A Theory of Supply Function Choice and Aggregate Supply

Research Question

Modern macroeconomic models of aggregate supply universally restrict firms to price-setting — committing to a price and supplying whatever quantity the market demands. Flynn, Nikolakoudis, and Sastry ask: what happens if instead firms choose any supply function, a mapping that describes the price charged at each quantity of production? The paper develops the first general-equilibrium, macroeconomic theory of supply function choice and characterizes its implications for the slope of aggregate supply, monetary non-neutrality, and time-varying inflation-output tradeoffs.

Methodology

The paper proceeds in two stages. In partial equilibrium, a single monopolistic firm with constant-returns-to-scale technology and constant-elasticity demand faces log-normal uncertainty about demand shifters, the aggregate price level, real marginal costs, and the stochastic discount factor. The firm chooses a non-parametric supply function — any implicit mapping f(p,q) = 0 — to maximize expected real profits. The paper shows that supply function choice is equivalent to conditioning price-quantity decisions on the realized nominal demand state z = ΨP^η. The authors prove (Theorem 1) that the optimal supply function is endogenously log-linear: log p = α₀ + α₁ log q, where the inverse supply elasticity α₁ is characterized in closed form.

In general equilibrium, the authors embed supply function choice in an otherwise standard monetary business cycle model (in the tradition of Woodford 2003a and Hellwig and Venkateswaran 2009), featuring a representative household demanding differentiated goods, a money supply following a random walk with time-varying volatility, and idiosyncratic shocks to productivity, wages, and demand. They guess and verify a log-linear equilibrium and derive a scalar fixed-point equation for the equilibrium supply elasticity (Theorem 3).

For quantification, the authors calibrate structural parameters (η = 8 from Hottman et al. 2016 scanner data; γ = 0.11 from Gagliardone et al. 2023 Belgian firm data; κ^M = 0.29 calibrated to match an average aggregate supply slope of 0.11 from Hazell et al. 2022) and estimate time-varying uncertainty via a GARCH model of quarterly US data on GDP growth, inflation, and real marginal cost growth from 1960 Q1 to 2024 Q4. Idiosyncratic demand uncertainty is set proportional to aggregate TFP uncertainty using the proportionality factor R = 6.5 from Bloom et al. (2018).

Main Findings

Optimal supply function. The optimal firm-level supply function is log-linear with inverse supply elasticity α₁ determined by the relative variances and covariances of demand, the price level, and real marginal costs. Three comparative statics drive the macroeconomic results: (1) higher idiosyncratic demand uncertainty (σ²_Ψ) flattens the supply function toward price-setting, because a fixed price insulates profit markups against demand variation; (2) higher price-level uncertainty (σ²_P) steepens the supply function toward quantity-setting, because setting a fixed quantity allows relative prices to adjust; (3) lower price elasticity of demand (less elastic demand, more market power) flattens the supply function, conditional on a sufficient condition that holds in US data whenever η > 2.5.

From micro supply to aggregate supply. With fixed log-linear supply functions, the economy has a unique log-linear equilibrium with an AD/AS representation (Theorem 2). The slope of aggregate supply ε^S_t depends on ω₁ (the transformed inverse supply elasticity), κ^M (firms’ signal precision about the money supply), γ (income effects), and η (demand elasticity). Aggregate supply is maximally elastic — money is as non-neutral as possible — if and only if firms are pure price-setters (ω₁ = 0). Aggregate supply is perfectly inelastic — money is neutral — if and only if firms are quantity-setters (ω₁ = 1/η). A lower elasticity of demand flattens aggregate supply through general equilibrium strategic complementarities, a prediction opposite to the New Keynesian model.

Equilibrium supply slope and its determinants. The equilibrium ω₁ solves a fixed-point equation (Theorem 3) in which macroeconomic uncertainty shapes firms’ optimal supply functions, which in turn shape macroeconomic dynamics. Under the special case of balanced strategic interactions (ηγ = 1), the slope of aggregate supply has a clean closed form depending only on the ratio ρ_t = σ_{ϑ,t}/σ^M_{t|s} (idiosyncratic demand uncertainty relative to posterior monetary uncertainty). Critically, the equilibrium supply slope is invariant to the overall level of uncertainty — only the composition of uncertainty matters (Proposition 3). Even vanishingly small uncertainty can generate any level of monetary non-neutrality depending on uncertainty composition.

