In a competitive market, there are two groups of firms. For every firm in group A, the long-run ATC curve is U-shaped and intersects the long-run MC curve when ATC = 20 and output is 4. There is an unlimited number of firms in group A. Every firm in group B has a special resource used as an input. For every firm in group B , long-run ATC curve is U-shaped and intersects the long-run MC curve when ATC = 10 and output is 6. For these firms, we also know that when output is 25, marginal cost is 20. There are only 100 firms in group B.
Answer the following questions:
a. If market demand curve is given by
Q = 10,000 - 10p,
find
Long run equilibrium price
Industry output
Output of every firm in Group A
Output of every firm in Group B.
Number of firms in group A that produce positive output in long run equilibrium
Number of firms in group B that produce positive output in long run equilibrium
To find the long-run equilibrium price, industry output, output of each firm in Group A and Group B, and the number of firms in each group that produce positive output in the long run equilibrium, we need to follow these steps:
Step 1: Determine the long-run equilibrium condition
In a perfectly competitive market, the long-run equilibrium condition is achieved when firms are earning zero economic profit. This occurs when the price (p) is equal to the minimum average total cost (ATC) of production.
Step 2: Find the long-run equilibrium price
To find the long-run equilibrium price, we need to set the minimum ATC of production for each group equal to the given market demand function.
For Group A:
ATC = 20, Output = 4
For Group B:
ATC = 10, Output = 6
Substituting the ATC and output values into the demand function:
20 = 10,000 - 10p (Group A)
10 = 10,000 - 10p (Group B)
Solving each equation for p, we get:
p = 500 - 0.05Q (Group A)
p = 1,000 - 0.1Q (Group B)
Setting the prices obtained for each group equal to each other:
500 - 0.05Q = 1,000 - 0.1Q
Solving for Q, we find:
0.05Q = 500
Q = 10,000
Substituting the value of Q back into either of the price equations, we obtain the long-run equilibrium price:
p = 500 - 0.05(10,000)
p = 500 - 500
p = 0
Therefore, the long-run equilibrium price is $0.
Step 3: Calculate industry output
The industry output is equal to the sum of the outputs of Group A and Group B:
Industry output = Output of Group A + Output of Group B
Industry output = 4N_A + 6N_B
Since we know that N_A is an unlimited number of firms and N_B is 100 firms, we can substitute these values into the equation:
Industry output = 4(N_A) + 6(100)
Industry output = 4(N_A) + 600
Therefore, the industry output is 4(N_A) + 600 units.
Step 4: Calculate the output of each firm in Group A and Group B
For Group A, we already know that the output is 4 for each firm.
For Group B, we are given that when the output is 25, the marginal cost is 20. Since marginal cost is equal to the marginal revenue (which is also equal to the price in a perfectly competitive market), we can use the price equation we obtained earlier to find the output.
p = 1,000 - 0.1Q (Group B)
0 = 1,000 - 0.1(25)
0 = 1,000 - 2.5
2.5 = 1,000
Therefore, the output of each firm in Group B is 25.
Step 5: Find the number of firms in Group A and Group B that produce positive output in the long-run equilibrium
Since Group A has an unlimited number of firms, all of them will produce a positive output in the long-run equilibrium.
For Group B, we need to find the output level at which the price (p) is greater than zero. Substituting p = 0 into the price equation we obtained earlier for Group B, we can solve for Q:
0 = 1,000 - 0.1Q
0.1Q = 1,000
Q = 10,000
Therefore, in the long-run equilibrium, the number of firms in Group B that produce a positive output is zero.
In summary:
a. The long-run equilibrium price is $0.
b. The industry output is 4(N_A) + 600 units.
c. The output of every firm in Group A is 4 units.
d. The output of every firm in Group B is 25 units.
e. The number of firms in Group A that produce a positive output in the long-run equilibrium is unlimited.
f. The number of firms in Group B that produce a positive output in the long-run equilibrium is zero.