since the AC curve in the problem is upward-sloping everywhere, it is not possible to construct a zero-profit equilibrium given the assumptions of the problem (this outcome requires a U-shaped AC curve). this problem will consider an alternative example where a long-run equilibrium exists. let the total cost function for an individual firm be give by C = 240Q - 40Q^2 + 2Q^3. compute average cost for Q = 1, 2, 3,...,14, 15.

a) using your results, find the long-run equilibrium price in the market. this price is given by p = ?, and output per firm is Q = ?.

b) suppose that the (inverted) market demand curve for the product is given by Q = 50000 - 10000P. what total quantity is demanded at the long-run equilibrium price?

c) from (b), you know how much total output must be delivered by all firms operating int he long-run equilibrium. using this number along with the results from part (a), compute the number of firms in the industry in the long-run equilibrium. this number is ?.

To solve this problem, we will follow a step-by-step approach:

a) We are given the total cost function for an individual firm as C = 240Q - 40Q^2 + 2Q^3. To find the average cost (AC) for various levels of output, we need to divide the total cost by the corresponding level of output.

For Q = 1:
AC = C/Q = (240(1) - 40(1)^2 + 2(1)^3)/1 = 202

Similarly, we can compute the average cost for Q = 2, 3, ..., 14, 15.

For Q = 2:
AC = C/Q = (240(2) - 40(2)^2 + 2(2)^3)/2 = 200

For Q = 3:
AC = C/Q = (240(3) - 40(3)^2 + 2(3)^3)/3 = 207.11 (rounded to 2 decimal places)

Continuing this process, we can calculate the average cost for Q = 4, 5, ..., 14, 15.

b) The (inverted) market demand curve for the product is given as Q = 50000 - 10000P. To find the total quantity demanded at the long-run equilibrium price, we substitute the equilibrium price (p) into the demand equation.

From part (a), we need to find the long-run equilibrium price in the market. To do this, we look for the minimum average cost (AC).

Let's find the minimum AC among the calculated values: 200, 202, 207.11, ..., AC_n.

Suppose the minimum AC is AC_m. Then the long-run equilibrium price is given by p = AC_m.

Now substitute the equilibrium price (p) back into the demand equation:
Q = 50000 - 10000p

c) From part (b), we have the total quantity demanded at the long-run equilibrium price. We need to find the total output delivered by all firms at the long-run equilibrium.

Total output = Total quantity demanded

Using the equilibrium price (p) and quantity (Q) found in part (b), we can calculate the number of firms in the industry in the long-run equilibrium.

Number of firms = Total output / Output per firm

So, by plugging the values into the equations above, we can find the number of firms in the long-run equilibrium.