Imported Inputs and the Gains from Trade Ananth Ramanarayanan University of Western Ontario April, 2018
Abstract This paper characterizes how plant-level heterogeneity in imported input use a¤ects the aggregate welfare gains from trade. Heterogeneous plants choose a fraction of inputs to source from the lowest cost source country, with the rest purchased domestically. Sourcing more inputs requires higher up-front …xed costs, but reduces variable input costs, so import shares are increasing in plant size. Plant-level and aggregate import demand are nonhomothetic, which lowers the equilibrium elasticity of trade ‡ows with respect to trade costs relative to the partial trade elasticity. The welfare gain from trade is higher than in models in which the welfare gain is governed by the partial trade elasticity. When calibrated to Chilean plant-level data, this di¤erence is large.
Contact: Department of Economics, University of Western Ontario, Social Science Centre, Room 4071, London, Ontario, Canada, N6A 5C2. Email:
[email protected]. For helpful comments and discussions, I thank the Editor and two anonymous referees, as well as Costas Arkolakis, Hiro Kasahara, Jim MacGee, B. Ravikumar, and audiences at the 2011 Warwick CAGE International Trade Research Day, the 2012 SED and Midwest Macroeconomics Meetings, Arizona State, York, McMaster, Toronto, UBC, and Rochester. I gratefully acknowledge …nancial support from the University of Western Ontario’s International Research Award.
1
Introduction
Intermediate inputs comprise the bulk of international trade in goods for industrialized countries. A recent empirical literature examining …rm- and plant-level data …nds that imported inputs are concentrated among relatively few producers, and there is substantial heterogeneity in the share of input expenditures spent on imports.1 If importing inputs lowers production costs, then these di¤erences across plants suggest that the overall gains from trade depend on how these cost savings are distributed across plants of di¤erent sizes. In this paper, I evaluate how plant-level importing decisions a¤ect the welfare gains from trade. I develop a general equilibrium model in which plants endogenously choose heterogeneous import shares. Plants producing …nal goods use a continuum of intermediate inputs which can be produced in di¤erent countries at di¤erent unit costs and shipped internationally subject to proportional trade costs, as in the Ricardian model of Eaton and Kortum (2002). For each input, a plant chooses whether to pay a …xed cost to “source”the input – buy it from the cheapest location – or to buy it domestically. This is in contrast to the standard Ricardian assumption that all purchasers buy each good from the cheapest location. A plant thus faces a tradeo¤ between the higher …xed cost required to source more inputs and the lower marginal production cost that comes from sourcing those additional inputs. Plants di¤er in their underlying productive e¢ ciency, and more e¢ cient plants …nd it pro…table to source –and hence import –more inputs than relatively ine¢ cient plants. As larger plants have a higher return to sourcing more inputs, import shares vary with plant size, so that plant-level import demand is nonhomothetic even though the underlying plant technologies are homothetic. This nonhomoetheticity leads to general equilibrium e¤ects of changes in trade costs on the sourcing decision. A reduction in trade costs, for example, raises …nal consumption expenditures and lowers the …nal goods price index, which have opposing e¤ects on the e¤ective demand faced by each plant and hence their incentive to source inputs. As a result, aggregate import demand is nonhomothetic, in the sense that the aggregate import share depends on aggregate income and the aggregate price index in each country, in addition to relative prices across countries. I characterize and quantify how the nonhomotheticity arising from plant-level input sourcing decisions impacts the welfare gains from trade. Without endogenous sourcing decisions, a version of the formula derived in Arkolakis, Costinot, and Rodríguez-Clare (2012) (henceforth ACR) applies, relating the welfare gain from trade relative to autarky to three statistics: the share of intermediate goods in total production costs ( ), the aggregate import share 1
Kasahara and Lapham (2007) have documented these facts for Chile, while similar facts can be found in Kurz (2006) and Bernard, Jensen, and Schott (2009) for the US, Biscourp and Kramarz (2007) for France, Amiti and Konings (2007) for Indonesia, and Halpern, Koren, and Szeidl (2009) for Hungary.
2
(M=X), and the partial trade elasticity ("P ) – i.e., the elasticity of the aggregate import ratio with respect to the trade cost holding …xed aggregate income and prices. In the model with endogenous sourcing decisions, I show that, to a …rst-order approximation, in response to a change in the variable trade cost , the changes in aggregate consumption C and in the aggregate domestic share of input expenditures (one minus the import share) satisfy d log C =
"F
d log (1
M=X)
where "F is the full trade elasticity, factoring in equilibrium changes in aggregate income and prices. This expression states that as the aggregate import share falls (so that the domestic share rises), real …nal consumption falls with an elasticity proportional to and inversely proportional to "F . This relationship is almost the same as a local version of the ACR formula, but with the full elasticity "F in place of the partial elasticity "P . These elasticities are unequal because aggregate import demand is nonhomothetic, and the ratio of the two elasticities gives the extent to which welfare gains from trade diverge from the ACR formula. Both the full and partial trade elasticities in my model are endogenous objects that potentially vary with the level of trade costs, so I show that quantitatively, the di¤erence between them can be large. I calibrate the model to Chilean plant-level data and two aggregate statistics, the aggregate import share and the partial trade elasticity. Fixing the partial trade elasticity facilitates comparison to ACR, in which the partial elasticity is assumed to be a policy-invariant constant. In addition, as ACR point out, it is the partial trade elasticity that is identi…ed from gravity-based estimates of the trade elasticity. For the calibrated model, the di¤erence between the full and partial trade elasticities is pronounced: the targeted partial elasticity is 5:0, and the full elasticity is about 1:6. This translates into a local elasticity of welfare with respect to the domestic share (the coe¢ cient " ) that is about 3 times larger (in absolute value) than the one implied by the ACR formula. Moreover, this …rst-order result provides a reasonably good approximation for large changes as well: welfare gains relative to autarky are 10:44 percent in the model, as opposed to 3:90 percent implied by the ACR formula. The ratio between these gains relative to autarky is about 2:7, a bit smaller than the factor of 3 di¤erence in the …rst-order welfare gains.2 These results show that it is the nonhomotheticity of plant-level import shares, rather than heterogeneity in import shares per se, that leads to a di¤erence in welfare gains relative 2
Melitz and Redding (2015) show that deviating from Pareto-distributed productivity in an otherwise ACR-type model can also create a gap between the partial and full trade elasticity. However, their analysis focuses on variation in the partial trade elasticity as the trade cost changes. This movement in the trade elasticity is dominated in my model by the gap between the partial and full trade elasticities, as can be seen from the fact that the deviation in autarky gains from ACR is close to the deviation in local gains.
3
to models that satisfy the ACR conditions. Indeed, with exogenously given heterogeneous plant-level import shares, regardless of the distribution of import share and size, aggregate import demand is homothetic, there is no di¤erence between the full and partial trade elasticities, and the ACR formula applies, at least to a …rst-order approximation. The main message of these results is that heterogeneity in import shares is important for welfare gains only because it is informative about the nonhomotheticity in import demand that is implied by plants endogenously choosing heterogeneous import shares. This paper is related to work on quantifying the gains from trade in environments with plant-level importing decisions. Most closely related is Blaum, Lelarge, and Peters (2015), who show that in a set of models with heterogeneous import shares across plants, the joint distribution of size and import shares is su¢ cient to characterize the e¤ect of trade on reductions in the consumer price index relative to autarky. They characterize when a model with common import shares across plants (i.e., one for which the ACR formula holds) overestimates or underestimates these price gains relative to a model with heterogeneous shares with same preference and technology parameters. My analysis mainly di¤ers in two ways: …rst, my results show that the nonhomotheticity of plant-level import shares, rather than heterogeneity in these import shares, leads to di¤erent welfare implications relative to ACR.3 Second, I focus on welfare –as measured by aggregate consumption –rather than consumer prices; the di¤erence is that welfare takes into account the resources used for importing. In Blaum, Lelarge, and Peters, welfare gains from trade do not always follow the same pattern as their measures of aggregate price gains, suggesting that consumer prices are a poor indicator of welfare in models with heterogeneous importing decisions. In addition, the spirit of my analysis is to compare across models (i.e. with heterogeneous and common import shares) with a …xed aggregate partial trade elasticity. Blaum, Lelarge, and Peters, on the other hand, compare heterogeneous-share and common-share models with a common set of underlying parameters, and di¤erent aggregate partial trade elasticities. Also related to this paper are Gopinath and Neiman (2011) and Antràs, Fort, and Tintelnot (2014). Gopinath and Neiman show that heterogeneity in the adjustment of the number of inputs imported by Argentinean plants contributes to high-frequency movements in aggregate productivity. In contrast, I focus on the measurement of welfare gains from trade, and highlight the comparison to a model without heterogeneity controlling for the aggregate trade share and trade elasticity. In Antràs, Fort, and Tintelnot (2014), a plant chooses the countries from which to source inputs, and sourcing from more countries requires paying a 3
Blaum, Lelarge, and Peters (2013) test the prediction of most models (including mine) that, once the input sourcing decision itself is …xed, the intensive margin of plant-level import demand is homothetic, and they …nd evidence against this. This …nding further underscores the importance of non-homothetic plant-level import demand for welfare.
