Productivity Parameters

This section outlines the procedure for estimating and calibrating the following productivity parame-ters: the dispersion parameters of the Frechet distributions of task productivity (θa) and sectoral TFP (θ), the adoption parameter (β) that governs convergence of task productivity, the sectoral output elas-ticities of different tasks (ζ^k_a), and, ultimately, the diffusion parameters ( ˜ηa) that govern the extent of dynamic scale economies for different tasks.

I calibrate the dispersion parameters externally using estimates from the literature. Specifically, I set the trade elasticityθ to 4 (Simonovska and Waugh(2014)) andθa to 1.13 (Burstein et al.(2015)).

Moreover, I calibrateζ^k_a to the sector- and occupation-specific wage share of tradable sectors in the United States in 1970.³². This leaves the rest of this section to the estimation ofβand ˜ηa.

4.1.1 Comparative Advantage

Several international GE models -including the EK-style model outlined above- deliver a gravity equa-tion of the form:

lnπ_nm,t^k =δ^k_m,t+µ^k_n,t−θlnτ_nm,t^k (30) whereπ^k_nm,t is the import share of countrynfrom country min sector k. δ_m,t^k is an exporter-specific fixed effect in sectork,µ^k_n,tis an importer-specific fixed effect, andθlnτ_nm,t^k covers the sector-specific effect of trade costsτ_nm,t^k between the two countries that affect the trade share with elasticityθ.

Ifτ_ni,t^k = τ_in,t^k ∀n 6= iandτ_nn^k ′,t = 1 ∀n = n^′ (Head and Ries(2001)), one can recover trade costs using³³

τ_ni,t^k = [^π

kni,t

π_nn,t^k π_in,t^k

π^k_ii,t]^−2θ1 (31)

In an Eaton-Kortum type model (e.g. Caliendo and Parro (2015)), the exporter- and

importer-32For details on occupational groups and micro-data used for wage sector- and occupation-specific wage shares, see section5and AppendixD

33I choose to specify trade costs symmetrically to be consistent with the counterfactual exercise in section7.1. An alterna-tive would be to specify trade costs as a sector- and year-specific log-linear function of distance variables (e.g. Levchenko and Zhang(2016),Bartelme et al.(2019)). Using this specification, estimated comparative advantage terms or spillover parameters are not qualitatively different.

specific fixed effects take the form

δ_m,t^k =−θlnc^k_m,t/ ¯c^k_t +lnT_n,t^k / ¯T_t^k (32)

µ^k_n,t =lnΦ^k_n,t/ ¯Φ^k_n,t (33) where a bar over a variable refers to its global (unweighted) mean. The exporter-specific fixed effect thus captureseffectiveunit costs of countrymin sectork, which reflect the country’s comparative ad-vantage in that sector.³⁴ The importer-specific fixed effect captures the ’multilateral resistance’ in sec-torkof countrynsuch thatΦ^k_n,t = (∑_i^N₌₁T_i,t^k (c^k_i,tτ_ni,t^k )^−θ)⁻¹.Hanson et al.(2015), followingEaton et al.

(2012), recast this gravity equation to allow for zero trade flows by assuming that in each industry-country pair only a finite number of firms make productivity draws. As a result, equation30 holds only in expectation:

E[π_nm,t^k ] = ^exp[δ_m,t^k −θlnτ_nm,t^k ]

exp[µ^k_n,t] ⁽³⁴⁾

Combining the gravity and unit cost equations, and expressing them relative to the global mean yields an estimating equation

lnE[π^k_nm,t] =lnT_m,t^k / ¯T_t^k−θlnc^k_m,t/ ¯c^k_t

| {z }

exporter fixed effectδ_m,t^k

+^Φk_n,t/ ¯Φ^k_n,t

| {z }

importer fixed effectµ^k_n,t

−¹

2ln π_ni,t^k π^k_nn,t

π_in,t^k

π_ii,t^k +ν_nm,t^k (35)

whereν^k_nm,tis a mean-zero misspecification term. I estimate equation35separately for each sector and year under the constraint that the exporter and importer fixed effects each sum up to zero. I use both log-linear OLS and PPML (Silva and Tenreyro,2006).³⁵

4.1.2 Spillovers

Changes in a country’s sector-specificeffectiveunit costs ˜c^k_n,tcan be expressed as

∆ln ˜c^k_n,t =^∆lnc^k_n,t−^∆lnT_n,t^k (36)

34Note that in a world without trade costs (τ_ni,t^k = 1∀n,i ∈ N), the exporter fixed effects are a measure of revealed comparative advantage (RCA), i.e. for any two countriesnandmand any two sectorskandk^′, ln ^RCAkn,t/RCAk

′ n,t

RCA^k_m,t/RCA^k_m,t^′ = ^δkn,t/δk

′ n,t

δ^k_m,t/δ^k_m,t^′ . See also Appendix sectionB.

