L11 | Macro Paper Warehouse

Competitive Advertising and Pricing

Mon, 01 Jan 0001 00:00:00 +0000

Hwang, Kim, and Boleslavsky study how firms in an oligopoly simultaneously choose prices and advertising strategies, where advertising is modeled as the choice of how much product information to disclose to consumers. The paper extends the canonical Perloff-Salop (1985) random-utility discrete-choice framework — in which n firms engage in Bertrand competition for a consumer whose value for each product is independently drawn from a common distribution F — by endogenizing the information environment: each firm may choose any mean-preserving contraction (MPC) of F as its advertising strategy, with no structural restriction on feasible content. This full flexibility, drawn from the information design literature, allows each firm to choose the consumer’s effective value distribution, ranging from full information (choosing F itself) to complete concealment (a degenerate distribution at the mean). The model is silent on advertising costs, which are assumed to be zero throughout.

The central result is that intense competition forces firms to provide precise product information. Formally, the full information equilibrium — in which every firm chooses F — exists in the advertising game (the subgame in which prices are fixed symmetrically) if and only if F^(n-1) is convex over its support. Because F^(n-1) represents the distribution of the consumer’s best outside option, convexity means the consumer likely faces an attractive alternative, incentivizing each firm to maximize the chance of offering the highest possible value. Crucially, this convexity condition is guaranteed to hold when n is sufficiently large, regardless of the shape of F, because the power function x^(n-1) becomes more convex as n rises. This establishes that under sufficiently intense competition, full information disclosure is the unique symmetric equilibrium.

The general equilibrium advertising strategy G* — which governs cases where full information is not an equilibrium — satisfies two necessary and sufficient conditions: (i) (G*)^(n-1) is convex over the support of G*, and (ii) for almost all values in the support, G* either coincides with F (where the MPC constraint binds, preventing further dispersion) or (G*)^(n-1) is locally linear (where the firm is locally risk-neutral and has no incentive to alter its distribution). The paper proves existence and uniqueness of G* for any F satisfying the stated regularity conditions (density positive, continuously differentiable, bounded, with finitely many peaks). When F has log-concave density, a unique symmetric pure-price equilibrium (p*, G*) exists in the full game.

The paper demonstrates that strategic advertising has ambiguous implications for prices and consumer welfare. Strategic advertising necessarily reduces social surplus through information loss, since consumers select suboptimal products with positive probability when G* differs from F. However, it compresses the support of the value distribution relative to F, which — by a new result (Proposition 3) — tends to lower the equilibrium price. Offsetting this, strategic advertising also redistributes marginal consumers in ways that may raise or lower the price. In the duopoly case with power distributions F(v) = v^alpha on [0,1], strategic advertising lowers the market price if and only if alpha > 1/sqrt(2) (approximately 0.7071), and raises consumer surplus if and only if alpha > 0.7928.

The paper examines three extensions: (1) a binding consumer outside option, (2) multi-unit (k-out-of-n) demand, and (3) asymmetric firms with two types. In all three cases, full information cannot be a strict equilibrium for any finite n under the relevant structural condition, yet the equilibrium distribution G* converges pointwise to F as n tends to infinity, preserving the paper’s core asymptotic insight.

Q: What is the main research question? A: The paper asks how much product information firms will voluntarily disclose when they compete both on price and advertising content in an oligopoly. Unlike the monopoly literature, the oligopoly context creates strategic interdependencies — each firm’s optimal disclosure depends on rivals’ disclosure choices — that the paper characterizes fully.

Q: How is advertising modeled, and why use mean-preserving contractions? A: Each firm’s advertising strategy is modeled as a choice of any mean-preserving contraction (MPC) of the true value distribution F. An MPC preserves the expected value but reduces dispersion, capturing the idea that a firm can selectively conceal information (moving toward a degenerate distribution) but cannot fabricate value dispersion beyond what F allows. Because consumers are risk-neutral and buy based on expected values net of prices, this MPC formulation captures full flexibility in information design without loss of generality.

Q: What is the precise necessary and sufficient condition for the full information equilibrium in the advertising game? A: The full information equilibrium — in which every firm chooses F — exists if and only if F^(n-1) is convex over its support [v, v̄]. The “only if” direction follows from Lemma 1: in any equilibrium, (G*)^(n-1) must be convex, so if F^(n-1) is not convex, F is not an equilibrium. The “if” direction follows because a convex F^(n-1) makes each firm locally risk-loving, so no MPC of F yields a higher payoff than F itself.

Q: Why does sufficiently intense competition force full information disclosure? A: For any distribution F with positive, continuously differentiable, bounded density f with bounded derivative f’, the second derivative of F^(n-1) satisfies F(v)^(n-1)’’ >= (n-1)F(v)^(n-3)[(n-2)epsilon^2 - M], where epsilon = min f(v)^2 > 0 and M = max |f’(v)| < infinity. This expression is strictly positive for n sufficiently large, so F^(n-1) is convex and the full information equilibrium exists. Economically, with many competitors each firm wins the consumer only when it offers the highest possible value, so providing full information is optimal.

Q: What are the two necessary and sufficient properties characterizing the general equilibrium advertising strategy G?* A: First (Lemma 1), (G*)^(n-1) must be convex over the support of G* — this prevents any firm from profitably concentrating mass to reduce dispersion. Second (Lemma 2), for almost all values in the support, either G* = F locally (the MPC constraint binds, preventing further dispersion) or (G*)^(n-1) is locally linear (the firm is locally risk-neutral and indifferent over distributions with the same local mean). Theorem 1 proves these two conditions are both necessary and sufficient, and that G* is unique for any F satisfying the stated regularity conditions.

Q: What structure does G take when F^(n-1) has strictly quasi-concave density?* A: By Corollary 2(1), there exists a cutoff v* in [v, v̄] such that G*(v) = F(v) for v <= v* (full information below the cutoff) and (G*)^(n-1) is linear above v*. As n increases, v* rises, meaning the region of full disclosure expands, and G* increases in convex order — so consumers receive strictly more information. One immediate implication is that consumer surplus strictly increases in n: consumers benefit both from more options and from more accurate information about each product.

Q: What happens when F^(n-1) is concave? A: By Corollary 3, when F^(n-1) is concave, (G*)^(n-1) is linear over the entire support, with lower bound v. In the illustrative Example 1 (truncated exponential with n=2), this yields G* = U[0, 2*mu_F] — a uniform distribution on an interval whose upper bound is twice the mean of F.

Q: Does strategic advertising raise or lower equilibrium prices, and consumer surplus? A: Both effects are ambiguous and depend on the shape of F. Strategic advertising compresses the support of the value distribution (since G* is an MPC of F), which by Proposition 3(1) tends to lower equilibrium prices. But it also reshapes the distribution of marginal consumers, which may raise or lower prices. In the power distribution example (n=2, F(v) = v^alpha on [0,1]), strategic advertising lowers the market price if and only if alpha > 1/sqrt(2) ≈ 0.7071, and raises consumer surplus if and only if alpha > 0.7928. Thus even with deadweight loss from information suppression, consumers can be better off under strategic advertising than under forced full disclosure.

Q: What does Proposition 3 contribute about equilibrium prices in the Perloff-Salop model? A: Proposition 3 delivers two results about how the distribution of marginal consumers (integral (F^(n-1))’ dF) determines equilibrium prices. First, the measure of marginal consumers decreases if F is proportionally stretched over a larger support, confirming that longer support raises equilibrium prices. Second — presented as novel — among all distributions with support in [v, v̄], the power distribution F(v) = ((v-v)/(v̄-v))^(2/n) minimizes the measure of marginal consumers, corresponding to the maximum equilibrium price. The key property is that marginal consumers are uniformly distributed under this power distribution, and any deviation from uniformity allows a “flattening” adjustment that increases the measure of marginal consumers and lowers the price.

Q: Under what condition does the full game (price plus advertising) have a unique symmetric pure-price equilibrium? A: Theorem 2 states that log-concavity of the density f is sufficient for existence and uniqueness of a symmetric pure-price equilibrium (p*, G*) as characterized in Theorems 1 and 2. Log-concavity ensures that the equilibrium distribution G* has a convex-linear structure (as in Corollary 2), which preserves log-concavity of each firm’s profit function even under compound deviations (simultaneous changes to both price and advertising strategy), making the first-order conditions sufficient for global optimality.

Q: Can strategic advertising create or destroy pure-price equilibria relative to the Perloff-Salop benchmark? A: Yes, both directions are possible. When F^(n-1) is convex (so G* = F), equilibrium existence in the Perloff-Salop (PS) model is necessary but not sufficient for existence in the full model, because compound deviations (changing both price and advertising) may be profitable even when pure price deviations are not. Conversely, when G* differs from F, the changed distribution of marginal consumers can sustain an equilibrium in the full model even when none exists in PS. Appendix E of the paper provides a specific example of the latter phenomenon.

Q: What happens with a binding consumer outside option? A: Proposition 4 shows that a full information equilibrium never exists in the advertising game when the consumer has a binding outside option (p* in (v, v̄)). The firm’s value function acquires a discrete jump at p* due to the indicator 1_{v >= p*}, making it optimal to pool mass around p* rather than disclose fully. Nevertheless, Proposition 5 proves that G* converges pointwise to F as n tends to infinity, because the jump of size F(p*)^(n-1) vanishes exponentially fast as n grows.

Q: Does the full information result survive multi-unit demand? A: No. Proposition 6 shows that with k > 1 units demanded (out of n products), the full information equilibrium never exists for any finite n or F. The reason is that phi’(v; F) — the firm’s marginal value of offering value v — is zero at v̄ when k > 1, so the firm can profitably pool values near the top of the support. However, Proposition 7 shows that G* converges pointwise to F as n tends to infinity (with k fixed), preserving the asymptotic full information result.

Q: What happens with asymmetric firms differing in their value distribution supports? A: Proposition 8 shows a sharp dichotomy. If both firm types share the same upper bound of their value supports (v̄_1 = v̄_2), the full information equilibrium exists whenever both F_1^(n1-1) and F_2^(n2-1) are convex. If the supports have different upper bounds (v̄_1 < v̄_2), the full information equilibrium never exists regardless of n_1 and n_2, because type-2 firms face a downward kink in their winning probability at v̄_1 and always have an incentive to pool mass there. The authors conjecture that G*_1 and G*_2 still converge to F_1 and F_2 asymptotically but do not prove this due to technical complexity.

Q: How does this paper relate to Ivanov (2013)? A: Ivanov (2013) also uses the Perloff-Salop framework and shows that full information is an equilibrium when n is sufficiently large, but restricts advertising to rotation-ordered strategies (in the sense of Johnson and Myatt, 2006). The present paper imposes no structural restriction and strengthens Ivanov’s result by: (a) providing a necessary and sufficient condition for the full information equilibrium (not just a sufficient condition for large n); (b) fully characterizing G* when full information is not an equilibrium; and (c) demonstrating robustness across multiple model variants.

Q: What policy implication does the ambiguity result carry? A: The paper warns against assuming that mandating full information disclosure is unambiguously consumer-beneficial. While strategic advertising creates deadweight loss through information suppression, it can simultaneously compress support and alter the marginal consumer distribution in ways that lower equilibrium prices significantly. The power distribution example (alpha > 0.7928) shows consumers can be strictly better off under strategic advertising than under forced full disclosure. This ambiguity is a cautionary tale for disclosure regulation.

Mean-Preserving Contraction (MPC): A distribution G_i is an MPC of F if it has the same mean as F but less dispersion (in the sense of second-order stochastic dominance). In the paper, each firm’s feasible advertising strategies are exactly the set MPC(F) — this captures all informationally feasible disclosures without structural restriction on content.

Advertising Game: A restricted subgame of the full market game in which firms choose their advertising strategies G_i taking the symmetric price as given. An equilibrium in the advertising game is a necessary condition for equilibrium in the full game. The advertising game’s equilibrium uniquely pins down G* independently of the price level (under the baseline model without binding outside option).

