This is going to be really long, but I want to see if my answers are correct. This is problem number 10.10 in my Intermediate Microeconomics book. A perfectly competitive painted necktie 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. The minimum average cost is $10 per unit. Total market demand is given by Q=1500-50P.
a. What is the industry's long-run supply schedule?
b. What is the long-run equilibrium price? The total industry output? The output of each firm? The profits of each firm?
c. The short-run total cost curve associated with each firm's long-run equilibrium output is given by
STC=.5q^2-10q+200
where SMC=q-10. Calculate the short-run average and marginal cost curves. At what necktie 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 painted neckties become more fashionable and the market demand function shifts upward to Q=2000-50P. 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 part b.
g. What is the new long-run equilibrium for the industry?
Yesterday, a person under the user-name of "timmy" posted a very similiar question. Questions a and b were addressed in that post.
c) You are given the short run MC curve. AC is simply TC divided by Q.
So, SAC = .5Q - 10 + 200/Q
To find the minimum, take the first derivitive, then set the equation to zero. Hint: I get Q=20.
d) the short run supply curve for a firm is simply its MC curve. The industry supply curve is the sum of the firm supply curves, (ignoring any constants) I get supply is N*Q - 10, where N=number of firms (50 in this example)
Take it from here.
thank you economyst. I have been doing it correctly :) except for industry supply curve as 50q-500. Is that correct?
The industry supply curve could be written as Q = 500+50P
Alternative, using some algebra, industry supply curve is also: P = Q/50-10
(where Q = industry supply)
Check it out, solve for P when 500 + 50P = 1500-50P
(my bad, I used a big Q instead of a small q in my original post.
Could you please clarify how did you derive the answer for (a) supply schedule = 500+ 50P? Thanks,
SB
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For industry supply I got P = 50Q - 500?
intermediate micro economic
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a. To find the industry's long-run supply schedule, we need to determine the quantity of output that each firm in the industry would supply at different prices in the long run. In a perfectly competitive market, each firm will enter or exit based on the profitability of operating at a given price.
In this case, the long-run average cost (LRAC) is minimized at an output of 20 units per firm. We know that the minimum average cost is $10 per unit. Therefore, each firm will supply 20 units at any price above or equal to $10.
b. To find the long-run equilibrium price, we need to determine the price at which the total industry output equals the market demand. Given that the market demand function is Q = 1500 - 50P, we can substitute Q with the total industry output to get:
1500 - 50P = industry output
To find the equilibrium price, we set the industry output equal to the sum of the outputs of all firms. Since each firm produces 20 units, the total industry output is the number of firms (N) multiplied by 20 units.
Industry output = N * 20
By substituting this into the demand function, we can solve for the equilibrium price:
1500 - 50P = N * 20
50P = 1500 - N * 20
P = (1500 - N * 20) / 50
The equilibrium price is given by this equation. The total industry output is N * 20, and the output of each firm is 20 units. To find the profits of each firm, we need more information about their cost structure.
c. The short-run total cost curve associated with each firm's long-run equilibrium output is given by STC = 0.5q^2 - 10q + 200.
To find the short-run average cost curve, we divide the short-run total cost (STC) by the corresponding output (q):
SAC = STC / q
Substituting the total cost function, we have:
SAC = (0.5q^2 - 10q + 200) / q
Simplifying, we get:
SAC = 0.5q - 10 + 200/q
To find the short-run marginal cost curve, we take the derivative of the short-run total cost (STC) with respect to output (q):
SMC = d(STC) / dq
Taking the derivative of the total cost function, we have:
SMC = d(0.5q^2 - 10q + 200) / dq
SMC = q - 10
To find the necktie output level at which short-run average cost reaches a minimum, we need to set the derivative of the short-run average cost (SAC) equal to zero and solve for q:
d(SAC) / dq = 0.5 - 200/q^2 = 0
200/q^2 = 0.5
q^2 = 400
q = 20
Thus, the necktie output level at which short-run average cost reaches a minimum is 20 units.
d. To calculate the short-run supply curve for each firm, we need to determine the quantity of output that each firm would supply at different prices in the short run. In the short run, each firm will supply output as long as the price is greater than or equal to its short-run average variable cost (SAVC).
Given that the short-run total cost curve is STC = 0.5q^2 - 10q + 200, and the output level at which short-run average cost reaches a minimum is 20 units, we can calculate the short-run average variable cost (SAVC) by dividing the short-run total cost (STC) by output (q):
SAVC = STC / q
Substituting the values, we get:
SAVC = (0.5(20)^2 - 10(20) + 200) / 20 = 10
Therefore, each firm will supply output as long as the price is greater than or equal to $10. The short-run supply curve for each firm is a horizontal line at a price of $10.
To find the industry short-run supply curve, we add up the supply of all firms at each price. Since each firm supplies 20 units, the industry short-run supply curve is a vertical line at a quantity of 20 units.
e. If painted neckties become more fashionable and the market demand function shifts upward to Q = 2000 - 50P, the equilibrium price and output in the very short run (when firms cannot change their outputs) will still be determined by the original demand and supply conditions.
Therefore, the long-run equilibrium price and output will not change in the very short run when firms cannot adjust their outputs.
f. In the short run, we can use the industry short-run supply curve we calculated earlier to recalculate the answers to part b.
To find the short-run equilibrium price, we need to determine the price at which the total industry output equals the market demand. Given the new market demand function Q = 2000 - 50P, we can substitute Q with the industry output (N * 20) and solve for the equilibrium price:
2000 - 50P = N * 20
50P = 2000 - N * 20
P = (2000 - N * 20) / 50
The equilibrium price is given by this equation. The total industry output is N * 20, and the output of each firm is still 20 units. The profits of each firm will depend on the cost structure of the industry, which we do not have enough information about.
g. To find the new long-run equilibrium for the industry after the shift in demand, we need to analyze how the entry or exit of firms will occur to restore the long-run equilibrium.
If painted neckties become more fashionable and the market demand function shifts upward to Q = 2000 - 50P, the equilibrium price and output in the long run will change. More specifically, the higher demand will create an incentive for firms to enter the industry. As new firms enter, the industry supply will increase, leading to a decrease in price until the long-run equilibrium is reached again.
To determine the new long-run equilibrium, we need more information about the cost structure of the industry and the number of potential entrants. Without these details, we cannot provide a specific answer about the new long-run equilibrium.