NOT FOR PUBLICATION
Theory Appendix for “The China Syndrome”
Small Open Economy ModelIn this appendix, we develop a general equilibrium model of how increased import competition from
China affects employment and wages in a U.S. commuting zone, which we treat as a small open
economy. Productivity growth in China and global reductions in trade barriers facing China cause
the country’s exports to expand. As a commuting zone faces greater competition from China in the
U.S. market and in other markets in which its firms sell goods, demand for CZ output contracts,
causing CZ wages to fall. As long as the CZ is running a current-account deficit, there is a resulting
shift in employment out of traded goods and into non-traded goods. Initially, we ignore the impact
of changes in China on wages and income levels outside of a CZ, focusing on the direct effects ofrising productivity/falling trade costs in China on a commuting zone, which operate through making
the CZ’s goods less competitive in its export markets. Below, we consider a two-economy model(e.g., for the U.S. and China), in which the same qualitative results obtain. Hsieh and Ossa (2012)
model the effects of productivity growth in China in full global general equilibrium.
The total supply of labor in CZ i is Li , where labor may be employed in traded goods or in nontraded goods. We assume that there is no migration between commuting zones (making the model
short to medium run in nature). Allowing CZ labor supply to be an elastic function of the wage is a
simple extension of the model. Demand for goods is given by a Cobb-Douglas utility function, withshare γ of expenditure going to traded goods and share 1 − γ going to non-traded goods. There isa single non-traded good which is manufactured under the production function,
XN i = LηN i ,(1)where LN i is labor employed in non-traded goods and the coefficient η ∈ (0, 1) indicates there isdiminishing marginal returns to labor in production (due, e.g., to short-run constraints on expanding
production capacity). Profit maximization in the non-traded good implies that
Wi = ηPN i Lη−1
Ni ,

(2)

where Wi is the wage and PN i is the price of the non-traded good in commuting zone i. Becauseof diminishing returns in non-traded production, any shock that expands employment in the sector
will tend to push down wages in the commuting zone. (Alternatively, we could consider (1) as animplicit function for the production of leisure and (2) as arising from utility maximization, requiringthat wages equal the marginal utility of leisure.)
Market clearing for the non-traded good requires that,PN i XN i = (1 − γ) (Wi Li + Bi ) ,1(3)

where Bi is the difference between expenditure and income in commuting zone i (i.e., Bi > 0 implies
that CZ i is running a current-account deficit).1 We treat the trade imbalance as given (due to US
macroeconomic conditions) and investigate how its magnitude affects CZ labor-market adjustment.
With balanced trade for a commuting zone, a positive shock to productivity in one of China’s export
sectors generates changes in the CZ wage and non-traded good price that re-equilibrate imports andexports. These adjustments keep total CZ employment in the traded sector from declining (althoughemployment shifts out of the traded sector with positive Chinese productivity growth and into othertraded sectors). With imbalanced trade a positive shock to Chinese export productivity reduces
employment in CZ traded goods and increases employment in non-traded goods.2
Traded goods are produced by firms in a monopolistically competitive sector (Helpman and Krugman, 1985).3 There are two traded-good sectors, indexed by j, where consumers devote a share ofspending γ/2 on each. It is straightforward to extend the model to multiple traded-good sectors (asin Hanson and Xiang, 2005); doing so does not change the qualitative results. Each of the Mij firms
in sector j is the unique producer of a differentiated product variety. The labor used to produce any
individual variety in sector j is given by,
lij = αij + βij xij ,(4)

where for sector j αij is the fixed labor required to produce positive output, βij is the labor required
to produce an extra unit of output, and xij is the quantity of the variety produced. αij and βij(which are identical across firms within CZ i) reflect sectoral productivity in a commuting zone andtherefore determine comparative advantage. For each traded sector j, demand for product varieties
is derived from a CES sub-utility function, such that total demand for output of an individual
variety, xij , is the sum over demand in each destination market k, xijk , given by,
xij =