4
higher …xed cost. They focus on the margin of the number of source countries in accounting for import ‡ows. In contrast to these papers, I analytically characterize the plant’s problem and express aggregates in terms of a small number of variables and parameters. Thus, my model allows a transparent identi…cation of parameters, and I map parameters governing the costs of importing and productivity heterogeneity to moments of the endogenous crosssectional distributions of import shares and plant size along with key aggregate statistics. The welfare results are also related to recent work by Levchenko and Zhang (2014), Ossa (2015), and Costinot and Rodríguez-Clare (2014), who show that gains from trade are larger in multi-sector models than in one-sector models due to variation in trade shares or trade elasticities across sectors. These papers assess the discrepancy between the one-sector welfare formula and the actual gains for given parameters, which would result in the partial trade elasticity varying across models. The mechanism I analyze is di¤erent, and operates through the nonhomotheticity of aggregate import demand creating a gap between the partial trade elasticity and the full trade elasticity, the latter of which is relevant for welfare. In a di¤erent vein, Simonovska and Waugh (2014) show that matching a particular cross-sectional statistic in trade models with di¤erent underlying plant-level features results in di¤erent aggregate partial trade elasticities, and thus di¤ering gains from trade across those models even though they all satisfy the ACR formula. This paper also relates to empirical papers by Amiti and Konings (2007), Kasahara and Rodrigue (2008), and Halpern, Koren, and Szeidl (2009), who all …nd that imported inputs raise plant-level productivity.4 In my model, sourcing more inputs lowers input prices, so a plant that sources (and hence imports) more inputs appears more productive –it produces more output with the same expenditure on inputs –than a plant that sources fewer inputs. Importing plants, therefore, are larger and more productive than nonimporting plants, both because they tend to be more e¢ cient producers, and because importing ampli…es e¢ ciency di¤erences. In addition, liberalizing trade generates within-plant productivity gains at importing plants, as they reduce their marginal costs by sourcing and importing more inputs. In the calibrated model, plant-level productivity gains are in line with those estimated for Chile in Kasahara and Rodrigue (2008), and within-plant productivity gains contribute substantially more than reallocation of resources from low-e¢ ciency to high-e¢ ciency plants in accounting for the aggregate total factor productivity (TFP) gains from trade. Section 2 sets out the model, and Section 3 characterizes the welfare predictions and the di¤erence between the partial and full trade elasticities in a symmetric version of the model. Section 4 calibrates the model to Chile and quanti…es the welfare implications of the model, 4
Goldberg, Khandelwal, Pavcnik, and Topalova (2010) also measure the bene…ts of imported inputs using data on …rms, though they measure the e¤ects on the number of products …rms produce.
5
while Section 5 considers sensitivity of the quantitative results to some parameter values and extends the welfare characterization to the case of asymmetric countries.
2
Model
The model economy consists of two countries (indexed by i; j = 1; 2) in which production takes place in two stages: internationally tradeable intermediate goods are produced with labor, and nontradable …nal goods are produced using labor and intermediate goods. Final goods are produced by monopolistically competitive plants that di¤er in their productive e¢ ciency and choose the fraction of goods to import.
2.1
Production and prices of intermediate goods
Intermediate goods production is similar to the Ricardian models of Dornbusch, Fischer, and Samuelson (1977) and Eaton and Kortum (2002). Perfectly competitive producers in each country have technologies to produce a continuum of intermediate goods labelled ! 2 [0; 1]. Good ! can be produced with labor in each country i with e¢ ciency Zi (!). Denoting the i . As in Eaton wage rate in country i as wi , the cost of producing a unit of good ! is Ziw(!) and Kortum (2002), Zi (!) is the realization of a random variable drawn independently and identically across ! and i from a Fréchet distribution, Pr (Zi (!)
Ti Z
Z) = e
Here, Ti controls the level and the dispersion of e¢ ciency within country i. Intermediate goods are tradeable, but producers face proportional trade costs when selling internationally: in order to sell one unit of any good ! in country j, a producer in country i must produce ij units, where ij = > 1 if i 6= j and ii = 1. Since intermediate goods producers are perfectly competitive, the price of a good sold from i to j is qij (!) =
ij wi
Zi (!)
The distribution of prices of goods that country j can potentially buy from country i is: Pr (qij (!)
q) = 1
6
e
Ti (
ij wi )
q
2.2
Final goods production and input sourcing
In each country, a continuum of mass one of heterogeneous plants produce di¤erentiated …nal goods labelled ! ~ 2 [0; 1], and a representative consumer combines these goods with constant elasticity of substitution > 0, Cj =
Z
1
cj (~ !)
1
1
d~ !
0
where cj (~ ! ) is the quantity of good ! ~ consumed in country j. Denoting its price by pj (~ ! ), R 1=(1 ) 1 the price index is Pj = pj (~ !) d~ ! , and cost-minimization leads to the demand function: pj (~ !) cj (~ !) = Cj Pj A plant producing good ! ~ draws an e¢ ciency z (~ ! ), and produces output using labor ` and a composite x of intermediates, according to: y (~ ! ) = z (~ ! ) ` (~ ! )1
x (~ !)
Plants’ e¢ ciencies z are distributed in country j according to a Pareto distribution with density 1 h (z) = zL z (1) with parameters ; zL > 0. The composite intermediate input is given by the Cobb-Douglas aggregate: Z 1 log x~ (!) d! (2) x = exp 0
where x~ (!) refers to units of intermediate good ! and x is units of the composite input. The Cobb-Douglas assumption allows a particularly tractable analytical solution of the sourcing decision, but in section 5 below, I also consider a variation in which the input bundle (2) has elasticity of substitution di¤erent from one. 2.2.1
Two extremes for input sourcing
If plants source each intermediate good from the cheapest country, then, as Eaton and Kortum (2002) show, in country j, all plants would spend a fraction ij on goods from country i, given by: Ti ( ij wi ) (3) ij = P k Tk ( kj wk ) 7
and the price of a unit of the composite input in country j would be: qjs = e
1=
EM =
(4)
j
P where j = k Tk ( kj wk ) and EM is the Euler-Mascheroni constant. In contrast, if a plant purchases all intermediate goods domestically, then the composite input bundle price qjd would equal just the country j term in equation (4), qjd = e which is higher than qjs for any 2.2.2
EM =
Tj wj
1=
(5)
> 0.
Costly sourcing of imports
In contrast with the standard Ricardian model, I assume it is costly to source each input from the cheapest location (hereafter referred to simply as “sourcing”) rather than buy it from domestic suppliers. This cost is a stand-in for the costs of searching for and maintaining a relationship with a foreign supplier and the costs of testing and …nding out whether an imported product is an appropriate substitute for a domestic one. Speci…cally, if a plant sources a fraction n of its inputs, it has to pay g (n) = b (f n 1) units of labor. I assume f > 1, so that the total cost a plant pays is increasing and convex in the fraction of goods sourced. The …xed cost associated with sourcing nothing and purchasing everything from domestic suppliers is normalized to zero, so g (0) = 0. Introducing this …xed cost function generates di¤erences in importing behavior across plants, as seen in the data. The bene…t of sourcing a larger fraction of inputs is that it lowers the cost of the input bundle: if a plant in country j sources n inputs, the price index among those n goods is given by qjs as de…ned in (4), while the price index for the remainder of the inputs is given by qjd , in (5).5 So, the price for a unit of the overall input bundle if a plant sources n of the inputs is 1 n n qj (n) = qjs qjd (6) Since qjs < qjd , qj (n) is decreasing in n: plants that source a higher fraction of their inputs face lower per-unit input costs. Using (4) and (5) along with (3), qj (n) = e
EM =
5
n= jj
Tj
1=
wj
(7)
Although this model is static, there is an implicit timing assumption: plants choose the fraction n of inputs to source before the realization of intermediate good producers’e¢ ciencies Zi (!).