Full Information Equilibrium: An equilibrium of the advertising game in which every firm chooses the true underlying distribution F as its advertising strategy. This corresponds to complete, unobstructed product disclosure. The paper’s central result is that this equilibrium exists if and only if F^(n-1) is convex over its support.

Convexity of F^(n-1): The key distributional condition governing advertising equilibria. F^(n-1) is the distribution of the consumer’s best alternative among (n-1) rivals’ products. Convexity of F^(n-1) means its density is increasing, signaling a likely attractive outside option, which makes each firm risk-loving and induces full disclosure. This convexity is guaranteed for n sufficiently large.

Locally Linear (G)^(n-1):* A region of the equilibrium distribution where (G*)^(n-1) has constant slope, making the firm locally risk-neutral. Over such a region, the firm is indifferent among all distributions with the same local mean, and the equilibrium G* need not coincide with F — it is only required to be an MPC of F on that interval. This alternating structure (coinciding with F on strictly convex regions; linear elsewhere) fully characterizes G*.

Marginal Consumers: In the Perloff-Salop pricing formula, the equilibrium price p* = (1/n) / integral [(G*(v)^(n-1))’ dG*(v)]. The integrand (G*(v)^(n-1))’ * g*(v) is the density of consumers who are indifferent between a given firm’s product and their best alternative at value v. A larger measure of marginal consumers implies lower equilibrium prices through greater competitive pressure.

Compound Deviation: In the full game, a deviation by a firm that changes both its price p_i and its advertising strategy G_i simultaneously, rather than varying only one dimension. The possibility of compound deviations is what distinguishes equilibrium existence conditions in the full model from those in the standard Perloff-Salop model, even when G* = F.

Customer accumulation, returns to scale, and secular trends

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks how rising returns to scale in production contributed to three concurrent U.S. secular trends since 1980: declining business dynamism, rising markups, and growing firm expenditures on customer acquisition. The author constructs a firm dynamics model in the Hopenhayn (1992) tradition with endogenous entry and exit, heterogeneous markups, and customer accumulation grounded in directed search in the product market. Firms compete for customers through both prices and selling activities; larger firms gain a competitive edge when returns to scale rise because their marginal costs fall more than those of smaller firms—even though the technological shift is uniform across firms. This demand-based channel triggers winners-and-losers dynamics and the rise of superstar firms.

The empirical foundation rests on Compustat data for U.S. publicly traded firms (1977–2014) and Business Dynamics Statistics (BDS) for aggregate and sector-level dynamism measures. Production-function estimation using Ackerberg, Caves, and Frazer (2015) augmented with sales-share controls documents that aggregate returns to scale rose from approximately 1.0 in 1980 to approximately 1.05 by 2014—a within-sector increase, not a reallocation effect. Over the same period, the cost-weighted markup rose by 42%, the firm entry rate fell by 33%, the excess reallocation rate fell by 29%, and selling costs relative to production costs rose by 60%–90% depending on the measure used.

The model is calibrated to 1980 steady-state moments (firm life-cycle patterns, markups, entry and reallocation rates). A 5% increase in returns to scale—matching the empirical estimate—accounts for: a +15 percentage point rise in the average cost-weighted markup (vs. +42% in the data); a 33% decline in the entry rate (exactly matching the data); a 21% decline in the reallocation rate (vs. 29% in the data); and a 23% increase in selling costs relative to production costs (vs. 60%–90% in the data). The model also generates a 53% rise in the share of firms aged 11 years or older (vs. 50% in the data) and a 58% decline in the employment share of firms aged 5 years or younger (vs. 56% in the data), closely tracking the aging of the U.S. firm population. Firm-level responsiveness to productivity shocks declines by 0.08 in the model, versus about 0.01 in Compustat and 0.09 in Decker et al. (2020).

Sector-level panel regressions with sector fixed effects confirm the model’s directional predictions: within-sector increases in returns to scale are associated with lower entry rates (coefficient −2.89, significant at 1%), lower reallocation rates (−1.16, significant at 1%), higher markups (+3.15, significant at 1%), and higher selling costs relative to production costs (+1.85 for the advertising-based measure; +8.52 for adjusted SG&A).

A key scope condition is that the model yields a constrained-efficient allocation: directed search and full internalization of returns to scale imply decentralized equilibrium efficiency, making the paper a laboratory for assessing how far efficient firm responses to technological change can explain the secular trends without invoking market failures. The model fits the post-2000 transition dynamics better than the 1980s–1990s period, and explains a substantial but incomplete share of the trends, suggesting complementary—possibly inefficient—forces also contributed.

Q: What is the core mechanism through which rising returns to scale generate winners-and-losers dynamics?

A: The marginal cost of production under increasing returns to scale (alpha > 1) is MC(z,n) = l(n,z)^(1−alpha) × (1/alpha) × (W/e^z), which depends on firm size l(n,z). A uniform rise in alpha rotates the marginal cost schedule clockwise by firm size: larger firms see a proportionally larger cost reduction than smaller firms, even though the technological change is identical across all firms. Because firms compete for the same pool of customers, this asymmetric cost advantage allows large firms to offer lower prices while sustaining higher margins, attracting customers away from small firms. The result is a demand-based channel that generates winners-and-losers dynamics and increases market concentration.

Q: How does the model capture customer accumulation, and why is it central to the paper’s argument?

A: The model introduces directed search in the product market, where firms post advertisements and customers—including those already matched with a firm—choose which submarket to enter by trading off offered utility against matching probability. A constant-returns-to-scale matching function governs match creation; in submarket with tightness theta, customers match with probability m(theta) = theta(1+theta)^(−1) and firms attract customers with probability q(theta) = (1+theta)^(−1). The customer accumulation motive creates an investment-harvest trade-off: firms can either post high promised utility (low prices) to grow their customer base or extract surplus through high prices. Rising returns to scale amplify large firms’ ability to resolve this trade-off favorably, linking the technological change directly to markup dynamics, entry incentives, and selling expenditures.

Q: What is the directed search framework’s role in ensuring equilibrium uniqueness and efficiency?

A: The author introduces firm-side commitment contracts—specifying price, separation probability, and continuation utility contingent on productivity realizations—combined with directed search. Because search is directed on both sides and firms fully internalize returns to scale, the decentralized equilibrium is constrained-efficient. This delivers uniquely determined heterogeneous prices in equilibrium (solving the indeterminacy problem common in customer-market models) and establishes the paper’s efficient-mechanism benchmark: it tests how far profit-maximizing firm responses to technological change—without any market failure—can account for the secular trends.

Q: How are prices structured in the model, and what life-cycle pattern do they generate?

A: Each firm charges two distinct prices in each period: one to incumbent customers (the same for all incumbents, since they are identical conditional on being attached to the same firm) and one to newly acquired customers (which varies based on the promised utility in the submarket searched). Firms that are expanding their customer base offer greater promised utility and therefore charge lower prices to attract customers; firms harvesting their existing base charge higher prices. Because firms enter small and grow, this dynamic generates a price life cycle: young firms invest via low prices and mature firms harvest through higher prices, which the model reproduces as a rising markup pattern over the firm life cycle—an untargeted moment the model fits well.

Q: What does the calibration target and what untargeted moments does the model reproduce?

A: The model is calibrated to 1980 using: the number of employees of entrant firms (pinning entry customer base n_e), employees of age-5 firms (pinning convex cost chi_1), share of firms aged 11+ years (pinning chi_2), average firm size (operating cost f), entry rate (entry cost kappa), excess reallocation rate (exit shock delta), and average cost-weighted markup (linear cost c). Untargeted moments reproduced include: a sales-weighted markup of 0.28 (vs. 0.25 in De Loecker et al. 2020), endogenous customer turnover of approximately 9% (vs. 15% in Gourio and Rudanko 2014), and an elasticity of customer base shrinkage to price of 0.08 (within the 0.01–0.16 range from Paciello et al. 2019). The model also matches markup and selling-cost life-cycle patterns that are typically overlooked.

Q: How large is the quantitative contribution of the 5% rise in returns to scale to each secular trend?

A: Comparing the 1980 steady state (alpha = 1) to the 2014 steady state (alpha = 1.05): the average cost-weighted markup rises by 15% in the model versus 42% in the data; the entry rate declines by 33% in the model, exactly matching the data; the reallocation rate declines by 21% in the model versus 29% in the data; and selling costs relative to production costs rise by 23% in the model versus 60%–90% in the data. The model thus explains a substantial share of each trend while leaving a residual requiring additional mechanisms.

Q: How does the model explain the aging of U.S. firms, and how well does it match the data?

A: The winners-and-losers mechanism shifts activity toward larger, older firms, which mechanically ages the firm population. The model generates a 53% increase in the share of firms aged 11 years or older (vs. 50% in the data) and a 58% decline in the employment share of firms aged 5 years or younger (vs. 56% in the data). This aging arises because rising returns to scale increase the cost of customer acquisition, acting as a barrier to entry that disproportionately hurts new, small firms while allowing large incumbents to remain viable at lower productivity thresholds.

Q: What is the channel through which rising returns to scale reduce business dynamism specifically?

A: The unequal reduction in marginal costs intensifies competition for customers and raises customer acquisition costs. This operates through two simultaneous effects on the exit threshold: (i) lower marginal costs allow large firms to remain viable at lower productivity levels despite higher customer acquisition costs; and (ii) heightened competition forces smaller firms to require higher productivity to survive in a market that has become increasingly costly to operate in. Higher customer acquisition costs therefore function as an endogenous barrier to entry, reducing the entry rate and the reallocation of resources across firms.

Q: Does the model attribute the secular trends entirely to efficient firm behavior, and what does it conclude about residual explanations?

A: No. The model is explicitly designed as a constrained-efficient benchmark, and the paper finds that while rising returns to scale account for a substantial share of the trends—particularly in magnitude—the transition dynamics show a less accurate fit before the 2000s. The author concludes that complementary mechanisms, likely involving inefficiencies (such as market power from horizontal product differentiation or barriers to entry beyond those captured by the model), played a significant role in the earlier evolution of these trends and in the portion of the trends not explained by the efficient channel.

Q: What evidence supports the rising returns to scale finding, and what are its limitations?

A: Production-function estimation using the Ackerberg-Caves-Frazer method with sales-share controls on Compustat data shows returns to scale rising from approximately 1.0 in 1980 to approximately 1.05 by 2014, driven primarily by within-sector increases rather than reallocation toward high-returns sectors. A translog production function finds limited evidence of heterogeneous increases across firm sizes within Compustat. However, Compustat predominantly covers large publicly traded firms; smaller firms outside the sample may have experienced minimal or no increase in returns to scale. If technology adoption involves fixed costs, the aggregate impact could be larger than estimated, meaning the quantitative exercises likely represent a conservative lower bound.

Q: How does the paper relate to and extend the directed search literature in product markets?

A: The paper builds on Gourio and Rudanko (2014) and Roldan-Blanco and Gilbukh (2020), where customers are locked in once matched, by introducing labor-search tools from Schaal (2017) to allow: (i) incumbent customer switching between firms at rates of 10%–25% annually (Gourio and Rudanko 2014), and (ii) a non-zero price sensitivity of incumbent customers (Paciello et al. 2019). It also allows firms to invest in demand through selling expenditures, which prior directed search models in product markets typically abstracted from, making it possible to study how technological changes affect customer reallocation and firms’ cost structures jointly.

Customer capital: The stock of customers a firm has accumulated through prior selling and pricing decisions; treated as a state variable that firms invest in (by offering low prices and spending on advertisements) or harvest from (by charging high markups), with a customer turnover rate estimated at 10%–25% annually in the literature.