Quantitative results — United States over time. The model’s estimated slope of aggregate supply shows sharp variation since 1960. The slope is relatively flat and stable during the 1960s, the Great Moderation (1991–2007), the Great Recession (2008–2019), and the recovery from the Great Recession. It spikes dramatically during the 1970s oil crisis and the post-Covid inflation of the 2020s. Compared to Ball and Mazumder (2011), the model qualitatively matches the steepening during 1973–1984 (+58% in the model) vs. the data’s +175%, and a subsequent flattening of −25% vs. −32% in the data during 1985–2007. Compared to Cerrato and Gitti (2022), the model accounts for approximately 4/5 of the steepening between the pre-Covid and post-Covid periods (+112% model vs. +145% data). For the Hazell et al. (2022) comparison, the model accounts for approximately 1/2 of the estimated flattening from 1978–1990 to 1991–2018.

Quantitative results — Cross-country. Using OECD annual data from 1960–2019, the model’s predicted slope of aggregate supply is not positively correlated with the average level of inflation across countries. For countries with the highest inflation rates, the model predicts a negative slope of aggregate supply, driven by very high correlation between price-level uncertainty and real marginal cost uncertainty. The model-predicted slope correlates positively with the reduced-form regression coefficient of inflation on real output growth across countries, even after instrumenting for demand. This predictive power is over and above what can be explained by the level or volatility of inflation alone.

Scope Conditions

All results are derived under log-normality of uncertainty, which ensures the log-linear structure of optimal supply functions. The quantification relies on GARCH-estimated uncertainty and treats idiosyncratic demand uncertainty as proportional to aggregate TFP uncertainty. The model abstracts from microeconomic nominal price stickiness (though the authors show in Appendix B that Calvo-style sticky prices can be incorporated). The baseline model requires the equilibrium condition on firm beliefs to be consistent (rational expectations). Multiple equilibria of the scalar fixed-point are possible in principle, bounded by at most five log-linear equilibria (Proposition 2).

Q&A

Q1: What is wrong with assuming price-setting or quantity-setting as a primitive restriction on firm behavior?

A: Price-setting and quantity-setting are two isolated, generically non-optimal points in the larger space of supply functions. Corollary 2 establishes that price-setting is optimal only in the limit as idiosyncratic demand uncertainty becomes unboundedly large (σ²_Ψ → ∞), while quantity-setting is optimal only in the limit as price-level uncertainty becomes unboundedly large (σ²_P → ∞). In a macroeconomic environment where both sources of uncertainty are present in comparable magnitudes, both extreme policies perform poorly and the analyst who imposes either inadvertently restricts firms’ strategies in ways that have large macroeconomic consequences — for example, making money neutral under quantity-setting even when information frictions are present, or making the slope of aggregate supply invariant to demand elasticity under price-setting.

Q2: What is the formal equivalence between supply function choice and conditioning on realized demand?

A: The firm’s problem of choosing a supply function f(p,q) = 0 ex ante is mathematically equivalent to choosing a price-quantity plan (p(z), q(z)) indexed by the nominal demand state z = ΨP^η (Equation 4 in the paper). After the supply function is set, the firm produces where the supply function intersects the demand curve, which pins down the market-clearing outcome as a function of z. Choosing the supply function ex ante is therefore the same as choosing z-contingent prices and quantities without any parametric constraint. This links the model to rational expectations equilibrium in the spirit of Lucas (1972): firms use the demand for their product as a noisy signal to update beliefs and set their optimal price and quantity in response to realized demand conditions.

Q3: How is the optimal inverse supply elasticity α₁ derived, and what is the 2SLS interpretation?

A: Because the optimal supply function allows the firm to set a z-contingent price, the first-order condition at each realized demand state z = t equates expected marginal revenue and expected marginal cost (Equation 7). Under log-normality, this yields a log-linear relationship log p = α₀ + α₁ log q. The elasticity α₁ equals the ratio (d log p / d log z) / (d log q / d log z) = Cov[log z, log p**] / Cov[log z, log q**], where p** and q** are the full-information optimal price and quantity (Equation 9). This is formally equivalent to a 2SLS regression: the firm estimates how its optimal price should change with its optimal quantity, using the nominal demand state z as an instrument for the optimal quantity. The supply function is steep if nominal demand strongly predicts movements in the full-information optimal price (large reduced-form coefficient); it is flat if nominal demand primarily predicts movements in the full-information optimal quantity (large first-stage coefficient).