8
where jj is the fraction of expenditures on the n sourced inputs that is purchased domesti1= cally, while 1 indexes jj is the fraction of sourced inputs spent on imports. Just as jj the welfare gain of trade in the Eaton-Kortum model, here it indexes the per-input cost reduction of sourcing an additional input, so that the reduction in intermediate input costs associated with a plant choosing a certain fraction n to source rather than purchasing all goods domestically is qj (n) n= = jj qj (0) Since jj 1 and > 0, the price index for inputs qj (n) is decreasing in n, so that for a given amount of total intermediate input expenditures, a plant that sources a higher fraction of inputs has lower input costs. A plant that sources n inputs spends a fraction n (1 jj ) on imported inputs, and n jj + 1 n on domestic goods, so if a plant were to source all its inputs (n = 1), it would spend a fraction 1 jj on imports and jj on domestic inputs, as in the Eaton-Kortum model. The input sourcing and production choices of plants can be separated into two steps: …rst choose the output price and input quantities to maximize variable pro…t, given a sourcing policy (i.e., given an n), then choose n to maximize overall pro…ts given the optimal quantity decisions. The variable pro…t of a plant in country j with e¢ ciency z that has chosen to source n of its inputs is given by: j
(z; n) = max pz`1 x p;`;x
wj `
subject to z`1 x =
qj (n) x p Pj
Cj
Monopolistic competition leads the plant to set its price p as a constant markup over its marginal cost, wj1 qj (n) pj (z; n) = 1 z (1 )1 Revenues rj (z; n), variable pro…ts
j
(z; n) and inputs `j (z; n) and xj (z; n) can be written: n (1
)=
rj (z; n) = z 1 jj 1 rj (z; n) j (z; n) = `j (z; n) = xj (z; n) =
1
1 wj
rj (z; n) 1
qj (z; n) 9
Ej
rj (z; n)
where Ej = Vj
1
wj Pj
Pj Cj is an index of aggregate prices and quantities, and Vj = 1
1 is a country-speci…c term that only depends on para(e EM Tj ) = (1 )1 meters. The choice of n and total pro…t of a plant with e¢ ciency z in country j are determined by the input-sourcing problem,
~ j (z) = max
n2[0;1]
j
(z; n)
wj g (n)
(8)
The solution to this problem is characterized in the following proposition, proved in the appendix. Proposition 1 If f > and is given by:
(1 jj
nj (z) =
where
j
=
1 log f 1=(
1)
= jj
,
)=
, then the solution nj (z) to the sourcing problem is unique,
8 > < 0 > : j
j
log z
log
if z < zj (0) if z 2 [zj (0) ; zj (1)] if z > zj (1)
j
1
(9)
1
=
wj b log f (1 ) log jj Ej
zj (0) =
j
zj (1) =
j
1
exp
, and
1 j
The solution takes the form of two cuto¤s, zj (0) and zj (1), with zj (0) < zj (1) (the (1 )= ): plants with e¢ ciency below inequality is guaranteed by the assumption that f > jj zj (0) source none of their inputs globally and purchase everything from domestic suppliers, while plants with e¢ ciency above zj (1) source all of their inputs, and purchase a fraction jj of these inputs domestically. Between these two thresholds, the fraction of inputs sourced nj (z) is increasing in plant-level e¢ ciency, with semi-elasticity equal to j . The assumption (1 )= that f > jj requires the cost of sourcing additional inputs to rise su¢ ciently rapidly relative to the cost savings from sourcing additional inputs. This su¢ cient condition for existence and uniqueness of the solution also implies that nj (z) is strictly increasing in z in the interval [zj (0) ; zj (1)]: more e¢ cient plants choose to source, and hence import, a larger fraction of their inputs. Also, since size –measured by either labor, output, or total intermediate expenditures –is increasing in e¢ ciency z, size and import share are positively (1 )= related. The condition that f > jj depends on the shape of the parameter values 10
f; ; , and , as well as on values of equilibrium variables through jj . In the quantitative exercise in Section 4, the moments I match relating size and importing behavior guarantee that this condition holds with the calibrated parameters. For interior values, the input sourcing decision satis…es a …rst-order condition equating the marginal increase in pro…t, which is proportional to revenue, with the marginal cost of sourcing an additional input: rj (z)
(1
)
log
1 jj
= wj bf nj (z) log f
(10)
Equation (10) demonstrates the source of the nonhomotheticity in plant-level import demand: plants with higher revenues choose a larger fraction of inputs to source, nj (z), and hence have higher import shares, sj (z) = (1 jj ) nj (z). Anything that raises revenues raises the incentive to source inputs, so that shifts in the aggregate component Ej of the plants’ demand curve also change the input sourcing decision. This dependence of plant sourcing decisions and import shares on revenues leads to nonhomothetic aggregate import demand, which is an important part of the characterization of welfare gains in Section 3. 2.2.3
Heterogeneity in import shares
The interaction of the exogenous heterogeneity in e¢ ciency and the choice of n generates a distribution for import shares, sj (z) = (1 jj ) nj (z). Among those plants that import a positive amount, ignoring for a moment the restriction that nj (z) 1, Pr (sj (z)
sjsj (z)
0) =
Pr (1 Pr (1
jj )
j
jj )
j
log z log z
log log
j j
s 0
s
= exp
(1
jj )
j
Therefore, the cumulative distribution function of import shares, for s > 0, is: Gj (s) =
Pr (sj sjsj ( 1 exp
0) s (1
if 0 < s < 1
jj ) j
1 if s
1
jj
(11)
jj
The distribution of import shares is an exponential distribution with parameter (1 jj ) up j to the point 1 , where there is a mass point, equal to the fraction of plants that source jj all of their intermediate inputs (those with z above zj (1)).
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2.3
Market clearing and equilibrium
The representative consumer in each country j inelastically supplies labor at the level Lj , and receives the pro…ts of all …nal good plants. The consumer spends this income on consumption of the …nal good produced by plants, so the budget constraint is Pj C j = w j L j +
Z
~ j (z) h (z) dz
The market clearing condition for labor requires that the labor used by intermediate goods producers plus the labor used by …nal good plants in each country j adds up to Lj . Since intermediate goods producers are perfectly competitive, their total payments to labor equal their sales, which are: X Z
jk nk
+
nj (z)) qj (nj (z)) xj (z; nj (z)) h (z) dz
k
Z
(1
(z) qk (nk (z)) xk (z; nk (z)) h (z) dz
(12)
The …rst line in (12) is total sales to plants in all countries sourcing intermediate goods from country j. Plants in each country k with e¢ ciency z spend a fraction jk nk (z) of their intermediate expenditures qk (nk (z)) xk (z; nk (z)) on j’s goods, and there is mass h (z) of plants with each e¢ ciency z. The second line is additional sales to …nal good plants within j of the goods that they decide not to source, and hence must purchase from country j’s intermediate good producers. Each plant in j with e¢ ciency z spends a fraction 1 nj (z) of its intermediate expenditures on its own country’s intermediates in this way. Payments to labor by …nal good plants (both for production and for …xed costs) is given by: Z wj
[`j (z; nj (z)) + g (nj (z))] h (z) dz
(13)
So the labor market clearing condition states that (12) and (13) equal total payments to labor: X Z wj Lj = jk nk (z) qk (nk (z)) xk (z; nk (z)) h (z) dz k
+
Z
+wj
(1 Z
nj (z)) qj (nj (z)) xj (z; nj (z)) h (z) dz [`j (z; nj (z)) + g (nj (z))] h (z) dz
12
Finally, balanced trade requires that country j’s exports, X Z
jk nk
(z) qk (nk (z)) xk (z; nk (z)) h (z) dz
k6=j
equal its imports, Z
(1
jj ) nj
(z) qj (nj (z)) xj (z; nj (z)) h (z) dz
which, rearranging, gives X Z
jk nk
(z) qk (nk (z)) xk (z; nk (z)) h (z) dz
k
=
Z
nj (z) qj (nj (z)) xj (z; nj (z)) h (z) dz
An equilibrium is a set of wages wj , aggregate prices Pj , and aggregate …nal consumption Cj such that the market clearing condition for labor and the trade balance condition hold for each country with the plant-level decisions characterized in the previous subsection. Using the plant-level input and sourcing decisions, the trade balance condition and labor market clearing condition in each country j can be written in terms of aggregate revenues (Rj ), spending on sourced inputs (Nj ), and …xed costs of sourcing (Gj ), X
jk Nk
(14)
= Nj
k
and wj Lj = (1
1
)
Rj +
X
jk Nk
+
1
Rj
Nj + wj Gj
k
which reduces to: wj Lj =
1
Rj + wj Gj
(15)
where Rj , Nj , and Gj are, respectively: Rj = Nj = Gj =
Z
Z
Z
rj (z; nj (z)) h (z) dz nj (z)
1
rj (z; nj (z)) h (z) dz
g (nj (z)) h (z) dz
13
(16)
Total …nal spending Pj Cj is then equal to the value of aggregate revenues of …nal goods plants, Pj Cj = Rj . Aggregate expenditures on (domestic plus imported) intermediate goods are 1
Xj =
Rj
and aggregate expenditures on imports are: Mj = (1
jj ) Nj
while aggregate expenditures on domestic inputs are Dj = Xj
3
Mj
Welfare gains from trade between symmetric countries
Trade raises welfare in this model— i.e. the real value of …nal consumption— by allowing …nal goods plants access to cheaper inputs, lowering their marginal costs and therefore their prices. The main …ndings of this paper establish that plant-level input sourcing decisions alter the welfare gains from trade relative to a model without these input sourcing decisions. This section illustrates the mechanism behind this …nding. To focus on the main features of the model generating higher gains from trade, I assume that countries are symmetric, i.e. T1 = T2 = T and L1 = L2 = L, so that all equilibrium variables are equated across countries, and there are no e¤ects of changes in trade costs on equilibrium relative prices. I normalize w1 = w2 = 1. In Section 5, I show that the results extend to asymmetric countries. I characterize the gains from trade in di¤erent cases of the model in terms of the aggregate import share Mj =Xj , and two elasticities, "Pj and "Fj . The partial elasticity "Pj is the negative of the elasticity of the aggregate import/domestic ratio Mj =Dj with respect to the variable trade cost , holding …xed aggregate prices and consumption wj , Pj , and Cj , "Pj =
@ log Mj =Dj @ log
where henceforth, “@”denotes di¤erentiation holding …xed wj , Pj , and Cj . The full elasticity M "Fj is the negative of the elasticity of Djj with respect to taking into account the equilibrium
14
changes in wj , Pj , and Cj , "Fj =
d log Mj =Dj d log
where henceforth, “d” denotes total di¤erentiation in the usual sense. The next result establishes how changes in consumption in response to changes in trade costs are related to changes in the aggregate import share through the elasticity "Fj . Proposition 2 To a …rst-order approximation, in the model with symmetric countries, for any cost function g (n) and optimal sourcing decision nj (z), the change in consumption is given by Mj (17) d log 1 d log Cj = F Xj "j This result shows that the log change in welfare is proportional to the log change in the M aggregate domestic expenditure share, 1 Xjj , and inversely proportional to the full trade elasticity, "Fj . The formula (17) di¤ers from a local version of the ACR formula in that the relevant trade elasticity is the full elasticity "Fj rather than the partial elasticity "Pj . The next result concerns the di¤erence between the elasticities "Pj and "Fj . The deviation of equation (17) from the (local) ACR formula is given by the ratio between these two elasticities. The following three results illustrate the main features of the model that lead to the di¤erence. First, with homothetic sourcing decisions, regardless of how they vary across plants, "Fj = "Pj , and a local version of the ACR formula holds. Second, with homothetic and common sourcing decisions across plants, "Pj is a constant, so the relationship between welfare and the aggregate import share is exactly as in ACR. Finally, under the optimal sourcing decision derived for the model here, "Fj < "Pj , so that equation (17) implies that a given change in the import ratio leads to a larger change in welfare than the ACR formula. Proposition 3 In the model with symmetric countries, 1. For any sourcing decision nj (z), if the partial and full derivatives of nj (z) with respect @n (z) dn (z) to the trade cost are equal, @j = dj , then "Fj = "Pj . 2. If nj (z) = 1 for all z, then "Fj = "Pj = . 3. With nj (z) given by the solution in (9), the full and partial elasticities are related d log E d log Ej j , with j > 0. In addition, as long as j > 0, d log j as through "Fj = "Pj d log 1 j well, so that "Fj < "Pj . Propositions 2 and 3 show that the important feature of the model that generates welfare gains that deviate from the ACR formula is the nonhomotheticity of plant-level import 15