Directed search in the product market: A market structure in which both firms and customers choose which submarket (indexed by the promised utility level) to enter, trading off match probability against terms; delivers constrained-efficient equilibrium and uniquely determined heterogeneous prices.

Investment-harvest trade-off: The firm’s dynamic choice between offering high promised utility (low prices, low current markups) to grow the customer base versus extracting surplus through high prices from an existing customer base; shaped by the firm’s current size, productivity, and the cost structure implied by returns to scale.

Returns to scale (alpha): The curvature of the production function y = e^z × l^alpha; equals 1.0 under constant returns and approximately 1.05 by 2014 in the empirical estimates; the paper’s central technological change parameter, whose rise disproportionately reduces marginal costs for larger firms.

Winners-and-losers dynamics: The reallocation of customers and market share from small to large firms triggered by the asymmetric cost advantage large firms obtain when returns to scale rise; the demand-based channel through which superstar firms emerge.

Cost-weighted markup: The average markup aggregated using each firm’s costs as weights, as opposed to sales-weighted markup; the primary measure of market power used in the paper, rising by 42% in the data between 1980 and 2014.

Constrained-efficient allocation: An equilibrium outcome in which, given the frictions present (search-and-matching in the product market), no social planner operating under the same constraints could improve welfare; the paper uses this as a benchmark to assess how far efficient firm responses explain secular trends without invoking market failures.

Selling costs relative to production costs: The ratio of customer acquisition expenditures (advertising or adjusted SG&A) to cost of goods sold; rose by 60%–90% in the data between 1980 and 2014 and by 23% in the model’s steady-state comparison.

Firm idiosyncratic risk and productivity investment: Macroeconomic implications

Mon, 01 Jan 0001 00:00:00 +0000

This paper quantifies how idiosyncratic firm-level risk affects aggregate output, TFP, and firm life-cycle growth in an environment where firm productivity evolves endogenously through risky investment. The paper embeds endogenous productivity investment into a Lucas span-of-control model with risk-averse firm owners and endogenous entry and exit, and studies the effects of mean-preserving increases in the variance of returns to productivity investment. A mean-preserving increase in the variance of firm productivity shocks that raises the firm exit rate by 10% (from 0.10 to 0.11) is estimated to cause a 0.73% decline in output, a 0.38% decline in measured TFP, and a 3.69% decline in firm productivity investment; these elasticities remain approximately constant in the empirically relevant range. The driving force is that risk-averse firm owners reduce their risky productivity investment as variance rises; if capital financing constraints are present—as is common in developing economies—these effects are amplified and the increase in uncertainty may also slow firm life-cycle growth. Previously circulated as “Uncertainty, Firm Lifecycle Growth, and Aggregate Productivity.”

Summary based on a working paper version, AI-assisted and human-reviewed. See the linked published article for the authoritative version.

In depth

Q1. What distinguishes this paper from standard models of firm misallocation?

Unlike the bulk of firm misallocation literature (Hsieh-Klenow 2009; Gopinath et al. 2017; Sraer-Thesmar 2023), which takes firm productivity as exogenous, this paper models productivity as an endogenous outcome of risky investment, so that idiosyncratic uncertainty affects allocative efficiency not only through selection effects but also through its discouragement of productivity investment by risk-averse owners. The paper incorporates endogenous productivity investment into a standard Lucas span-of-control model, allowing the model to capture how higher uncertainty reduces the incentive to invest in productivity, on top of any selection effects from the exit option.

Q2. What are the two opposing effects of higher idiosyncratic risk?

Higher idiosyncratic firm-level risk has two opposing effects on aggregate productivity: (i) a selection effect—a mean-preserving increase in variance leads to stronger selection and raises the productivity of survivors while reallocating exiters to alternative productive uses—that tends to raise average productivity; and (ii) a productivity investment effect—risk-averse owners reduce risky productivity investment in response to higher variance—that tends to reduce aggregate productivity and firm life-cycle growth. The paper shows quantitatively that the productivity investment effect dominates in the baseline calibration, so that higher idiosyncratic risk reduces output and TFP despite positive selection effects.

Q3. What are the main quantitative findings?

A mean-preserving increase in the variance of firm productivity shocks calibrated to raise the firm exit rate by 10% (from 0.10 to 0.11) results in a 0.73% decline in output, a 0.38% decline in measured TFP, and a 3.69% decline in firm productivity investment; these elasticities remain approximately constant in the empirically relevant range. The exit-rate increase from 0.10 to 0.11 is also associated with a 7.5% increase in the job destruction rate and a 14.6% increase in the standard deviation of firm growth rates—the latter is less than one-fifth of the increases in these risk measures observed when comparing India or Mexico to the U.S.

Q4. How do capital financing constraints interact with the results?

When firms face capital financing constraints—as is common in developing economies—the negative effects of higher idiosyncratic risk are amplified and the increase in uncertainty may also slow firm life-cycle growth. The mechanism is that constrained firms must rely more heavily on internal financing, making risk-averse owners even more sensitive to increases in variance. The paper implies that the macro-financial implications of idiosyncratic risk are more severe in developing economies where both idiosyncratic risk levels and financing constraints are greater—consistent with cross-country patterns of firm growth dynamics.

Key concepts

productivity investment : endogenous spending by firms on activities that shift their productivity process; in the model, this investment exposes firm owners to idiosyncratic risk via the innovation in the productivity process; the key margin through which higher uncertainty reduces aggregate productivity and output. mean-preserving increase in variance : a statistical experiment that increases the spread of the distribution of returns to productivity investment while leaving the mean unchanged; used here to isolate the pure risk effect on firm behavior and aggregate outcomes from any change in expected returns. span-of-control model : the Lucas (1978) model of firm size distribution with decreasing returns to scale in the entrepreneurial input; used as the production environment; extended here by adding endogenous productivity investment and endogenous entry and exit.

Firm Responses and Wage Effects of Foreign Demand Shocks with Fixed Labor Costs and Monopsony

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. The paper asks three related questions in the context of Belgium, a small open economy: (1) What do firms’ responses to demand shocks reveal about their cost structures? (2) What are the worker and wage impacts of foreign demand shocks? (3) How sensitive are the aggregate wage effects of foreign demand shifts to firms’ cost structures and imperfect competition in the labor market?

Data. The analysis combines administrative micro-data from Belgium for 2002–2014, provided by the National Bank of Belgium. The linked dataset covers 995,739 firm-year observations from private, non-financial firms with at least one FTE employee, and integrates: (a) a Business-to-Business (B2B) VAT transactions registry capturing all annual domestic firm-to-firm sales above €250; (b) customs records and intra-EU declarations for imports and exports at the 8-digit product level; (c) annual accounts containing data on sales, labor costs, intermediate inputs, capital, and firm characteristics; and (d) employer-employee matched data from the Belgian social security administration (BCSS) for a random sample of 500,000 workers in firms with 10 or more FTE employees, covering 2003–2014.

Identification Strategy. To isolate variation in firms’ sales driven by foreign demand rather than supply-side factors, the authors construct a firm-specific foreign demand instrument following Hummels et al. (2014) and Dhyne et al. (2021). The instrument is the weighted average of changes in world import demand facing a firm, using lagged export shares as weights and excluding Belgian imports from the world import measure. Crucially, the instrument captures both direct foreign demand exposure (for exporters) and indirect exposure through the domestic production network — including the foreign demand shocks passing through to upstream domestic suppliers via buyer-supplier links. Firm and industry-year fixed effects control for time-invariant heterogeneity and industry-level trends.

Key Empirical Facts. Within-firm analysis over four-year windows finds that intermediate input purchases respond nearly proportionally to changes in sales (slope coefficient 0.82), while labor costs respond less than proportionally (slope coefficient 0.57). The less-than-proportional response of labor costs — with the employment slope of 0.48 and the average wage slope of 0.09 — is consistent with sizable fixed overhead costs in labor inputs and upward-sloping labor supply curves. Output prices co-move more with input prices than with average wages, consistent with labor constituting a smaller share of variable costs than intermediate inputs.

IV Estimates of Firm Responses. In response to a foreign demand shock inducing a 10 percent instantaneous increase in a firm’s sales, the firm’s cumulative sales over four years increase by approximately 7.6 percent (balanced panel). Over the same four-year horizon, total input purchases increase by about 7.0–7.8 percent, while labor costs increase by only 3.5–4.1 percent — a substantially less-than-proportional response. Roughly one-quarter of the labor cost change comes from changes in average wages rather than employment changes. Domestic input purchases increase by 5.3–6.0 percent, indicating that firms pass on a large share of foreign demand shocks to their domestic suppliers.

Structural Parameters. The implied IV estimate of the labor cost elasticity with respect to sales is 0.53 (standard error 0.08), statistically significantly below one. The implied elasticity of total input purchases is 1.05 (standard error 0.15), close to one, so the fixed share of intermediate inputs is approximately zero. The labor supply elasticity estimated from the ratio of wage and employment responses is approximately 3.9 in the full sample and 2.3 in the stayer subsample; the implied wage markdown is 21 percent and 30 percent respectively. Incorporating upward-sloping labor supply into equation (15), the estimated share of total labor inputs that is fixed overhead is approximately 53 percent. By comparison, the fixed share of total costs (labor and intermediate inputs combined) is approximately 29 percent in Belgium — higher than the 18–22 percent found in U.S. data (De Loecker et al. 2020) and the 20 percent found in U.S. manufacturing plants (Ederhof et al. 2021).

General Equilibrium Counterfactuals. The authors parameterize and solve a small open economy general equilibrium model with monopsonistic competition in labor markets, monopolistic competition in product markets, and fixed and variable labor and intermediate input costs. Using the Dekle-Eaton-Kortum (2007) “hat algebra” technique, they simulate a 5 percent increase in foreign tariffs on all Belgian exports and compare four counterfactual economies: (1) baseline Belgium with fixed costs and imperfect labor market competition (ε = 3.9); (2) fixed costs and perfectly elastic labor supply (ε = ∞); (3) no fixed costs with imperfect competition; (4) no fixed costs and perfectly competitive labor markets.

Main Findings on Wages. In the baseline Belgian economy, a 5 percent increase in foreign tariffs produces a 4.9 percent fall in the average real wage. With fixed costs but perfectly elastic labor supply, the real wage falls by 4.8 percent — nearly identical. With upward-sloping labor supply but no fixed costs, the real wage falls by only 3.0 percent; without fixed costs and with perfectly competitive labor supply, the fall is only 2.8 percent. The paper concludes that fixed overhead costs in labor substantially amplify real wage declines, while incorporating upward-sloping labor supply appears quantitatively less consequential for aggregate wage outcomes. Standard models that assume no fixed costs and perfectly elastic labor supply — the typical modeling choice in the trade literature — may substantially understate (by roughly 43–75 percent of the true effect) the aggregate wage decline from a negative foreign demand shock.

Mechanism. Fixed overhead costs reduce labor’s share of variable costs. When labor is a smaller share of variable costs, output prices are less sensitive to changes in wages. With a fixed aggregate labor supply, the economy must lower prices through wage reductions to restore equilibrium after a negative demand shock; the required wage decline is larger when fixed labor costs are taken into account. The findings are robust to adjustment cost specifications, a nested logit extension of the labor market model, and controlling for location-year fixed effects and import price changes.

Layer 2 — Q&A

Q1: What two motivating empirical facts about Belgian firms does the paper establish? A1: First, within-firm four-year changes show that intermediate input purchases respond nearly proportionally to changes in sales (slope coefficient 0.82), while labor costs respond less than proportionally (slope coefficient 0.57). The labor cost response decomposes into an employment slope of 0.48 and a wage slope of 0.09. Second, output prices co-move more strongly with input (intermediate goods) prices than with average wages, consistent with labor constituting a smaller share of variable costs than intermediate inputs.

