q_s = 50(q - 10)\). - In the very short run, after an increase in demand, the new price rises to $30. - The industry adjusts output and number of firms to reach a new long-run equilibrium at the minimum average cost, ensuring firms make zero economic profit.

Step by step solution

01

Recall the conditions for long-run equilibrium

In a perfectly competitive industry, firms in the long run will make zero economic profit. The long-run supply will be horizontal at minimum average cost.
02

Identify the minimum average cost

The minimum average cost, given in the problem, is $10 per unit.
03

Derive the long-run supply schedule

Since the supply will be horizontal at minimum average cost, the long-run supply schedule will be: \(q_s = \infty\) when \(P = 10\), \(q_s = 0\) when \(P < 10\). Essentially, the industry's long-run supply is perfectly elastic at \(P = \$10\). ##Part b: Finding the Long-run Equilibrium##
04

Set the quantity demanded equal to quantity supplied in the long run

In the long run, equilibrium occurs when the quantity demanded is equal to the quantity supplied. Thus, we have: \(Q = 1,500 - 50P = n \times 20\), where \(n\) is the number of firms in the industry and \(20\) is the output per firm.
05

Solve for the equilibrium price

We are given that the price in the long run must be equal to the minimum average cost, which is $10. So: \(P* = 10\)
06

Substitute the equilibrium price into the demand function

Using the demand function \(Q = 1,500 - 50P\), plug in the equilibrium price: \(Q* = 1,500 - 50(10) = 1,000\)
07

Determine the output per firm

The output per firm is given in the problem as 20 units, so: \(q* = 20\)
08

Calculate the number of firms in the industry

We know that \(Q* = n \times 20\), and we have already calculated \(Q* = 1,000\). Therefore, we can find the number of firms \(n\): \(n = \frac{Q*}{q*} = \frac{1,000}{20} = 50\)
09

Determine the profits of each firm

In the long run, firms in a perfectly competitive industry make zero economic profit, so: \(\pi = 0\) The long-run equilibrium is as follows: - Equilibrium price: \(P* = \$10\) - Total industry output: \(Q* = 10,000\) - Output of each firm: \(q* = 20\) - Number of firms: \(n = 50\) - Profits of each firm: \(\pi = 0\) ##Part c: Short-run Total Cost, Average Cost, and Marginal Cost##
10

Write down the short-run total cost function

We are given that the short-run total cost function is: \(C(q) = 0.5q^2 - 10q + 200\)
11

Calculate the short-run average cost function

Short-run average cost function is the total cost divided by the quantity of output, so: \(AC(q) = \frac{C(q)}{q} = \frac{0.5q^2 - 10q + 200}{q} = 0.5q - 10 + \frac{200}{q}\)
12

Calculate the short-run marginal cost function

Short-run marginal cost function is the derivative of the total cost function with respect to the quantity of output, so: \(MC(q) = \frac{dC(q)}{dq} = q - 10\)
13

Find the minimum short-run average cost

To find the minimum short-run average cost, set the derivative of the average cost function with respect to quantity equal to zero and solve for q: \(\frac{dAC(q)}{dq} = 0\) Using calculus, it can be found that: Minimum short-run average cost occurs at \(q = 20\). ##Part d: Short-run Supply Curve##
14

Determine the individual firm's supply in the short run

In the short run, a firm will produce at the quantity where its marginal cost equals the market price, given that this is above its shutdown point. The firm's short-run supply curve will be derived from its marginal cost curve, given by: \(MC(q) = q - 10\)
15

Determine the industry short-run supply

The industry short-run supply curve will be obtained by summing up the supply of all individual firms in the industry. Since the marginal cost of each firm is the same, the industry supply curve can be obtained by multiplying the supply of each firm by the number of firms in the industry (\(n = 50\)): \(Q_s = 50(q - 10)\), where \(q\) is the output of a firm ##Part e: Equilibrium in the Very Short Run##
16

Write down the new market demand function

We are given that the market demand function shifts upward: \(Q_d = 2,000 - 50P\)
17

Assume that there is no change in firm output in the very short run

As firms cannot change their output in the very short run, output for each firm will remain constant at \(q = 20\).
18

Determine the new market equilibrium

Set the quantity demanded equal to the quantity supplied in the very short run: \(Q_d = Q_s\) So \(2,000 - 50P = 50(q - 10)\) Substitute \(q = 20\): \(2,000 - 50P = 50(20 - 10)\)
19