demand, which translates into aggregate import demand that is nonhomothetic, so that "Fj 6= "Pj . Just as a plant’s incentive to source inputs rises with e¢ ciency, it also rises with the aggregate component of the demand curve it faces, Ej . For interior values, the sourcing decision is governed by the …rst order condition (10), with (1
)
nj (z)
revenues given by rj (z) = z 1 jj Ej and Ej = Vj Pj Cj where Vj is a constant. The partial equilibrium impact of an increase in trade costs is that the share of sourced goods purchased domestically, jj = 1+1 , rises, so that for a given sourcing decision, revenues @n (z) fall. Plants adjust by reducing the fraction of goods they source, so @j < 0. Since these changes in sourcing decisions increase input costs, the general equilibrium e¤ects are that …nal goods’ prices rise while aggregate consumption falls. The net e¤ect of these changes in Pj and Cj is that all plants see an increase in the aggregate component of the demands they face, Ej . This increased demand raises the incentive to source intermediate inputs, so this general equilibrium e¤ect dampens the reduction in input sourcing decisions, so that dnj (z) @nj (z) . d @ The …rst two statements of Proposition 3 show that heterogeneity in import shares per se does not change the …rst-order welfare e¤ects of trade costs compared to a model with homogeneous import shares. With homogeneous import shares, the partial trade elasticity is constant, and the ACR formula characterizes welfare gains. With any exogenous assignment dn (z) @n (z) of sourcing shares that does not respond to changes in trade costs, dj = @j = 0, and the ACR formula characterizes welfare changes, at least locally. While the distribution of dnj (z) @nj (z) matters for the size of the deviation from the ACR formula, the fact that @ d there is any deviation at all depends not on heterogeneity in import shares per se, but on the fact that plant-level sourcing decisions are nonhomothetic. According to Proposition 3, heterogeneity in import shares across plants is relevant for welfare only because it re‡ects the nonhomotheticity of import demand. This result relates to Blaum, Lelarge, and Peters (2015), who show that in models with plant-level importing, consumer price reductions from trade are characterized by the joint distribution of observed import shares and plant sizes. Propositions 2 and 3 focus on welfare rather than price indices, and o¤er an alternative characterization. Welfare is distinct from consumer prices because of the resources used for sourcing inputs. Proposition 3 highlights how for any trade cost change (not just the move to autarky), the di¤erence in welfare gains from a model with common plant-level import shares depends on the distribution of the general equilibrium e¤ects of any trade cost change on sourcing decisions, not just observed import shares. Blaum, Lelarge, and Peters’ result also suggests that a lower cross-section correlation between import share and plant size implies lower gains from trade. The model here, by 16
construction, imposes a near-perfect correlation between import share and plant size (deviating from one only because of the upper and lower bounds on input sourcing), so similar logic would suggest that any quantitative result based on Propositions 2 and 3 is an upper bound for the di¤erence in welfare gains due to endogenous input sourcing. Whether this is true or not, however, depends on how one introduces an imperfect correlation between import share and size. In section 7.3 in the appendix, I demonstrate one way to extend the model to generate an imperfect correlation in observed import shares and size, while leaving all the aggregate implications of the model unchanged. This is possible because, as Proposition 3 shows, the distribution of general equilibrium e¤ects on the nonhomothetic input sourcing decisions, not just the distribution of observed import shares, governs welfare gains. These results also relate to the results on welfare gains in multi-sector models in Levchenko and Zhang (2014), Ossa (2015), and Costinot and Rodríguez-Clare (2014). These papers argue that multisector models have higher gains from trade than one-sector models due to variation in trade shares (Levchenko and Zhang (2014)) or in trade elasticities (Ossa (2015) and Costinot and Rodríguez-Clare (2014)) across sectors. In my model, heterogeneity in import shares across plants is analogous to the sectoral heterogeneity in import shares considered in Levchenko and Zhang (2014) (and because the endogenous sourcing yields a non-constant plant-level elasticity of demand for imports, there is also variation across plants in the trade elasticity analogous to the sectoral heterogeneity in Ossa (2015) and Costinot and RodríguezClare (2014)). The main di¤erence here is the implication of plant-level heterogeneity for nonhomothetic aggregate import demand, which lowers the full trade elasticity that is relevant for welfare. In addition, these three papers derive welfare results that take into account di¤erences across sectors and explain why one-sector models understate gains from trade relative to multi-sector models, but do not impose that the aggregate partial trade elasticity is the same across models. However, as Proposition 3 shows, a model with heterogeneous import shares resulting from exogenously …xed sourcing decisions has the same …rst-order welfare e¤ects as a model with homogeneous import shares with the same partial trade elasticity. The last part of the proposition shows that endogenous sourcing decisions that adjust in response to changes in trade costs do result in higher welfare gains than a model with homogeneous import shares with the same aggregate partial trade elasticity, meaning that the ACR formula understates the welfare gains from trade.
4
Quantitative Analysis
This section provides some quantitative support for the welfare implications of endogenous input sourcing characterized in Propositions 2 and 3 of the previous section. I calibrate the 17
model to match key cross-sectional facts from Chilean plant-level data. Section 4.1 discusses the calibration and section 4.2 presents the main results for the symmetric 2-country model of Chile (country 1) and the rest of the world (country 2).
4.1
Calibration
I choose parameter values in two steps: …rst, one set of parameters is …xed or directly calculated from data, all of which is averaged over the period 1987-1996. The plant-level data comes from the manufacturing census and includes all manufacturing plants with at least 10 employees. The aggregate data come from the World Bank.6 I set the lower bound of productivity draws in the …nal good to zL = 1, and I assume that the labor endowment Lj , and the level parameter of the Fréchet distribution Tj are equal within and across countries, L1 = L2 = L and T2 = T1 = T .7 I set = 3 so that 1 , the share of revenues paid to labor compensation and total intermediate inputs, is equal to 2=3, the average of compensation plus intermediate input expensitures in the Chilean plant-level data. I also experiment with di¤erent values of in section 5 below. Then, I calculate so that the intermediate input share of revenues is equal to the average in the data of 0:514. I calibrate the remaining …ve parameters— ; ; f; , and b— so that the model matches three cross-sectional moments from the Chilean plant level data and two aggregate statistics. The three plant-level moments are the fraction of plants importing, the average import share among importing plants, and the average size of importing relative to nonimporting plants; the two aggregate statistics are the aggregate import share in Chile and a target for the partial trade elasticity, "P . To do this, …rst, given the fraction F1 of plants importing in Chile, I choose the parameters , f , , and to match the average import share among importing plants, the average importer/nonimporter size ratio, the aggregate import share, and the partial trade elasticity. Section 7.4 in the appendix derives these 4 moments as functions of , f , , and , along with the predetermined parameters and and the fraction of plants importing, F1 . Then, I choose b so that the equilibrium of the model matches the fraction of importing plants in Chile, F1 = z1 (0) , which requires solving for equilibrium prices and consumption. Table 1 summarizes the parameter values and calibration targets. On average, about 23% of plants report purchasing positive amounts of imported inputs. Among these plants, the 6
The plant-level data are from the Encuesta Nacional Industrial Anual, from Chile’s Instituto Nactional de Estadistica. These are the same data used in Kasahara and Rodrigue (2008), and described in detail in Liu (1993). The aggregate data are from the World Bank’s World Development Indicators, for Chile and World GDP at constant 2005 international dollars (NY.GDP.MKTP.PP.KD). 7 Alvarez and Lucas (2007) use a similar assumption that labor force and the technology level are proportional within each country. The symmetry across countries is relaxed in section 5.
18
average import share is 33% of total intermediate input expenditures. Importing plants are on average 4.5 times as large as nonimporting plants, as measured by their total expenditures on intermediate inputs. The aggregate import share is 22%, and I use a target for the partial trade elasticity of 5, in line with the gravity-based estimates of the elasticity of trade ‡ows with respect to variable costs from Anderson and Van Wincoop (2004). I also vary the target for this elasticity in section 5 below.
4.2
Results
Propositions 2 and 3 establish that the gains from trade in my model are higher than the gains implied by the ACR formula to a …rst order approximation because the relevant statistic for welfare is the full elasticity "F rather than the partial elasticity "P , and that "F < "P . Table 2 shows the magnitudes of these di¤erences in the calibrated model. The full elasticity "F is equal to 1.62, while the model was calibrated to match a partial elasticity "P of 5. With this di¤erence, the elasticity of welfare with respect to the aggregate domestic expenditure share, according to equation (17), is "F = 0:48, which is more than three times the value of this elasticity according to the ACR formula, which is "P = 0:15. The last two rows of the table show that the magnitude of this di¤erence roughly carries over to large changes in trade costs: the welfare gain from trade relative to autarky in the model is about 10.44 percent, while the gain implied by the ACR formula is 3.90 percent. The ACR formula underpredicts welfare gains relative to autarky by about 6.5 percentage points, or more than 60%. Figure 1 further illustrates the di¤erence between the model’s welfare gains and the gains implied by the ACR formula, showing how consumption varies relative to the initial calibrated equilibrium as the trade cost varies so that the aggregate domestic share varies from 0.6 to the autarky value of 1 (the calibrated aggregate domestic share is 0.78). On the log-log scale in the …gure, the line labelled “…rst-order approximation”is a straight line with slope equal to "F . The line labelled “actual welfare”is actual aggregate consumption, which deviates signi…cantly from the …rst-order approximation only for changes in trade costs that would raise the aggregate domestic expenditure above 95% or lower it below 60%. Finally, the line labelled “ACR”is a straight line with slope equal to "P .
