Markups: A Search-Theoretic Perspective

Q1. What two theories of market power does the paper compare, and how do they differ at root?

The paper contrasts the Dixit–Stiglitz (1977) monopolistic-competition framework, in which market power comes from product differentiation, with the search-theoretic framework of Butters (1977), Varian (1980) and Burdett–Judd (1983), in which market power comes from buyers’ limited choice sets. In Dixit–Stiglitz, “every seller is a monopolist of its own product variety,” and the size of markups “is determined by the substitutability of different varieties in the buyers’ utility function.” In the search-theoretic framework, by contrast, “a seller has market power not because it carries a good that has no perfect substitutes, but because (some) buyers do not have every seller in their choice set due to informational frictions … or physical frictions,” so markups are instead “determined by the distribution of the size of buyers’ choice sets.” Menzio motivates the second view with retail examples (e.g., the same bottle of Heinz ketchup sold at many stores at different markups), where it strains credulity that buyers see one store’s bottle as a poor substitute for the identical bottle elsewhere.

Q2. What is the equilibrium markup formula when all sellers are identical?

With homogeneous sellers, a seller at quantile x of the price distribution charges a gross markup μ(x) = 1 + (u/c − 1)·e^(−λ(1−x)), the product of a monopoly markup and a rank-dependent discount factor. Here u is the buyer’s valuation, c the common marginal cost, and λ the Poisson coefficient for the number of sellers a buyer contacts — “the average number of sellers with which a buyer is in contact, and, in this sense, … a measure of the extent of competition in the market.” The term u/c − 1 is “the net markup for a monopolist.” The discount factor e^(−λ(1−x)) “is equal to 1 for the seller at the top of the price distribution” (no discounting) and falls to its minimum e^(−λ) for the seller at the bottom; a higher λ makes markups decline more steeply down the price ranking. The equilibrium price distribution and its support are derived in closed form (F(p) and the lowest price p_ℓ = c + e^(−λ)(u − c)), and the equilibrium is shown to exist, be unique, and be efficient (Proposition 1).

Q3. Why are markups positive and dispersed even when goods are perfect substitutes and technology is identical?

Markups are positive because search frictions leave some buyers “captive” — in contact with only one seller — which forces equilibrium profits, and hence prices, strictly above marginal cost; markups are dispersed for the same reason there is price dispersion in these models — non-captive buyers prevent any mass point in the price distribution. As Menzio puts it, “sellers meet a positive measure of buyers that are captive, in the sense that these buyers cannot purchase from any other seller,” so “prices must be strictly above marginal cost”; simultaneously, the positive measure of non-captive buyers “implies that the price distribution cannot have any mass points above marginal cost.” The two facts together require sellers to post different prices and therefore charge different markups, despite identical goods and identical technology.

Q4. In the homogeneous-seller case, how do markups relate to a seller’s price and size?

With identical sellers, markups are increasing in a seller’s price and decreasing in a seller’s size. Because μ(x) and the posted price p(x) both rise with rank x while quantity sold q(x) = bλ·e(−λx) falls with x, “markups are increasing in the seller’s price” and “decreasing in the seller’s size.” Menzio notes this is the opposite of “Marshall’s second law of demand,” and that it implies larger sellers face a higher elasticity of demand. He stresses this counterfactual pattern (empirically, larger firms tend to charge higher markups) is exactly why the paper goes on to add cost heterogeneity.

Q5. What changes when sellers differ in marginal cost?

With heterogeneous marginal costs, the markup formula gains an extra term reflecting that higher-ranked (higher-cost) firms put less competitive pressure on a seller, and equilibrium markups need no longer be decreasing in size — they can be increasing, decreasing, or hump-shaped. A seller’s price is a strictly increasing function of its cost (Lemma 3), so its rank in the price distribution equals its rank in the cost distribution. The generalized markup (eq. 3.22) adds, to the monopoly-times-discount term, “the additional markup that the seller can charge because the firms ranked above it in the price distribution produce at higher marginal cost,” with the excess cost of nearer-ranked firms weighted more heavily. Using a phase-diagram (nullcline) analysis, Menzio shows the markup function μ(x) can be strictly increasing, strictly decreasing, or hump-shaped in rank depending on parameters. The heterogeneous-cost equilibrium is again shown to exist, be unique, and be efficient (Proposition 2).

Q6. What is the “anything-goes” theorem, and why does it matter?