Solve for the new price

Solve for P: \(2,000 - 50P = 500\) \(P = \frac{2000-500}{50} = 30\) In the very short run, the new equilibrium price is \(P = \$30\). ##Part f: Recalculating Equilibrium in the Short-run## Using the industry short-run supply curve \(Q_s = 50(q - 10)\), we can recalculate the answers to part b with the new demand curve. ##Part g: New Long-run Industry Equilibrium## To find the new long-run equilibrium in the industry, repeat the steps from part b. Firms will adjust their output and enter or exit the market as necessary to bring the industry back to long-run equilibrium where firms are making zero economic profit at the minimum average cost.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose the demand for frisbees is given by \\[ g=100-2 P \\] and the supply by \\[ Q=20+6 P \\] a What will be the equilibrium price and quantities for frisbees? b. Suppose the government levies a tax of \(\$ 4\) per frisbee. Now what will be the equilibrium quantity, the price consumers will pay, and the price firms will receive? How is the bur den of the tax shared by buyers and sellers? c. How would your answers to parts (a) and (b) change if the supply curve were instead \\[ Q=70+P ? \\] What do you conclude by comparing these two cases?

Wheat is produced under perfectly competitive conditions. Individual wheat farmers have U-shaped, long-run average cost curves that reach a minimum average cost of \(\$ 3\) per bushel when 1,000 bushels are produced. a. If the market demand curve for wheat is given by \\[ Q_{o}=2,600,000-200,000 P \\] where \(Q p\) is the number of bushels demanded per year and \(P\) is the price per bushel, in long-run equilibrium what will be the price of wheat, how much total wheat will be demanded, and how many wheat farms will there be? b. Suppose demand shifts outward to \\[ Q_{D}=3,200,000-200,000 P \\] If farmers cannot adjust their output in the short run, what will market price be with this new demand curve? What will the profits of the typical farm be? c. Given the new demand curve described in part (b), what will be the new long-run equi librium? (That is, calculate market price, quantity of wheat produced, and the new equi librium number of farms in this new situation.) d. Graph your results.

Suppose that the long-run total cost function for the typical mushroom producer is given by \\[ T C=w q^{2}-10 q+100 \\] where \(q\) is the output of the typical firm and \(w\) represents the hourly wage rate of mushroom pickers. Suppose also that the demand for mushrooms is given by \\[ Q=-1,000 \mathrm{P}+40,000 \\] where (Pis total quantity demanded and Pis the market price of mushrooms. a. If the wage rate for mushroom pickers is \(\$ 1,\) what will be the long-run equilibrium out put for the typical mushroom picker? b. Assuming that the mushroom industry exhibits constant costs and that all firms are iden tical, what will be the long-run equilibrium price of mushrooms, and how many mush room firms will there be? c. Suppose the government imposed a tax of \(\$ 3\) for each mushroom picker hired (raising total wage costs, \(w,\) to \(\$ 4\) ). Assuming that the typical firm continues to have costs given by \\[ T C=w q^{2}-10 g+100 \\] how will your answers to parts (a) and (b) change with this new, higher wage rate? d. How would your answers to (a), (b), and (c) change if market demand were instead given by \\[ Q=-1,000 \mathrm{P}+60,000 ? \\]

A perfectly competitive market has 1,000 firms. In the very short run, each of the firms has a fixed supply of 100 units. The market demand is given by \\[ Q=160,000-10,000 P \\] a Calculate the equilibrium price in the very short run. b. Calculate the demand schedule facing any one firm in this industry. c. Calculate what the equilibrium price would be if one of the sellers decided to sell noth ing or if one seller decided to sell 200 units. d. At the original equilibrium point, calculate the elasticity of the industry demand curve and the elasticity of the demand curve facing any one seller. Suppose now that in the short run, each firm has a supply curve that shows the quantity the firm will supply \((

Suppose that the demand for stilts is given by \\[ Q=1,500-50 P \\] and that the long-run total operating costs of each stilt-making firm in a competitive industry are given by \\[ T C=0.5 q^{2}-10 q \\] Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is given by \\[ Q_{s}=0.25 \mathrm{a} \\] where \(w\) is the annual wage paid. Suppose also that each stilt-making firm requires one (and only one) entrepreneur (hence, the quantity of entrepreneurs hired is equal to the number of firms). Long-run total costs for each firm are hence given by \\[ T C=0.5 q^{2}-10 q+w \\]. a. What is the long-run equilibrium quantity of stilts produced? How many stilts are pro duced by each firm? What is the long-run equilibrium price of stilts? How many firms will there be? How many entrepreneurs will be hired, and what is their wage? b. Suppose that the demand for stilts shifts outward to \\[ Q=2,428-50 P \\] Answer the questions posed in part (a). c. Because stilt-making entrepreneurs are the cause of the upward sloping long-run supply curve in this problem, they will receive all rents generated as industry output expands. Calculate the increase in rents between parts (a) and (b). Show that this value is identi cal to the change in long-run producer surplus as measured along the stilt supply curve.
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free