Transcript
Measuring Openness to Trade
Michael E. Waugh New York University and NBER
B. Ravikumar Federal Reserve Bank of St. Louis and Arizona State University October 21, 2015
ABSTRACT ———————————————————————————————————— This paper derives a new measure of openness that quantifies the potential gains to trade as a simple function of data. Using a standard, multi-country trade model we measure a country’s potential welfare gain from the world moving to a frictionless trade regime. A country’s openness-potential depends upon only three statistics: the county’s home trade share, its income level, and the trade elasticity. Quantitatively, we find that the world—in welfare terms—is effectively in autarky and that a substantial portion of possible gains arise from setting the trade barriers in all countries to the levels of developed countries. —————————————————————————————————————————-
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1. Introduction How open is a country to trade? How large are the welfare gains from trade? These questions have been typically answered by computing the welfare cost of autarky—the change in real income due to a change from the observed equilibrium to autarky. Arkolakis, Costinot, and Rodriguez-Clare (2012) show that this calculation in a large class of models takes a simple form: a country’s home trade share (i.e., one minus the country’s import penetration ratio) taken to the power of the inverse of the trade elasticity. In this paper, we deliver a measure of openness that quantifies the potential gains to trade. We define a country’s openness potential as the the change in real income due to a change from the observed equilibrium to the efficient, frictionless trade equilibrium. Within a standard model of trade, we show that a country’s potential takes a simple form: it is proportional to
λii Yi
1 1+θ
(1)
where λii is country i’s observed home trade share, Yi is country i’s observed real GDP, θ is the trade elasticity.1 Relative to the standard welfare cost of autarky calculation, the key feature of our measure is that it encodes how a country’s potential depends upon its technology and endowments as summarized by its observed GDP. This distinction is important because two countries may have the same welfare cost of autarky, but vastly different potentials. Empirically, this is the pattern in the data. The welfare cost of autarky is similar across countries, but poor countries have the largest potential to gain from trade. We derive our measure by embedding the multi-country trade model developed by Eaton and Kortum (2002) into a neoclassical growth model. Each country is endowed with a stock of capital and a labor force. Both factors are immobile across countries. There is a continuum of tradable goods. The distribution of productivity over the continuum belongs to the Fr´echet family with countries differing in the centering parameter, but having a common shape parameter. International trade is subject to barriers in the form of iceberg costs. All markets are competitive. The amount of trade between any two countries, in equilibrium, depends on wages, rental rates, and trade costs. Our openness potential roughly describes how much can each country gain by moving from a current world with trade costs to a frictionless world. Operationally, this involves computing the change in GDP per worker from its current observed level to a new level in the frictionless world. To measure a country’s openness potential, we proceed in two steps. We first provide a closed1
The constant of proportionality is not-country specific.
2
form expression for a country’s real GDP per worker, no matter what the trade costs are. This expression depends upon the country’s home trade share, technology parameter, labor and capital endowments, and two non-country-specific parameters—the trade elasticity and capital’s share in production. This expression shows how a country with a larger technology parameter or capital endowment per worker will be relatively richer. We can use the above expression for GDP per worker in the frictionless world and the fact that the home trade share is a function of country-specific endowments and technology parameter and the non-country-specific parameters. To solve for each country’s home trade share in the frictionless world we need to know the country-specific technology parameters. In the second step, we measure the country-specific technology parameters. This measurement is easily done using a modified development accounting approach based on the neoclassical growth model (see, e.g. Hall and Jones (1999) or Caselli (2005)). The combination of development accounting plus the characterization of country’s real GDP per worker in the frictionless world allows us to eliminate the endogenous trade variables and infer a country’s opennesspotential. Our approach yields the surprising result that a country’s openness-potential is completely summarized by its observed home trade share and the level of GDP as shown in (1). That is, details such as its capital stock, labor endowment, technology, etc. are not necessary in our calculation of the openness-potential. The important qualitative prediction of our measure is how a country’s potential depends on its current level of income. As (1) makes clear, holding all else constant, a country whose observed real GDP is relatively low has relatively more to gain from trade. In other words, the poorest countries have the most openness-potential. This prediction of our measure highlights our contribution relative to the standard welfare cost of autarky calculation. The welfare cost of autarky calculation implies that if a rich country and a poor country have the same home trade share, then the rich and the poor experience the same loss from a move to autarky i.e., the welfare gain is the same across countries. Missing from this inference, however, is the fact that the poor country has more potential to gain from trade. Thus, while the rich and the poor country have the same welfare cost of autarky, the poor country is a more closed economy in comparison to its potential relative to the rich country. We quantify our measure using data on trade shares, endowment, and GDP directly from the Penn World Tables. We find two important results. First, the world is effectively in autarky. That is the distance between the observed equilibrium and autarky is small relative to the large potential gains. Second, trade has large cross-country distributional consequences. Because all countries are relatively close to autarky, but low technology/poor countries have the most potential to gain, which implies that the potential gains can eliminate cross-country income 3
inequality. Frictionless trade is obviously an extreme case, so we complement our theoretical results with a quantitative assessment of the gains associated with plausible changes in trade costs. We keep our quantitative work simple with a relatively parsimonious description of the barriers to trade—one trade cost per country. This structure of trade costs allows us to calibrate our model without any more data than that available in the Penn World Table. And, it provides one parameter per country to succinctly summarize the trade frictions. With our calibrated trade frictions, we ask how much of the potential can be achieved if the world had the same trade cost as that of the U.S. The key result from this exercise is that while it delivers about one-fourth of the overall gains, on average, this counterfactual delivers nearly all the reduction in cross-country income inequality that the frictionless trade economy would.
2. Model We outline the environment of the multi-country Ricardian model of trade introduced by EK. We consider a world with N countries, where each country has a tradable final-goods sector. There is a continuum of tradable goods indexed by j ∈ [0, 1]. Within each country i, there is a representative consumer of size Li . This consumer supplies labor inelastically in the domestic labor market and also owns physical capital Ki that is inelastically supplied to the domestic capital market. This consumer also enjoys the consumption of a CES bundle of final tradable goods with elasticity of substitution ρ > 1: Ui =
Z
1
xi (j)
ρ−1 ρ
0
dj
ρ ρ−1
.
