Consider a monopolistically competitive market for automobiles. N symmetric firms provide differentiated varieties of cars, with each firm producing a single variety. Each firm faces the following demand for their variety:

Q=S[1/N-B(P-Pavg)]

This implies marginal revenue:

MR=P-Q/SB

Here, S represents market size, P is the firm's price, Pavg is average price in the market, and B measures the responsiveness of consumers to changes in price. Suppose B = 0.1 and S = 100 million.
Firms face the following cost function:

TC(Q)=F+cQ

Where F represents fixed costs of production and c is the constant marginal cost of production. Each firm has a fixed cost of $10 million and a constant marginal cost of $5,000.

a. Show in a single diagram how marginal cost, total cost, average total cost vary with output within a firm. Explain in words how increasing returns to scale are depicted in the diagram, and why increasing returns to scale exist here.

b. Suppose firms are able to charge a 10% markup on marginal cost. If this is so, how many units must firms sell in order to break even in the market? Explain.

For the remainder of the question, ignore part b.

c. What is the relationship between average total cost and the number of varieties in the market? Between price and number of varieties? Explain intuitively, and using the equations from the model. Explain how these two relationships are used to solve for the number of firms in the market.

d. In equilibrium, how many firms will operate in the market? What price will they charge? How many units will they sell?

e. Suppose the country opens to trade with the rest of the world. The rest of the world has identical cost and demand structures, but has market size of 125 million. Under free trade, how many firms will operate? What price will they charge? Explain in words why price and variety have changed in this way.

f. How are gains from trade different in this model from those discussed in the Ricardian model?