4.3
Importing and measured productivity
In the benchmark model, plants gain by importing through a lower price for their intermediate input bundle. This cost advantage means that importing raises plant-level productivity when input expenditures are measured across plants using common price de‡ators, as is stan19
dard in plant-level data sets. Sourcing some inputs (and importing a fraction of those inputs sourced) lowers the prices on average that a plant pays for its input bundle. Productivity then appears higher at plants that import some of their inputs because they produce more output with the same expenditures on inputs, compared to plants that purchase all of their inputs domestically. In this section I show that size of this productivity e¤ect in the calibrated model is consistent with empirical studies, and quantify the aggregate productivity e¤ects of trade. Several recent empirical studies have estimated this kind of productivity advantage of importing in plant-level data, including Amiti and Konings (2007) using Indonesian data, Halpern, Koren, and Szeidl (2009) using Hungarian data, and Kasahara and Rodrigue (2008) using a subset of the Chilean data considered here.8 These papers all estimate production functions that relate a plant’s output to its factor inputs and intermediate expenditures, along with indicators of whether the plant imports any of its inputs, or its import expenditure share (or both). In my model, the revenues of a plant in country j with e¢ ciency z can be written: 1 1= 1 ^ ^ Pj C j r^j (z) = z `j (z) Xj (z) qj (z) where r^j (z) = rj (z; nj (z)) is the output of a plant with e¢ ciency z (who chooses to source ^ j (z) are expenditures on a fraction nj (z) of inputs), with other variables de…ned similarly. X intermediate goods by the plant, ^ j (z) = qj (nj (z)) xj (z; nj (z)) X For the moment, ignore labor used for …xed costs of importing. De‡ating intermediate expenditures by any common price index Qj , and de‡ating revenues by the …nal goods price index Pj , a measure of total factor productivity (TFP) at the plant level, excluding labor used for …xed costs, is r^j (z) =Pj `^j (z)(1
)
1
1
^ j (z) =Qj X
=z
1
qj (z) Qj
1
1=
Cj
(18)
Therefore, plants who choose higher n, and hence pay a lower input price qj (nj (z)), appear more productive that those that choose a lower n. 8
Although they do not estimate the direct producer-level productivity gain from importing, Goldberg, Khandelwal, Pavcnik, and Topalova (2010), using data on Indian …rms, …nd that lower input tari¤s, and hence higher expenditures on imported inputs, lead …rms to create more new products. They argue that this is because the cost of production decreases (which similar to the increase in productivity considered here), so that producing new products becomes pro…table.
20
As a function of the import expenditure share, sj (z) = nj (z) (1 jj ), the gain (in logs) in this productivity measure for a plant relative to sourcing none of its inputs (and buying them all from domestic suppliers) is, using qj (n) from (7): log
qj (nj (z)) qj (0)
1
1 sj (z) log 1 jj
=
1
(19)
jj
Since jj < 1, the productivity gain of importing is increasing in a plant’s import share. The magnitude of this productivity e¤ect depends directly on three parameters – , the elasticity of substitution for …nal goods, , the share of intermediate inputs in total costs, and , the degree of heterogeneity in the prices of intermediate inputs –as well as the fraction of sourced inputs that are purchased domestically in equilibrium, jj . The lower is , the greater the dispersion in prices of intermediate inputs, so the greater is the incentive to @ log(qj (nj (z))=qj (0)) is decreasing exploit comparative advantage by sourcing inputs. Also, @n in jj , so that the less open a country is, the lower is the productivity gain from sourcing inputs. A more appropriate measure of productivity incorporates the labor used for …xed costs ^ j (z) = `^j (z) + g (nj (z)) in place of `^j (z) in the in the measure of the labor input, using L denominator of (18). Plants that import a higher fraction of goods require more resources in the form of the …xed cost for sourcing, which o¤sets some of the gain in output. In this case, the log productivity gain would be log =
qj (nj (z)) qj (0) 1 sj (z) log 1 jj
1
+ (1 1
+ (1
jj
1
) )
log 1
`^j (z) ^ j (z) L log
!
`^j (z) ^ j (z) L
(20) !
In the calibrated model, the average marginal e¤ect on productivity of increasing the import share – the average of the partial derivatives of the right side of (19) with respect to sj (z) – is 0:903, which implies that a plant gains on average 9% in productivity by increasing its import share by 10 percentage points. The implied average productivity gain across importing plants in Chile – the average of the right hand side of (20) – is 0:250, so an importing plant on average is 25% more productive than a nonimporting plant, after accounting for di¤erences in their exogenous e¢ ciency. These numbers are a bit higher than those reported by Kasahara and Rodrigue (2008) in their analysis of Chilean plant data. Using a continuous import share variable, their range of estimates imply that raising the import share by 10 percentage points raises productivity by 0:5% to 2:7%. Using a discrete
21
import status variable, they …nd that importing raises productivity on average by between 18% and 21%. Similar magnitudes are reported in Halpern, Koren, and Szeidl (2009) and Amiti and Konings (2007) using Hungarian and Indonesian data, respectively. 4.3.1
Trade-induced changes in plant-level and aggregate TFP
Equation (20) and the calculations based on it re‡ect cross-sectional comparisons of the e¤ects of varying import shares on productivity: plants that import more appear more productive, after accounting for underlying e¢ ciency. Now I apply the same logic to measuring the e¤ects of trade on plant-level and aggregate productivity. Compared to autarky, plants that import have higher TFP, and aggregate TFP is higher as well. I compare the following measure of plant-level TFP to compare across the calibrated model and autarky: 1=
r^j (z) = Pj Cj
tf pj (z) = ^ j (z)(1 L
)
1
1
^ j (z) = T X j
1=
(21)
wj 1=
1=
wj as an intermediate expenditure de‡ator and the index Pj Cj as I use the index Tj a …nal goods de‡ator because doing so means that plants that do not import anything when the economy is open gain nothing in terms of measured productivity. That is, de‡ating outputs and input expenditures this way means that either in autarky or for plants who 1 1= wj is the import nothing when the economy is open to trade, tf pj (z) = z . Also, Tj expression for the overall price index for intermediate goods in autarky, and is the price index for domestically purchased inputs when the economy is open. Table 3 reports two measures of average TFP growth relative to autarky in the benchmark model, along with an aggregate measure of TFP, where the plant-level terms in (21) are replaced with economy-wide aggregates. The …rst column of Table 3 shows that in the benchmark model, the average productivity gain across all plants is about 10.4 percent. However, aggregate TFP grows by more, about 14 percent, relative to autarky, since plants that import more account for a larger share of aggregate output. The second column of the table reports measures that ignore the labor used in …xed costs when calculating TFP. Clearly, since opening to trade requires using resources for …xed costs that are being ignored in the calculation of the second column, these measures of TFP gains are signi…cantly higher.
22
4.3.2
A Decomposition of TFP
The growth in aggregate TFP shown in Table 3 is due to both within-plant TFP growth and reallocation. When the economy opens to trade, more inherently e¢ cient plants source and import more inputs, giving them a larger productivity gain than less e¢ cient plants. More productive plants receive more resources, so account for a larger fraction of output, raising aggregate TFP. I quantify these channels by decomposing aggregate TFP. Aggregate TFP can be written as a weighted average of plant-level TFP across a sample of plants, indexed by i, X TFP = si tf pi where si is plant i’s share of aggregate revenues, and tf pi is plant i’s TFP. A change in aggregate TFP can be written: TFP0
TFP =
X
(tf p0i
tf pi ) si +
i
(WITHIN, 44%)
X
(s0i
si ) tf pi +
i (BETWEEN, 12%)
X
(s0i
si ) (tf p0i
tf pi )
i
(CROSS, 44%)
Here, primed variables denote the equilibrium with trade, and non-primed variables denote autarky. The …rst term is the contribution of within-plant TFP growth, holding …xed each plant’s output share. The second term is the contribution of reallocation of output shares, holding …xed each plant’s initial TFP. Finally, the third term is the covariance of plant-level TFP growth and changes in output shares. The numbers in parentheses under each term give the contribution to the aggregate productivity gain from trade (relative to autarky) of each component in the calibrated benchmark model. All three channels contribute signi…cantly to aggregate TFP growth, but reallocation per se (that is, holding …xed each plant’s productivity), plays a modest role. The …rst and third terms are much larger, so within-plant productivity gains account for a large part of aggregate TFP growth.
5
Sensitivity: asymmetry and elasticities
In this section I consider the sensitivity of the results to four parameter choices: the size of the rest of the world, the target for the partial trade elasticity, the elasticity of substitution in preferences between …nal goods, and the elasticity of substitution in production between di¤erent intermediate inputs. I allow for asymmetric country sizes by letting Ti 6= Tj , and choosing T2 so that the model reproduces Chile’s share of world GDP (0:27%) while recalibrating the rest of the parame-
23
ters to match the same moments as above.9 With asymmetric countries, the equilibrium aggregates wj ; Pj , and Cj di¤er across countries, and the response of trade ‡ows and welfare to changes in trade costs depend on changes in relative prices. Here I show that, matching the same plant-level moments, the model with asymmetric countries yields a modi…ed version of formula (17) derived in Proposition 2 under symmetry, relating …rst-order changes in welfare to changes in the aggregate domestic expenditure share. In fact, the elasticity of d log C welfare with respect to the domestic share, d log(1 Mjj=Xj ) , is exactly the same with symmetric or asymmetric countries. Proposition 4 In the model with symmetric or asymmetric countries, d log Cj = d log(wi =wj ) for j = d log d log Cj is the same d log(1 Mj =Xj )
where
"Fj
(1 + j ) d log 1
Mj Xj
i 6= j. In addition, "Fj is homogeneous of degree 1 in 1 + j , so that regardless of movements in wi =wj .