(2)
To produce quantity xi (j) in country i, a firm employs a Cobb-Douglas production function combining capital and labor with factor shares α and 1 − α and productivity zi (j). Country i’s productivity for good j is, in turn, the realization of a random variable (drawn independently for each j) from its country-specific Fr´echet probability distribution: Fi (zi ) = exp(−Ti zi−θ ).
(3)
The country-specific parameter Ti > 0 governs the location of the distribution; higher values of it imply that a high productivity draw for any good j is more likely. The parameter θ > 1 is common across countries and, if higher, generates less variability in productivity across goods in each country. Having drawn a particular productivity level, a perfectly competitive firm from country i in4
curs a marginal cost to produce good j of riα wi1−α /zi (j), where wi is the wage rate and ri is the rental rate of capital in country i. Shipping the good to a destination n requires a per-unit iceberg trade cost of τni > 1 for n 6= i, with τii = 1. We assume that cross-border arbitrage forces effective geographic barriers to obey the triangle inequality: For any three countries i, k, n, τni ≤ τnk τki . Below, we describe equilibrium prices, trade flows, aggregate output, and the welfare cost of autarky. Prices. Perfect competition implies that the price of good j from country i to destination n to be equal to the marginal cost of production and delivery: pni (j) =
τni riα wi1−α . zi (j)
(4)
So, consumers in destination n would pay pni (j), should they decide to buy good j from i. Consumers purchase good j from the least-cost supplier; thus, the actual price consumers in n pay for good j is the minimum price across all sources k: pn (j) = min
k=1,...,N
pnk (j) .
(5)
The pricing rule and the productivity distribution allow us to obtain the following CES exact price index for each destination n: Pn =
−1 γΦn θ
where
Φn =
" N X k=1
In the above equation, γ = Γ such that θ > ρ − 1.
θ+1−ρ θ
1 1−ρ
#
Tk (τnk wk )−θ .
(6)
is the Gamma function, and parameters are restricted
Trade Flows. To calculate trade flows between countries, let Xn be country n’s expenditure on final goods, of which Xni is spent on goods from country i. Since there is a continuum of goods, the fraction of income spent on imports from i, Xni /Xn , can be shown to be equivalent to the probability that country i is the least-cost supplier to country n given the joint distribution of productivity levels, prices, and trade costs for any good j. In the equations below, we do not include proportionality constants that are not country-specific since such constants will not enter into our welfare gain calculations or our measure of openness. The expression for the share of expenditures that n spends on goods from i or, as we will call
5
it, the trade share, λni , is: Ti (τni riα wi1−α )−θ Xni = PN . λni := α 1−α −θ Xn ) k=1 Tk (τnk rk wk
(7)
Expressions (6) and (7) allow us to relate trade shares to trade costs and the price indices of each trading partner via the following equation: λni = λii
Pi τni Pn
−θ
,
(8)
where λii is country i’s expenditure share on goods from country i, or its home trade share. While well known, this point is worth reiterating: expression (8) is not particular to EK’s model. Several popular models of international trade relate trade shares, prices and trade costs in the same exact manner. These models include Anderson (1979), Krugman (1980), Bernard, Eaton, Jensen, and Kortum (2003), and Melitz (2003), when parametrized as in Chaney (2008). Aggregate GDP per Worker. A feature of this model (and other trade models) is that it has a convenient representation of real gross domestic product (GDP) per worker that is similar to a standard one-sector growth model with a total factor productivity term and capital-labor ratio raised to a power term. The key difference is that measured TFP is endogenous and depends on the country’s home trade share. This simple connection between trade and the tools of development accounting is something that we will exploit continually in this paper. To arrive at this representation, a couple of steps are needed. First, with competitive factor markets, the rental rate on capital is pinned down by the following relationship between the α wi ki−1 , where k is the aggregate capital to labor wage rate and the capital to labor ratio: ri = 1−α ratio. Second, combining this relationship with (7) yields an expression for each country’s home trade share. λii = PN
ki−α wi
−θ
Ti
−α −θ k=1 Tk (τnk ki wi )
.
(9)
Third, a rearrangement of (9) provides an expression for the real wage −1 1 wi = Tiθ λiiθ kiα , Pi
(10)
in which wages, deflated by the aggregate price index, are a function of each country’s technology parameter, its home trade share and its capital-labor ratio. Abstracting from the role of capital, this is the same expression discussed extensively in Arkolakis, Costinot, and RodriguezClare (2012) that relates home trade shares and the real wage and, hence, the welfare cost of
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autarky. Finally, market clearing conditions allow us to connect the real wage in (10) with real GDP per worker. Balanced trade and equating production of the aggregate commodity with total factor payments gives yi =
w i ri k i + . Pi Pi
(11)
And then using (10) and the observation above that the wage-rental ratio is proportional to the capital-labor ratio gives yi = Ai kiα ,
1
where
−1
Ai = Tiθ λiiθ .
(12)
Real GDP per worker in the model is expressed in the exact same way as in the standard onesector growth model with a TFP term and capital-labor ratio raised to a power term. The one difference is that measured TFP contains an endogenous trade factor, λ−θ ii , and an exogenous θ domestic factor, Ti . Several comments are in order regarding (12). First, the balanced trade assumption may seem strong, but it’s not. A similar expression, but with an additive term representing the difference between the balanced and imbalanced trade equation, is easily derivable. For now we simply abstract from this distinction. Second, given that the expressions for trade shares are not specific to the EK model, this implies there is nothing unique about (12) and its association with the EK model. Many trade models share the same “gravity equation” (i.e. equation 7), so inverting the home trade share (as we did when going from (9) to (10)) delivers the same expression for real wage irrespective of the micro-details of trade. In some ways, this observation is the essence of the isomorphism result in Arkolakis, Costinot, and Rodriguez-Clare (2012). The expression in (12) is useful for its close connection to the use of the (closed-economy) growth model in accounting exercises such as Hall and Jones (1999) and Caselli (2005) to measure TFP.2 This connection provides a method to identify a country’s technology parameter. Identifying a country’s technology parameter is important, because the set of technology parameters and observed endowments are sufficient to completely characterize the distribution of income per worker in the frictionless economy. Welfare Cost of Autarky. Given the representation of Real GDP per worker in (12), it is straightforward to compute the change in real income due to a change from an observed equilibrium 2
This “openness-adjusted” measure of total factor productivity has been noted before in Waugh (2010) and in Finicelli, Pagano, and Sbracia (2013).