Q4: How do uncertainty and demand elasticity shape the firm’s optimal supply function in partial equilibrium?

A: Three key comparative statics apply when the supply function is upward-sloping. (1) Greater price-level uncertainty (σ²_P increases) steepens α₁ toward quantity-setting: not knowing competitors’ prices makes aggressive dynamic pricing attractive because it allows the firm’s relative price to adjust ex post. (2) Greater idiosyncratic demand uncertainty (σ²_Ψ increases) flattens α₁ toward price-setting: demand uncertainty favors a fixed price to keep the markup over real marginal costs constant, accommodating demand with quantity variation. (3) A lower price elasticity of demand (more market power, lower η) flattens α₁: more market power reduces the cost of setting the “wrong” price, reducing the benefit of dynamic pricing. Corollary 1 provides a sufficient condition — σ_{M,P} ≥ 0, 2ησ_{M,P} + σ_{M,Ψ} ≥ σ_{P,Ψ}, and α₁ ≥ 0 — under which ∂α₁/∂η > 0, implying greater market power flattens supply; the paper verifies this condition holds in US data whenever η > 2.5.

Q5: How does the model generate an aggregate supply and demand representation from supply function choices?

A: Theorem 2 establishes that, given any fixed log-linear supply functions with slope ω₁,t, there is a unique log-linear equilibrium. In this equilibrium, the price level and real output are jointly determined by an aggregate demand curve — shifting with the money supply but not productivity — and an aggregate supply curve — shifting with productivity but not the money supply. The inverse elasticity of aggregate supply is ε^S_t = γ(κ^M_t + ω₁,t(η − 1/γ)(1 − κ^M_t)) / ((1 − ω₁,t η)(1 − κ^M_t)), derived from aggregating firm-level pricing decisions. The slope depends on ω₁,t (micro supply), κ^M_t (signal precision about money), γ (income effects), and η (demand elasticity). An aggregate demand shock of ∆ log M raises the price level by ε^S_t ∆ log M / (ε^D_t + ε^S_t) and raises real output by ∆ log M / (ε^D_t + ε^S_t), where ε^D_t = γ is the inverse elasticity of aggregate demand.

Q6: What is the equilibrium fixed-point equation and why can there be multiple equilibria?

A: Theorem 3 shows that the equilibrium transformed inverse supply elasticity ω₁,t solves a quintic polynomial fixed-point equation (Equation 29) that depends on the variances of idiosyncratic demand shocks (σ²_ϑ,t), posterior uncertainty about productivity (σ^A_{t|s}), and posterior uncertainty about money (σ^M_{t|s}). Multiple equilibria can arise because of a self-reinforcing feedback: if firms set steep supply functions, prices respond more to demand, which raises price-level volatility, which in turn makes quantity-setting more attractive, further steepening supply functions. Proposition 2 establishes existence of at least one log-linear equilibrium and at most five. Idiosyncratic productivity and factor price uncertainty do not enter the fixed-point equation because the variance of real marginal costs per se does not affect optimal supply function choice — only the covariance of marginal costs with demand and the price level matters.

Q7: What determines the slope of aggregate supply in the special case of balanced strategic interactions (ηγ = 1)?

A: Under ηγ = 1 — where strategic complementarities from relative price effects exactly offset strategic substitutabilities from aggregate consumption effects — the slope of aggregate supply has the closed-form expression ε^S_t = γ(κ^M_t / (1 − κ^M_t))(1 + 1/(γ²ρ²_t κ^M_t)) where ρ_t = σ_{ϑ,t}/σ^M_{t|s} is the ratio of idiosyncratic demand uncertainty to posterior monetary uncertainty (Corollary 5). Aggregate productivity uncertainty drops out entirely because firms do not use the demand state to infer aggregate productivity when strategic interactions are balanced. As ρ_t → ∞ (idiosyncratic demand dominates), the slope converges to the price-setting value γκ^M_t/(1 − κ^M_t). As ρ_t → 0 (monetary uncertainty dominates), the slope goes to infinity, corresponding to quantity-setting and monetary neutrality.

Q8: What is the role of total uncertainty versus the composition of uncertainty?