Menzio proves (Theorem 3) that any twice-continuously-differentiable markup function μ(x) > 1 can be generated as an equilibrium of the search-theoretic model, given an appropriate contact intensity λ and cost distribution c(x).* Concretely, for any target markup schedule there is a λ and a quantile cost function c(x) (given in closed form) that deliver it as the equilibrium outcome. The consequence is sharp: “the search-theoretic model of market power can rationalize any pattern of markups observed in the data,” so “markup data cannot be used to reject the search-theoretic model.” Combined with the fact that the Dixit–Stiglitz model can reproduce the same markups, both theories are consistent with any markup evidence — which is the crux of the paper’s identification argument.

Q7. Can a Dixit–Stiglitz model reproduce these markups, and at what cost?

Yes — a Dixit–Stiglitz model can always reproduce the search model’s markups, but only with reduced-form buyer preferences that depend on the search model’s deep parameters (λ, u, c, b) and are therefore unstable. Menzio constructs the buyer utility function v(q) (its marginal utility solves a differential equation, eq. 2.24) that makes a Dixit–Stiglitz seller choose the same markups and quantities as in the search model. That reduced-form utility has v’(q) decreasing (so varieties look like imperfect substitutes, rationalizing positive markups) and an elasticity of demand that rises with q (rationalizing markups that fall with size). Critically, “the reduced-form utility function depends on the parameters of the search-theoretic model” and so “is unstable, in the sense that changes in the environment and counterfactual experiments lead to changes in the reduced-form utility function” — meaning any policy or counterfactual exercise that holds these preferences fixed “would not produce valid predictions,” i.e., is subject to the Lucas critique.

Q8. Why would reading these markups through the Dixit–Stiglitz lens give the wrong welfare and policy conclusions?

Because in Dixit–Stiglitz positive and heterogeneous markups signal inefficiency and call for subsidies, whereas the search-theoretic equilibrium that generated those very markups is efficient. Through the Dixit–Stiglitz lens, positive net markups imply “sellers produce an inefficiently small quantity,” and heterogeneous markups imply misallocation across sellers, leading an analyst to “recommend the introduction of consumption subsidies” and “finely-tuned production subsidies that reallocate inputs and consumption from low to high-markup sellers.” “None of these welfare and policy implications are, however, correct, since the equilibrium of the search-theoretic model … is efficient.” The root of the error is the demand curve’s interpretation: the quantity q(p) − q(c) a seller does not sell is, in Dixit–Stiglitz, lost gains from trade (an inefficiency), but in the search model it is “equally valuable trades that the buyers make with other sellers,” and so is not an inefficiency.

Q9. What determines the level and shape of the markup distribution?

For a log-uniform cost distribution (Theorem 4), markups decrease with the extent of competition λ, increase with the buyers’ valuation u, decrease with the highest marginal cost c_h, and increase with the rate κ at which marginal costs decline across sellers; the sign of the markup–size relationship flips at parameter thresholds. Specifically, the markup function is strictly decreasing in rank x (markups rising with size) when competition is weak (λ below a cutoff λ*), constant when λ = λ*, and strictly increasing in x (markups falling with size) when λ > λ*; analogous thresholds u* and κ* govern the slope’s sign as u and κ vary. The intuition: when λ is low, sellers rarely compete for the same buyers and low-cost sellers face little pressure, so markups are high and higher for low-cost (large) sellers; when λ is high, low-cost sellers are pushed toward marginal-cost pricing while high-cost sellers — facing no pressure from above — retain markups near u/c_h. Menzio notes the monotone-level results (markups decreasing in λ and c_h, increasing in u and in κ(x) = c’(x)/c(x)) generalize beyond the log-uniform family to arbitrary cost distributions, while the slope-sign results are stated for the log-uniform case.

Q10. What is the bottom-line claim for macroeconomics?

Markup data alone are insufficient to draw conclusions about the welfare, policy, and counterfactual consequences of market power; identifying those consequences requires evidence on the source of market power — product differentiation versus search/information frictions. The paper frames this as “a cautionary note to the macroeconomic literature that uses the Dixit–Stiglitz framework to model market power and markups” — a literature spanning monetary policy (e.g., Blanchard–Kiyotaki 1985; Christiano, Eichenbaum and Evans 2005; Golosov and Lucas 2007), misallocation and aggregate TFP (Hsieh and Klenow 2009), and the gains from trade (Krugman; Melitz 2003). In Dixit–Stiglitz estimations, markup heterogeneity is “quantitatively important” for the welfare cost of inflation in sticky-price models (Galí 1995), the gains from trade (Dhingra and Morrow 2019), and the cost of market power (Boar and Midrigan 2024); Menzio’s point is that “neither the level nor the dispersion of markups observed in the data are necessarily symptomatic of any inefficiency.”