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to autarky, i.e., the welfare cost of autarky. In autarky, a country purchases all goods from itself implying that λii equals one. Since technology and endowments are fixed, the change in real income change from an observed equilibrium to autarky is −1
λiiθ .
(13)
which is the same formula derived in Arkolakis, Costinot, and Rodriguez-Clare (2012). The power of (13) is its simplicity. Give a researcher two statistics—the home trade share and the trade elasticity—and one can compute the welfare cost of autarky. The weakness is that it treats all countries in the same way. That is, conditional on an observed level of trade, the distance in welfare terms is the same for all countries—rich or poor. Missing from this inference is the role that technology and endowments play in determining a country’s possibilities. What we mean by this is that a country’s “distance to the efficient frontier” will differ depending upon the country’s technology and endowments, even conditional on an observed level of trade. Understanding the possibilities is important because while two countries may have the same welfare cost of autarky, one may be more or less closed in comparison to its potential. The next section derives a measure of openness-potential and shows how to measure it using easily available data.
3. Openness Potential This section describes several results that are useful in deriving a metric of a country’s openness potential. We first describe a country’s income level in the frictionless economy. We then combine this measure with a development accounting approach which allows us to measure a country’s potential gain as a function of observable data. Finally, we derive a openness metric which computes a country’s distance between autarky and the frictionless trade equilibrium. 3.1. Openness Metrics Several observations allow us to compute a country’s income level in frictionless trade. First, in frictionless trade, a country’s home trade share equals its nominal GDP as a share of world GDP. Second, frictionless trade implies that price differences are equalized and, thus, nominal GDP is the same as real GDP. A country’s home trade share is Li y FT λFii T = PN i FT , k=1 Lk yk 8
(14)
where yiFT is a country’s real GDP per worker in frictionless trade. Combining (14) with the general representation of real GDP per worker in (12) provides a method to solve out for a country’s income level in frictionless trade. Proposition 1 summarizes this result. Proposition 1 (GDP Per Worker in a Frictionless Economy.) GDP per worker in the frictionless trade economy is yiFT = Ω(L, T, k)
Ti Li
1 1+θ
αθ
ki1+θ ,
(15)
where Ω(L, T, k) =
N X
Lk
k=1
Tk Lk
1 1+θ
αθ 1+θ
kk
! θ1
.
(16)
Proposition 1 says much about a country’s position in the income distribution in a frictionless economy. A country with a larger technology parameter or capital endowment will have larger GDP per worker in a frictionless economy. This is intuitive—a country’s position in the distribution of income will reflect its advantages in technology or capital. A country’s labor endowment also affects its level of GDP per worker in a frictionless economy. Holding all else fixed, equation (15) implies that smaller countries will have higher levels of output per worker relative to larger countries. Thus, this model (and other trade models) displays a peculiar form of weak scale effect. This scale effect is peculiar in the sense that it works in the opposite direction of weak scale effects in endogenous growth models with larger countries experiencing higher income levels. Alvarez and Lucas (2007) suggest one way to kill these scale effects by assuming that the labor force is proportional to the technology parameter. The key problem with this solution is that it implies a counterfactual relationship between measures of TFP from (12) and labor endowments. Ramondo, Rodr´ıguez-Clare, and Sabor´ıoRodr´ıguez (2012) provide an alternative solution by modeling the structure of trade within a country. The Ω term in (16) summarizes the effects of all other countries on the income level of a country. Because Ω is common to all countries, it does not affect a country’s relative position income level—it simple scales everything up or down. The inner term of Ω is a weighted sum of the country-specific terms in (15). This has an implication for the covariance between size and endowments and technology and their impact on the levels of GDP in the frictionless economy. If the rich countries in frictionless economy are big countries, then the level effects will be larger than if the rich countries are small countries.
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Finally, a country’s potential income level is measurable from readily available data, e.g., the Penn World Table. Equation (8) and development accounting procedures provide estimates of a country’s technology parameter. Specifically, we measure the technology parameter T as Ti = yi ki−α
θ
λii .
(17)
The term in the first set of brackets is the standard “Solow residual.” Take the Solow residual to the power θ and scale it by a country’s home trade share and one has measured a country’s technology parameter. Substitution of (17) into (15) provides a country’s income level in a frictionless world as a function of observable data. The potential change in real income a country could experience in the frictionless trade economy directly follows from this expression. Proposition 2 summarizes this result. Proposition 2 (Measuring Openness Potential) A country’s level of GDP per worker in the frictionless trade economy is yiFT
λii ˜ = Ω(Y, λ) Yi
1 1+θ
yi
(18)
with ˜ Ω(Y, λ) =
N X
Yk
k=1
λkk Ykk
! 1θ 1 1+θ
.
(19)
The change in real income from a country’s observed income level to the frictionless trade economy is 1 1+θ yiFT λii ˜ . = Ω(Y, λ) yi Yi
(20)
Equation (18) expresses a country’s income level in the frictionless economy as a function of the country’s statistics today: its home trade share, total GDP, and GDP per worker. A richer country with less trade or smaller size will have larger income level in the frictionless economy. A interesting feature of (18) is how capital and labor endowments cancel out of the equation with only the home trade share and GDP mattering. This says that the reason why a country is rich or poor does not matter, what matters is just that it is rich or poor. Thus, the importance of a country’s endowments is completely summarized by the country’s income level.3 ˜ term also takes an interesting form as well. The inner part of Ω ˜ is a GDP weighted sum of each country’s The Ω 1 1+θ specific gain term λYiii . 3
10
Dividing a country’s potential income level by to its current income level determines the potential gains in (20). Equation (20) provides a welfare metric regarding how far a country is from the frontier as a function of several statistics. The key insight from (20) is the dependence of a country’s distance to the frontier on its current income level. Holding fixed the volume of trade, the poorer a country is the more potential it has. A simple back of the envelope calculation reveals the potential that trade has to reduce income inequality. Assume that home trade shares and labor endowments are the same across countries. Equation (18) implies that trade has the potential to reduce the variance in log income 2 θ . Quantitatively, with a θ of four, trade has the potential to reduce income by a factor 1+θ inequality by 36 percent.