Q2: How does the instrument for foreign demand shocks capture indirect exposure through production networks? A2: The instrument for firm k is a weighted average of changes in world import demand, where the weights reflect both the firm’s own direct export shares across countries and products and the firm’s indirect export exposure through its domestic buyers’ export shares. The term H̃_{kn,t-1} captures the share of firm k’s total sales purchased by firm n directly and indirectly through all upstream chains. This means even non-exporting firms receive a non-zero instrument through their sales to directly-exporting firms. In fact, non-directly-exporting firms sell on average nearly 10 percent of their output indirectly to foreign markets.

Q3: What is the estimated magnitude of the labor supply elasticity facing Belgian firms, and what does it imply for wage markdowns? A3: In the full main estimation sample (balanced panel), the IV estimate of the firm-specific labor supply elasticity is approximately 3.9, implying a wage markdown of about 21 percent relative to the marginal revenue product of labor. In the stayer subsample (incumbent workers only, holding workforce composition fixed), the estimated labor supply elasticity is approximately 2.3, implying a markdown of about 30 percent. The paper can reject perfect competition (infinite elasticity, zero markdown) at a significance level of 0.06 in the full sample and 0.001 in the stayer sample using the closure method.

Q4: What is the estimated labor cost elasticity with respect to demand-driven sales changes, and what does it imply about fixed labor costs? A4: The IV estimate of the labor cost elasticity with respect to sales is 0.528 (standard error 0.085), statistically significantly below one. If labor supply were perfectly elastic, this would directly imply a fixed labor cost share of approximately 47 percent. Incorporating the estimated upward-sloping labor supply curve through equation (15), the model implies that approximately 53 percent of total labor inputs are fixed overhead. For context, occupational data from Belgium’s 2014 Structure of Earnings Survey shows that clerical support workers and managers together account for 21 percent of total earnings, and adding professionals raises this to 51 percent — broadly consistent with the estimated fixed share.

Q5: What does the estimated elasticity of input purchases with respect to sales imply about fixed intermediate input costs? A5: The IV estimate of the elasticity of total input purchases with respect to sales is 1.050 (standard error 0.150), close to one. The implied fixed share of total intermediate inputs is therefore approximately zero. However, there is substantial heterogeneity by input type: purchases from the manufacturing sector (roughly half of all input purchases) have an elasticity close to one, whereas service-sector inputs (roughly 30 percent of total input purchases) have an implied fixed cost share of approximately 36 percent, with a size-weighted average cumulative response of 4.3 percent against a total cumulative sales increase of 6.7 percent.

Q6: How does the paper rule out alternative explanations for the less-than-proportional response of labor costs? A6: The paper considers three main alternatives. First, adjustment costs: even in the presence of labor adjustment costs, under a homothetic constant-returns production function a permanent shock should eventually produce a proportional labor response. The paper focuses on four-year cumulative responses where firm responses change little after the first couple of years, and shows identification of fixed costs holds even in models with quadratic or Calvo-style adjustment costs. Second, a non-homothetic CES production function without fixed costs: Appendix B.3 shows that such a specification predicts that if the labor cost elasticity is below one, the input purchase elasticity must be above one — at odds with the data, which shows the input purchase elasticity is close to one while the labor cost elasticity is well below one. Third, variable markups: a uniform markup change would reduce both elasticities proportionally, not create the large gap between labor cost and input purchase elasticities observed.

Q7: Why are firms’ domestic suppliers affected by foreign demand shocks, and how large are the pass-through effects? A7: Firms pass on foreign demand shocks to their domestic suppliers through buyer-supplier production network links. When a foreign demand shock increases a firm’s sales by 10 percent instantaneously, its domestic input purchases increase cumulatively by approximately 5.3–6.0 percent over four years. Total input purchases increase by 7.0–7.8 percent over the same period; the difference between total and domestic input purchases reflects service inputs (which have smaller responses) and the composition of imported versus domestic inputs.

Q8: What is the aggregate real wage effect of a 5 percent increase in foreign tariffs on Belgian exports in the baseline model? A8: In the baseline counterfactual representing the actual Belgian economy (with fixed overhead costs and labor supply elasticity ε = 3.9), a uniform 5 percent increase in foreign tariffs on all Belgian exports produces a 4.9 percent fall in the average real wage. The median firm reduces output by 3.8 percent, marginal costs by 4.8 percent, and wages by 7.9 percent. The fall in wages is driven by a general equilibrium mechanism: since the foreign price is exogenous and trade balance must hold, wages are the key adjusting margin.

Q9: How much does the modeling of fixed overhead costs versus imperfect labor market competition matter for the aggregate wage counterfactual? A9: Fixed overhead costs account for nearly all of the amplification relative to the standard model. With fixed costs but perfectly elastic labor supply, the real wage falls 4.8 percent — almost identical to the 4.9 percent in the baseline. Without fixed costs but with the estimated upward-sloping labor supply, the fall is only 3.0 percent. Without either, the fall is 2.8 percent. Thus, incorporating fixed overhead costs in labor raises the estimated wage decline by approximately 1.9 percentage points, while incorporating imperfect labor market competition adds only about 0.1 percentage points. The paper concludes that fixed overhead costs, not monopsony, are the essential feature for accurately predicting tariff impacts on wages.

Q10: What is the mechanism by which fixed overhead costs amplify the aggregate wage decline from a negative demand shock? A10: Fixed overhead costs reduce the share of labor in firms’ total variable costs. When labor constitutes a smaller fraction of variable costs, output prices are less sensitive to changes in wages. With aggregate labor supply fixed, the economy restores equilibrium after a negative demand shock by reducing prices through wage cuts. To achieve the same magnitude of price reduction when labor is a smaller fraction of variable costs, wages must fall by a larger amount — amplifying the aggregate wage impact. Fixed overhead costs in labor also make foreign inputs relatively more important in variable costs, as shown empirically in Appendix D.1.

Q11: Is the conclusion about the relative importance of fixed costs versus labor market imperfections robust to alternative specifications of the labor market? A11: Yes. The paper extends the model to a nested logit structure for worker preferences (following Lamadon et al. 2022), which allows Belgium to contain multiple labor markets (defined as industry-region nests), permits heterogeneous markdowns across markets, and is still identified from the data. Empirically, incorporating multiple labor markets and heterogeneous markdowns does not quantitatively alter the aggregate counterfactual predictions for the wage effects of foreign demand shocks.

Q12: Are heterogeneous responses to the foreign demand shock observed across exporters, importers, and domestic-only firms? A12: The paper finds no systematic differences in the elasticities of labor cost and input purchases between firms that trade internationally and those that do not. This implies that exporters and importers have higher absolute fixed costs (consistent with fixed export and import costs) but comparable fixed cost shares — since these firms tend to be larger and thus spread higher absolute fixed costs over larger output volumes.

Q13: Do the findings about fixed overhead costs extend beyond foreign demand shocks? A13: Yes. The paper shows in Appendix D.4 that a uniform 5 percent reduction in the productivity of all Belgian manufacturing firms generates qualitatively and quantitatively similar conclusions: fixed overhead costs amplify the predicted wage effects of domestic productivity shocks, while imperfect competition in the labor market matters to a lesser but still meaningful extent.

Key Concepts

Fixed Overhead Costs (Fixed Labor Costs / Fixed Intermediate Input Costs): In the paper’s model, each firm has firm-specific fixed overhead input requirements for labor (denoted ℓ̄_k^f) and intermediate inputs (denoted q̄_k^f) that must be satisfied regardless of the firm’s output level. These fixed requirements are separate from the variable inputs used in production. Fixed labor costs may reflect administration, worker management, facility maintenance, and other tasks that do not directly translate into output. Fixed intermediate input costs include waste management, accounting services, and electricity payments that occur irrespective of sales. The share of total labor inputs that is fixed is identified by how much less than proportionally labor costs respond to demand-driven changes in sales.

Monopsonistic Competition in the Labor Market: The paper models each firm as facing an upward-sloping firm-specific labor supply curve arising from workers’ heterogeneous idiosyncratic preferences over non-wage firm attributes (amenities). Because workers’ idiosyncratic tastes are private information, firms cannot price-discriminate and thus face an increasing marginal cost of labor. Each firm is infinitesimal within the aggregate labor market but has wage-setting power at the firm level. This gives rise to a constant-elasticity firm-level labor supply curve ℓ_k = A_k w_k^ε, where ε is the labor supply elasticity facing the firm.

Wage Markdown: The firm’s equilibrium wage is marked down relative to the marginal revenue product of labor by the factor ε/(1+ε), which is less than one when ε is finite. With a labor supply elasticity of 3.9, the implied markdown is approximately 21 percent; with a supply elasticity of 2.3 (stayer sample), the markdown is approximately 30 percent. Perfect competition corresponds to ε = ∞ and a markdown of zero.

Labor Cost Elasticity: The elasticity of a firm’s total labor cost with respect to a demand-driven change in the firm’s sales, as derived from the model’s comparative statics (equation 15). This elasticity depends on both the variable share of labor inputs (ℓ_k^v / ℓ_k) and the labor supply elasticity ε. It lies strictly between zero (all labor fixed) and one (all labor variable), and is declining in ε for a given variable share. The paper estimates this elasticity at 0.528 via IV, implying substantial fixed overhead in labor.

Total Foreign Demand Shock: The firm-level measure of foreign demand used as an instrument, defined as the weighted average of changes in world import demand (excluding Belgium) across country-product pairs, where the weights reflect both the firm’s own lagged direct export shares and its indirect exposure through the domestic production network (via the Leontief inverse matrix H̃). This measure captures both direct exporter exposure and indirect upstream exposure for non-exporting firms that supply to exporters.

Indirect Export Exposure: The share of a firm’s output that reaches foreign markets indirectly through sales to domestic buyers who subsequently export. Defined recursively: the total export share of firm k equals its direct export revenue share plus the sum over all domestic buyers of the product of k’s revenue share from that buyer and the buyer’s own total export share. Even non-direct-exporting firms sell on average approximately 10 percent of their output indirectly to foreign markets in the Belgian data.

Dekle-Eaton-Kortum Hat Algebra: A technique for solving general equilibrium counterfactuals in trade models by expressing all outcomes as proportional changes (“hats”) relative to the observed equilibrium, without needing to recover the underlying structural parameters. The paper uses this approach to compute counterfactual wages under alternative tariff scenarios, holding fixed the observed firm-level expenditure shares from the reference year (2012) while allowing parameters such as productivity and technology weights to vary across counterfactual economies to rationalize identical observed firm-level observables.

Worker Rents: In the monopsony model, inframarginal workers earn rents defined as the excess return over what would be required to make them indifferent between employers. These rents arise because firms cannot price-discriminate across workers with heterogeneous amenity valuations. The additional rents accruing to workers from a demand-driven increase in firm sales decompose into: (1) wage increases for incumbent workers multiplied by current employment, (2) rents for new hires (the excess of their wage bill over the amount required to induce them to switch to the expanding firm), and (3) a correction term related to the fraction of the labor cost increase borne by expanding employment rather than wages.

Heterogeneous innovations and growth under imperfect technology spillovers

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. Jo and Kim ask two related questions: (1) How do firms use different types of innovation when learning others’ technology takes time? (2) How does this process alter the aggregate implications of firm innovation, particularly in the context of increasing competition?

Model. The paper develops a discrete-time infinite-horizon endogenous growth model with multi-product firms pursuing two types of innovation — “own-innovation” (improving existing product quality) and “creative destruction” (entering new product markets by displacing incumbents) — subject to a novel friction called “imperfect technology spillovers.” The friction takes the specific form of lagged learning: creative destruction builds on the one-period-lagged technology of the target market’s incumbent, while only the incumbent can observe the current frontier technology level. This one-period lag creates a technology gap (Δ = q_t / q_{t−1}) between the incumbent’s frontier and the level available to rivals. Four possible technology gap values arise in equilibrium: Δ₁ = 1 (no gap), Δ₂ = λ (one successful own-innovation), Δ₃ = η (one successful creative destruction), and Δ₄ = η/λ. The step sizes satisfy λ² > η > λ, meaning a single creative destruction improves quality more than a single own-innovation, but two consecutive own-innovations dominate a single creative destruction.

