Alchian–Allen Effect

Alchian–Allen effect (often summarized colloquially as "shipping the good apples out" or the Third Law of Demand) is a foundational theorem of neoclassical price theory within business and economics theory demonstrating how per-unit transaction, transportation, or tax charges alter the quality composition of consumer demand. Formulated in 1964 by American economists Armen A. Alchian and William R. Allen in their celebrated textbook University Economics, the proposition states that if a fixed per-unit fee t > 0 (such as per-pound shipping freight, a specific excise tariff, or an opportunity cost of time) is added equally to the prices of two substitute commodities of different quality grades—a high-grade premium variety priced at p₁ and a low-grade standard variety priced at p₂ < p₁—the relative price of the premium variety decreases in the distant or taxed market. Consequently, holding real purchasing power constant via the Slutsky equation, consumers in distant markets substitute toward a higher ratio of the premium good to the lower-grade good than consumers located at the point of production.

The theorem was originally inspired by a puzzling observation in agricultural markets of the American Pacific Northwest: residents of Washington State routinely complained that the finest, highest-grade Washington apples were exported to distant East Coast markets such as New York and Chicago, leaving mostly smaller or second-grade apples in local Washington grocery stores. Rather than reflecting producer conspiracy or regional differences in consumer tastes, Alchian and Allen showed that this quality sorting is an inescapable arithmetic consequence of relative prices under per-unit transport costs.

Algebraic Compression of the Relative Price Ratio

Consider two quality grades of the same physical commodity produced in a single origin region: a high-quality (choice) variety x₁ with local origin price p₁ and a low-quality (standard) variety x₂ with local origin price p₂, where:

Suppose shipping one physical unit (one pound or one crate) of either variety to a distant destination market incurs an identical per-unit freight charge t > 0, because a freight carrier charges by weight or volume rather than by the intrinsic culinary quality of the apple inside the crate. In the destination market, the delivered prices are P₁ = p₁ + t and P₂ = p₂ + t. Comparing the destination relative price ratio P₁ / P₂ with the origin relative price ratio p₁ / p₂ yields:

Differentiating the relative price of the premium variety with respect to the per-unit fee t confirms that any increase in t strictly lowers the relative price of the high-grade good:

For example, if a choice apple costs p₁ = 0.20 dollars and a standard apple costs p₂ = 0.10 dollars in Washington orchards, purchasing one choice apple in Washington requires sacrificing p₁/p₂ = 2.0 standard apples. If refrigerated rail freight to New York costs t = 0.10 dollars per apple, the New York delivered prices become P₁ = 0.30 dollars and P₂ = 0.20 dollars; in New York, one choice apple costs only P₁/P₂ = 1.5 standard apples—a 25% drop in the relative opportunity cost of quality.

Cross-Price Elasticities with the Numeraire Composite Good

Shortly after Alchian and Allen popularized the proposition, critics led by John P. Gould and Joel Segall (1969) pointed out a subtle mathematical complication in multi-good general equilibrium theory and consumer demand theory. In a pure two-good world containing only high-grade apples (x₁) and low-grade apples (x₂), the Slutsky substitution matrix is a 2 × 2 negative semi-definite matrix, so a drop in P₁/P₂ unambiguously increases the Hicksian compensated ratio x₁^h(P, u) / x₂^h(P, u). However, in a realistic economy with at least three goods—high-grade apples (x₁ at price p₁ + t), low-grade apples (x₂ at price p₂ + t), and an outside composite numeraire good x₃ ("all other goods" at price p₃ = 1, which is not subject to the apple transport charge t)—an increase in t raises both P₁/p₃ and P₂/p₃ while lowering P₁/P₂.

In their landmark 1978 Journal of Political Economy article "Shipping the Good Apples Out: The Alchian and Allen Theorem Reconsidered", Thomas E. Borcherding and Eugene Silberberg derived the exact three-good Hicksian comparative-statics derivative for the quality ratio x₁^h / x₂^h with respect to the per-unit charge t, holding utility u constant. Using homogeneity of degree zero of Hicksian demands (∑_j=1³ ε_ij = 0, where ε_ij ≡ (P_j / x_i^h)(∂x_i^h / ∂P_j) is the compensated cross-price elasticity of good i with respect to price j), Borcherding and Silberberg expressed the three-good comparative-statics condition as:

Specifically, because P₁ > P₂ and the direct two-good substitution term between the close substitutes x₁ and x₂ is strongly positive, the Alchian–Allen effect holds unambiguously whenever the high-grade and low-grade varieties are close substitutes for each other and exhibit approximately equal compensated cross-elasticities with respect to the outside composite good (ε₁₃ ≈ ε₂₃, as satisfied identically whenever utility is weakly separable between the quality-differentiated product group {x₁, x₂} and outside goods x₃).