Using our openness measure (20) and the welfare cost of autarky in (13), we construct a summary statistic Λi that summarizes a country’s distance between autarky and the frictionless trade. Proposition 3 summarizes the result. Proposition 3 (Openness Metric) A country’s distance between autarky and frictionless trade is 1/θ
Λi =
1 − λii , 1 1+θ 1/θ λii ˜ − λii Ω(Y, λ) Yi
(21)
where this statistic has the following properties: (i) it lies between zero and one, (ii) equals zero in autarky, (iii) equals one in frictionless trade. This openness metric is useful for putting into perspective the relative magnitude of the welfare cost of autarky and openness potential in (20). For example, a country’s welfare cost of autarky may appear to be large, but (21) places this number in context relative to its openness potential. So while a country may have a relatively high welfare cost of autarky, it may still be relatively far from frictionless trade. Proposition 3 provides an openness metric that encodes this feature. 3.2. Quantifying the Gain From Trade The key feature of (20) and (21) is that all the data required to compute the potential gains and a country’s relative openness metric is GDP data, home trade shares, and the trade elasticity. Below we discuss the country-specific measures the trade elasticity that we use. 3.2.A. Cross-country Data and the Trade Elasticity Output per worker, the number of workers are standard measures used in development and growth accounting exercises. In our computations, we use the expenditure side of output measure because this measure of real GDP treats trade balances with how we treat them in the 11
1000 COM
Percent Increase in GDP Per Worker
900 800
LBR
GNB GMB
CAF 700
400 300 200 100 0 1/128
COD
DJI BTN
VCT
MDV LCA BLZ SLE TGO BMU MNE LSO RWA SWZ MRT MWINER KGZ BRB SUR LAO GIN BEN COG MNG GNQ ATG TJK MDA FJI MLI BFA ARM NAM MDG ZMB MOZ GEO TCD ALB MUSMKD GAB BRN BHS PRY KHM SEN BWATTO MAC HND NPL UGA BOL MLT PAN URY CMR AZE TKM ZWE TZA BIH CYP ISL JAM CIV ETH BHR KEN GHA AGOSDN YEM JOR CRI LVA LBN DOM GTM OMN SRB SYR LKA ECU TUN LTU UZB BGR HRV IRQ BLR KWT SVN MAR NZL KAZ PER EST BGD SVK CHL ROU VEN NOR VNMNGA ISR FIN IRL COL PRT EGY PHL UKR DNK PAK CZE GRC ZAFARG HUN SWE CHE AUT POL SAU MYS THA IDN IRN AUS LUX TUR TWN BRA KOR MEX ESP CAN RUS NLD ITA FRA GBR IND JPN DEU CHN USA
BDI
600 500
CPV
1/64
1/32
1/16 1/8 1/4 Log GDP Per Worker (Data, USA = 1)
1/2
1
QAT
2
Figure 1: Openness Potential Versus Output Per Worker Data model (i.e., where exports and imports are deflated together and not separately as in production side measures of GDP). See, for example, the discussions in Feenstra, Heston, Timmer, and Deng (2004) and Waugh (2010) for a more detailed explanation. We measure a country’s home trade share λii , as one minus a country’s ratio of imports to GDP at current prices. A complicating issue with our approach is that imports are largely intermediates and measured in gross terms, while GDP is a value added measure. One approach to correct this mismatch is to “gross up” GDP by a multiplier that represents intermediates share in value added. If this multiplier is constant across countries, the only effect would be on the level of potential gains and not their distributional consequences. Another alternative would be to explicitly model intermediates directly as in Eaton and Kortum (2002), Alvarez and Lucas (2007), or Waugh (2010) and construct measures of trade penetration using gross production rather than value added. Measures of home trade shares using gross production would only affect the level and the distributional consequences if they varied in a systematic way across countries. We focus on the year 2005 which is the the benchmark year for the PWT 8.1. We drop any countries with missing data. We also drop the few countries (e.g. Belgium, Panama, etc.) that have an imports to GDP ratio which is larger than one. This leaves 160 countries in the sample. As a baseline we set θ equal to four. This is consistent with the estimates from Simonovska 12
Table 3: Openness Potential Data
Frictionless
Autarky
Relative Openness
Mean
0.33
1.28
0.27
0.03
Standard Deviation
1.20
1.04
1.16
0.92
90/10 Ratio
25.2
14.8
22.6
11.8
Note: The first three columns report statistics for Real GDP per worker in the data and the model. The last column reports the openness metric in (21.
and Waugh (2014). Simonovska and Waugh (2014) also provides an extensive discussion on other estimates from the literature. A short summary is that a variety of different methods and estimation approaches point to a plausible range of between three and five. As discussed above, the potential reduction in income inequality decreases with a larger θ. 3.2.B. Openness Potential Figure 1 plots a countries openness potential in (20) (in percent change terms) versus the logarithm of country’s observed of output per worker. Below we make several observations. First, the change in the level of income for all countries is large. To provide one measure, Table 3 shows that the average income level in the data relative to the frictionless economy is nearly a factor of four. Eye-balling the overall level of gains in Figure 1 reveals a similar observation. Second, poor countries have substantially more potential than rich countries. Figure 1 shows the negative relationship between the potential gains and a country’s observed income level. This relationship implies that trade has a large potential to increase income inequality. Table 3 reports summary measures of income inequality: the standard deviation of log income per worker and the ninety-ten percentile ratio. Both these metrics of cross-country income inequality decline substantially. The reason the potential gains from trade are larger for poor countries is because they do not trade more than rich countries. In fact, poor countries trade systematically less then rich countries.Iin the context of (20), this observation implies that the potential gains from trade will be larger the poorer a country is. To formalize this argument, consider the slope coefficient from a projection of (20) on GDP per worker data. The sign of this slope coefficient corresponds with correlation coefficient between
13
50 MLT
45
ISL
Percent Increase in GDP Per Worker
SUR 40
FJI JAM
VCT
35
NLD HUN BHS SVN
LCA 30
SVK
LSO JOR BLZ
25
BMU CYP AUT DNK CHE
BHR LTU CZE SWE BIH IRL MYS DEU LVA FIN CPV DJI HRV ISR CRI BRB 15 TTO PRT TWN SWZ MRT BGR CAN MDV NZL HND GTM LBR MDA MKD MNE MUS GBR ESP FRA NAM THA MAC MOZ ITA LBN 10 GRC SEN ROU CIV POL KOR MAR BLR SRB MEX ALB TUN COMVNM NOR CHL GINZMB GNQ STP COG PHL QAT TUR AUS OMN BWA PRY GEO ARM SAU MLI URY AGO MWI MNG ECU BTN USA BRN UKRIRQ PAN KHM TGO KAZZAF DOM KEN 5 AZE SLEGNB GHA MDG KGZ JPN GMB SYR BOL LKA CODBDI YEM KWT LAO ETH NER TZA TJK BFA CHN PER VEN BEN IDN CMR COL ARG GAB NGA SDN BGD BRA CAF ZWE EGY RWA UGA TKM RUS IRN IND PAK NPL UZB TCD 0 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 Log GDP Per Worker (Data, USA = 1) 20
1/128
Figure 2: Welfare Cost of Autarky vs GDP Per Worker Data (20) GDP per worker data or Corr
1 log 1+θ
λii Yi
, log yi .
