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 = 400 - 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, we need to equate the market supply and demand curves. In the long run, firms in a competitive market adjust their output levels until they produce at minimum average total cost (ATC) and all economic profits are driven to zero.
To calculate the industry output, we need to find the quantity where market supply = market demand. In this case, the market demand curve is given by Q = 400 - 10p.
1. Equating market supply and demand:
Q = 400 - 10p (market demand)
Q = Q(group A) + Q(group B) (market supply)
Since group A has an unlimited number of firms and group B has 100 firms, the market supply can be expressed as:
Q = Q(group A) + 100 (market supply)
2. Let's determine the output of every firm in group A:
We know that the long-run ATC curve intersects the long-run MC curve for group A at an ATC of 20 and output of 4. Therefore, at this point, MC = ATC = 20.
Since MC = ATC in the long run, we can find the equation of the long-run MC curve for group A:
MC = 20 (equation 1)
3. The output of every firm in group B:
We are given that the long-run ATC curve intersects the long-run MC curve for group B at ATC = 10 and output = 6. At this point, MC = ATC = 10.
We are also given that when output is 25, MC = 20 for group B.
Using this information, we can find the equation of the long-run MC curve for group B:
MC = 10 (equation 2)
20 = 10 + b*(25 - 6)
20 = 10 + 19b
19b = 10
b ≈ 0.5263
So, the equation of the long-run MC curve for group B is given by:
MC = 10 + 0.5263*(output - 6) (equation 3)
4. Number of firms producing positive output in long-run equilibrium:
In the long run, firms produce at minimum ATC, which is the lowest point on their U-shaped ATC curves. For group A, the ATC curve intersects the MC curve when ATC = 20 and output is 4. Comparing this with equation 1, we can see that the minimum ATC for group A is 20. Therefore, any firm in group A that produces at an output greater than 4 will have costs higher than the industry minimum and will exit the market.
For group B, the ATC curve intersects the MC curve when ATC = 10 and output is 6. Comparing this with equation 3, we can see that the minimum ATC for group B is 10. Therefore, any firm in group B that produces at an output greater than 6 will have costs higher than the industry minimum and will exit the market.
Let's solve for the equilibrium price and industry output:
1. Equating market supply and demand:
Q = 400 - 10p (market demand)
Q = Q(group A) + 100 (market supply)
2. Let's determine the output of every firm in group A:
MC = 20 (equation 1)
3. The output of every firm in group B:
MC = 10 + 0.5263*(output - 6) (equation 3)
4. Number of firms producing positive output in long-run equilibrium:
For group A, any firm producing at an output greater than 4 will exit the market.
For group B, any firm producing at an output greater than 6 will exit the market.