The Evolution of the Fashion Industry: Consider the market for clothes and suppose there exist 100 different €œstyles€ that can be produced and can be arranged (and equally spaced) on a circle. Among the billions of consumers of clothes, each has an ideal style somewhere n that circle (either at one of the 100 styles that can potentially be produced or in between two of those). Styles become less appealing the farther they are from the consumer€™s ideal. For simplicity, suppose that the marginal cost of producing clothes of any style is constant (once the fixed cost of starting production has been paid), and suppose that a firm that comes into the industry must pay the fixed entry cost for each style it wants to produce.
A: Suppose first that only a single firm operates in the industry (and produces one of the 100 styles) and that the fixed cost of starting production is sufficiently high for no second firm to wish to enter.
(a) Explain how the firm in the industry can be making positive economic profit but the firms outside would make negative economic profit by entering.
(b) Over the decades, the price of the equipment necessary for producing clothes has fallen€”thus lowering the fixed entry cost into the clothing industry. When the costs fall to the point where the second firm enters, where on the circle would you expect that firm to locate its clothes?
(c) What would happen to the price of clothing assuming the two firms are price competitors?
(d) Suppose entry costs have fallen sufficiently for 100 different firms to be in the clothing industry. Now suppose entry costs fall further and firms continue to be price competitors. How low would entry costs have to fall for another firm to enter the market (assuming only 100 clothing €œstyles€ can potentially be produced)?
(e) Suppose that an avalanche of new ideas has made all clothing styles on the circle€” not just the initial 100 €” possible to produce. As entry costs fall, how many new entrants would you expect when the next firm finds it profitable to enter?
(f) Beginning with the case where the industry first consists of 100 firms, would you expect price to fall as entry costs fall even before any additional competitors enter the industry (assuming that existing firms can credibly announce their price before new firms have to make a decision on whether or not to enter)?
(g) Suppose entry costs disappear altogether. What happens to price?
B: (Part B of this exercise is not directly related to part A but rather offers you a chance to go through solving the €œcircle model€ with a slight modification from the version used in the text.) In our treatment of the €œcircle model€ in Section 26B.3, we assumed that the cost consumer n ˆˆ [0,1] incurs from consuming a product with characteristic y ˆˆ [0,1] (rather than her ideal of n) increases linearly with the distance between n and y €” i.e. the cost was α| n ˆ’ y|. In our treatment of the Hotel ling €œline€ model, we instead assumed that this cost increases with the square of the distance€”i.e. the cost was α (n ˆ’ y) 2.
(a) Consider the second stage of the €œcircle model€ game €” i.e. the stage at which N firms have entered in the first stage having equally spaced their products on the product characteristic circle (of circumference 1). Assume that every point y on the circle contains one consumer n whose ideal point is y. What is the farthest that any consumer n€™s ideal point will lie from the closest firm€™s product?
(b) Suppose that all firms other than firm i charge a price p and suppose firm is product characteristic is yi = 0. Denote by n the consumer who is indifferent between consuming from firm i and adjacent firm j (with firm j producing yj) assuming firm i charges price pi . Given that the consumer€™s total cost from consuming a particular product includes both the price she has to pay and the cost of consuming away from her ideal, what has to be true about the total cost n incurs when shopping at firm i versus firm j ? Express this in an equation and solve it for n.
(c) Given that there are N (equally spaced) firms in the industry, what is y j (when yi = 0)? Substitute this into your expression for n. What is the demand Di (pi, p) that firm i face? Explain.
(e) Since all firms end up charging the same price in equilibrium, what is the equilibrium price pˆ— (N) in terms of c, α and N given that N firms have entered in stage 1 of the €œcircle game€?
(f) Assuming that firms have to pay a fixed cost FC to enter the circle market in stage 1 of the game, how many will enter (given they forecast pˆ— in the second stage)? Denote this as Nˆ—. What is the equilibrium price that will emerge as a result?
(g) Now consider the problem a social planner who wants to maximize efficiency faces when deciding how many firms to set up on the circle. Suppose the planner sets the number of firms at N. Explain why the cost consumers incur from not consuming at their ideal is

(h) What is the socially optimal number of firms Nopt that the planner would set up? How does it compare to the equilibrium number of firms Nˆ— €” and what has to be true for the two to converge to one another?

Transcribed Image Text:

17/2N) 2 dx. 2N fo