(22)
A simple way for this correlation to be negative is if home trade shares are largely uncorrelated with real GDP. A way for this correlation to take a large negative sign is if home trade shares are negatively correlated with real GDP. The latter case is the pattern in the data. The correlation between log home trade shares and log gdp per worker is −0.39 and statistically different from zero. This correlation can be seen in Figure 2 since the welfare cost of autarky is just an inverse function of the home trade share. Given that gdp per worker is strongly correlated with total GDP, these observations imply that poor countries have more to gain from trade than rich countries. 3.2.C. Welfare Cost of Autarky Figure 2 provides a point of comparison. It plots the welfare cost of autarky as measured by (13). The third column of Table 3 reports summary statistics. Several observations. First, rich countries have larger welfare costs of autarky. Consistent with the discussion above, because home trade shares are negatively correlated with income level, this implies the welfare 14
cost of autarky is positively correlated with income level. Rich countries trade more and, thus, have the most to loose from closing to trade. This pattern has the implication that international trade is modestly increasing cross-country income inequality. The inequality statistics in Table 3 illustrate this point with the standard deviation and ninety-ten ratio decreasing in the closed economy. The losses from autarky, however, are very modest. For example, the average loss across all countries is 13 percent. Moreover, these loses become tiny when compared to the potential gains seen in Figure 1. This observation suggests that relative to the potential gains, the world is quite close to autarky. Our openness metric in 21 formalizes this observation. 3.2.D. Openness Metric Our openness metric in (21) maps a country’s observed position into a value between zero and one. If a country has the value near one, then the country is close the frictionless trade benchmark, if a country has the value near zero it is close to autarky. Figure 3(a) and 3(b) both plot the same openness metric, but on with different scales; Figure 3(a) plots the whole scale between zero and one and Figure 3(b) zooms in. Figure 3(a) shows the world is effectively in autarky. Many countries lie near zero with notable exceptions being Germany and the Netherlands. The summary statistics in Table 3 quantify this observation with the average openness level being 0.03 or 97 percent away from the frictionless trade benchmark. Figure 3(a) and 3(b) show that rich countries are relatively more open. This is not because they have large welfare costs of autarky, but because their welfare cost of autarky relative to their potential is large. The US is a good example of this pattern. The US is not exceptional in the size of its welfare cost of autarky—yet the US is among the top ten most open countries in the world by our openness metric. The insight is that the cost of autarky for the US is not large, but relative to its potential, it is a very open country. To generalize the observation, note that the correlation between log GDP per worker and the welfare cost of autarky is 0.38. The correlation of log GDP per worker and our relative openness metric is fifty percent larger at 0.57. This implies that rich countries are systematically closer to their potential than poor countries, and, hence measured to be more open countries.
4. Openness and Trade Frictions This section asks several follow up questions. First, how do these openness measures relate to the underlying frictions that impede trade? Second, what are plausible potential gains and how do they relate to the frictionless gains? The difficulty in asking these questions is that we must 15
1
Opennes Metric (0 = Autarky, 1 = Frictionless)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
NLD DEU EST HUN 0.1 AUT USA CHE DNK GBR MYS SWE ATG SVKSVN CZEMLT FRA CAN ISL TWN ESP ITA IRL JAM FIN KOR ISR LTU BHS PRT FJITHAJOR CYP SUR MEX BHR POL CHN HRV JPN BIH LVA GRC NZL TUR BGR CRI LSO BMU AUS VCT LCA ROU GTM SAU VNM NOR TTO BLZ MAR PHL CHL BLR HND LBN ZAF TUN BRB SRB MKD MUS UKR CIV MAC MRT SWZ MDA CPV NAM DJI SEN BRA MOZ IDN OMN MNE KAZ ALB IND ECU RUS IRQ ZMB AGO VEN DOM IRNGAB LBRMWI ARG BWA GNQ URY PRY GIN COL KEN SYR KWT LKAEGY GEO NGA PERMDV PAN GHA ARM COG AZE KHM BRN QAT MLI YEM BGD PAK MNG BOL MDG TZA COD BDIETH COM CMR TGO KGZ SDN BTN SLE NER BFA LAO TJK UGA BEN STP ZWE GMB GNB TKM NPL UZB CAF RWA TCD 0 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 Log GDP Per Worker (Data, USA = 1)
(a) Openness Metric Λi vs GDP Per Worker Data
0.2
Opennes Metric (0 = Autarky, 1 = Frictionless)
0.18
NLD
0.16 DEU 0.14
EST
0.12
HUN
0.1
AUT CHE USA GBR MYS SWE SVN DNK ATG SVK CZE 0.08 FRA CAN ISL TWN ESP MLT ITA IRL JAM FIN 0.06 KOR ISR LTU BHS PRT FJITHAJOR CYP SUR MEX BHR CHN HRV JPN BIH POL LVA 0.04 GRC NZL TUR BGR CRI LSO BMU AUS VCT LCA ROU GTM SAUNOR VNM TTO BLZ MAR PHL CHL BLR HND LBN BRB ZAF TUN SRB MKD MUS UKR CIV MAC MRT SWZ MDA QAT 0.02 CPV NAM DJI SEN IND BRA MOZ IDN OMN MNE KAZ ALB ECU RUS MDV IRQ ZMB AGOPRY VEN DOM IRN LBR ARG GNQ URY GIN COL SYR KWT LKAEGY GEO PER BWA PAN GHANGA ARM COGAZE KHM BRN MLIKEN YEM BGD PAK MNG BOL MDG TZA COD ETHMWINER COM CMR TGO KGZ SDN BTN SLE LAO STP GAB TJK UGA BEN ZWE GMB GNB BDI TKM NPL UZB CAF RWABFA TCD 0 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 Log GDP Per Worker (Data, USA = 1)
(b) Zoomed In
Figure 3: Openness Metric Λi vs GDP Per Worker Data 16