Key Mechanisms. The learning friction generates two novel mechanisms. First, the “market-protection effect”: incumbents with a technology advantage (Δ > 1) intensify own-innovation to widen the gap and protect their product lines when competitive pressure rises. Formally, own-innovation probability is highest for Δ₂ products and declines monotonically (z₂ > z₃ > z₄ > z₁), and ∂z₂/∂x > ∂z₃/∂x > 0 while ∂z₁/∂x < 0, conditional on value coefficients. Second, the “technological barrier effect”: higher overall own-innovation and creative destruction intensity widens the average technology gap across products, reducing rivals’ conditional probability of successfully taking over a product market. This is distinct from the standard Schumpeterian effect (lower expected future profits) and from the escape-competition effect in step-by-step models (which apply only to neck-and-neck, single-product firms).

Data and Empirical Strategy. The empirical analysis combines the USPTO PatentsView database, the Longitudinal Business Database (LBD), the Longitudinal Firm Trade Transactions Database (LFTTD), the Census of Manufactures (CMF), Compustat, and NBER-CES data, covering the universe of U.S. patenting firms from 1976 to 2016, with main analyses from 1982 to 2007. Own-innovation is proxied by the self-citation ratio of patents (the ratio of self-citations to total backward citations); creative destruction by new products added and low-self-citation patents. Exogenous competitive pressure comes from China’s WTO accession in 2001, instrumented by the industry-level NTR tariff gap (the gap between non-NTR and NTR rates in 1999) following Pierce and Schott (2016).

Empirical Findings. Pre-shock (1982–1999): patents with lower self-citation ratios (closer to creative destruction) have significantly longer backward citation gaps (coefficient −2.29 to −2.59, p < 0.01 across specifications), confirming that learning others’ technology takes more time. Creative-destruction-type patents also have higher market value (Kogan et al. stock return measure) and scientific value (forward citations), with self-citation ratio negatively associated with both (e.g., coefficient on self-citation for market value: −0.289 without firm FE; −0.110 with firm FE, p < 0.01). Conditional on patenting, higher self-citation ratios are negatively associated with employment growth (coefficient −0.256, p < 0.05), number of industries added (−0.158, p < 0.05), and products added (−0.274, p < 0.01).

Post-shock (DID): foreign competition had no statistically significant effect on overall patent counts, but firms with above-average innovation intensity in industries with high NTR gaps significantly increased their self-citation ratio — indicating a shift toward own-innovation. The triple-interaction coefficient is 0.795 (p < 0.01) with baseline controls. For a firm with average lagged innovation intensity (0.18) in an industry with an average NTR gap (0.291), this corresponds to a 4.2 percentage point increase in the seven-year growth rate of the self-citation ratio, representing a 15.0% increase relative to the average growth rate of 28.2 percentage points. Consistent with the technological barrier effect, firm entry rates are lower in industries with higher TFPR-skewness-based technological barriers (coefficient −0.012 to −0.016, p < 0.05).

Quantitative Analysis. Calibrated to the U.S. manufacturing sector in 1992, the model matches six target moments including average number of products (2.3), products added (0.3), firm entry rate (7.6%), average productivity growth (1.9%), high-growth-firm employment growth (22.5%), and import penetration (15.3%). Creative destruction contributes approximately 1.88 times more to growth per unit than own-innovation (step size ratio 0.075/0.04). The aggregate R&D-to-sales ratio (untargeted) is 4.6% in the model vs. 4.1% in data.

A counterfactual increasing outside entrants by 83% (matching the rise in import penetration from 15.3% to 25.1% between 1992 and 2007) generates a 1.51% increase in aggregate creative destruction arrival rate x, but firm-level creative destruction probability falls 1.33% and startup creative destruction also falls 1.33%. The aggregate R&D-to-sales ratio falls 1.6% and creative destruction R&D intensity falls 1.2%. Average domestic productivity growth declines 11.0%, with growth from creative destruction falling 13.0% and growth from domestic startups falling 1.7%. The total mass of domestic firms falls 6.4%.

In economies with creative destruction costs 80 times higher than the U.S. baseline, the same competitive pressure shock raises rather than lowers total R&D (by 1.0%), but domestic growth still falls 9.7%, because the marginal decline in creative destruction impedes the growth contribution and firm entry even when aggregate innovation spending rises.

Layer 2 — Q&A

Q1: What is the key friction that distinguishes this model from the existing multi-product firm literature (e.g., Klette and Kortum 2004; Akcigit and Kerr 2018)?

A: The key friction is “imperfect technology spillovers,” modeled as lagged learning: creative destruction can only build on the one-period-lagged technology of the target product (q_{j,t−1}), while the product’s current owner observes the frontier technology (q_{j,t}). In models without this friction — such as Akcigit and Kerr (2018) — rivals can instantly learn and copy frontier technology, so firms have no technological advantage and cannot protect their markets. In the current model, own-innovation by the incumbent widens the gap between q_{j,t} and q_{j,t−1}, creating a barrier that a rival must overcome even after successful creative destruction. This makes own-innovation an endogenous function of the technology gap, a feature absent from existing multi-product firm frameworks.

Q2: Why does the model predict that own-innovation increases with the technology gap up to a point, then decreases?

A: From Corollary 1, the ordering z₂ > z₃ > z₄ > z₁ reflects competing forces. Products with gap Δ₂ = λ gain the most from additional own-innovation in terms of reducing the probability of losing the product line (equation 2), so own-innovation is highest there. Products with Δ₃ = η or Δ₄ = η/λ already have substantial technological advantages from prior creative destruction, so the marginal value of own-innovation in reducing market loss probability is lower. Products with Δ₁ = 1 have no advantage at all: if a rival succeeds in creative destruction, the incumbent loses the product regardless of own-innovation (equation 1), so z₁ is lowest. Beyond a certain gap level, the incumbent is sufficiently protected that additional own-innovation has diminishing returns in deterrence.

Q3: What is the market-protection effect formally, and for which products is it strongest?

A: The market-protection effect (Corollary 2) is the positive response of a firm’s own-innovation to an increase in the aggregate creative destruction arrival rate x, conditional on the value coefficients A₁ and A₂ being fixed. It is strongest for products with Δ₂ = λ (∂z₂/∂x is the largest and positive), positive but weaker for Δ₃ = η (∂z₃/∂x > 0), of ambiguous sign for Δ₄ = η/λ, and negative for Δ₁ = 1 (∂z₁/∂x < 0). The asymmetry reflects the asymmetric payoff to own-innovation across gap levels: for Δ₂ products, successful own-innovation can turn a losing situation into a winning one because it shifts the technology gap from Δ₁ to Δ₂ from the rival’s perspective, effectively defeating the rival’s creative destruction attempt. This mechanism provides a micro-foundation for why frontier firms (like Google or NVIDIA) keep innovating intensely despite their technological leads, a pattern the standard step-by-step model cannot explain.

Q4: What is the technological barrier effect and how does it differ from the Schumpeterian effect?

A: The technological barrier effect refers to the reduction in rivals’ incentive for creative destruction caused by an increase in the average technology gap across product lines. When incumbents do more own-innovation or when outside firms do more creative destruction, the distribution of technology gaps shifts rightward (density at Δ₁ falls; density at Δ₂, Δ₃, Δ₄ rises). This raises the average technology barrier rivals must overcome to successfully take over a product market, reducing the conditional takeover probability x^{takeover} and the expected value of creative destruction B. In the U.S. counterfactual, the technological barrier effect accounts for 17.0% of the total change in the aggregate creative destruction rate x and 15.0% of the change in startup creative destruction x_e. In contrast, the Schumpeterian effect refers to the reduction in expected future profits from owning a product due to increased displacement risk (through the value coefficient A₂), a mechanism present in standard quality-ladder models. Both operate simultaneously but the technological barrier effect is a novel feature of this framework.

Q5: How is own-innovation vs. creative destruction measured empirically, and what validates this measure?

A: The self-citation ratio (the share of a patent’s backward citations that cite the same assignee’s earlier patents) is used as the primary measure: a higher ratio indicates greater reliance on the firm’s own prior knowledge, hence a higher probability that the innovation improves an existing product line (own-innovation). This is validated empirically in three ways. First, patents with lower self-citation ratios have significantly larger backward citation gaps (coefficient −2.29 to −2.59 across fixed-effect specifications on 728,721 observations), consistent with creative destruction requiring more time to learn others’ technology. Second, lower self-citation patents have higher market value and scientific value (forward citations), consistent with η > λ (creative destruction contributes more per event to quality). Third, firm-level regressions show that lower self-citation ratios are associated with higher employment growth, more products added, and more industries entered, consistent with creative destruction contributing more to firm expansion.

Q6: How does the DID identification strategy work, and what are the main results?

A: The identification exploits the removal of trade policy uncertainty (TPU) after China’s WTO accession in 2001. The treatment variable is the industry-level NTR gap (the gap between non-NTR and NTR tariff rates in 1999): industries with larger gaps experienced a larger reduction in uncertainty and thus a greater increase in Chinese import competition. The DID compares patenting firms across periods (1992–1999 vs. 2000–2007) and across high- vs. low-NTR-gap industries, with a triple interaction for firm-level innovation intensity (lagged five-year average patents per employee, normalized within two-digit NAICS). The main finding (Table 4): the NTR gap × Post interaction has no significant effect on overall patent counts (coefficient 0.238 without controls, standard error 0.237), but the triple interaction (NTR gap × Post × innovation intensity) has a positive and significant effect on the growth rate of the self-citation ratio (0.732 without controls, p < 0.05; 0.795 with baseline controls, p < 0.01). This implies that innovation-intensive firms in high-competition industries shifted their composition toward own-innovation, while overall patenting was unchanged — consistent with an offsetting rise in own-innovation and fall in creative destruction.

Q7: What are the aggregate growth effects of increasing competitive pressure in the calibrated model?

A: Using an 83% increase in outside entrants (matching the 1992–2007 rise in import penetration from 15.3% to 25.1%), average domestic productivity growth falls 11.0%. Decomposing: growth from domestic own-innovation falls 11.4%, growth from domestic creative destruction falls 13.0%, and growth from domestic startups falls 1.7% (Table 9). The aggregate R&D-to-sales ratio falls 1.6% and the creative destruction R&D intensity falls 1.2%, indicating that the decline in creative destruction R&D outweighs the rise in own-innovation R&D. The total mass of domestic firms falls 6.4% and the average number of products per firm falls 5.5%.

Q8: How do results differ in economies with high creative destruction costs vs. the U.S.?

A: When creative destruction costs (χ̃) are set 80 times higher than the U.S. baseline, the initial equilibrium has much lower creative destruction: R&D-to-sales ratio is 1.39% (vs. 4.58% in U.S.), creative destruction R&D intensity is 8.6% (vs. 63.9%), average number of products is 1.0 (vs. 2.3), and average domestic productivity growth is 1.4% (vs. 1.9%). Under the same competition shock, total R&D actually rises by 1.0% in this high-CD-cost economy (because own-innovation increases more than creative destruction falls, given the already low baseline of creative destruction), in contrast to the −1.6% in the U.S. However, domestic growth still falls 9.7% even in this economy, driven by reductions in creative destruction by incumbents and startups combined with a decline in the mass of domestic incumbents. This result holds even with a fixed firm mass (Table E5), confirming the mechanism is not solely due to entry/exit dynamics.

Q9: What is the technological barrier effect’s quantitative contribution to the decline in creative destruction?

