The short-run cost curve for each firm's long run equilibrium output is C=y^2-20y+400. Calculate the short-run average and marginal cost curves. At what output level does short-run average cost reach a minimum? I already know the MC and the SRA is TC/Q. I got the minimum as 10. Is that right?

b) Calculate the short-run supply curve for each firm and the industry short-run supply curve? Total market demand is y=2500-20p. Each firm has an identical cost structure such that long-run avg. cost is minimized at an output of 20. The minimum avg. cost is $30.

the short-run average cost is

TAC=y^2/y-20y/y+400/y
=y-20+400/y
The marginal cost curve is the first derivative of the cost curve.
so,mc=2y-20
short run average cost reaches it's minimum when smc=sAc or the derivative of SAC=0
SO,2y-20=y-20+400/y

To calculate the short-run average cost (SAC) and marginal cost (MC) curves, we need to differentiate the given cost function, C(y) = y^2 - 20y + 400, with respect to y.

The short-run average cost (SAC) is calculated by dividing the total cost (TC) by the output (Q). Thus, SAC = TC/Q.

1. To find the short-run total cost (TC), we multiply the cost function by output: TC = C(y) * Q.
Since C(y) = y^2 - 20y + 400, the short-run total cost becomes: TC = (y^2 - 20y + 400) * Q.

2. Now, we can find the short-run average cost (SAC):
SAC = TC/Q
= (y^2 - 20y + 400) * Q / Q
= y^2 - 20y + 400.

3. To calculate the short-run marginal cost (MC), we differentiate the total cost function (TC) with respect to output (Q):
MC = d(TC)/dQ.
Since TC = (y^2 - 20y + 400) * Q, we differentiate this with respect to Q:

d(TC)/dQ = (d(y^2 - 20y + 400)/dy) * Q
= (2y - 20) * Q
= 2yQ - 20Q.

So, the short-run average cost (SAC) curve is given by SAC = y^2 - 20y + 400, and the short-run marginal cost (MC) curve is given by MC = 2yQ - 20Q.

To determine the output level at which the short-run average cost (SAC) reaches a minimum, we can find the minimum point of the SAC curve by taking its derivative and setting it equal to zero.

dy/dSAC = 2y - 20 = 0
2y = 20
y = 10

Therefore, the output level at which the short-run average cost (SAC) reaches a minimum is y = 10.

b) To calculate the short-run supply curve for each firm and the industry short-run supply curve, we need to find the price at which each firm is willing to supply different quantities of output.

Given the market demand function: y = 2500 - 20p,
we know that each firm has a cost structure where the long-run average cost (LRAC) is minimized at an output of 20, with a minimum average cost of $30.

From this information, we can calculate the short-run supply curve for each firm and the industry short-run supply curve:

1. To find the short-run supply curve for each firm, we need to determine the price at which each firm is willing to supply different quantities of output.

Given the long-run average cost (LRAC) minimum at an output of 20 and a minimum average cost of $30, we can determine the marginal cost (MC) at this output level.

MC = SAC = y^2 - 20y + 400 (as we calculated previously)
MC = (20)^2 - 20(20) + 400 = 400 - 400 + 400 = 400

Therefore, the short-run supply curve for each firm is:

MC = 400, for y ≥ 20 (output greater than or equal to 20)

2. To find the industry short-run supply curve, we need to determine the combined quantity supplied by all firms at different price levels.

At a price level (p), we know that the total quantity demanded in the market is given by:
Qd = 2500 - 20p

To find the industry short-run supply, we sum up the quantity supplied by each firm when the price (p) is greater than or equal to the minimum average cost ($30) associated with an output level of 20.

Since MC = SAC for each firm, the quantity supplied by each firm is given by:
Qs = Qd, Q ≥ 20
= 2500 - 20p, Q ≥ 20

Therefore, the industry short-run supply curve is:
Qs = 2500 - 20p, Q ≥ 20 (output greater than or equal to 20)

Please note that these calculations are based on the given cost and demand functions. Make sure to verify the accuracy of the data and check any assumptions if required.