now take an explicit stand on the underlying frictions. 4.1. Calibration Calibrating trade models of this type present the challenge in that there are a large number of trade costs and technology parameters to discipline. In fact, this observation illustrates the value added of the welfare cost of calculations in 13, 20, or 21—they are model consistent, yet they do not require explicit stands on (potentially) hard to infer parameters. Our objective is to have a relatively parsimonious description of the barriers to trade, and add as little extra data that we have worked with so far. To achieve these objectives we reduce the parameter space such that each country faces only one trade cost, τi , to import from all other countries. And this one number will succinctly summarize the frictions that each country face to trade. To calibrate these trade frictions, we use the following procedure. First, we infer the technology parameters using our development accounting equation in (8). Unlike our results in Proposition 2, we must use data on capital stocks and capital shares to invert the technology parameters. Recall, that this was not necessary before as a country’s total GDP was a sufficient statistic for the role that endowments play. Given the model’s structure resulting in equation (12), we want α to be consistent with the exercises in the income accounting literature. To do so, we set α equal to 1/3. Gollin (2002) provides an argument for setting α equal to 1/3 by calculating labor’s share for a wide crosssection of countries and finding it to be around 2/3 with no systematic variation across income levels. Given this value, we then use the capital stock measures for the year 2005 from the PWT 8.1. Given the technology parameters, we then chose τi s to such that the model in equilibrium exactly fits each country’s observed home trade share. The Appendix describes the algorithm used. This procedure results that the equilibrium of the model exactly match the observed distribution of output per worker across countries and each country’s home trade share. This also implies that the frictionless trade equilibrium of our calibrated economy will be the same as the results presented above. Calibrated Trade Costs. Table 2 presents statistics summarizing the calibrated trade costs. Figure 4 plots each country’s calibrated trade cost versus its income level. There are several important observations to make. First, an important aspect of the calibrated trade costs are that they are large. The mean and the median trade cost are about seven and six and a half. The trade elasticity plays an important role in determining the size. With a larger θ, the inferred trade costs would be smaller. 17
Table 2: Calibrated Trade Costs Mean τ
Median τ
All Countries
6.99
6.59
Rich
4.92
4.64
Poor
9.06
8.44
Note: Rich is the set of countries above the median GDP per worker. Poor is the set below.
Second, the trade costs are substantially larger for poorer countries. Table 2 illustrates this point by reporting the trade costs for those “poor” countries which are defined by those countries below median GDP per worker; “rich” countries are those above the median. By this demarcation, poor countries have trade costs that are almost twice as large as those in rich countries. Figure 4 further illustrates this point with a strong, downward sloping relationship between the calibrated trade costs and income level. The intuition for both these two observations lies in Section 3.2. Relative to the frictionless benchmark the observed levels of trade are small. Thus, to reconcile the small levels of trade, the model needs large frictions. Moreover, poor countries, have the most to gain relative to frictionless trade—thus, one infers that poor countries require even larger frictions to reconcile their relatively low levels of trade. The next observation makes this connection between our openness metric and the trade costs event tighter. Third, the calibrated trade costs correlate highly with our relative openness metric in (21). In fact, they move almost one-for-one with each other. Figure 5(a) illustrates this observation by plotting the log of our relative openness measure in (21) versus the log of the calibrated trade costs. Figure 5(a) illustrates very clearly that country’s with measures of higher relative openness have low trade costs and vice versa. The correlation between these two measures is −0.94. As a point of comparison, Figure 5(b) plots the welfare cost of autarky in (13), versus the calibrated trade costs. The calibrated trade costs are more modestly correlated with the welfare cost of autarky with a correlation coefficient of −0.48. Similar to the discussion above, the US is a good example. The US is measured to have a low welfare cost of autarky, yet it also has very low trade costs. This incongruence disappears when using our relative openness metric: the US is one of the most open countries and it has some of the lowest trade costs. A concern with these claims is that these correlation are spurious or mechanical, e.g., relative 18
16
Log Trade Cost
8
4
2
1 1/128
GNB GMB COM NPL BTN SLE BEN TKM UZB NER TGO TJK LAO BFAUGA KGZ ZWE GAB LBRMWI MNG MDG CMR COG SDN MDV MLI TZA BOL ARM ETH CPV GEO DJI BRN GIN KHM MNEGNQ AZE YEM PAN PRY BWA ZMB GHA URY SWZ KEN ALB MRT VCT LKA BGD EGY BLZ SYR NAM MDA DOM MOZ BRB LCA PER AGO PAK SEN NGA MAC KWT MKD BMU ECU IRQ COL MUS OMN VEN LSO CIV HND ARG KAZ IRN SRB TTO LBN TUN SUR BLR RUS FJI GTM UKR IND IDNMAR BRACRI ZAF BIH BHS ATG LVA CHL BGR VNM PHL CYP NOR BHR ROU NZL MLT ISL HRV JAM JOR SAU LTU GRC AUS TUR POL ISR PRT FIN IRL SVNJPN THA MEX SVK CHN KORDNK CZE SWE EST TWN MYS ESP ITA CHE AUT CAN HUN FRA GBR USA LUX NLD DEU
BDI
COD
TCD STP
CAF RWA
1/64
1/32
1/16 1/8 1/4 Log GDP Per Worker (Data, USA = 1)
1/2
1
QAT
2
Figure 4: Trade Costs openness is correlated with GDP as are trade costs etc. This is not the case. For example, after conditioning on income level, the correlation between relative openness and the trade costs is −0.88. In contrast, after conditioning on income level, the correlation between the welfare cost of autarky and the trade costs declines to −0.32. This observation is important because is tells us that the simple statistic in (21) summarizes well the underlying frictions that a country faces (in addition to measuring how open a country is). In contrast, the welfare cost of autarky is not a good summary statistic of the underlying frictions in the economy. 4.2. Counterfactual: Toward the US Trade Cost We now perform several exercises to understand plausible, intermediate levels of trade costs. The first exercise endows all countries with the same trade costs that the US faces (which is approximately 2.25) and then computes the new equilibrium. To be clear, this is a simultaneous change and general equilibrium effects are taken into account as a new equilibrium is computed. The idea behind this exercise is to use the US trade costs a plausible benchmark without altering the technological considerations for moving goods across space. Table ?? reports the summary statistics. The move to the US trade costs gives rise to large gains.
