Broda–Weinstein Import Variety Welfare Model

The Broda–Weinstein import variety welfare model is a landmark theoretical and empirical framework in international trade, price index theory, and business and economics theory developed by Christian Broda and David E. Weinstein in their 2006 Quarterly Journal of Economics article, "Globalization and the Gains from Variety." A central prediction of Krugman's (1979, 1980) new trade theory and the Dixit–Stiglitz (1977) monopolistic competition model is that international trade raises consumer and producer welfare not only by lowering the prices of existing goods through comparative advantage, but also by expanding the set of available product varieties ("love of variety"). Yet conventional government import price indices (such as the Bureau of Labor Statistics Laspeyres/Törnqvist import price index) track price changes only over a fixed basket of continuing goods, omitting the welfare value of newly introduced foreign varieties. Building on Robert C. Feenstra's (1994) theoretical constant-elasticity-of-substitution (CES) price index for new and disappearing varieties, Broda and Weinstein (2006) estimated over 30,000 disaggregated elasticities of substitution (σ_g) across 7-digit Tariff Schedule of the United States (TSUSA, 1972–1988) and 10-digit Harmonized System (HS, 1990–2001) goods using Generalized Method of Moments (GMM) identification from exporter supply-and-demand heteroskedasticity. Defining a variety in the Armington (1969) sense as a discrete good–country pair (such as French Bordeaux wine vs. Australian Shiraz), they documented that the number of imported varieties entering the United States more than tripled from 74,667 in 1972 (from 7,731 good categories) to 259,215 in 2001 (from 16,390 good categories). Accounting for this unmeasured expansion of import variety coupled with category-specific elasticities σ_g lowered the variety-corrected U.S. import price index by 1.2 percentage points per year—a cumulative 28.0% reduction in the true cost of imports over 1972–2001, corresponding to an equivalent-variation welfare gain of 2.6% of U.S. GDP in 2001.

Love of Variety in New Trade Theory vs. Statistical Price Measurement

In the Dixit–Stiglitz monopolistic competition model and Krugman's (1980) new trade model, a representative consumer's sub-utility over a symmetric set of N_t differentiated varieties with elasticity of substitution σ > 1 is:

If every variety sells at price p and total expenditure on imports is E_t = N_t p x, the minimum expenditure per unit of utility (the theoretical CES import price index) is:

Because σ > 1 implies 1 / (1 − σ) = −1 / (σ − 1) < 0, an increase in the number of available varieties N_t lowers the true price index P_t (raising real income E_t / P_t = N_t^1/(σ−1)(E_t / p)) even if the unit price p of every variety remains unchanged. Moreover, the magnitude of this variety welfare gain is governed inversely by σ − 1:

• When varieties are close substitutes (σ → ∞, homogeneous commodities like crude oil or unwrought aluminum), 1 / (σ − 1) → 0: adding a new country source yields negligible variety gain.
• When varieties are strongly differentiated (σ close to 1, such as specialized machinery, pharmaceuticals, or fine wine), 1 / (σ − 1) is large: adding new country-good varieties yields substantial welfare gains.

Despite the theoretical importance of this mechanism (emphasized by Romer, 1994, who showed that ignoring newly traded varieties understates the welfare costs of trade barriers), official statistical agencies construct aggregate import price indices using matched-model prices of common (continuing) varieties I = I_t ∩ I_t−1 that are traded in both period t − 1 and period t, omitting the entry of new varieties (I_t \ I_t−1) and the exit of disappearing varieties (I_t−1 \ I_t).

Three-Tier Utility Specification

Broda and Weinstein (2006) specified a three-tier nested constant-elasticity-of-substitution (CES) utility (or aggregate production) function for the U.S. economy:

1. Top Tier (Domestic Composite vs. Aggregate Import Composite): Aggregate U.S. utility U_t at time t is a CES aggregate of the composite domestic good D_t and the composite imported good M_t with macro elasticity of substitution κ > 1:

1. Middle Tier (Aggregation Across Imported Good Categories g ∈ G): The composite imported good M_t aggregates consumption M_g,t across all disaggregated imported good categories g ∈ G (7-digit TSUSA or 10-digit HS product codes) with cross-good elasticity γ > 1 (with γ = 1 in the geometric benchmark):