35The correlation between the estimated fixed effects of these two methods is always higher than 0.99. In the rest of the paper I use the OLS estimates whenever referring to results or computations based on these fixed effects

Using the definition of unit costs and arbitrage in the labor market gives

∆ln ˜c^k_n,t =^∆lnwn,t−

A

∑

a=1

ζ^k_a∆lnT_n,t^a −^∆lnT_n,t^k (37) In the endogenous growth model outlined above, we can express the growth of the task produc-tivity levelT_n,t^a in log changes as

∆lnT_n,t^a = β0+ ¹

θaη˜alnL^a_n,t−1+ (β−1)lnT_n,t^a +ǫ^a_n,t (38) where β0 is a constant, η_n,t^a is a diffusion parameter that differs by task, country and time. ǫ_n,t^a is a shock specific to task a in countrynat time t. Plugging this expression into equation37 on the right-hand-side, and the definition of the exporter fixed effect on the left-hand-side, while expressing all to the global unweighted mean, yields an estimating equation:

1

θ∆δ^k_n,t = −β0

|{z}

constant

−¹ θa

A

∑

a=1

˜

ηaζ^k_alnL^a_n,t−1/Ln,t−1+ (β−1)¹

θδ_n,t^k ₋₁+γn,t+ǫ˜_n,t^k (39) whereγn,tis a fixed effect that captures∆lnwn,t+ (1−β)lnwn,t−1, and ˜ǫ^k_n,tis a an error term capturing

∑_a^A₌₁ζ^k_aǫ^a_n,t−^∆lnT_n,t^k + (1−β)lnT_n,t^k ₋₁.³⁶ The only unknown param of interest are ˜ηa, and β. ¹_θ∆δ_n,t^k and¹_θδ^k_n,t−1can be constructed using a value ofθand gravity estimates. I first estimate equation39for 10 year periods using OLS.

This naive estimation method raises endogeneity issues, however. A potential concern could be that producers in fast-growing sectors preemptively increase production in periodt−1, anticipating productivity growth between t and t−1. Such anticipatory behavior would generate a correlation between ζ^k_aL_n,t^a ₋₁ and ζ^k_aǫ_n,t^a or between ζ^k_aL^a_n,t−1 and ∆lnT_n,t^k . To circumvent these kind of supply side concerns, I rely on foreign demand shocks that generate variation in the task employment share L^a_n,t/Ln,t.

Note that the task employment share has a model-implied equivalent in the weighted average of exports, i.e.

L^a_n,t/Ln,t = ^∑

K

k=1ζ^k_a∑_n^N′=1X_n^k′n,t

∑^K_k=1∑_n^N′=1X^k_n′n,t

(40) Using the gravity equation in section9, exports of countrynin sectorkcan be expressed as (relative

36Note that under the assumption that both T_n,t^k andǫ_n,t^a are orthogonal to ζ^k_athe regression coefficient on ζ^k_aln^L_L^an,t−1_n,t−1 etersyields consistent estimates of_θ¹

aη˜_ausing OLS.

to the global mean):

N

∑

n^′6=n

X_n^k′_n,t=δ^k_n,t·

N

∑

n^′6=n

E^k_n^′_,tµ^k_n^′_,t(τ_n^k′_n,t)^−θ

| {z }

FMA^k_n,t

(41)

where δ^k_n,t captures effective unit costs relative to the global mean and E_n^k′,t is total expenditure of countryn^′in sectork. While the exporter fixed effect reflects the effect of supply side factors in country non its exports, the second term captures foreign demand. I will refer to this sum of foreign demand factors as Foreign Market Access (FMA) (Bartelme et al.(2019)).

I can now construct a synthetic measure of task employment share that is only driven by demand shocks. Removing supply side variation by setting effective unit costs to the global mean:

(L_n,t^a /Ln,t)^FMA = ^∑

Kk=1ζ_a^k·FMA^k_n,t

∑^K_k=1FMA^k_n,t (42) I also estimate equation39using this demand shock driven measure of task employment. Finally, I run IV 2SLS using(L^a_n,t/Ln,t)^FMAas instruments forL_n,t^a /Ln,t.

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