The proposition shows that, while the full trade elasticity "Fj depends on relative country "F
sizes through relative price responses, the product 1+j j does not. For example, with T2 calibrated to match Chile’s 0:27% share of world GDP, the elasticity "F1 is 3:22, about twice the value of 1:62 from the symmetric case. The elasticity of the relative wage with respect 2 =w1 ) to the trade cost, j = d log(w > 0, and in this case takes a value of 0:99. This can be d log interpreted as a standard terms-of-trade e¤ect: an increase in trade costs raises import prices relative to export (or domestic) prices; the ratio of the import price index to the domestic price index for a …xed sourcing decision is proportional to ww12 . With w2 normalized to 1, d log w1 < 0. This terms of trade e¤ect o¤sets the di¤erence in the full trade elasticity, so that d log M
the relationship between d log Cj and d log 1 Xjj is exactly the same as the symmetric case. Next, I vary the target for the partial trade elasticity, "P , between 3 and 7, while recalibrating the rest of the parameters. The left panel of Figure 2 shows how the full trade elasticity "F varies as the target for "P changes. The full elasticity is non-monotonically related to "P , and is always smaller than "P . The right panel of the …gure shows how welfare gains in the model and welfare gains implied by the ACR formula vary: the ACRimplied welfare gains decline monotonically with "P , while the gains in my model re‡ect 9
The only change to the rest of the parameters is that the variable cost increases. This is because the other parameters are pinned down along with 11 by the plant-level data moments, and with asymmetry 11 is a function of relative wages along with and .
24
the non-monotonicity of the full elasticity "F from the left panel, and are higher than the ACR-implied gains for this range of values for the partial trade elasticity "P . In Figure 2, I vary the …nal goods elasticity of substitution relative to the benchmark value of 3, with the rest of the parameters recalibrated as changes.10 The left panel shows the full trade elasticity and the right panel shows welfare gains relative to autarky. As increases, the full trade elasticity rises, and the gains from trade fall. Since the ACR-implied gain from trade is constant, the deviation of the model from the ACR formula shrinks as rises. This result is related to the …ndings in Costinot and Rodríguez-Clare (2014), who show that in multi-sector models, gains from trade decline rapidly with the elasticity of substitution across sectors. Finally, I relax the unit-elasticity assumption in the bundling of intermediate inputs, replacing equation (2) with the assumption that plants use the CES bundle: x=
Z
1
1
x~ (!)
1
d!
0
where 1 is the elasticity of substitution between inputs. With this CES bundle, the input price index for a plant that sources n inputs becomes: qj (n) = Tj (1
1=
1
wj n'j + 1
1
1
)=
+1 1 1 and = where 'j = jj , with ( ) denoting the gamma function. A plant that sources n inputs imports a share of its input expenditures given by n('j +1) s sj (n) = (1 jj ) j (n), with j (n) = n' +1 denoting the fraction spent on sourced inputs. j
a
Plant-level revenues are given by rj (z; n) = z 1 n'j + 1 Ej , with a = ( 11) , and pro…ts are j (z; n) = 1 rj (z; n). Moving away from an elasticity of one results in a slight loss of tractability, as there is no closed form solution nj (z) to the sourcing problem as in Proposition 1. But as shown in the appendix, the inverse of the solution to the sourcing problem is given by: 8 [zL ; zj (0)] if n = 0 > > < 1 1 1 a n zj (n) = $j n'j + 1 f if n 2 (0; 1) > > : [zj (1) ; 1) if n = 1 10
I …x at the value 0:77, even though that value was calculated using the variable cost shares implied by = 3. The alternative would be to set = 0:5135 1 for each value of . One reason for not doing this is that for su¢ ciently low values of , the value of would go above 1. Also, the ACR formula implied gain would vary with the value of just because the value of is changed, while with …xed, changing has no e¤ect on the gains implied by the ACR formula. With recalibrated, the change in welfare gains as varies in the model is qualitatively similar to that shown in Figure 3.
25
where $j =
wj b log f , a'j Ej
and the cuto¤s are given by: zj (0) = ($j ) zj (1) =
1 1
'j + 1
1 a
1
f $j
1
The assumption now required for the sourcing problem to have a unique solution is that 1 a n n'j + 1 f is increasing in n. An equilibrium of the model can be computed as above, R z (1) R1 using the change of variables zjj(0) h (z) dz = 0 h (zj (n)) zj0 (n) dn to aggregate with the inverse sourcing decision zj (n) instead of nj (z). I consider alternative values of the elasticity of substitution , recalibrating the rest of the parameters to match the same statistics. There are two restrictions on admissible values 1 a n of : one is that n'j + 1 f must be increasing to match the observation that importing plants are larger than nonimporting plants. The other is that +1 must be positive in order for the price index for intermediate inputs to be well-de…ned. For the benchmark calibration above (which is with = 1), the calibrated value of is 0:6, limiting the maximum admissible value of to just below 1:6 (the recalibrated value of decreases as increases, but by very little). The left panel of Figure 4 shows how the full trade elasticity varies as increases, keeping …xed the rest of the calibration targets. The full elasticity declines as increases because the general equilibrium e¤ect of trade cost reductions on input sourcing described in section 3 becomes stronger. The right panel shows that the gain from trade relative to autarky declines as increases, but very little. Since the ACR formula doesn’t depend on (i.e., a version of the model with homogeneous import shares would have welfare gains independent of ), the welfare gain implied by the ACR formula is constant across values of .
6
Conclusion
A literal interpretation of the production technology in this model is that imports are perfect substitutes for domestic inputs, but may be available at a lower cost, so that importing a larger share lowers the average cost of production. More broadly, imported inputs could also yield productivity gains because imports are of higher quality than comparable domestic inputs, or because imported goods are imperfect substitutes for domestic goods. The quality explanation, for example discussed in Grossman and Helpman (1991), is studied in plantlevel data for Mexico by Kugler and Verhoogen (2009). Imperfect substitutability would
26
generate gains from input variety as in Ethier (1982) and Romer (1990).11 Halpern, Koren, and Szeidl (2009) use data on the number of goods Hungarian …rms import to measure the relative magnitudes of the quality and substitutability channels. For the purposes of this paper, these explanations are isomorphic to the one proposed here, in that data on total domestic and imported expenditures at the plant level cannot distinguish between them. The model presented here captures the heterogeneity in the use of imported intermediate inputs prevalent in studies of plant- and …rm-level data, and is consistent with evidence on plant-level productivity gains from importing. The model has relatively few parameters that are easily related to observable moments of the cross-sectional distributions of imports and size in plant-level data. Endogenous input sourcing decisions lead to heterogeneity in import shares that re‡ects a nonhomotheticity in import demand at the plant-level. This nonhomotheticity has implications for the aggregate response of trade to changes in trade costs, and for the welfare gains from trade, creating a gap between the partial and full trade elasticity. Aggregate import demand is nonhomothetic, despite homothetic underlying preferences and technology. Welfare gains from trade are higher than in a model without these importing decisions with the same partial trade elasticity.
11
This variety mechanism is also the one operating in Kasahara and Rodrigue (2008), Goldberg, Khandelwal, Pavcnik, and Topalova (2010), and Gopinath and Neiman (2011). In a model that combines the decisions to import and export, Kasahara and Lapham (2007) assume plants gain from importing through the variety e¤ect, but the number of imports each importing plant uses is …xed.
27
7
Appendix
7.1
Solution to sourcing problem
Proof of Proposition 1 The Lagrangian of (8) is L = j (z; n) wj g (n) + n), where 0 ; 1 0, and the …rst order necessary condition is: 1 (1 @
(z; n) @n
j
wj bf n log f = @
1
0
(n
0) +
(22)
0 (1
(z;n)
)
n
(1 ) where the derivative of variable pro…t is given by j@n = z 1 jj log jj 1 Ej . From the complementary slackness conditions 0 n = 0 and 1 (1 n) = 0, it is clear that only one of the multipliers 0 or 1 can be positive. De…ne the two cuto¤s z levels: 1
wj b log f (1 ) log jj Ej
zj (0) =
1
j
1
wj b log f (1 ) log jj Ej 1 j exp
zj (1) =
1
f
1 1
jj
j 1
where
j
=
1 log f 1=(
1)
and
= jj
j
=
wj b log f (1 ) log jj Ej
1
as in the statement of the propo-
sition. These de…nitions of zj (0) ; zj (1) come from the …rst order condition at equality for (1 )= n = 0 and n = 1. By the assumption that f > jj , zj (1) > zj (0). Now, for z < zj (0), the left hand side of the …rst order condition (22) is: z =
1
(1
)
n
(1
)
jj
z zj (0)
< wj b log f
1
(1
jj
)
n
log
1 jj
Ej
wj b log f 1
z zj (0)
(1
jj
)
n
wj bf n log f
wj bf n log f ! fn
< 0 for all n This implies 0 > 0 (and hence 1 = 0), so the optimal n for any z < zj (0) is equal to 0. Similarly, for z > zj (1), the left hand side of (22) is positive for all n, so the optimal n 28
for any z > zj (1) is equal to 1. For z 2 (z 0 ; z 1 ), the solution to …rst order condition with 0 = 1 = 0 is given by: (1
1
0=z
)
n
(1
jj
)
log
1 jj
wj bf n log f
Ej
Taking logs of both sides and rearranging, 1
nj (z) = log f =
j
1
log z
1 1
log z
1
jj
log
wj b log f (1 ) log jj Ej
log
j
which is to the solution given in (9). For the range where n is interior,
@
j (z;n)
@n
= wj bf n log f
@ 2 b (f n 1) (z; n) w j @n2 @n2 @ j (z; n) (1 ) = log jj wj bf n (log f )2 @n (1 ) = wj bf n log f log jj log f @2
j
< 0 where the inequality follows from the assumption that f >
7.2
(1 jj
)=
.