A: In the U.S. counterfactual (Table 8 and associated decomposition), 17.0% of the total change in the aggregate creative destruction arrival rate x and 15.0% of the total change in startup creative destruction x_e are attributable specifically to the technological barrier effect — that is, to the shift in the technology gap distribution µ(Δℓ) holding all else equal. The conditional takeover probability x^{takeover} declines from 73.2% to 73.0%. The density at Δ₁ (the easiest gap to overcome) falls 0.4%, while densities at Δ₃ and Δ₄ rise 1.1% and 1.4% respectively, driven by increased creative destruction by outside firms and intensified own-innovation by incumbents.

Q10: What are the policy implications the paper draws from its framework?

A: The paper argues that policies evaluating innovation should account for composition, not just aggregate R&D levels or patent counts. Increased overall innovation driven by defensive own-innovation contributes less to economic growth than creative destruction and restricts firm entry — so it is less beneficial than it appears. In low-creativity economies (e.g., European economies with high regulatory barriers to creative destruction), increased foreign competition may raise aggregate R&D while still lowering domestic growth, misleading policymakers who track only total innovation spending. The model also suggests that the mixed empirical findings in the competition-innovation literature (Aghion et al. 2005; Bloom et al. 2016; Autor et al. 2020) can be reconciled by accounting for compositional shifts: the net effect of competition on total innovation is ambiguous because it raises own-innovation for technologically advantaged firms while reducing creative destruction for all firms.

Key Concepts

Imperfect Technology Spillovers: The novel friction introduced in this paper, modeled as lagged learning: firms attempting creative destruction can only access the one-period-lagged technology of the target product market (q_{j,t−1}), while the incumbent product owner observes and can improve from the current frontier (q_{j,t}). This asymmetry creates a persistent technological advantage for incumbents and enables strategic defensive innovation.

Own-Innovation: R&D investment by a firm to improve the quality of its existing product lines. Successful own-innovation raises product quality by a step size λ > 1. Own-innovation does not require learning others’ technology and, in the model, constitutes the incumbents’ defensive margin against creative destruction. At the aggregate level, it contributes more to total growth than creative destruction because it succeeds more frequently, but per successful event it contributes less (λ < η).

Creative Destruction: R&D investment enabling a firm to enter a new product market by displacing the incumbent. Successful creative destruction improves the lagged quality of the target product by a step size η > λ, where λ² > η > λ. It requires learning the incumbent’s one-period-lagged technology, takes longer to develop (evidenced empirically by longer backward citation gaps), and contributes more to firm growth and product expansion per event than own-innovation.

Technology Gap (Δ): The ratio of a product’s current-period technology to its previous-period technology (Δ_{j,t} = q_{j,t}/q_{j,t−1}). This gap summarizes the technological advantage the incumbent holds in a product market under imperfect spillovers. Four values are possible in equilibrium: Δ₁ = 1, Δ₂ = λ, Δ₃ = η, Δ₄ = η/λ. The gap determines both the incumbent’s own-innovation incentive and the rival’s probability of successfully completing a product takeover conditional on creative destruction.

Market-Protection Effect: The mechanism by which incumbents with a technological advantage (Δ > 1) increase own-innovation in response to heightened competitive pressure (an increase in the aggregate creative destruction arrival rate x). This effect is maximized for products with Δ₂ = λ and positive but diminishing for Δ₃. It is absent for Δ₁ = 1 products (where own-innovation cannot prevent displacement) and is formally distinct from the escape-competition effect in step-by-step innovation models, which applies only to neck-and-neck single-product firms.

Technological Barrier Effect: The reduction in rivals’ incentive for creative destruction caused by an increase in the average technology gap across the economy’s product lines. When incumbents intensify own-innovation and/or when outside creative destruction increases, the distribution of technology gaps shifts toward higher Δ values, reducing the conditional probability that a rival successfully takes over any given product market. This feedback mechanism endogenously suppresses creative destruction and firm entry beyond what the Schumpeterian effect alone would predict.

Self-Citation Ratio: The share of a patent’s backward citations that cite patents previously owned by the same firm. Used in the paper as a continuous proxy for the likelihood that a patent represents own-innovation vs. creative destruction: a ratio of 1 (100% self-citations) implies 100% probability of own-innovation; a ratio of 0 implies 100% probability of creative destruction. This measure follows Akcigit and Kerr (2018) and is validated in the paper against learning time, quality, and firm growth outcomes.

NTR Gap (Trade Policy Uncertainty Shock): The industry-level difference between non-NTR (column 2) and NTR (column 1) U.S. tariff rates in 1999, used as an instrument for the exogenous increase in Chinese competitive pressure following China’s WTO accession and the U.S. granting of Permanent Normal Trade Relations (PNTR) in 2002. Industries with larger NTR gaps experienced a greater reduction in trade policy uncertainty and thus a larger increase in competitive pressure from foreign firms.

The Illiquidity of Water Markets

Mon, 01 Jan 0001 00:00:00 +0000

Donna and Espín-Sánchez investigate whether a market (sequential English auction) or a non-market institution (fixed quota) more efficiently allocates an intermediate good — irrigation water — when some buyers are liquidity constrained. The setting is Mula, a city in southeastern Spain, where farmers used an unregulated water auction continuously from 1244 until August 1, 1966, when the institution was replaced by a fixed quota system. This 700-year natural experiment, combined with the fact that water demand for a given crop is pinned down by the crop’s production function rather than by farmer wealth, allows the authors to separately identify liquidity constraints from unobserved heterogeneity in productivity.

The empirical context has four features the authors exploit. First, the pre-1966 auction was entirely unregulated, so price differences directly reflect valuations without the confounds of regulatory changes. Second, water is an intermediate good for apricot production; conditional on plot area, tree count, and crop type, demand is determined by the apricot tree’s biological water requirements — not by the farmer’s wealth — so wealthy and poor farmers growing the same bulida apricot variety share the same underlying demand up to an idiosyncratic productivity shock. Third, farmers are classified as wealthy if they held positive urban real estate (non-agricultural wealth) in 1955 tax records; wealthy farmers’ average annual urban rental income (5,702 pesetas) far exceeded their average annual water expenditure (500 pesetas, rising to 1,619 in the highest-expenditure year, 1963), supporting the assumption that wealthy farmers were never liquidity constrained. Fourth, the 1966 institutional shift to quotas — under which each farmer received a fixed water allotment (tanda) every three weeks proportional to plot size, paying only a small annual maintenance fee after the critical season — provides the counterfactual.

The authors build a structural dynamic demand model with three key features: storability (irrigation raises soil moisture, creating intertemporal substitution between periods because water evaporates partially), liquidity constraints (poor farmers cannot always afford water during the critical season when prices peak), and weather seasonality (the critical season, corresponding to apricot fruit growth stages II–III and the Early Post-Harvest period, spans roughly weeks 18–32 and is when trees most need water). Farmers are forward-looking and form expectations about future prices and rainfall. The model’s production function, drawn from the agricultural engineering literature (Torrecillas et al., 2000; Allen et al., 2006), transforms soil moisture into apricot output via a transformation rate parameter gamma, a hydric stress coefficient, and a seasonal dummy.

Demand parameters are estimated using a two-step conditional choice probability (CCP) estimator (Hotz et al., 1994) on wealthy farmers only, then projected onto poor farmers’ welfare calculations. The sample consists of 24 single-crop apricot farmers observed in weekly auction records from January 1955 to July 1966, embedded in a market with over 500 total participants.

The main finding is that the institutional change from auction to quota increased total efficiency. Welfare increased by 23.4 real pesetas per farmer per tree, a 6 percent increase in total apricot production relative to the market. This gain arises because: (1) farmers were relatively homogeneous in productivity (small idiosyncratic shocks), so the primary source of misallocation was not productivity heterogeneity but wealth heterogeneity; (2) liquidity constraints prevented poor farmers from purchasing water during the critical season when their valuation was high, causing them instead to buy earlier (at lower prices but with partial evaporation loss) or later (when their trees had already experienced hydric stress); and (3) the apricot production function is concave in water, so uniform quota allocation is more efficient than market allocation when farmers are approximately homogeneous. The paper provides the first empirical demonstration that liquidity constraints can reverse the standard efficiency ranking of markets over quotas.

Q: What is the core research question? A: The paper asks whether a free market (water auction) or a non-market institution (fixed quota) more efficiently allocates an intermediate good when some buyers are liquidity constrained. The theoretical ranking is ambiguous when agents are heterogeneous in both productivity and wealth, making this an empirical question. The authors find that quotas dominated the auction in the specific Mula setting.

Q: What was the historical water market in Mula and when did it end? A: From 1244 to 1966 — over 700 years — Mula farmers used a sequential ascending-price (English) auction to allocate river water. The auctioneer sold water in discrete units called cuartas (each representing 3 hours of canal flow, or approximately 432,000 liters), holding 40 units per weekly Friday session. Farmers paid in cash on auction day. On August 1, 1966, the farmers’ union (Sindicato de Regantes) replaced the auction with a fixed quota system, having secured a credit line to purchase water property rights share by share.

Q: How did the quota system work, and how did it eliminate liquidity constraints? A: Under the quota, each plot of land received a fixed water allotment (tanda) every three weeks, proportional to plot size. Farmers paid only a small annual maintenance fee to the Sindicato at year-end, after the critical season harvest. Because payment occurred after farmers collected harvest revenue, no farmer was liquidity constrained under the quota. The fee was substantially lower than the per-unit average price under the market.

Q: How do the authors identify liquidity constraints separately from unobserved heterogeneity in productivity? A: The key insight is that water is an intermediate good whose demand is determined by the apricot tree’s biological production function, not by farmer wealth. Two farmers growing the same bulida apricot variety with the same number of trees should have the same water demand up to an idiosyncratic shock. The authors use wealthy farmers (those with positive urban real estate in 1955 tax records) to estimate preferences, under the assumption that wealthy farmers are never liquidity constrained. They then verify that outside the critical season, wealthy and poor farmers purchase similar amounts of water; the purchasing divergence appears only during the high-price critical season, consistent with a cash constraint rather than a preference difference.

Q: What empirical evidence shows poor farmers were liquidity constrained rather than simply less interested in water? A: Poor farmers display a bimodal purchasing pattern inconsistent with the apricot tree’s biological water needs: they buy water before the critical season (when prices are low) in anticipation of not being able to afford it during the critical season, and again after the critical season (when prices fall) to prevent their trees from withering from dehydration. Wealthy farmers, by contrast, delay purchases strategically to the critical season when trees most need water (weeks 18–32). Regression analysis confirms that wealthy farmers purchase significantly more water per tree during the critical season than poor farmers growing identical bulida apricots, while the difference outside the critical season is not statistically significant.

Q: How were wealthy farmers defined and why does their wealth validate the non-constrained assumption? A: A farmer is defined as wealthy if the value of their urban real estate (from 1955 urban tax records) is positive, and as poor if it is zero. Urban real estate constitutes non-agricultural wealth uncorrelated with the apricot production function. Wealthy farmers’ average annual urban rental income was 5,702 pesetas, while their average annual water expenditure was only 500 pesetas (rising to 1,619 pesetas in 1963, the highest-expenditure sample year). This large gap supports the assumption that wealthy farmers could always afford water purchases.

Q: What is the model’s treatment of soil moisture dynamics and why does it matter? A: Soil moisture (M_it) evolves according to an agricultural engineering formula: it increases with rainfall and irrigation purchases (each unit adding 432,000 liters divided by plot area) and decreases via evapotranspiration (ET), subject to a full-capacity ceiling (FC) and a permanent wilting point (PW) lower bound. This storage structure creates intertemporal substitution — water purchased early partially substitutes for future purchases, but at a cost (evaporative loss). The dynamics mean poor farmers who pre-buy water before the critical season lose some of that investment to evaporation, generating a real efficiency loss relative to the quota that delivers water closer to when it is biologically needed.