X

xijk =k−σj
X PijkγEkk1−σΦjk j 2,(5)where Pijk is the delivered price in market k of a variety in sector j produced in commuting zone
i, Ek is total expenditure in market k, and the term Φ1−σ
jk , which is a function of the price index,
Φjk , for traded goods in sector j and market k, captures the intensity of competition in a particularmarket. The parameter σj > 1 is the elasticity of substitution between any pair of varieties in j.
Under monopolistic competition, the price of each variety is a constant markup over marginal cost,σjPijk =
βij Wi τijk
(6)
σj − 1
where τijk ≥ 1 is the iceberg transport cost of delivering one unit of a good in sector j from
commuting zone i to market k. We assume that free entry in each sector drives profits to zero,1

Implicitly, China’s non-traded good is the numeraire.
The invariance of non-traded employment to trade shocks under balanced trade is due to the assumption of
Cobb-Douglas preferences (similar results hold in a two-country model, meaning that the small-country assumption
is not driving this outcome).
3
Our results generalize to other settings that have a “gravity” structure, as in Arkolakis, Costinot, and RodriquezClare (2011).
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implying that the level of output of each variety is xij = αij (σj − 1) /βij (adjustment in sectoral
output and employment occurs at the extensive margin, through changes in the sector number ofvarieties/firms, Mij ). The final equilibrium condition is that labor supply equals labor demand:
Li = LN i + LT i ,where LT i =(7)P

j Mij lij is total employment in traded goods.

The sectoral price index plays an important role in the analysis for it is the channel through which
competition from China affects a CZ. For each sector j, this index is given by,
"
Φjk =

#
X

1−σ
Mhj Phjk j

1
1−σj

,

(8)

h

where Mhj is the number of varieties produced by region h and Phjk is the price of goods from region
h sold in market k. Log differentiating (8), and defining x̂ ≡ ∆ ln x = ∆x/x, we obtain for eachsector j,Φ̂jk = −1 X
φhjk Âhjk ,σj − 1(9)hwhere φhjk ≡ Mjh Phjk xhjk /Pl Mlj Pljk
 xljk is the shareof region h in purchases of sector j goods bymarket k and Âhjk ≡ M̂hj − (σj − 1) Ŵh + β̂hj + τ̂hjk is the log change in the “export capability”

of region h in market k, determined by changes in the number of varieties region h produces (M̂hj ),its wages (Ŵh ), its labor productivity (β̂hj ), and its trade costs (τ̂hjk ). The price index for sector jgoods in market k declines if China has an increase in the number of varieties that it produces, a
reduction in its marginal production costs, an increase in its factor productivity, or a reduction inits trade barriers (each of which causes ÂCjk to rise, where C indexes China).
To solve the model, we plug (1) into (3), and (for each j) (4) and (6) into (5), which produces asystem of five equations in five unknowns, Wi , PN i , LN i , and Mij for j = 1, 2.4 After performing
these substitutions and log differentiating the five equations, we end up with the following system:
Ŵi = P̂N i − (1 − η) L̂N i ,η L̂N i = ρi Ŵi + L̂i + (1 − ρi ) B̂i − P̂N i ,L̂i = 1 −X

δij  L̂N i +j

σ Ŵi =XkX

δij M̂ij ,

j

h
i X
X
X
θijk Êk + (σj − 1) Φ̂jk =θijk Êk −θijk
φhjk Âhjk , j = 1, 2
k

4

k

(10)

h

For simplicity, we exclude the equation for adjustment in imported varieties. Because of the small-country
assumption, changes in imports are determined by the outcomes of other equations in the system and do not affectother variables.

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where for commuting zone i ρi ≡ Wi Li / (Wi Li + Bi ) is the initial share of labor income in total
expenditure, δij ≡ Mij lij /Li is the initial share of traded sector j in total employment, and θijk ≡P
xijk / l xijl is the initial share of market k in the total shipments of sector j goods. Because theoutput of each variety is fixed, labor used in each variety, lij , is fixed; all adjustment in sectoral
employment occurs through changes in the number of firms, Mij , as seen in the third line of (10).
By assumption, for commuting zone i the only changes in the Êk terms in (10) occur in China, where
we treat ÊC = ρC ŴC + (1 − ρC ) B̂C as exogenous, and in CZ i itself, where Êi = ρi Ŵi + (1 − ρi ) B̂iand we treat Ŵi as endogenous and B̂i as exogenous. As a trade shock causes wages in a commuting
zone to change, the CZ’s demand for its own goods will change, which will in turn generate further
adjustments in wages. Relatedly, for commuting zone i the only changes
in the Âhjk termsin (10) are
for China, where for each sector j we treat ÂCj = M̂Cj − (σj − 1) ŴC + β̂Cj + τ̂Cj