1. Bottom Tier (Aggregation Across Country Varieties c ∈ I_g,t Within Good g): Within each disaggregated good category g ∈ G, the sub-utility index M_g,t is a non-symmetric CES function over the active set of exporting countries c ∈ I_g,t ⊂ C supplying good g in year t, with good-specific elasticity of substitution σ_g > 1 and country-good taste/quality weights d_g,c,t > 0:

where m_g,c,t is the physical quantity of good g imported from country c in year t. Minimizing expenditure ∑_c p_g,c,t m_g,c,t to achieve M_g,t = 1 yields the minimum unit-cost function ϕ_g,t(I_g,t, d_g,t) = [∑_c∈Ig,t d_g,c,t (p_g,c,t)^1−σ_g]^1/(1−σ_g), and the expenditure share of country c in total imports of good g is:

Sato–Vartia Price Index for Common Continuing Varieties

When the set of supplying countries for good g is restricted to the common continuing subset I_g ≡ I_g,t ∩ I_g,t−1 with constant taste weights d_g,c,t = d_g,c,t−1 = d_g,c, the Sato (1976) and Vartia (1976) Theorem establishes that the ratio of minimum unit costs over I_g is a geometric mean of individual country price relatives (the Conventional Matched-Model Price Index P_g^conv), requiring no knowledge of the unknown taste weights d_g,c or σ_g:

where w_g,c,t are the Sato–Vartia logarithmic ideal weights computed from expenditure shares s_g,c,t(I_g) within the continuing subset I_g:

Feenstra's (1994) Price Index for New and Disappearing Varieties

Now allow new country varieties to enter in period t (I_g,t \ I_g ≠ ∅) and old varieties to exit after period t − 1 (I_g,t−1 \ I_g ≠ ∅), provided the overlapping continuing set is non-empty (I_g = I_g,t ∩ I_g,t−1 ≠ ∅).

Theorem (Feenstra's CES Price Index with Variety Change, 1994): The true CES price index π_g,t ≡ ϕ_g,t(I_g,t, d_g) / ϕ_g,t−1(I_g,t−1, d_g) equals the conventional Sato–Vartia continuing-goods price index P_g(I_g) multiplied by the Feenstra Lambda Ratio raised to the power 1 / (σ_g − 1):

where λ_g,t and λ_g,t−1 denote the fraction of total good-g expenditure in period t and period t − 1, respectively, devoted to the common continuing varieties c ∈ I_g:

Proof and Economic Intuition of the Feenstra Lambda Ratio

Summing the CES expenditure share s_g,c,t = d_g,c [p_g,c,t / ϕ_g,t(I_g,t)]^1−σ_g over the continuing subset c ∈ I_g gives λ_g,t = [ϕ_g,t(I_g) / ϕ_g,t(I_g,t)]^1−σ_g, or ϕ_g,t(I_g,t) = ϕ_g,t(I_g) · (λ_g,t)^1/(σ_g−1). Dividing period t by period t − 1 yields π_g,t = P_g(I_g) (λ_g,t / λ_g,t−1)^1/(σ_g−1). Notice two key properties of Feenstra's formula:

1. Expenditure-Share Weighting of New vs. Exiting Varieties: λ_g,t / λ_g,t−1 weights every entering and exiting country variety by its quality-adjusted expenditure share rather than counting raw country numbers. If new exporting countries capture 30% of period-t expenditure (λ_g,t = 0.70) while exiting countries accounted for only 2% in t − 1 (λ_g,t−1 = 0.98), then λ_g,t / λ_g,t−1 = 0.714 < 1, lowering π_g,t below the conventional index P_g(I_g).
2. Aggregation Across All Imported Goods and Welfare Gain: Aggregating across all imported good categories g ∈ G using Sato–Vartia geometric weights w_g,t yields the aggregate Variety-Corrected Import Price Index Π^M = P_conv^M × Λ_t, where Λ_t = ∏_g∈G (λ_g,t / λ_g,t−1)^{w_g,t/(σ_g−1)} is the Aggregate Import Bias Index. Weighting Λ_t by the U.S. import expenditure share w_M,t gives the equivalent-variation welfare gain as a percentage of GDP:

Double-Differenced Import Demand and Export Supply System

To estimate σ_g > 1 for every disaggregated good g (14,541 7-digit TSUSA goods in 1972–1988 and 16,390 10-digit HS goods in 1990–2001) without external country-good instrumental variables, Broda and Weinstein followed Feenstra (1994) and Rigobon (2003) by exploiting panel double-differencing and heteroskedasticity of supply and demand shocks across exporting countries:

1. Import Demand Equation: Taking first differences Δ_t = ln(·_t) − ln(·_t−1) of the CES expenditure share s_g,c,t and differencing relative to a reference country k (Δ^kx_g,c,t ≡ Δx_g,c,t − Δx_g,k,t) eliminates the good-level price index ϕ_g,t:

1. Export Supply Equation: Each exporting country c supplies good g along an upward-sloping inverse export supply curve with export supply elasticity ω_g ≥ 0, which after double-differencing gives:

The Feenstra–Hansen–Singleton Orthogonality Parabola

Under the identifying assumption that relative demand shocks ε_g,c,t^k and relative supply shocks δ_g,c,t^k are contemporaneously uncorrelated (E_t[ε_g,c,t^k · δ_g,c,t^k] = 0), multiplying the demand error and the supply error together yields a linear relation among the second moments (variances and covariances over time t) of observed relative prices and relative shares for each country c:

where E_t[u_g,c,t] = 0, and the reduced-form coefficients (θ_1,g, θ_2,g) map into (σ_g, ω_g):

Because different exporting countries c = 1, …, C_g experience different variances of supply and demand shocks (σ_ε,c² / σ_δ,c² varies across countries c, i.e., heteroskedasticity across exporters), Weighted Generalized Method of Moments (GMM) across at least 3 exporting countries uniquely identifies both σ_g > 1 and ω_g ≥ 0.

• Elasticities by Disaggregation and Rauch (1999) Product Differentiation: When goods are aggregated to coarse 3-digit SITC categories, the median elasticity of substitution across varieties is 2.2; when disaggregated to 10-digit HS categories, the median elasticity rises to 3.1 (1990–2001) and 3.7 (1972–1988). Grouping goods by James Rauch's (1999) classification confirmed that differentiated manufactures have lower median elasticities (σ_g = 2.9) than exchange-traded commodities (σ_g = 3.5 to 4.5).
• Magnitude of the Import Price Bias and GDP Welfare Gain: Between 1972 and 1988, unmeasured variety growth lowered the true U.S. import price index by 19.7% relative to the conventional matched-model index (−1.4 percentage points per year); between 1990 and 2001, it lowered the import price index by an additional 8.3% (−0.8 percentage points per year). Across the entire 1972–2001 period, the variety-adjusted U.S. import price index fell 28.0% faster than the unadjusted official import price index (−1.2 percentage points per year). Weighting this 28.0% cumulative import price drop by the U.S. import share of GDP (~10% in 2001) yields a total equivalent-variation welfare gain of 2.6% of U.S. GDP in 2001 (worth approximately 260 billion U.S. dollars per year in 2001 dollars).

Why does Feenstra's (1994) formula use expenditure share ratios (λ_g,t / λ_g,t−1) instead of simply counting the number of varieties (N_t / N_t−1)?

In Krugman's textbook model, all varieties are assumed to be symmetric (identical quality d_g,c = 1 and identical market share 1/N_t), in which case λ_g,t / λ_g,t−1 reduces to N_t−1 / N_t. In real trade data, however, varieties are asymmetric: China entering a 10-digit footwear category and capturing 25% of U.S. imports creates far more consumer surplus than a tiny exporter entering with 0.001% market share. Feenstra's ratio λ_g,t / λ_g,t−1 weights every entering and exiting country variety by its revealed quality-adjusted expenditure share. While the raw count of imported varieties rose by 247% (from 74,667 to 259,215), the expenditure-weighted Feenstra variety shift was a much more realistic 13.5% (1 − 0.865), because newly entering varieties are on average smaller than incumbent varieties.

Why is it necessary to estimate a separate elasticity of substitution (σ_g) for every 10-digit good instead of using a single macro elasticity like σ = 5?

Because the variety welfare term (λ_g,t / λ_g,t−1)^1/(σ_g−1) is nonlinear and convex in 1/(σ_g − 1), goods with low elasticities (σ_g ∈ (1.2, 2.5)) generate disproportionately large variety gains, and variety growth is empirically correlated with σ_g. Replacing the 30,000 heterogeneous σ_g estimates with a single constant elasticity across all goods would mismeasure both the level and the sectoral distribution of welfare gains due to Jensen's Inequality.

How can Broda and Weinstein identify both supply and demand elasticities without external instrumental variables?

Following Feenstra (1994) and Rigobon (2003), identification comes from heteroskedasticity across exporting countries. For a given 10-digit good g, the orthogonality condition E_t[ε_g,c,t^k · δ_g,c,t^k] = 0 defines a parabola in the (σ_g, ω_g) parameter space whose coefficients depend on the second moments Var(Δ^k ln p), Var(Δ^k ln s), and Cov(Δ^k ln p, Δ^k ln s) of country c. Because different exporting countries experience different exchange-rate and productivity shock variances relative to demand shock variances, their country-level parabolas intersect at a unique point (σ_g > 1, ω_g ≥ 0).

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