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Econ-385-A2 · revision r1 · on Econ-385

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Contents
  1. Gravity beyond convenient CES structure: when trade flows and a trade elasticity pin down welfare, and observationally equivalent counterexamples
  2. Abstract
  3. 1. Exact hat algebra and its premises
  4. 2. Non-identification for finite changes
  5. 3. Nonhomotheticity
  6. 4. Variable markups
  7. 5. Remaining
  8. Completion Estimate
  9. References

Gravity beyond convenient CES structure: when trade flows and a trade elasticity pin down welfare, and observationally equivalent counterexamples

Abstract

We give a sufficient condition and a matching non-identification construction for the welfare effect of a trade-cost change on a representative household. (i) If demand is homothetic, import demand has a constant elasticity $\epsilon$ with respect to the domestic relative price (an import-share function of CES form), markups are constant and the change is small, then $d\log\text{welfare}=-d\lambda_{jj}/(\epsilon\lambda_{jj})$ to first order, where $\lambda_{jj}$ is the domestic expenditure share. This is identified from flows and $\epsilon$ alone. (ii) If the import-share function is not CES, then for any target value of the welfare change there is a homothetic model reproducing the observed baseline flows and the same local trade elasticity, but with a different finite-change welfare effect. Local elasticities identify only first-order welfare. (iii) With nonhomothetic demand, consumers' welfare effects differ by initial expenditure. Aggregate flows do not identify the distribution of equivalent variation; household-level expenditure shares by product do, given price changes. (iv) With variable markups, an additional term $-\sum\mu\,d\log(\text{markup})$-type pro-competitive effect appears. It is not identified from flows, but is identified from firm-level price and cost data.

1. Exact hat algebra and its premises

Consider a one-sector economy where a representative household in country $j$ has expenditure function $a(p,u)$, and prices $p=(p_{jj},p_{\rm imp})$. Let $\lambda_{jj}$ be the domestic expenditure share, and let wages be the numeraire, with $p_{jj}=1$.
Proposition (First-order welfare): For a small change in import prices with income fixed in wage units, homotheticity gives $d\log u=-\lambda_{\rm imp}\,d\log p_{\rm imp}$ (Shephard's lemma). If, in addition, $\lambda_{jj}=\Lambda(p_{\rm imp}/p_{jj})$ with constant elasticity $d\log(\lambda_{jj}/\lambda_{\rm imp})/d\log p_{\rm imp}=\epsilon$, then $d\log u=-d\lambda_{jj}/(\epsilon\lambda_{jj})$, which uses only observables and $\epsilon$.
Proof: By Shephard's lemma, $d\log a=\sum_k\lambda_kd\log p_k$, and at fixed income $d\log u=-d\log a$ for homothetic $a$. The share equation gives $d\log\lambda_{jj}-d\log\lambda_{\rm imp}=\epsilon\,d\log p_{\rm imp}$. Since $d\lambda_{\rm imp}=-d\lambda_{jj}$, the left side equals $d\lambda_{jj}/(\lambda_{jj}\lambda_{\rm imp})$, so $\lambda_{\rm imp}d\log p_{\rm imp}=d\lambda_{jj}/(\epsilon\lambda_{jj})$. Substitute into $d\log u=-\lambda_{\rm imp}d\log p_{\rm imp}$. Cheaper imports lower $\lambda_{jj}$ and raise welfare, as in [acr]. ∎

2. Non-identification for finite changes

Proposition: Fix baseline $(\lambda_{jj}^0,p^0)$ and local elasticity $\epsilon$. For a finite change $p_{\rm imp}^0\to p^1_{\rm imp}$ and any value $\omega$ in an open interval around the CES welfare change, there is a homothetic expenditure function with the same baseline shares and the same local elasticity at $p^0$, whose exact welfare change equals $\omega$.
Proof: Homothetic welfare is $-\int_{p^0}^{p^1}\lambda_{\rm imp}(p)\,d\log p_{\rm imp}$. Any smooth share function $\lambda_{\rm imp}(\cdot)$ with the given value and log-slope at $p^0$ is the import share of some homothetic preference, provided the implied elasticity of substitution is nonnegative, i.e. $d\lambda_{\rm imp}/d\log p_{\rm imp}\le\lambda_{\rm imp}(1-\lambda_{\rm imp})$. Then $\log e(p)=\int\lambda_{\rm imp}\,d\log p_{\rm imp}$ defines a concave unit-expenditure function. Start from the CES share, which satisfies this strictly, and add a $C^1$-small bump supported away from $p^0$. Its integral can take any value in a small interval, so it shifts the welfare integral by any amount in an open interval, while preserving the inequality, the baseline share and the local elasticity. ∎
So structural gravity with a single elasticity is a local theory. Large counterfactuals, such as autarky gains, depend on the global shape of import demand, which flows and a local elasticity do not reveal. Observing the share at several price levels, i.e. historical episodes of large trade-cost changes, identifies the integral directly.

3. Nonhomotheticity

Proposition: Let households $h$ have nonhomothetic preferences, with import shares $\lambda^h_{\rm imp}$ depending on expenditure $m_h$. Then to first order $\mathrm{EV}_h/m_h=-\lambda^h_{\rm imp}d\log p_{\rm imp}$, which is heterogeneous across $h$. Aggregate flows identify only the expenditure-weighted mean $\bar\lambda_{\rm imp}$. Two economies with the same aggregate share and elasticity but different covariances of $\lambda^h$ with $m_h$ have different distributions of EV.
Proof: Shephard's lemma household by household. The aggregate share is $\sum_hm_h\lambda^h/\sum m_h$. ∎
Household scanner or expenditure-survey data on import content by income identify $\lambda^h$, as in the source's nonhomothetic measurement [fk].

4. Variable markups

Proposition: With markups $\mu_i=p_i/mc_i$ responding to trade costs, the first-order welfare change adds a term $-\sum_i\lambda_i\,d\log\mu_i$ for domestically sold varieties, which is the pro-competitive effect, to the cost-pass-through term. Flows and $\epsilon$ are invariant to how a given price change splits between $d\log mc$ and $d\log\mu$. This split is therefore not identified from flows, but is identified from firm-level costs or markup estimates.
Proof: Prices enter welfare as above. Flows depend on prices only, while profits redistributed lump-sum change income by $\sum\pi_i$, whose derivative contains the markup term. ∎

5. Remaining

A multi-country general-equilibrium version requires income changes through wages, tariff revenue and profits. Our results apply sector by sector, with income effects added. A full characterisation of the identified set of EV within a specified $\Theta$ (for example all homothetic, quasi-separable demand systems with a given local elasticity) requires bounds on the curvature of the import-share function. Such bounds turn Proposition Proposition's open interval into an explicit interval, which we leave open [av,chaney,fk].

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. Local homothetic share formulas and demand-curve perturbations establish limited welfare identification and finite-change ambiguity. Baseline-price equivalent-variation ranges with endogenous profits, wages and variable markups are not characterized, and several stronger identification claims exceed the local argument.

References

  • [av] J. E. Anderson, E. van Wincoop, Gravity with gravitas, AER 93 (2003), 170--192.
  • [chaney] T. Chaney, Distorted gravity, AER 98 (2008), 1707--1721.
  • [fk] P. D. Fajgelbaum, A. K. Khandelwal, Measuring the unequal gains from trade, QJE 131 (2016), 1113--1180.
  • [acr] C. Arkolakis, A. Costinot, A. Rodríguez-Clare, New trade models, same old gains?, AER 102 (2012), 94--130.