Q: What are the two sources of potential inefficiency the authors identify? A: The first is inefficiency due to heterogeneity: if farmers differ in ex-post productivity (captured by idiosyncratic shocks epsilon_it), allocating water to a less productive farmer at a given moment is wasteful. Markets correct this inefficiency (they direct water to highest-valuation buyers) while quotas do not. The second is inefficiency due to decreasing marginal returns (DMR): because the production function is concave in water, giving water to a farmer with already-high soil moisture is less productive than giving it to a farmer with low moisture. Quotas naturally avoid DMR inefficiency by allocating uniformly; markets with liquidity constraints exacerbate DMR inefficiency by directing scarce critical-season water to wealthy farmers who may have already accumulated moisture from prior purchases.

Q: What is the main quantitative result of the welfare analysis? A: Switching from the market auction to the fixed quota system increased welfare by 23.4 real pesetas per farmer per tree, representing a 6 percent increase in total apricot production relative to the market counterfactual. This is computed as the difference in yearly mean welfare per tree per farmer (net of irrigation costs, excluding water expenditures which are transfers) between the quota and market allocations using the estimated structural model.

Q: Under what conditions is a quota more efficient than a market with liquidity constraints? A: Quotas dominate markets when three conditions hold simultaneously: (1) farmers are relatively homogeneous in productivity (so the market’s advantage of directing water to high-valuation buyers is small), (2) liquidity constraints are significant (so the market misallocates water away from constrained high-valuation farmers), and (3) the production function is concave in water (so uniform allocation is efficient when farmers are homogeneous). The authors find all three conditions hold in Mula. Conversely, markets dominate quotas when heterogeneity in productivity is large relative to heterogeneity in wealth.

Q: How is the transformation rate parameter gamma estimated and interpreted? A: The transformation rate gamma measures how soil moisture above the permanent wilting point converts into apricot output (in pesetas) during the critical season, via the production function h() = gamma * (M_it - PW) * KS(M_it) * Z(w_t). It is identified from variation in purchasing patterns across seasons and variation in moisture across farmers within the same season. The preferred specification (column 3 of Table 3) yields gamma_L = 0.05. With average moisture per tree (accounting for the hydric stress coefficient) of 873.93 during the critical season, a farmer earns on average 29.09 pesetas per tree per week during the critical season, or 407.25 pesetas per tree per year.

Q: How does ignoring liquidity constraints bias demand estimates? A: If one estimates demand using the full sample (poor and wealthy farmers pooled), a decrease in demand during the critical season when prices rise conflates two effects: (1) the standard price effect (fewer farmers have valuations above the price) and (2) the liquidity constraint effect (some farmers with valuations above the price still cannot buy because they lack cash). Attributing the second effect to price sensitivity overstates the demand elasticity, biasing its absolute value upward.

Q: What robustness checks do the authors provide against unobserved heterogeneity? A: The authors provide four pieces of evidence that wealthy and poor farmers do not have systematically different underlying preferences: (1) wealthy and poor farmers are not geographically sorted into different locations (both groups appear in subareas 1, 2, 4, and 7); (2) wealthy and poor farmers grow the same bulida apricot variety; (3) outside the critical season, wealthy and poor farmers purchase statistically similar amounts of water; and (4) the purchasing divergence is significant only during the critical season when prices are high, precisely the pattern predicted by the liquidity constraint mechanism.

Q: What are the policy implications for water allocation in developing countries? A: The paper implies that before introducing water markets in regions where farmers may be liquidity constrained, policymakers should assess the magnitude of those constraints. If liquidity constraints are significant and farmers are relatively homogeneous in productivity, a quota system or a market supplemented with credit provision may deliver higher efficiency than a pure market. The standard presumption that markets outperform quotas can reverse when poor farmers cannot access credit to purchase water at the times they most need it.

Q: How does this paper relate to Che et al. (2013)? A: Che, Gale, and Kim (2013) assume agents consume at most one unit with linear utility and find that markets always dominate quotas, though some non-market mechanisms with resale outperform markets. Donna and Espín-Sánchez extend this framework by allowing multiple discrete units, a concave utility function, and intertemporal dynamics. Under these extensions, the efficiency ranking between markets and quotas is theoretically indeterminate, and the authors show empirically that quotas can dominate markets. Both papers agree that non-market mechanisms with resale outperform both markets and simple quotas.

Liquidity constraint (paper’s sense): A farmer is liquidity constrained when they lack sufficient cash to purchase water at the prevailing auction price, even if their valuation (marginal productivity of water) exceeds that price. In Mula, poor farmers without urban real estate income faced this constraint during the critical season when prices peaked, because they had already spent their harvest proceeds from the prior year and lacked access to credit markets.

Soil moisture (M_it): The state variable measuring water accumulated in a farmer’s plot, computed using the agricultural engineering evapotranspiration formula. Moisture increases with rainfall and irrigation purchases (each auction unit contributing 432,000 liters divided by plot area) and decreases via evapotranspiration. It is bounded below by the permanent wilting point (PW) — below which trees die — and above by field capacity (FC). Moisture creates intertemporal substitution in demand.

Critical season: The period corresponding to apricot fruit growth stages II and III and the Early Post-Harvest (EPH) period, spanning approximately weeks 18–32 (early May to early August). This is when the bulida apricot tree transforms water into fruit at the most rapid rate, when water demand peaks biologically, and when auction prices rise to their highest levels. It is the season during which liquidity constraints are binding.

Transformation rate (gamma): The parameter in the apricot production function that measures the rate at which excess soil moisture (above the permanent wilting point) converts into apricot output (measured in real pesetas) during the critical season. Estimated at gamma_L = 0.05 in the preferred specification (column 3). It is identified from cross-seasonal variation in purchasing patterns and cross-farmer variation in moisture levels.

Inefficiency due to decreasing marginal returns (DMR): One of two sources of allocation inefficiency identified in the paper. It arises when a farmer with already-high soil moisture receives water, yielding less additional output than if that water had gone to a farmer with lower moisture, given the concavity of the production function. Quotas avoid this inefficiency by allocating uniformly; markets with liquidity constraints exacerbate it by directing critical-season water to wealthy farmers who may have accumulated moisture from earlier purchases.

Cuarta (quarter): The unit of water sold at Mula auctions, representing the right to use water flowing through the main channel for three hours. At approximately 40 liters per second of flow, each cuarta carried approximately 432,000 liters of water. Water rights and land rights were held independently; farmers who participated in auctions owned only land, while waterlords separately owned canal usage rights.

Conditional choice probability (CCP) estimator: The two-step estimation procedure used to recover demand parameters from wealthy farmers’ purchasing choices. In Step 1, transition probability matrices for observable state variables (moisture, week, price, rainfall) are computed and CCP is estimated via multinomial logit. In Step 2, the value function is forward-simulated using these transition matrices and parameters are estimated by GMM, following Hotz et al. (1994).

Who's Afraid of the Minimum Wage? Measuring the Impacts on Independent Businesses Using Matched U.S. Tax Returns

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question

This paper asks how independent (pass-through) businesses in the United States accommodate minimum wage increases — specifically whether they reduce employment, compress profits, pass costs through to customers, or exit — and what happens to the low-earning workers and business owners affected by these adjustments.

Data and Methodology

The authors construct a novel linked firm-worker-owner panel dataset from the universe of U.S. tax returns, covering approximately 235,000 pass-through firms (S-corporations, partnerships, and LLCs) per year in highly exposed industries over 2010–2019. “Highly exposed” industries are defined as those where at least 15% of workers earned below the full-time equivalent of the federal minimum wage ($15,080 per year) in 2013. The dataset links annual business income tax returns to the individual income tax returns and W-2 information reports of all workers and owners.

The causal identification strategy exploits the six state minimum wage increases that took effect in 2014 (California, Connecticut, Delaware, Michigan, Minnesota, and New Jersey) relative to 24 states that did not change their wage floors at any point from 2012–2018. The empirical workhorse is a panel difference-in-differences event study (Equation 1), augmented by DFL re-weighting (DiNardo et al., 1996) to improve comparability of treatment and control firms on observables. The analysis covers cumulative effects through 2018, by which point the average minimum wage across treatment states had risen 30.6%.

Main Findings

Employment: The average exposed independent firm does not meaningfully reduce employment. The authors estimate an own-wage elasticity of -0.209 (s.e. = 0.0112). Employment adjustments manifest as moderately lower hiring rather than layoffs of existing workers. Reduced hiring is wholly concentrated among teenagers and very part-time jobs paying less than $3,900 annually (with 67% earning less than $1,000 per year).
Worker earnings: Despite the hiring reduction, low-earning workers employed at exposed independent firms experience average earnings gains of approximately $2,000 per year by 2018, relative to comparable workers in untreated states. Young individuals aged 20–26 without a 2013 job earn roughly $4,000 more per year by 2018; teenagers without a 2013 job gain approximately $1,000 per year. Workers in these groups are no less likely — and in some cases slightly more likely — to be employed five years after the minimum wage increase.
Wage bills: Average wage bills among surviving treated firms rose 7.03% (s.e. = 0.0153) by 2018. Earnings gains are concentrated among workers earning $15,600–$35,000 annually, with no evidence of reduced earnings for higher-paid workers. The 7% average wage bill increase amounts to only 1.4% of 2013 firm revenues, easing pass-through.
Revenue and profits: Revenues of surviving treated firms grew approximately 2.1% more than control firms by 2018. On average, this revenue increase fully offsets the higher wage bill, yielding a small net profit increase of roughly $3,360 (s.e. = $1,123) per owner by 2018, or about 2.7% of mean 2013 owner income.
Firm exit: On average across all highly exposed industries, minimum wages increased the five-year exit probability by 0.9 percentage points (s.e. = 0.0029), relative to a baseline raw exit rate of approximately 29%. Exit effects are driven entirely by restaurants: by 2018, restaurants in treated states were 1.85 percentage points (s.e. = 0.0039) more likely to have exited, while the exit response for non-restaurant exposed firms is a precisely estimated zero.
Heterogeneity by productivity within restaurants: Exit is concentrated entirely in the bottom productivity quartile (coefficient = 0.0254, s.e. = 0.0079), with no significant effect in the upper three quartiles. Profits among surviving small restaurants rise by $5,941 (s.e. = $1,546) by 2018 relative to 2013. Among small restaurants, the profit gains are larger for firms in the higher productivity quartiles (Q3: +$7,915; Q4: +$9,161). Surviving restaurants also increase non-labor input spending by 2.53% (s.e. = 0.0101), consistent with expanded output following competitor exits.
Entrant characteristics: Post-reform restaurant entrants in treatment states have higher wage bills (13.8% higher in logs), higher revenues (4.0% higher), higher value-added (8.4% higher), and higher productivity (net income/revenue ratio 2.24 percentage points higher) than entrants in control states, indicating the minimum wage raises the productivity floor for new entrants.
Owner outcomes after exit: Owners of small restaurants forced out by the minimum wage are significantly less likely to own an independent business five years later, but earn no less on average in wages plus business income. Policy-induced exiters are significantly less likely to report negative incomes, suggesting substitution away from risky or marginally profitable business ownership.

Theoretical Framework

The authors present a Cournot competition model with heterogeneous firm productivity and fixed production costs. A minimum wage cost shock raises marginal costs, narrowing margins for all firms. Firms whose cost increases exceed the market price increase cannot cover fixed costs and exit. Remaining firms gain higher markups and larger market shares as demand is reallocated from exiting firms. Selection on ex-ante productivity (the least productive firms exit) limits the distortion to market quantity and amplifies profit gains among productive survivors. The model predicts profit increases only in markets with firm exit, which matches the data: profits rise among restaurants (where exit occurs) but not among retailers (where exit is a precisely estimated zero).