A: Proposition 3 establishes a striking invariance result: if all standard deviations in the economy are scaled by a common factor λ > 0, the equilibrium supply elasticity and slope of aggregate supply are unchanged. The equilibrium outcomes depend only on the ratios of different sources of uncertainty, not their absolute magnitudes. This sharply distinguishes the model from menu-cost models, in which any increase in uncertainty unambiguously raises the benefit of price adjustment and steepens aggregate supply. A corollary is that idiosyncratic productivity uncertainty has no effect on the slope of aggregate supply in the supply function model, whereas it would steepen aggregate supply in Golosov-Lucas menu-cost models. Moreover, even a vanishingly small level of uncertainty can generate any level of monetary non-neutrality, because the equilibrium supply elasticity is discontinuous at zero uncertainty (ε^S_t (0) = {∞} while ε^S_t (λ) is bounded for any λ > 0).

Q9: How does market power (demand elasticity) affect the slope of aggregate supply, and why does this differ from the New Keynesian prediction?

A: In the supply function model, a lower elasticity of demand (more market power, lower η) flattens aggregate supply by reducing general-equilibrium strategic complementarities. When other firms raise their prices following a demand shock, a given firm faces higher relative demand; the strength of this effect is parameterized by η. With supply functions (ω₁,t ≠ 0), this relative demand increase generates an additional price response, so higher η steepens aggregate supply. Crucially, this effect is exactly zero if and only if firms are pure price-setters (ω₁,t = 0) — meaning the prediction that market power affects aggregate supply is absent from price-setting models. This is the opposite of the New Keynesian prediction: in Woodford (2003b) with decreasing returns to scale, a higher elasticity of demand (less market power) steepens the Phillips curve, because more elastic demand amplifies the quantity response to price changes and thereby the marginal cost response to nominal cost shocks.

Q10: How does the model rationalize the steepening of aggregate supply in the 1970s and 2020s?

A: The GARCH estimates of macroeconomic uncertainty show abrupt increases in inflation uncertainty during the 1970s oil crisis period and after the Covid-19 shock in the 2020s. In the model, a spike in aggregate price-level uncertainty (σ²_P increases) causes firms to choose steeper supply functions — closer to quantity-setting — endogenously. This steepens the aggregate supply curve so that demand shocks have larger nominal effects and smaller real effects. Quantitatively, relative to the base period, the model predicts a steepening of +58% during 1973–1984 and +112% during 2021–2023. The empirical comparisons are +175% (Ball and Mazumder 2011, 1973–1984) and +145% (Cerrato and Gitti 2022, 2021–2023). The model thus accounts for the direction and rough order of magnitude of both episodes but not their full extent. The quarterly time series of model-implied ε^S_t has a correlation of 0.93 with one-quarter-ahead inflation uncertainty and 0.62 with the quarterly level of inflation.

Q11: How does the cross-country evidence help distinguish the model from alternatives based on the level of inflation?

A: The cross-country analysis uses OECD data from 1960–2019 to construct country-level model-implied slopes of aggregate supply using the same structural parameters (η = 8, γ = 0.11, κ^M = 0.29) and country-specific GARCH uncertainty estimates from a one-lag VAR. The key finding is that the model-implied slope is not positively predicted by average inflation across countries (Panel A of Figure 5) — in fact, for the highest-inflation countries such as Chile, Israel, and Mexico, the model predicts a negative slope of aggregate supply, reflecting high correlation between price-level uncertainty and real marginal cost uncertainty. By contrast, the model-implied slope correlates positively with the reduced-form regression coefficient of inflation on real output growth (Panel B), and this positive correlation is also found using a model-derived instrument isolating exogenous monetary variation. This implies that relative uncertainties, not the mean or volatility of inflation per se, help account for cross-country heterogeneity in inflation-output tradeoffs beyond the predictions of Ball et al. (1988).

Q12: How can supply functions be integrated into larger linearized macroeconomic models?