Q11. Does the paper claim the search-theoretic model is the correct one?

No — the paper explicitly does not argue that the search-theoretic model is closer to the truth than monopolistic competition; it makes the “more modest, but not unimportant” claim that two sensible, well-established models fit the same markup data yet imply very different welfare, policy, and counterfactual conclusions. Menzio notes both theories “are likely to be overly simplified descriptions of the world,” and that the existence of still other models generating the same markups “only strengthens” the point. The constructive takeaway he poses is an empirical identification question: “How much of the downward sloping demand curve facing a seller is due to the heterogeneity in buyer’s outside options and how much is it due to preferences?”

Q12. Does the argument extend beyond product markets?

Yes — the same logic applies to the labor market: in the Burdett–Mortensen (1998) search model one can derive a closed-form formula for equilibrium markdowns that are positive even when employers are perfect substitutes to workers, are heterogeneous even with identical technology, and may be increasing, decreasing, or constant in firm size, with the equilibrium again efficient. Menzio concludes that “the same caution that I recommend using when interpreting markups should be applied to the interpretation of markdown data.”

Q13. What are the scope conditions, and what does the paper not do?

The results are theoretical, derived under unit buyer demand, constant returns to scale, and a Poisson process for the number of sellers each buyer contacts; the closed-form comparative statics of Theorem 4 assume a log-uniform marginal-cost distribution; and the paper offers no empirical calibration or estimation. Menzio notes the efficiency result depends on the model’s assumptions — relaxing unit demand or adding externalities could make the equilibrium inefficient — but argues this does not weaken the core identification point. A companion paper (Menzio 2024b, NBER WP 33253) shows the efficiency of the search-theoretic equilibrium extends to a general-equilibrium setting with endogenous firm entry. The paper’s contribution is an analytical characterization and a cautionary/identification argument, not a quantitative welfare estimate.

Key concepts

Search-theoretic model of imperfect competition : The Butters (1977)/Varian (1980)/Burdett–Judd (1983) framework in which sellers carry identical (perfectly substitutable) goods, and market power arises because buyers contact only a random subset of sellers — so some buyers are “captive” to a single seller. Markups are determined by the distribution of buyers’ choice-set sizes, not by preferences over differentiated varieties.

Dixit–Stiglitz monopolistic competition : Any model in which each seller is the sole producer (monopolist) of its own variety, sets its price, and is too small to affect the aggregate; the size of markups is governed by the substitutability of varieties in buyers’ utility (CES, VES, translog, or Kimball preferences all qualify in the paper’s usage).

Gross / net markup : The gross markup μ is the ratio of a seller’s posted price to its marginal cost (p/c); the net markup is μ − 1.

Captive vs. non-captive buyers : A captive buyer is in contact with only one seller and so cannot shop around (the source of strictly positive markups); a non-captive buyer is in contact with several sellers and buys from the cheapest (the source of price dispersion and the absence of mass points in the price distribution).

λ (extent of competition) : The coefficient of the Poisson distribution governing how many sellers a buyer contacts — equivalently the average number of contacts per buyer; higher λ means more competition and lower markups.

Reduced-form preferences / Lucas critique : The buyer utility function a Dixit–Stiglitz modeller would infer to rationalize the search model’s markups; because it depends on the search model’s deep parameters (λ, u, c, b), it shifts whenever the environment or policy changes, so counterfactuals computed holding it fixed are invalid.

Efficiency (of the equilibrium) : The search-theoretic equilibrium maximizes the sum of buyer and seller payoffs — every contacted buyer buys (since valuation u exceeds cost c) and, with heterogeneous costs, buys from the lowest-cost contacted seller — so the positive, dispersed markups are not symptoms of any inefficiency.

Markdown : The labor-market analogue of a markup — the gap between a worker’s marginal product and the wage — which in the Burdett–Mortensen (1998) search model has the same qualitative properties (positive, heterogeneous, size-dependent, efficient) as product-market markups here.