Cost Pass-Through

When input costs rise—whether from tariffs, commodity price spikes, or new taxes—how much of that increase reaches consumers? The answer is neither obvious nor uniform. It depends on the shape of the demand curve, not merely its slope or elasticity at the current price. Under the most common textbook assumption (linear demand), exactly half of any cost shock is passed through. But under other demand forms, the pass-through rate can exceed 100%, a phenomenon known as overshifting. This page develops the theory, formalizes the key results of Weyl and Fabinger (2013), and provides interactive tools to explore how demand curvature determines the consumer burden of cost changes.

The Pass-Through Problem

Consider a monopolist setting a profit-maximizing price $p^*$ given demand $q(p)$ and constant marginal cost $c$ . If marginal cost rises by a small amount $\Delta c$ , the new optimal price will be $p^* + \Delta p$ . The pass-through rate is the ratio of the price change to the cost change.

Definition — Pass-Through Rate

The pass-through rate (or rate of cost pass-through) measures the fraction of a marginal cost change that is reflected in the profit-maximizing price:

\rho = \frac{dp^*}{dc}

When $\rho = 1$ , the firm passes through the full cost increase. When $\rho < 1$ , it absorbs part of the shock. When $\rho > 1$ , the firm over-shifts: the price rises by more than the cost increase.

The pass-through rate is a central quantity in public economics (for understanding tax incidence), trade policy (for predicting tariff effects), and industrial organization (for merger analysis). As Weyl and Fabinger (2013) emphasize, it is also the key sufficient statistic linking cost shocks to changes in consumer surplus, producer surplus, and deadweight loss.

Linear Demand: The 50% Benchmark

Start with the most familiar case: linear demand $q = a - bp$ . A monopolist facing this demand and constant marginal cost $c$ maximizes $\pi = (p - c)(a - bp)$ . The first-order condition yields:

p^* = \frac{a + bc}{2b} = \frac{1}{2}\left(\frac{a}{b} + c\right)

Differentiating with respect to $c$ :

\rho = \frac{dp^*}{dc} = \frac{1}{2}

Linear Demand Pass-Through

Under linear demand $q = a - bp$ , the pass-through rate of a monopolist is exactly $\rho = 1/2$ , regardless of the demand parameters $a$ and $b$ , or the level of marginal cost $c$ .

This result is remarkably clean: under linear demand, the consumer always bears exactly half of any cost increase, and the firm absorbs the other half. The intuition is that the demand curve’s slope is constant, so the marginal revenue curve has exactly twice the slope of the demand curve. A unit cost increase shifts the intersection of marginal revenue and marginal cost up by half a unit.

Tariff on Imported Goods

Suppose a retailer faces demand $q = 100 - p$ and marginal cost $c = \$20$ . The optimal price is $p^* = (100 + 20)/2 = \$60$ with quantity $q = 40$ and profit $\pi = 40 \times 40 = \$1{,}600$ .

A $\$10$ per-unit tariff raises marginal cost to $\$30$ . The new price is $p^{*\prime} = (100 + 30)/2 = \$65$ . The price rose by $\$5$ , exactly half of the $\$10$ tariff. Quantity drops to $q = 35$ and profit falls to $\$1{,}225$ . The government collects tariff revenue of $\$10 \times 35 = \$350$ .

The Weyl-Fabinger Framework

The 50% result for linear demand generalizes beautifully. Weyl and Fabinger (2013) show that the pass-through rate under any demand curve is determined by a single quantity: the elasticity of the slope of the inverse demand function, which they denote $E_s$ .

For a general demand function $q(p)$ , the monopolist’s first-order condition is:

q(p) + (p - c) \, q'(p) = 0

Totally differentiating with respect to $c$ and solving for $dp/dc$ , we obtain:

\rho = \frac{dp^*}{dc} = \frac{-q'(p)}{-2q'(p) - (p-c)\,q''(p)}

Dividing numerator and denominator by $-q'(p)$ and defining the curvature measure:

E_s = \frac{(p-c)\,q''(p)}{q'(p)}

Weyl-Fabinger Pass-Through Formula

Under monopoly pricing with a twice-differentiable demand function, the pass-through rate is:

\rho = \frac{1}{2 + E_s}

where $E_s$ is the elasticity of the slope of demand evaluated at the profit-maximizing price. Equivalently, using $ms(p)$ for the marginal surplus (the negative inverse demand slope), $\rho = 1/(1 + E_{ms})$ where $E_{ms}$ is the elasticity of marginal surplus.

The key insight is that the pass-through rate depends on $q''(p)$ , the curvature (second derivative) of the demand function, not merely its slope or elasticity. Three cases emerge:

Linear demand ( $q'' = 0$ ): $E_s = 0$ , so $\rho = 1/2$ .
Convex demand ( $q'' > 0$ , demand curves outward): $E_s > 0$ , so $\rho < 1/2$ . Convex demand dampens pass-through.
Concave demand ( $q'' < 0$ , demand curves inward): $E_s < 0$ , so $\rho > 1/2$ . Sufficiently concave demand can produce $\rho > 1$ .