The Alchian–Allen theorem has been confirmed across a wide spectrum of empirical microeconomic and international trade settings:

1. International Trade and the Gravity of Quality (Hummels and Skiba, 2004): Combining bilateral Harmonized System customs data with detailed ocean and air freight schedules alongside the gravity model of trade, David Hummels and Alexandre Skiba (2004) showed that international freight costs are predominantly per-unit (per-ton or per-cubic-meter) rather than purely ad valorem. Estimating the elasticity of average export unit values (FOB prices) with respect to bilateral freight costs across thousands of product categories, they confirmed that doubling freight distance raises average exported product quality by 12% to 25%, directly validating Alchian and Allen on a global scale.
2. Specific vs. Ad Valorem Commodity Taxation (Barzel, 1976): Yoram Barzel demonstrated that whereas an ad valorem sales tax (τ% of retail price) leaves the relative price p₁(1 + τ) / [p₂(1 + τ)] = p₁/p₂ unchanged, a specific per-unit excise tax (such as a tax of t dollars per pack of cigarettes, per gallon of gasoline, or per liter of wine) induces consumers and producers to upgrade along untaxed quality dimensions (longer cigarettes, higher-octane fuel, or vintage wines).
3. The "Iron Law of Prohibition" (Thornton, 1991): When a government prohibits an addictive substance (such as alcohol during the 1920–1933 U.S. Prohibition or illicit narcotics), smuggling, concealment, and legal enforcement risks impose a massive per-unit volume/weight penalty t on trafficking. By the Alchian–Allen effect, higher enforcement t compresses the relative price of concentrated, high-potency forms (distilled spirits vs. beer; fentanyl vs. opium), shifting black-market supply and demand toward higher-potency grades.

Why doesn't an ad valorem (percentage) tariff cause the Alchian–Allen effect?

An ad valorem tariff or sales tax multiplies both prices by the same factor (1 + τ), so the relative price ratio [p₁(1 + τ)] / [p₂(1 + τ)] = p₁ / p₂ remains completely invariant. Only an additive per-unit cost t > 0 (specific tariff, per-kilogram shipping charge, or fixed time cost per unit consumed) reduces the price ratio (p₁ + t) / (p₂ + t) below p₁ / p₂. Indeed, as Hummels and Skiba (2004) showed, while per-unit freight costs shift imports toward higher-quality goods, ad valorem tariffs shift imports in the opposite direction when combined with per-unit freight.

Could the income effect reverse the Alchian–Allen substitution effect in uncompensated Marshallian demand?

Because adding a transport cost t > 0 reduces consumers' real purchasing power, the uncompensated Marshallian demand change includes both the Hicksian substitution effect (which favors the higher-grade good x₁) and a real income effect (via the Slutsky equation). However, for any single product category such as apples, wine, or apparel that represents a small budget share of total household expenditure, the income term is negligible relative to the direct substitution elasticity between the two close quality grades; furthermore, comparing two distant export markets or comparing taxed vs. untaxed goods at constant real income isolates the pure Alchian–Allen substitution effect.

1. Alchian, A. A., & Allen, W. R. (1964). University Economics (1st ed.). Belmont, CA: Wadsworth Publishing Company. https://doi.org/10.1086/259017
2. Borcherding, T. E., & Silberberg, E. (1978). "Shipping the Good Apples Out: The Alchian and Allen Theorem Reconsidered." Journal of Political Economy, 86(1), 131–138. https://doi.org/10.1086/260651
3. Hummels, D., & Skiba, A. (2004). "Shipping the Good Apples Out? An Empirical Confirmation of the Alchian-Allen Conjecture." Journal of Political Economy, 112(6), 1384–1402. https://doi.org/10.1086/422562
4. Barzel, Y. (1976). "An Alternative Approach to the Analysis of Taxation." Journal of Political Economy, 84(6), 1177–1197. https://doi.org/10.1086/260507
5. Gould, J. P., & Segall, J. (1969). "The Substitution Effects of Transportation Costs." Journal of Political Economy, 77(1), 130–137. https://doi.org/10.1086/259502