Consider the following monopolistic competition setting: The marginal cost of production is c, the fixed cost of production is F. Each firm faces a demand curve q=Sn(p¯p)−σq=\frac {S}{n}(\frac {p}{\overline {p}})^{-\sigma } where S=9000S=9000, and n is the number of firms in the sector, p is the price charged by the firm and ¯p\overline {p} is the average price charged by other firms. From the firm’s profit maximization problem, derive the equilibrium price. (Hint: express equilibrium p as markup*marginal cost (c).) If different firms have different marginal costs but the same fixed cost, does the markup, defined as the ratio of price to marginal cost, vary by firm?

Question

BlackTom AI · Accepted Answer

The question asks us to derive the equilibrium price in a monopolistic competition setting and to assess whether the markup varies across firms when marginal costs differ.

First, restating the setup: each firm faces a residual demand of the form q = (S/n) (p / p̄)^{-σ}, where S and n are given, p is the firm’s price, and p̄ is the average price charged by other firms. Marginal cost is c and fixed cost is F. We are told to express the equilibrium price as a markup over marginal cost, p = markup × c, and to address whether the markup varies with firm-specific marginal costs if fixed costs are the same.

Now, step by step analysis of the options/possibilities:

- Deriving the firm’s marginal revenue and optimal price under isoelastic demand: If a firm faces q(p) = A p^{-σ} with A = S/n, then revenue is R(p) = p q(p) = A p^{1-σ}. The marginal revenue with respect to output, MR, can be obtained either by MR = dR/dq or by the standard isoelastic conduit MR = (1 - 1/σ) p when expressed in terms of price, but the direct calculus approach is cleaner here: MR as a function of price is MR(p) = dR/dp ÷ dq/dp, which yields MR = [(1 - σ) A p^{-σ}] / [-σ A p^{-σ-1}] = ((σ - 1)/σ) p.

- Setting MR equal to marginal cost to maximize profit: In profit maximization with constant marginal cost c, MR = MC gives ((σ - 1)/σ) p = c. Solving for p gives p = (σ/(σ - 1)) c. This is the standard result for a monopolistic competitor with isoelastic demand: the equilibrium price is a constant markup over marginal cost, independent of the number of firms n and of the fixed cost F, as long as the market shares and elasticities are exogenous and identical across firms.

- Interpreting the markup: The markup over marginal cost is p / c = σ/(σ - 1). This ratio is determined solely by the elasticity parameter σ and is independent of the firm’s absolute marginal cost level. Hence, p = markup × c with markup = σ/(σ - 1).

- Addressing the question about heterogeneous marginal costs with the same fixed cost: If different firms have different marginal costs c_i but share the same fixed cost F, the same MR = MC condition applies to each firm, yielding p_i = (σ/(σ - 1)) c_i for each firm i. Consequently, the markup defined as p_i / c_i equals σ/(σ - 1) for every firm, i.e., the markup does not vary by firm despite differences in c_i. The intuition is that, under isoelastic demand with constant elasticity, the optimal price for each firm scales proportionally with its marginal cost, preserving a uniform markup across firms as long as demand elasticity and competitive structure remain the same.

- Why this reasoning avoids common pitfalls: If one thought that a larger c_i would imply a smaller quantity or a different p̄ effect, that intuition would misinterpret the MR = MC condition under the given isoelastic residual demand. Because q(p) is isoelastic, MR scales proportionally with p, and the optimal p is proportional to c, not to the level of output. Also, since the markup is constant and the price responds to each firm’s own marginal cost, the average price p̄ across firms in equilibrium remains consistent with the individual p_i choices in a symmetric-in-structure market with many firms, and the functional form of p in terms of c_i holds firm-levelly.

In summary:
- The equilibrium price is p = (σ/(σ - 1)) c, i.e., a constant markup over marginal cost.
- If different firms have the same fixed cost but different marginal costs, the markup p_i / c_i remains constant at σ/(σ - 1) for all firms; thus, the markup does not vary by firm. The price for each firm simply scales with its own marginal cost.

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