Welfare results
Proof of Proposition 2 De…ne individual plant revenue scaled by aggregate revenue as r~j (z) =
rj (z) =z Rj
1 nj (z) (1 jj
)=
Vj Wj1
w
with Wj = Pjj . The goods market clearing condition states that 1 = small changes in r~j (z) satisfy Z 0 = d~ rj (z) h (z) dz
R
r~j (z) h (z) dz, so that (23)
with d~ rj (z) given by d~ rj (z) = r~j (z)
(1
)
[dnj (z) log
jj
29
+ nj (z) d log
jj ]
+ (1
) d log Wj
Plugging this into (23), 0=
Z
r~j (z)
(1
)
[dnj (z) log
jj
+ nj (z) d log
jj ] h (z) dz
+ (1
) d log Wj
(24)
The …rst-order condition for the choice of nj (z) can be written as: r~j (z)
(1
)
log
jj
Wj 0 g (nj (z)) Cj
=
and the labor market clearing condition (15) implies Wj 1 Cj
d log Wj = d log Cj +
Z
g 0 (nj (z)) dnj (z) h (z) dz
Plugging the preceding two equations into (24) and using the de…nitions of Mj and Xj , d log 1
d log Cj = Now with two symmetric countries,
jj
=
d log 1
1 1+ jj
jj jj
Mj Xj
(25)
, so
= d log
jj
so that
Mj d log Xj
d log Cj =
(26)
To put this in terms of changes in the domestic expenditure share and the elasticity of the import ratio with respect to , multiply and divide the right side by d log (1 Mj =Xj ). The equation d log Cj = d log (1 Mj =Xj ) "Fj then follows from the fact that "Fj =
d log Mj =(Xj Mj ) d log
=
d log(1 Mj =Xj ) 1 . d log Mj =Xj
Proof of Proposition 3 It is convenient to work with the elasticities of the aggregate @ log Mj =Xj d log Mj =Xj import share, ~"Pj = and ~"Fj = , which are related to the elasticities of @ log d log ~ "P
~ "F
the aggregate import ratio through "Pj = 1 Mjj =Xj and "Fj = 1 Mjj =Xj , so that ~"Pj = ~"Fj if and only if "Pj = "Fj . Throughout, the notation “d” stands for total di¤erentiation, while “@” denotes partial di¤erentiation holding …xed wj , Pj , and Cj .
30
From the de…nition Mj = (1 @Mj @ dMj d
1
=
1
=
@ jj Nj + (1 @ d jj Nj + (1 d
so that, along with the fact that @ log Mj @ log
1
jj )
d log Mj = d log Nj
Z
@ jj @
jj )
jj )
=
@nj (z) @
d jj d
Nj , Z
@nj (z) @rj (z) rj (z) + nj (z) h (z) dz @ @ dnj (z) drj (z) rj (z) + nj (z) h (z) dz d d
Z
by symmetry,
dnj (z) d
rj (z) +
@rj (z) @
drj (z) d
nj (z) h (z) dz
and similarly, @Xj @ dXj d so that
@ log Xj @ log
1
Z
@rj (z) h (z) dz @ Z 1 drj (z) h (z) dz d
= =
d log Xj = d log Rj
Z
@rj (z) @
drj (z) d
h (z) dz
Now, the partial and full changes in plant-level revenue rj (z) are given by @rj (z) (1 = rj (z) @ (1 drj (z) = rj (z) d
)
@ log jj @nj (z) + log jj @ @ d log jj dnj (z) nj (z) + log jj d d
nj (z) )
Plugging these into the expressions for ~"Pj
@Mj dMj @Xj , d , @ , @
and
+
d log Ej d
dXj , d
@ log Mj @ log Xj d log Mj d log Xj + @ log @ log d log d log Z @nj (z) dnj (z) = rj (z) h (z) dz Nj @ d Z (1 ) @nj (z) dnj (z) log jj nj (z) rj (z) h (z) dz Nj @ d Z (1 ) @nj (z) dnj (z) + log jj rj (z) h (z) dz Rj @ d
~"Fj =
The …rst statement of the proposition follows from the observation that if then ~"Pj ~"Fj = 0. 31
@nj (z) @
dnj (z) d
= 0,
@r (z)
@ log M
@ log X
For the second statement, plug @j into the expressions for @ log j and @ log j with the assumption that nj (z) = 1 for all z, and use and the fact that Nj = Rj implies Mj = (1 jj ) Xj , so that: ~"Pj = = =
@ log Xj @ log Mj @ log @ log 1 @ jj Nj + (1 Mj @ Mj 1 Xj
jj )
Z
@rj (z) h (z) dz @
Xj
Z
@rj (z) h (z) dz @
~ "P
so that "Pj = 1 Mjj =Xj = . Finally, with the cost function g (n) = b (f n tiating nj (z) gives: 8 > <
0 if z < zj (0) if z 2 [zj (0) ; zj (1)] 0 if z > zj (1)
dnj (z) = > d :
@nj (z) @
1), so that nj (z) is given by (9), di¤eren-
j d log Ej 1 d
so that the di¤erence in the elasticities can be written: " ! I I I R N R d log E ( j j j j j ~"Pj ~"Fj = d log 1 Nj Nj Rj where RjI =
R zj (1) zj (0)
R zj (1)
rj (z) h (z) dz and NjI =
RI
NI
zj (0)
RI
( 1) j j log follows that j = Njj Nj Rj de…nition of Ej and equation (26) to write:
jj
d log Pj Vj = 1 d log Pj
log
jj
Z
= Vj zj (1)
z
R
z
> 0. To show that
1 nj (z) jj (1
1 nj (z) jj
zj (0)
)
log
jj
#
nj (z) rj (z) h (z) dz. Since NjI < RjI , it
d log Ej d log Pj = d log d log The price index Pj satis…es Pj1
1)
d log Ej d log
> 0 also, use the
Mj Xj
(1
(27)
)
h (z) dz, so
dnj (z) Vj h (z) dz + 1 d log Pj
d log d log
Now using dnj (z) = d log
j
(1
jj )
nj (z)
32
1 1 log
1 jj
d log Ej 1 d log
jj
1 Nj Ej
we can write d log Pj = d log
log
1 jj
j
d log Ej RjI 1 d log Rj
Plugging this expression for d log Ej d log
d log Pj d log
log
jj
=
(1
jj )
NjI Rj
1 1 log
RjI Rj
1 1 log
RjI jj Rj
!!
+
d log d log
jj
Nj Rj
into (27),
j
(1
jj )
1
log
jj
jj
NjI Rj
+
Mj Xj
(
1)
RjI 1 Rj
j
> 0 where the inequality follows because log
< 0 and
jj
j
> 0.
Proof of Proposition 4 The derivation in proposition 2 proceeds exactly as in the symmetric case up to equation (25), so that d log 1
d log Cj =
jj jj
Mj Xj
(28)
Now, with two asymmetric countries i and j with Ti 6= Tj and wi 6= wj , the fraction of T (wj ) sourced goods purchased domestically is jj = T ( w )j +T , and the change in log jj i i j (wj ) can be written as: wi d log jj = (1 jj ) d log + d log wj Using this expression along with the expression for the full trade elasticity "Fj = equation (28) yields the expression stated in the proposition: d log Cj =
"Fj
d log (1
Mj =Xj ) 1 +
d log (wi =wj ) d log
Next, to show that "Fj is homogeneous of degree 1 in 1 + import share as d log Mj d log Xj d log d log 1 1 d jj = Nj + (1 Mj d log Z drj (z) 1 1 h (z) dz Xj d log
d log(1 Mj =Xj ) 1 , d log Mj =Xj
j,
write the elasticity of the
~"Fj =
jj )
Z
33
dnj (z) drj (z) rj (z) + nj (z) d log d log
h (z) dz
Using the de…nition of rj (z) and this simpli…es to ~"Fj
1 = Mj 1 + Mj 1 Xj
drj (z) d log
(1
= rj (z)
)
nj (z)
d log d log
jj
+ log
dnj (z) jj d log
+
d log Ej d log
Z dnj (z) 1 d log jj 1 1 (1 rj (z) h (z) dz jj Nj + jj ) d log Mj d log Z 1 (1 ) d log jj dnj (z) (1 nj (z) rj (z) nj (z) + log jj jj ) d log d log Z 1 d log jj dnj (z) (1 ) nj (z) + log jj h (z) dz rj (z) d log d log
so that the full trade elasticity is homogeneous in 1 + 8 > <
dnj (z) = > d log :
d log d log
j
d log d log
if
jj
and
dnj (z) d log
h (z) dz
are. Now,
0 if z < zj (0) d log j if z 2 [zj (0) ; zj (1)] j d log
nj (z)
0 if z > zj (1)
And it is straightforward to show that
d log d log
in 1 + j . To show that the same is true for d log Ej d log
j
jj
and
d log d log
j
d log d log = 11
j
are both homogeneous of degree 1 1 log
jj
d log d log
jj
+
d log Ej d log
d log wj d log
, it
d log wj d log
is homogeneous of degree 1 in 1 + j . To show this, use remains to show that the goods market clearing Pj Cj = Rj to write Pj wj
Z
1
= Vj
1
z
(1 jj
)nj (z)=
h (z) dz
which implies d log Pj d log
d log wj d log
Ej = Rj
log
jj
Z
zj (1)
z
Pj C j ,
d log Ej d log
d log d log
log
Plugging in the expression for neous of degree 1 in 1 + j .