A: Section 4.5 provides a general framework. For any model in which firms face a demand function q_it = d(p_it, z^D_it) and a value function V(p_it, q_it, z^V_it), log-linearization around a deterministic steady state yields an optimal pricing rule ˆp_it = ω₁,it ˆz^D_it (Equation 35) for some scalar ω₁,it determined by the covariance structure of the linearized model. The coefficients ω₁,it enter the standard representation of aggregate dynamics (McKay and Wolf 2023) through the ideal price index ˆP_t = ∫₀¹ ˆp_it di. The additional “rational expectations” restriction is that ω₁,it must be consistent with the equilibrium law of motion for prices. The paper argues that supply functions can thereby be embedded in the broad class of linearized DSGE models used for quantitative work, including models with decreasing returns, monopsony, endogenous markups, sticky prices, investment, and quality choice.

Q13: What are the implications of supply function choice for monetary policy discretion?

A: The model implies a thorny tradeoff for monetary policymakers. If a central bank wishes to maintain discretion — the ability to surprise private agents — this increases firms’ uncertainty about the money supply (higher σ²_M). Under balanced strategic interactions (ηγ = 1), greater posterior monetary uncertainty (σ^M_{t|s}) lowers the ratio ρ_t = σ_{ϑ,t}/σ^M_{t|s}, which flattens the aggregate supply curve (reduces ε^S_t) and thereby increases the real effect of monetary surprises. However, this also endogenously induces firms to set steeper supply functions — closer to quantity-setting — so that the aggregate supply curve steepens in response to the greater price-level uncertainty generated by such an environment. The paper therefore concludes that maintaining monetary policy discretion may be, at least partially, self-defeating.

Inverse supply elasticity (α₁): The percentage by which a firm increases its price in response to a one percent increase in production, characterizing the slope of the firm’s optimal supply function. It is endogenously log-linear and determined by the ratio of covariances relating the nominal demand state to the firm’s optimal price vs. optimal quantity under full information — formally equivalent to a 2SLS coefficient using nominal demand as an instrument.

Supply function: A mapping f(p, q) = 0 describing the locus of prices and quantities a firm commits to, as an implicit function over price-quantity pairs. Unlike price-setting (f depends only on p) or quantity-setting (f depends only on q), the general supply function allows prices to vary with realized demand, nesting both polar cases as limits of extreme uncertainty.

Nominal demand state (z): The composite variable z = ΨP^η that indexes the demand curve. Firms observing their own output market clearing can use z as a noisy signal for inference about the aggregate price level, real marginal costs, and monetary conditions. The supply function is formally equivalent to conditioning price-quantity choices on z.

Slope of aggregate supply (ε^S): The inverse elasticity of the aggregate supply curve in the AD/AS representation, measuring the relative within-period response of the price level versus real output to an aggregate demand shock. It depends on the slope of firm-level supply functions (ω₁) interacted with the information precision about the money supply (κ^M) and income effects (γ).

Transformed inverse supply elasticity (ω₁): The reparameterization ω₁ = α₁/(1 + ηα₁), where α₁ is the firm-level inverse supply elasticity and η is the price elasticity of demand. ω₁ = 0 corresponds to price-setting; ω₁ = 1/η corresponds to quantity-setting. The equilibrium value of ω₁ solves a fixed-point equation that maps macroeconomic uncertainty back into firms’ optimal supply function choices.

Balanced strategic interactions (ηγ = 1): A parametric special case in which strategic complementarities from aggregate demand externalities (parameterized by η) exactly offset strategic substitutabilities from wage pressure (parameterized by 1/γ). Under this condition, the slope of aggregate supply has a closed-form solution that depends only on the relative uncertainty about idiosyncratic demand vs. the money supply.

Relative uncertainty sufficient statistic (ρ_t): The ratio σ_{ϑ,t} / σ^M_{t|s}, measuring firms’ uncertainty about idiosyncratic demand shocks relative to posterior uncertainty about the money supply. Under balanced strategic interactions (ηγ = 1), ρ_t is the single sufficient statistic determining the equilibrium slope of aggregate supply. As ρ_t → ∞ (idiosyncratic demand uncertainty dominates), firms converge to price-setting and aggregate supply flattens; as ρ_t → 0 (monetary uncertainty dominates), firms converge to quantity-setting and aggregate supply becomes vertical.

Invariance to total uncertainty: A key property of the model: the equilibrium slope of aggregate supply is invariant to the overall scale of uncertainty (Proposition 3). Only the composition of uncertainty across idiosyncratic vs. aggregate sources and demand vs. productivity shocks matters. This distinguishes the model from menu-cost models, in which any increase in uncertainty raises the benefit of price flexibility and steepens aggregate supply regardless of uncertainty composition.