The Math of the Krugman‐Obstfeld Model of Monopolistic Competition with Internal Economies of Scale Assume that countries have a number (N) of firms producing similar heterogeneous products with identical fixed costs (FC) and constant marginal costs (MC). We assume each country has a predetermined market size (S). By assuming demand is not a function of price we make the math easier, but it does not change the fundamental result. Since firms have identical costs,each firm has a market share (1/N), and each produces Q = S/N. The Firm’s Average Costs Under Economies of Scale: The firm’s total cost = FC + VC, and VC = MC•Q, so Average Cost AC = FC/Q + MC. A typical graph of AC would show it as a downward‐sloping and flattening function of Q. This is what we expect with internal economies of scale. As a function of the number of firms: AC = FC / (S/N) + MC = (FC/S) •N + MC That is, the firm’s AC will always be above MC, and it rises with FC. The bigger the market, the more each firm can produce and the lower AC will be. The more firms, the less each firm produces and the higher AC will be. The Firm’s Profit‐Maximizing Price: The firm maximizes profits in competing for market share. We assume that the firm gains some amount (B) of its market share for every $1 its price is below the average market price (AVGP) for the other firms. (By the way, B equals 1/PN times the firm’s demand elasticity). That is: Market Share = 1/N + B•(AVGP ‐ P) If the firm’s P = AVGP, then market share = 1/N. If P is higher, market share is lower. Firm’s Demand Q = S•(market share) = S/N + S•B•AVGP ‐ S•B•P If we assume AVGP is independent of P, and then solve this equation for P, we get: P = 1/BN – Q /BS + AVGP The firm’s revenue is: REV = Q•P = Q/BN ‐ Q2 /BS + Q•AVGP With calculus, we take the first derivative to find marginal revenue MR: MR = 1/BN ‐ 2•Q /BS + AVGP This looks complicated until we realize that it looks a lot like P, so we substitute to get: MR =P – Q /BS If the firm is maximizing profit, then MR = MC. Solving for P, and substituting (Q/S) = (1/N), we find: P = MC + Q /BS = MC + (Q/S)/B = MC + (1/N)/B = MC + 1/BN The Monopolistically‐Competitive Equilibrium: So now we have solved for both AC and P as a function of N. That is: AC = MC + (FC/S) •N P = MC + 1/BN Under monopolistic competition, firms will enter the market as long as the next firm can make a profit. To find the equilibrium number of firms, we set AC = P and solve for N. The steps involved are: AC = P MC + (FC/S) •N = MC + 1/BN (FC/S) •N = 1/BN N2 = S / (B•FC) N = ට
ௌ
஻ൈி஼
If N is not an integer, we must round down because we can’t have fractions of a firm and the next firm would not enter because it would make a loss. Using the lower N, total profits would equal S•(P‐AC), and these profits equal zero if the original N was an integer. Since the market size is fixed, we can calculate any change in consumer surplus as S times any change in prices.