19
1/2 1/4
Log Relative Openness
1/8
LUX NLD DEU
1/16 1/32 1/64 1/128 1/256
EST HUN AUT CHE USAGBR MYS DNK SWE SVN ATG SVK CZE FRA CAN ISL TWN IRL ESP MLT ITA JAM KORFIN ISR BHS PRT LTU JOR FJISUR THA CYP MEX BHR CHN HRV JPNPOL GRC BIH LVA NZL TUR BGR LSO BMU AUS ROU CRI VCT LCA GTMTTO SAU VNM NOR BLZ MAR PHL CHL BLRLBN HND BRB ZAF TUN MUS UKR SRB CIVMKD MAC MRT SWZ DJI MDA CPV NAM SEN BRA MOZ IDN QAT OMN MNEMDV ALB IND ECU RUS KAZ IRQ ZMB AGO VEN DOM IRN ARG BWA GNQ URY PRY GIN LBR COL KEN SYR KWT LKA GEO NGA PER PAN GHA ARM COG AZE KHM BRN MLI BGDYEM PAK MNG EGY MWI BOL ETH MDG TZA COD CMR TGO KGZ SDN BTNCOM SLE NER BFA LAO GAB TJK UGA ZWEBEN GMB GNB STP BDI TKM UZBNPL CAF RWA
1/512
TCD
2
4
8
16
Log Trade Cost
(a) Openness Metric Λi vs Trade Cost τi
2 ATG
Log Welfare Cost of Autarky
EST
MLT ISL 1.41
NLD
DEU
1 2
HUN
JAM
SVN
SUR FJI BHS
VCT LCA BMU LSO BLZ
SVK CYP AUT JOR DNK CHE BHR LTU CZE SWE BIH MYS IRL LVA FIN CPV DJI ISR HRV CRI BRB TTO PRT TWN SWZ MRT BGR CAN MDVLBR NZL GTM HND MDA MKD MUS GBR ESP FRA NAM MNE MOZ ITA KORTHA LBNCIVMAC ROU MAR POL GRC BLR SRB SENALB TUN VNM COM NOR MEX TUR PHL ZMB CHL GIN GNQ STP QAT AUS COG OMN BWA PRY GEO ARM SAU ZAF MLI URY AGO MWI TGO MNG ECU BTN GMB USA UKR KAZ BRN PAN KHM DOM IRQ KEN AZE SLE GHA MDG GNB KGZ JPN SYR BOL LKA COD NER YEM KWT LAO ETH TZA TJK VEN BFA CHN PER BDI BEN IDN CMR GAB COL ARG NGA BGD BRA UGA SDN CAF ZWE RUSIRN PAK EGY TKM IND NPL RWA UZB TCD 4
8 Log Trade Cost
(b) Welfare Cost of Autarky vs Trade Cost τi
Figure 5: Openness Metrics and Trade Costs 20
16
Table 3: Counterfactual τ s Data
US τ
τ =1
yi
Openness Λi
yi
Openness Λi
yi
Openness Λi
Mean
0.33
0.03
0.54
0.25
1.28
1.0
Standard Deviation
1.20
0.92
1.05
0.21
1.04
—
90/10 Ratio
25.2
11.8
14.5
1.78
14.8
—
Note:
The average percent increase is almost 100 percent. Moreover, there is substantial heterogeneity, with the gains largely accruing to the poorest countries. Figure 6(a) illustrates this point by plotting the percent gain versus gdp per worker data. The second column of Table ?? summarizes this finding by showing that the standard deviation in income and the 90/10 ratio decline by large amounts. The third row of Table ?? provides a point of comparison by reporting the same statistics in the frictionless trade economy. The most important (and surprising) result is that that crosscountry inequality is nearly the same as in the US trade cost economy. In other words, the US trade friction delivers all the reduction in inequality that the frictionless trade equilibrium delivers. The key difference between row two and row three of Table ?? is the level of the gains. The difference between going from a trade friction of 2.2 to frictionless trade amounts to an additional 300 percentage point increase in the gains from trade. The difference between levels versus inequality reduction can be seen in Figure 6(b). Figure 6(b) plots the results from both the US trade friction economy and the frictionless trade economy with the same axis. The key thing to note is that the slope between output per worker and the gains are nearly the same. The only difference is the intercept or level.