as exogenous,

and in CZ i itself, where for each sector j, Âij = M̂ij − (σj − 1) Ŵi and we treat M̂ij as endogenous,in addition to Ŵij . As a China trade shock causes a CZ’s wage and number of firms to change, priceindexes in the markets that the CZ serves will change, generating further adjustments in its wages
and number of firms.5
Imposing the zero migration assumption that L̂i = 0 and rearranging the first two expressions in(10), we obtain the following representation of the system of equations in (10):P̂N i = Ŵi + (1 − η) L̂N i ,L̂N i = (1 − ρi ) B̂i − Ŵi ,L̂N i = −δ̃i1 M̂i1 − δ̃i2 M̂i2 ,Ŵi = ai1 Γ̂i1 + bi1 B̂i − ci1 M̂i1 ,Ŵi = ai2 Γ̂i2 + bi2 B̂i − ci2 M̂i2 ,where for sector j = 1, 2 we employ the following notational definitions: δ̃ij ≡ δij / (1 −(11)Pn δin ) is

the initial hratio of employment ini traded sector j to employment in non-traded goods, the quantityPΓ̂ij ≡ θijC ρC ŴC + (1 − ρC ) B̂C − k θijk φCjk ÂCj is the China trade shock facing CZ i in industry
j, and aij , bij , and cij are each positive constants that are functions of the model parameters or initialPP
sectoral employment or expenditure shares (aij ≡ [σj (1 − k θijk φijk ) + k θijk φijk − θiji ρi ]−1 ,Pbij ≡ aij θiji (1 − ρi ), and cij ≡ aij k θijk φijk ). In the first two lines of (11), we see that wage
shocks affect non-traded employment and non-traded prices only if trade is imbalanced (ρi 6= 1).
This outcome depends on the first two equations in (10), which applies to the model even if we allow
the country to be large enough to affect world prices, as is done below.
For CZ i, the China trade shock in sector
between increased demand byh j (Γ̂ij ) is the differenceiChina for the CZ’s exports, given by θijC ρC ŴC + (1 − ρC ) B̂C , and increased import competition
5

For notational simplicity, we assume that changes in China’s trade costs are common across its destination
markets–due,e.g., to its accession to the WTO–and that CZ i has no changes in its productivity or trade costs.

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from China in the markets in which the CZ sells goods, given byP

k θijk φCjk ÂCj . Growth in China’s

demand for CZ i’s exports will be smaller the smaller is the share of CZ output that is destined for
China (θijC ) and the more wage growth in China (ŴC > 0) is offset by growth in China’s currentaccount surplus (B̂C < 0). Import competition from China will be more intense the larger is theincrease in China’s export capabilities (ÂCj ) and the larger is China as a source of supply for thePmarkets that CZ i serves (captured by the term, k θijk φCjk ).
Solving the system in (11), we obtain changes in the endogenous CZ variables (ŴN i , L̂T i , L̂N i , P̂N i )as functions of model parameters and the exogenous shocks (Γ̂i1 , Γ̂i2 , B̂i ), where we show results forthe change in total employment in traded goods (rather that for individual traded sectors), given byP ˜P˜L̂T i = j δ̃ ij M̂ij , where δ̃ ij ≡ δij / l δil is the share of sector j in total traded-good employment
for CZ i. The solutions for the endogenous variables are:

 i
1 h
Ŵi =
ai1 ci2 δ̃i1 Γ̂i1 + ai2 ci1 δ̃i2 Γ̂i2 + bi1 ci2 δ̃i1 + bi2 ci1 δ̃i2 + (1 − ρi ) ci1 ci2 B̂i ,gi
L̂T i = i
1 − ρi h˜˜˜˜
ai1 ci2 δ̃ i1 Γ̂i1 + ai2 ci1 δ̃ i2 Γ̂i2 − (1 − bi1 )ci2 δ̃ i1 + (1 − bi2 )ci1 δ̃ i2 B̂i ,gi