Scope Conditions

Findings pertain to the short-to-medium run (up to five years post-legislation) of phased-in minimum wage increases averaging 30.6% in six U.S. states. The sample covers pass-through (independent) businesses in highly exposed industries. Longer-run effects may differ if entrants adopt production technologies that rely less on low-wage labor or incumbents reconfigure inputs. Border-county retailers appear to be less able to pass through costs than interior firms, suggesting product market competition is a key moderating factor.

Layer 2 — Q&A

Q1: Why do the authors focus on pass-through businesses rather than publicly traded corporations?

Pass-throughs (S-corporations, partnerships, and LLCs) comprise 78% of non-sole-proprietorship businesses and 79% of firms with fewer than 20 employees. They represent the majority organizational form for independent businesses in virtually all two-digit NAICS industry groups except utilities and enterprise management. Because minimum wage concerns are disproportionately raised on behalf of small independent businesses, and because most minimum wage workers in restaurants are employed at pass-throughs, studying pass-throughs directly addresses the policy debate. Additionally, pass-through tax returns link business income directly to the individual tax returns of each owner, enabling the authors to separately identify employee versus owner responses.

Q2: How do the authors define “highly exposed” industries and why does this matter for identification?

Highly exposed industries are defined as four-digit NAICS industries where at least 15% of workers earned below the full-time federal minimum wage equivalent ($15,080 per year) in 2013, using tax data to construct a proxy for minimum wage workers. The analysis focuses on these industries because minimum wage workers are extremely concentrated — the vast majority are in Leisure/Hospitality and Retail. Restricting to highly exposed industries allows the authors to estimate average effects within affected markets and conduct heterogeneity analysis across firm characteristics within those markets, including comparing firms with different baseline shares of low-earning workers that nonetheless all face the market-level cost shock.

Q3: How do the employment effects decompose into hiring versus retention?

The average firm subject to a higher wage floor does not lay off existing workers (the retention line is flat in event study estimates). By 2018, firms in treated states hire roughly one fewer worker on average than similar firms in control states, entirely through reduced hiring. This reduced hiring is wholly concentrated among teenagers in very part-time jobs: the missing hires consist entirely of workers who would have earned less than $3,900 annually, with 67% earning less than $1,000 per year. Simultaneously, workers already employed at exposed firms are 2 to 4 percentage points more likely to remain with their 2013 employer by 2016, with prime-age low-earning workers exhibiting the largest retention increases.

Q4: What happens to low-earning workers and young people in individual-level panels?

Low-earners (those earning below $25,000 in each year from 2012–2014) at exposed independent firms experience average earnings gains of approximately $2,000 per year by 2018 relative to similar workers in untreated states, including teenage low-earners. Young individuals aged 20–26 with no job in 2013 experience a relative earnings increase of approximately $4,000 per year by 2018; teenagers without jobs in 2013 gain approximately $1,000 per year. These workers are no less likely — and often slightly more likely — to be employed relative to their counterparts in control states, so the earnings gains are not offset by employment losses at the individual level.

Q5: What is the magnitude of the cost shock for firms and how does it compare to revenues?

By 2018, the average wage bill among surviving firms in treated states was 7.03% (s.e. = 0.0153) higher than comparable firms in control states. This is consistent with a back-of-envelope calculation: low-earning workers account for about 21% of wage bills at these firms, and states raised minimum wages by 30.6% on average (0.21 × 0.306 = 0.064). However, the 7% wage bill increase amounts to only approximately 1.4% of 2013 firm revenues, making cost pass-through relatively modest. Higher minimum wages have no discernible impact on pension contributions but slightly reduce deductions for other benefits including health insurance.

Q6: How do surviving firms finance the increased wage bill, and what happens to profits?

Surviving firms finance the wage increase primarily through higher revenues. By 2018, revenues of firms in treated states grew approximately 2.1% more than revenues of firms in control states. On average, this revenue increase outpaces the higher wage bill, resulting in a net profit increase of approximately $3,360 (s.e. = $1,123) per owner by 2018, representing about 2.7% of mean 2013 owner income. There is no evidence of redistribution from middle- or high-income workers within firms; wage bill increases are concentrated among workers earning $15,600–$35,000 annually, consistent with minimum wage spillovers to workers slightly above the statutory floor.

Q7: Why do restaurants experience exit effects but retailers do not?

The asymmetry stems from the intensity of low-wage labor in production. While low-earning workers account for a similar share of labor costs at restaurants (41.8%) and retailers (38.5%), labor costs overall are more than twice as large at restaurants relative to retailers. Wage bills account for 39% of variable costs and 27% of revenues at restaurants, but only 16% of variable costs and 13% of revenues at retailers. As a result, raising the minimum wage raises variable costs by 5.76% at restaurants. Non-restaurant exposed firms are able to fully pass through their smaller cost shock, yielding flat profits and neither employment nor exit impacts.

Q8: Why is firm exit concentrated in the lowest productivity quartile of restaurants rather than among the most exposed firms?

The Cournot framework predicts exits among firms with the lowest ex-ante productivity (highest marginal costs), the largest cost shock (highest share of low-wage labor per unit of output), or a combination. Empirically, productivity is the primary determinant: restaurants across all productivity quartiles use similar shares of low-earning workers (40–44% of wage bills for Q1 through Q4). Exit rises significantly only among restaurants in the bottom productivity quartile (coefficient = 0.0254, s.e. = 0.0079), with no significant effects in Q2–Q4. Among the lowest-productivity restaurants, those most dependent on low-earning labor face the largest exit rates.

Q9: How do the model’s predictions about profit heterogeneity match the data?

The Cournot model predicts profits should rise only in markets with firm exit (via increased margins and market share reallocation to survivors). This is exactly what the data show. Among restaurants, where exit is concentrated in the bottom productivity quartile, profits among surviving small restaurants rise by $5,941 (s.e. = $1,546) by 2018. Among small restaurants specifically, profit gains increase with productivity: Q3 restaurants gain $7,915 (s.e. = $3,326) and Q4 restaurants gain $9,161 (s.e. = $2,127), while Q1 and Q2 gains are statistically indistinguishable from zero. In non-restaurant exposed industries where the exit effect is a precise zero, profits are also flat — exactly as the model predicts.

Q10: What happens to the characteristics of new restaurant entrants after the minimum wage increase?

Post-reform restaurant entrants in treatment states are systematically more productive than entrants in control states. They have wage bills 13.8% higher (in logs), revenues 4.0% higher, value-added 8.4% higher, and productivity ratios (net income/revenue) 2.24 percentage points higher than new entrants in control markets. This implies the minimum wage raises the minimum viable productivity threshold for entrant restaurants, consistent with Sorkin (2015)’s insight that minimum wages shape the capital and technology choices of entering firms. The restaurant industry thus becomes more productive on average through both the exit of the least productive incumbents and the entry of more productive new firms.

Q11: How do worker transition patterns reflect the reallocation of output to surviving firms?

Workers at large independent businesses (top revenue quartile) are 3.52 percentage points more likely to remain with their 2013 employer in 2018 and 2.36 percentage points less likely to switch to another large firm. The large firms that retain more of their existing workforce also reduce their hiring of very part-time teenagers the most — in the top revenue quartile, firms shed roughly 4.5 employment relationships on average, comprising higher retention of 4.15 existing workers offset by reduced hiring of 8.67 very part-time teenage workers. Workers originally at smaller exposed firms are more likely to be found working at larger firms five years out, consistent with demand reallocation from exiting and shrinking small firms toward larger, more productive survivors.

Q12: What happens to owners of restaurants that exit due to the minimum wage?

Policy-induced exiters of small restaurants are significantly less likely to own an independent business five years later and less likely to receive all earnings from business ownership, relative to owners of restaurants that exited for other reasons in control states. However, their average incomes (wage income plus ordinary business income) are no lower. This income stability is partly explained by the fact that policy-induced exiters are significantly less likely to report negative incomes five years out, suggesting they substitute away from potentially risky or marginally profitable business ownership toward wage employment or other activities. The utility implications are ambiguous: these former owners may have preferred business ownership even if it did not yield higher income.

Q13: What is the role of product market competition in mediating pass-through, as evidenced by border-county analysis?

The border county robustness analysis reveals that product market competition is central to pass-through success. Retailers near state borders, where consumers can cross-state-border shop, face more elastic demand and are less able to finance the wage cost shock with new revenues, exhibiting reduced profits and higher exit rates (though estimates are imprecise). Further from the border, where the cost shock is more commonly felt by all potential substitutes (making market demand elasticity rather than firm demand elasticity the relevant parameter), results are very similar to the full-sample aggregate findings. This confirms that the common nature of the minimum wage cost shock — shared by all competing firms in the market — is a key reason firms can pass through costs to consumers.

Q14: How do the findings address the divide among independent business owners on minimum wage policy?

The heterogeneous outcomes rationalize why surveys consistently find business owners divided. Among restaurants, some owners (those operating the least productive small restaurants) face exit and loss of business ownership, while surviving productive restaurateurs see higher profits of $5,941–$9,161 per year. Among non-restaurant exposed businesses, owners are broadly unaffected in terms of profits and viability. Uncertainty about whether a given firm’s demand is elastic enough to bear cost pass-through — given that owners may be more familiar with the elasticity of firm-level demand from prior unilateral price changes, rather than the relevant market-level demand elasticity applying to a common cost shock — may broaden opposition to include even owners who would ultimately benefit.

Key Concepts

Pass-through businesses (independent businesses): Privately owned firms organized as S-corporations, partnerships, or LLCs, taxed by passing income through to the individual returns of owners rather than at the entity level. In 2015, these comprised 78% of non-sole-proprietorship U.S. businesses and 46% of employment. The paper uses “pass-through” and “independent business” interchangeably as the unit of analysis.

Highly exposed industries: Four-digit NAICS industries where at least 15% of workers earned below the annual full-time equivalent of the federal minimum wage ($15,080) in 2013, as measured in the authors’ administrative tax data. This threshold proxies the concentration of minimum-wage workers across industries and drives the sample selection for firm-level analysis.

Own-wage elasticity of employment: The estimated percentage change in employment at a firm associated with a given percentage change in the firm’s minimum wage. The authors estimate this as -0.209 (s.e. = 0.0112), reflecting the average effect across all exposed independent businesses, conditional on the firm’s industry, size, and local market characteristics.

DFL re-weighting (DiNardo-Fortin-Lemieux): A non-parametric reweighting procedure that adjusts the distribution of control-group firms to match the distribution of treatment-group firms on observables (specifically, two-year lagged value-added within three-digit NAICS industries). Used to improve pre-reform comparability of treatment and control firm samples without parametric functional form assumptions.

Firm productivity (in this paper’s sense): Measured as the ratio of net profits to revenues (net income/revenue) at the firm level in the base year 2013, used to assign firms to productivity quartiles for heterogeneity analysis. This is a firm-level profitability measure constructed from pass-through tax returns, not a total factor productivity estimate requiring production function estimation.

Firm exit: An indicator for a firm that filed a tax return in 2013 but did not file a return in a subsequent year t. The average one-year exit rate for highly exposed independent businesses is 5.2%; the cumulative five-year raw exit rate is approximately 29% across treatment and control states.

Cournot competition with heterogeneous productivity and fixed costs: The paper’s conceptual framework, in which N firms compete in quantities with asymmetric marginal costs (reflecting heterogeneous productivity), a common output price, and a fixed cost of production. Under this framework, a minimum wage cost shock narrows margins unevenly, induces exit among firms that cannot cover fixed costs, and generates both demand reallocation and market share gains for productive survivors — rationalizing simultaneous exit and profit increases in the same industry.

Common cost shock: The property that a minimum wage increase raises production costs for all firms employing low-wage workers in the same market simultaneously. Because all competing firms face higher costs, the relevant pass-through parameter is the elasticity of market demand rather than the (higher) elasticity of individual firm demand, facilitating cost pass-through to consumers and distinguishing minimum wages from unilateral price changes by a single firm.