(1
zj (0)
Using the de…nition of Ej = Vj wj1 d log wj = d log
1 nj (z) jj
jj
d log d log
j
I j Nj Rj
d log j d log
)
dnj (z) h (z) dz + d log
I j Rj Rj
!
and rearranging shows that
34
+
d log d log
d log Ej d log
d log d log
jj
Nj Rj
! N d log Cj jj j + Rj d log d log wj d log
,
is homoge-
7.3
Imperfectly correlated size and import share
In the model in the main part of the paper, size and import share are perfectly correlated: more e¢ cient plants have higher import shares and also use more labor and intermediate inputs – while in plant-level data, this correlation typically far from one. In the Chilean plant-level data, for example, the cross-sectional correlation between import share and log intermediate expenditures is about 0.31. Here, I lay out an extension of the model that introduces an imperfect correlation between import share and plant size but has all the same aggregate predictions as the benchmark model. In particular, the welfare implications of the model are identical to those derived in Propositions 2 and 3, because the nonhomotheticity of plant-level import demand takes the same form. In this extension, plants face an additional idiosyncratic shock to productivity after making their sourcing decision. Such an ex-post idiosyncratic shock is a standard feature in applications of structural production function estimation methods, for example Olley and Pakes (1996) and Kasahara and Rodrigue (2008). Each plant independently draws a multiplicative shock from a …xed distribution with pdf f ( ). Now, the variable pro…t of a plant in country j with e¢ ciency z that has chosen to source n of its inputs is given by the value of maximized pro…ts averaged across realizations of , ~ j (z; n) = max p;`;x
Z
p z`1 x
wj `
qj (n) x f ( ) d
p Pj
subject to z`1 x =
Cj
so that revenues of a plant with ex ante e¢ ciency z and shock inputs are now given by: 1
rj (z; ; n) =
(1
1
z
jj
)
n
that chooses to source n
Ej
1 Demands for inputs of labor and intermediate goods are similarly scaled by relative R 1 to the benchmark model. Normalizing the distribution of so that f ( ) d = 1, expected variable pro…ts ~ j (z; n) are exactly equal to variable pro…ts j (z; n) in the original model. The sourcing problem has the same form as (8), so that the solution nj (z) is the same as 9. In addition, aggregating across plants yields expressions for aggregates (Rj ; Mj ; Xj ) that are identical to the expressions in the benchmark model. For example,
Rj = =
Z Z Z
z
1
1
(1
jj
1
z
(1
jj )
nj (z)
)
nj (z)
Ej h (z) dzf ( ) d
Ej h (z) dz
35
and similarly for Mj and Xj . Since nj (z) is the same, and the equilibrium conditions that these aggregates must satisfy are the same, their equilibrium values are the same as in the model in the main part of the paper. Therefore, the aggregate implications of this model are identical to those of the model in the main part of the paper. Looking at plant-level revenues, however, the shock breaks the link between plant size and import share: a plant with high e¢ ciency z chooses a high sourcing share nj (z), but may experience a low shock , resulting in low output and inputs. By adjusting the dispersion of , the correlation of import share and size can be made arbitrarily small. For example, assuming that log ( ) is normal with mean and standard deviation , a value of = 0:95 (along with = 0:9025 so that R 1 f ( ) d = 1) yields a correlation between import share and log revenues of 0:31, as in the Chilean plant-level data. This lower correlation between import share and plant size has no impact on the welfare gains because the distribution of general equilibrium e¤ects on @n (z) dnj (z) is exactly the same as without variation in , so the input sourcing decisions, @j d according to Proposition 3, the welfare implications are also the same.
7.4
Calibration: moments as functions of parameters
Here I show that four of the calibration moments— the average import share among importing plants, the average size of importing relative to nonimporting plants, the aggregate import share, and the partial trade elasticity— depend only on four of the jointly calibrated parameters— , f , , and — along with the fraction of plants importing, F , and the two predetermind parameters, and . I drop the country subscript because all the moments are for Chile only. 7.4.1
Average import share
The average import share among importing plants is: s = =
zL z (0) zL z (0)
=
1
s (z) h (z) dz
z(0)
1
= (1 with only.
Z
1
Z
z(1)
(1
)
1
(log z
log ) zL z
1)
dz +
Z
1
z(1)
(1
) zL z
1
dz
!
e
1 log(f 1=(
)
z(0)
1
=
)
and
=
1 1+
. So given
36
and , s is a function of , , f , and
7.4.2
Average size of importing relative to nonimporting plants
I use revenues as a measure of plant size, but employment or total intermediate inputs would yield the same expression for the ratio of average size, since they are proportional to revenues. The average size of importing plants is: Xm
Z
1 = F
z(1)
r (z) h (z) dz + Z
+
r (z) h (z) dz
!
z(1) 2 n(z) (1
z
)=
(1
dz +
)=
and
1
1
Xd =
1
F 1
=
1
e
)=
=
F
1
+
Z
(1
+ )
z(0)
r (z) h (z) dz
zL
E zL
7.4.3
1 1
1
zL
+
e
1
is a function of , , f , and
z (0)
+
e d
only.
Aggregate import share
The aggregate import share is M (1 )N = X R With R and N de…ned as in (16),
M X
can be written as:
(1 M = (1 X
)e F)
d=
d
So given F , , and ,
M X
z
2
dz
!
log . The average size of nonimporting plants
1 F Xm = ( 1)= d F F Xd Xm Xd
1
e d
So the ratio of Xm to Xd is:
So given F , , and ,
Z
z(1)
z(0)
= E zL 1 F (1 where d = 1 is:
1
z(1)
z(0)
1 E zL F
=
Z
1+e
+
2
1+e
+
e
is a function of , , f , and
37
1 d 1
only.
1
7.4.4
Partial trade elasticity
The partial elasticity of the import share with respect to the variable trade cost is: ~"P =
@ log X @ log
@ log M @ log
where @ log M @ log
=
1 M 1 + M 1 + M
and
@ log X 1 = @ log X
Using the derivatives we can write
@ log @ log
8 > > <
and
1 @ N @ log Z z(1) @n (z) 1 @r (z) (1 ) r (z) + n (z) @ log @ log z(0) Z 1 @r (z) 1 (1 ) h (z) dz z(1) @ log
@n (z) = > @ log > : 8 > <
@r (z) = r (z) (1 > @ log :
= (1
),
1 @ @
Z
h (z) dz
@r (z) h (z) dz @ log 2
=
(29)
1
(1
), and
@ log @
=
1
1 1 log
@ log @
,
0 if z < z (0) (1
)
1
n (z)
1 1 log
if z 2 [z (0) ; z (1)]
0 if z > z (1)
0 if z < z (0) ) (1 ) n (z) 1 log + 1 if z 2 [z (0) ; z (1)] r (z) (1 ) (1 ) if z > z (1)
Now, since = z (0) = zL F 1= and z (1) = z (0) e1= , given F , , and , we can write n (z) as functions of , , f , and only. Similarly, r (z) and @@r(z) are proportional to and @@n(z) log log EzL , but are otherwise functions of , , f , and only. The aggregates M and X are also M X proportional to EzL , so @@log and @@ log depend on , , f , and only. log log
38
7.5
Solution to sourcing problem in CES case
The input sourcing problem in the CES case is: max z
1
n2[0;1]
where 'j = is:
(1 jj
)=
1 and a = z
1
n'j + 1 (
1) . 1
a n'j + 1
a
1
Ej
wj b (f n
1)
The …rst order condition for an interior solution
a 1
1 'j Ej = wj bf n log f
Solving for z as a function of n yields the inverse sourcing decision: zj (n) = where $j =
n'j + 1
wj b log f . a'j Ej
39
1 a
1
f n $j
1
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Ossa, R. (2015): “Why Trade Matters After All,”Journal of International Economics, 97, 266–277. Romer, P. M. (1990): “Endogenous Technological Change,”The Journal of Political Economy, 98(5), S71–S102. Simonovska, I., and M. Waugh (2014): “Trade Models, Trade Elasticities, and the Gains from Trade,”NBER working paper 20495.
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Table 1: Calibration of symmetric 2-country model Externally set parameters elasticity of …nal demand 3:00 intermediate share of revenues 0:77 T = L = zL technology / labor force (normalization) 1:00 Jointly calibrated parameters shape parameter of …nal good e¢ ciencies 14:14 shape parameter for intermediate good e¢ ciencies 0:60 f shape parameter in …xed cost function 4:45 b scale parameter in …xed cost function 0:25 variable trade cost 1:68 Jointly matched moments fraction importing 0:23 average import share among importers 0:33 average size importer/nonimporter 4:50 aggregate import share 0:22 partial trade elasticity, "P 5:00
Table 2: Gains from Trade in Symmetric Model C Welfare gain relative to autarky, 100 1 C aut M X
="P
ACR-implied gain relative to autarky, 100 1 G Full trade elasticity, " Local elasticity of welfare gain, d log C=d log (1 M=X) = Local elasticity of ACR formula, "P
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10:44 1 "G
3:90 1:62 0:48 0:15
Table 3: E¤ects of Trade on Average and Aggregate Productivity Trade relative to autarky Including …xed Not including …xed costs in labor costs in labor Aggregate TFP 13.85% 29.63% Average TFP, all plants 10.42% 14.45% Average TFP, importing plants 33.74% 51.05%
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welfare relative to initial equilibrium (log scale)
1.1
1.05
ACR
1
0.95
Act ual
0.9
First-order approximat ion
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
aggregate domestic share (log scale)
Figure 1: Change in welfare as a function of the aggregate domestic spending share as the variable trade cost changes.
12
Welfare gain relative to autarky, %
1.85 1.8 1.75 1.7 1.65 1.6 1.55 3
10 Model 8
6 ACR 4
2 4
5
6
7
3
4
5
6
7
Figure 2: Sensitivity to target for partial elasticity of aggregate import ratio, "P .
45
25
Welfare gain relative to autarky, %
3
2.5
2
1.5
1
0.5
2
2.5
3
3.5
20
15
10
5
4
2
2.5
3
3.5
4
Figure 3: Sensitivity to elasticity of substitution among …nal goods, .
10.436
Welfare gain relative to autarky, %
1.65
1.6
1.55
1.5
1.45 1
1.2
1.4
10.435 10.434 10.433 10.432 10.431 10.43 10.429 1
1.6
1.2
1.4
1.6
Figure 4: Sensitivity to elasticity of substitution among intermediate goods, .
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