L̂N i = i
1 − ρi h−ai1 ci2 δ̃i1 Γ̂i1 − ai2 ci1 δ̃i2 Γ̂i2 + (1 − bi1 )ci2 δ̃i1 + (1 − bi2 )ci1 δ̃i2 B̂i ,gi

P̂N i =1[(1 − fi ) ai1 ci2 δ̃i1 Γ̂i1 + ai2 ci1 δ̃i2 Γ̂i2gi+ (bi1 + (1 − bi1 )fi ) ci2 δ̃i1 + (bi2 + (1 − bi2 )fi ) ci1 δ̃i2 + (1 − ρi ) ci1 ci2 B̂i ](12)where gi = ci2 δ̃i1 + ci1 δ̃i2 + (1 − ρi ) ci1 ci2 > 0, fi = (1 − ρi )(1 − η) > 0, and 1 − bij > 0, j = 1, 2. Tosummarize how trade shocks in China affect a CZ, we present the following comparative statics:
∂ Ŵi

=

∂ Γ̂ij∂ L̂T i∂ Γ̂ij∂ L̂N i∂ Γ̂ij∂ P̂N i∂ Γ̂ij={j, l} = {1, 2}, {2, 1},

˜(1 − ρi )aij cil δ̃ ij≥ 0,gi{j, l} = {1, 2}, {2, 1},(1 − ρi )aij cil δ̃ij
≤ 0,gi

{j, l} = {1, 2}, {2, 1},(1 − fi )aij cil δ̃ij
≥ 0,gi

{j, l} = {1, 2}, {2, 1}.

=−=

aij cil δ̃ij
≥ 0,
gi

(13)

In traded sector j, productivity growth in China or a fall in China’s trade barriers imply that Γ̂ij < 0.
In (13), we see that the consequence of such a shock is a reduction in CZ nominal wages, a reduction
in CZ employment in traded goods, an increase in CZ employment in non-traded goods, and areduction in CZ prices of non-traded goods. The impact on wages is due to the decreased demand5for CZ goods in its export markets (including the broader U.S. economy). The impacts on tradedand non-traded employment depend on ρi < 1, meaning the CZ is running a current-account deficit.
Regardless of the shift in employment between traded and non-traded goods, within traded goods
there is a reallocation of employment out of sectors in which China’s productivity is expanding.
Why does the impact of productivity growth in China on CZ traded and non-traded employment
depend on the CZ’s trade balance? With balanced trade, productivity growth in China merely
reallocates CZ employment between traded sectors based on which sectors face a net increase inimport competition from China (CZ employment contracts) and which experience a net increase inexport demand by China (CZ employment expands). With imbalanced trade, increases in importcompetition are not offset by increases in export demand. The excess of imports over exports pushes
employment out of exports (relative to balanced trade), with non-traded goods being the residual
sector. The logic for a CZ also applies to the United States as a whole, meaning that a U.S. currentaccount deficit vis-a-vis China implies that greater import competition from China can cause U.S.
employment in traded-good sectors to contract on net.
In (12), changes in wages, traded-good employment and non-traded good employment are eachweighted averages of changes in trade shocks in each traded-good sector, where these weights arefunctions of the share of each traded sector in total employment. These expressions motivate ourmeasure of trade exposure in the empirical analysis.

Two Economy Model
A small open economy is a non-standard application of the monopolistic competition model. Typically, in such models all goods prices are endogenous, which is not the case in the application above
where we have arbitrarily shut down price adjustment in all economies except CZ i. To verify thatthe results we obtain are not special to this setting, we solve a two-economy model, in which we compress CZs into a single aggregate U.S. region. We then examine the impact of productivity growthin China on U.S. wages, traded employment, and non-traded employment. To keep the analysis
simple, we ignore trade barriers between the countries and assume the traded sector consists of asingle industry (producing many varieties). No qualitative results depend on these restrictions.
Following equations (1)-(3), (6), and (7), we have the following equilibrium conditions for the U.S.:
W = ηPN Lη−1
N ,PN LηN = (1 − γ) (W L + B) ,
P =