This taxonomy was foreshadowed by Bulow and Pfleiderer (1983), who first showed that the sign of the second derivative of demand determines whether pass-through exceeds or falls short of 50%.

When Firms Over-Shift

The most striking prediction of the theory is that pass-through can exceed 100%. Under constant-elasticity demand $q = A p^{-\varepsilon}$ , the optimal price is:

p^* = \frac{\varepsilon}{\varepsilon - 1} \, c

The markup ratio $\varepsilon / (\varepsilon - 1)$ is constant, so the pass-through rate is:

\rho = \frac{\varepsilon}{\varepsilon - 1} = \frac{p^*}{c} > 1 \quad \text{for all } \varepsilon > 1

Constant-Elasticity Overshifting

Consider a firm with constant-elasticity demand with $\varepsilon = 2.5$ and marginal cost $c = \$20$ . The optimal price is $p^* = (2.5 / 1.5) \times 20 = \$33.33$ .

If a $\$10$ cost shock raises $c$ to $\$30$ , the new price is $p^{*\prime} = (2.5/1.5) \times 30 = \$50.00$ . The price increased by $\$16.67$ —more than the $\$10$ cost increase. The pass-through rate is $\rho = 16.67/10 = 167\%$ .

This overshifting is not a market failure or a sign of excessive market power. It arises mechanically from the curvature of the demand function. Under constant-elasticity demand, the optimal markup is a fixed proportion of cost. When cost rises, both the absolute markup and the price rise proportionally, which means the price increase exceeds the cost increase. The practical implication is important: policymakers who assume that a tax or tariff will raise prices by at most the tax amount may substantially underestimate the consumer price impact in markets with log-convex demand.

Exponential demand $q = Ae^{-bp}$ provides another instructive case: the pass-through rate is exactly $\rho = 1$ (100%). The price rises by exactly the cost increase—neither more nor less—because the optimal markup $1/b$ is independent of cost.

Interactive: Cost Shock Sweep

The visualization below shows a supply-demand diagram for linear demand. Adjust the cost shock to see how the original and shifted marginal cost lines determine the old and new equilibrium prices. The shaded region between the two price levels shows the consumer surplus lost to the cost increase. Under linear demand, the price always moves by exactly half the cost shock.

Interactive: Curvature Comparison

Different demand functional forms produce dramatically different pass-through rates. The bar chart below compares four canonical demand curves subjected to the same cost shock. Notice that constant-elasticity demand produces a pass-through rate exceeding 100%, while linear demand always yields exactly 50%. Logit and exponential demand fall in between.

Tax and Tariff Implications

The pass-through framework has direct applications to tax policy and international trade. Anderson, de Palma, and Kreider (2001) extend the monopoly pass-through results to differentiated product oligopoly and show that tax incidence under imperfect competition depends critically on the functional form of demand, not just on the number of competitors or the level of concentration.

Key Policy Implications

Per-unit taxes vs. ad valorem taxes: Under linear demand, a per-unit tax and an ad valorem tax that raise the same revenue at the initial equilibrium have the same pass-through rate. Under constant-elasticity demand, an ad valorem tax is passed through at a higher rate because it amplifies the multiplicative markup structure.
Tariff burden estimation: Standard trade models that assume 100% pass-through overestimate consumer price increases for products with linear or convex demand, but underestimate them for products with log-convex (constant-elasticity) demand. Empirical estimates of tariff pass-through in consumer goods typically find rates between 20% and 70%, consistent with moderately convex demand.
Competition and pass-through: In competitive markets, supply-side pass-through approaches 100% as the supply curve flattens. The key result of Weyl and Fabinger (2013) is that the pass-through formula $\rho = 1/(2 + E_s)$ generalizes to imperfect competition by replacing $2$ with $1 + \theta$ , where $\theta$ captures the conduct parameter (1 for monopoly, approaching 0 for perfect competition).
Merger analysis: Antitrust authorities use pass-through rates to predict post-merger price increases. The Werden (1996) method for simulating merger effects relies directly on estimated pass-through rates. If demand curvature implies a high pass-through rate, the predicted price effect of a merger is correspondingly larger.

The deeper lesson is that demand curvature—a second-order property of the demand function—has first-order consequences for prices, welfare, and policy. Practitioners estimating demand functions for pricing or policy analysis should pay careful attention to the choice of functional form, since it directly determines the pass-through rate and thus the predicted consumer impact of any cost-side intervention.

References

Anderson, S. P., de Palma, A. & Kreider, B. (2001). “Tax Incidence in Differentiated Product Oligopoly.” Journal of Public Economics, 81(2), 173–192.
Bulow, J. I. & Pfleiderer, P. (1983). “A Note on the Effect of Cost Changes on Prices.” Journal of Political Economy, 91(1), 182–185.
Weyl, E. G. & Fabinger, M. (2013). “Pass-Through as an Economic Tool: Principles of Incidence under Imperfect Competition.” Journal of Political Economy, 121(3), 528–583.