σ
βW,
σ−1

L = LN + M l,

(14)

where we take China’s wage to be the numeraire (such that W is the U.S. wage relative to China’swage) and B is the difference between U.S. aggregate expenditure and U.S. aggregate income (equal6to the difference between China’s aggregate income and expenditure–i.e., B + B ∗ = 0) and isdominated in units of China’s wage. The final equilibrium condition is that supply equals demandfor each variety of traded goods:x=P −σ γ (W L + L∗ ).
M Φ1−σ + M ∗ Φ∗1−σ(15)

We implicitly treat l, labor used to produce each variety, as exogenous given that its value ispinned down by the zero-profit condition (i.e., l = ασ); zero profits also imply that x is fixed(x = α (σ − 1) /β). For China, there are a corresponding set of equilibrium conditions, where wedominate China values using an (∗). Because trade costs are zero, x/x∗ =(P/P ∗ )−σ , which together
with the price-equals-marginal cost conditions in the U.S. and China imply that W = (β ∗ /β)(σ−1)/σ ,or that the U.S.-China relative wage is a function of relative labor productivities in the two countries.
Combining the conditions in (14) with the corresponding ones for China and incorporating thesolutions for W, P , and P ∗ , we have a system with six equations and xi unknowns (PN , PN∗ , LN , L∗N ,M, and M ∗ ). We assume that the only shocks to the system are productivity growth in traded-good
production in China (β̂ ∗ < 0) and an increase in the U.S. trade deficit/China trade surplus (B̂ > 0).
Log differentiating, we have that Ŵ = σ̄ β̂ ∗ , where σ̄ ≡ σ−1
σ , implying that the U.S. relative nominal
wage declines in proportion to productivity growth in China.6 The other equilibrium conditions arethat:P̂N = σ̄ β̂ ∗ + (1 − η) L̂N ,P̂N∗ = (1 − η) L̂∗N ,P̂N = ρσ̄ β̂ ∗ + (1 − ρ) B̂ − η L̂N ,P̂N∗ = − (1 − ρ∗ ) B̂ − η L̂∗N ,L̂N = −
L̂∗N = −

δ
M̂ ,
1−δ

δ∗
M̂ ∗ ,
1 − δ∗(16)

where ρ = W L/ (W L + B) is the initial share of labor income in total U.S. expenditure, (1 − ρ∗ ) =B/ (L∗ − B) is the initial ratio of China’s trade surplus to its aggregate expenditure, δ = M l/L
is the initial share of U.S. employment in traded goods, and δ ∗ = M ∗ l∗ /L∗ is the initial share of
China’s employment in traded goods. Solving the system in (16) we obtain,L̂N = (1 − ρ) B̂ − σ̄ β̂ ∗ ≥ 0,
L̂∗N = − (1 − ρ∗ ) B̂ ≤ 0,
M̂ = −6
1−δ
(1 − ρ) B̂ − σ̄ β̂ ∗ ≤ 0,
δU.S. real wages may of course rise owing to lower prices for and increased numbers of Chinese varieties produced.

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1 − δ∗(1 − ρ∗ ) B̂ ≥ 0,δ∗P̂N = β̂ ∗ + (1 − η) (1 − ρ) B̂ − σ̄ β̂ ∗ S 0,
M̂ ∗ =

P̂N∗ = − (1 − η) (1 − ρ∗ ) B̂ ≤ 0.

(17)

It is again the case that productivity growth in the traded sector in China lowers U.S. employment intraded goods (M̂ <0) and raises U.S. employment in non-traded goods (L̂N > 0), where these results
are conditional on the U.S. running an aggregate trade deficit. There is an ambiguous effect on U.S.
non-traded prices. Increases in the magnitude of the U.S. trade deficit reinforce these changes.

References
Arkolakis, Costas, Arnaud Costinot, and Andres Rodriguez-Clare. 2012. “New Trade Models, SameOld Gains?” American Economic Review, 102(1), 94-130.
Hanson, Gordon, and Chong Xiang. 2004. “The Home Market Effect and Bilateral Trade Patterns.”
American Economic Review, 94: 1108-1129.
Helpman, Elhanan, and Paul Krugman. 1985. Market Structure and Foreign Trade. Cambridge,
MA: MIT Press.
Hsieh, Chang-Tai, and Ralph Ossa. 2011. “A Global View of Productivity Growth in China.” NBER
Working Paper No. 16778.

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