21
450 STP
Percent Increase in GDP Per Worker
400 350 COM
300
GNB GMB
LBR
250
CPV
CAF BDI 200 150 COD 100 50 0 1/128
DJI BTN
VCT MDV LCA BLZ
SLE TGO BMU MNE RWA LSO SWZ MRT MWINER KGZ BRB SUR MNG LAO GIN BEN COG GNQ TJK ATG MDA MLI BFA ARM FJI NAM MDG ZMB TCD MOZ GEO ALB MUSMKD GAB BRN BHS PRY KHM BWATTO SEN MAC HND NPL UGA BOL PAN URY MLT CMR TKM AZE ZWE TZA BIH CIV CYP ETH ISL JAM LVA KEN GHA AGOSDN YEM BHR CRI JOR LBN DOM GTM SYR OMN SRB LKA ECUBGR QAT UZB TUN LTU IRQ BLR HRV KWT MAR KAZ NZL PER SVN BGD VEN CHL ROU EST NOR VNMNGA SVK COL ISR EGY FIN IRL PAK PHL UKR GRC ZAFARG PRT CZE DNK SAU POL IRN IDN SWE AUS THA TUR HUN CHE AUT MYS BRA RUS TWN MEX KOR IND ESP CAN ITA JPN FRA GBR CHN NLD USALUX DEU 1/64 1/32 1/16 1/8 1/4 1/2 1 2 Log GDP Per Worker (Data, USA = 1)
(a) Counterfactual: US τ
1
Opennes Metric (0 = Autarky, 1 = Frictionless)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
VCT ATG COM STP CPV LCA LBRCAF DJI BLZ GNB BMU SUR GMB MDV LSO BTN FJI SWZ BDI MNEGNQ BHS SLE TGO MRT MLT ISL BRB RWA MWI NER MDA MNG KGZ GIN COG JAMNAM LAO BEN TJK ESTGAB MUS MLI ARM MKD LUXBRN MOZ MDG CYP MAC BFA KHM ZMB ALB GEO TTO HND COD BIH SEN PRY JOR BWA BHR TCD LVA PAN URY NPL KEN GHA BOL CIV CRI AZE SVN CMR TZA UGA LTU TKM ZWE GTM ETH LBN AGOSDN YEM SRB HRV OMN BGR SVK TUN ECU LKA SYR BLR NZL IRQDOM MAR UZB HUN IRL FIN KAZ ISR DNK PER VEN CZE GRC CHL PRT BGD VNMNGA AUTNOR KWT CHE SWE COLROU PHL UKR EGY PAK NLD ZAFARGMYS POL THA SAU TWN AUS TUR IDN IRN CAN KOR ESP MEX BRA ITA FRA RUS GBR DEU IND JPN CHN USA
QAT
0.1 0 1/128
1/64
1/32
1/16
1/8 Log Trade Cost
1/4
1/2
(b) Counterfactual: US τ vs. Frictionless Trade
Figure 6: Change in Output Per Worker 22
1
2
5. Conclusion
23
References A LVAREZ , F., AND R. J. L UCAS (2007): “General Equilibrium Analysis of the Eaton-Kortum Model of International Trade,” Journal of Monetary Economics, 54(6), 1726–1768. A NDERSON , J. E. (1979): “A Theoretical Foundation for the Gravity Equation,” American Economic Review, 69(1), 106–16. A RKOLAKIS , C., A. C OSTINOT,
AND
A. R ODRIGUEZ -C LARE (2012): “New Trade Models, Same
Old Gains?,” American Economic Review, 102(1), 94–130. B ERNARD , A., J. E ATON , J. B. J ENSEN ,
AND
S. K ORTUM (2003): “Plants and Productivity in
International Trade,” American Economic Review, 93(4), 1268–1290. C ASELLI , F. (2005): “Accounting for Cross-Country Income Differences,” in Handbook of Economic Growth, ed. by P. Aghion, and S. Durlauf, vol. 1. Amsterdam: Elsevier. C HANEY, T. (2008): “Distorted Gravity: The Intensive and Extensive Margins of International Trade,” American Economic Review, 98(4), 1707–1721. E ATON , J., AND S. K ORTUM (2002): “Technology, Geography, and Trade,” Econometrica, 70(5), 1741–1779. F EENSTRA , R. C., A. H ESTON , M. P. T IMMER ,
AND
H. D ENG (2004): “Estimating Real Produc-
tion and Expenditures Across Nations: A Proposal for Improving the Penn World Tables,” NBER Working Paper 10866,. F INICELLI , A., P. PAGANO , AND M. S BRACIA (2013): “Ricardian selection,” Journal of International Economics, 89(1), 96–109. G OLLIN , D. (2002): “Getting Income Shares Right,” Journal of Political Economy, 110(2), 458–474. H ALL , R., AND C. J ONES (1999): “Why Do Some Countries Produce So Much More Output Per Worker Than Others?,” Quarterly Journal of Economics, 114(1), 83–116. K RUGMAN , P. (1980): “Scale Economies, Product Differentiation, and the Pattern of Trade,” American Economic Review, 70(5), 950–959. M ELITZ , M. J. (2003): “The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity,” Econometrica, 71(6), 1695–1725. R AMONDO , N., A. R ODR´I GUEZ -C LARE ,
AND
M. S ABOR´I O -R ODR´I GUEZ (2012): “Trade, domes-
tic frictions, and scale effects,” Discussion paper, National Bureau of Economic Research.
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S IMONOVSKA , I., AND M. E. WAUGH (2014): “The elasticity of trade: Estimates and evidence,” Journal of International Economics, 92(1), 34 – 50. WAUGH , M. E. (2010): “International Trade and Income Differences,” American Economic Review, 100(5), 2093–2124.
25
Appendix Below, we walk through the derivation of some of the relationships that we exploit. What we want to do is find an expression for a country’s real GDP per worker in frictionless trade. First, we want walk through a balanced trade condition. In general the balanced trade condition is Li (wi + ri ki ) =
N X
Lk (wk + rk kk )λki
(23)
k=1
which says that total income payments in country i must equal the total purchases of i’s goods made by all countries. These country-specific purchases equal the income in country k times the expenditure share that country k spends on goods from country i, i.e. λki . Note the right hand side of (23) includes purchases of country i from itself. In frictionless trade, all countries purchase the same amount from country i. That is, λki = λii . This then simplifies the equation to be Li (wi + ri ki ) = λi
N X
Lk (wk + rk kk )
(24)
k=1
which states that the share of goods all countries purchase from country i must equal country is share in world GDP: Li (wi + ri ki ) . λii = PN k=1 Lk (wi + ri ki )
(25)
Then the final observation is to note that (up to some non-country specific scale factor), real GDP per worker corresponds with nominal GDP per worker in frictionless trade. Thus we have Li y FT λFii T = PN i FT . k=1 Lk yk
(26)
Then to solve for real GDP in frictionless trade we substitute this expression into 12, giving
yiF T = Ti
yiFT
=
1 θ
N X k=1
Li yiFT PN FT k=1 Lk yk Lk ykFT
! −1 θ
1 ! 1+θ
26
Ti Li
−1
λiiθ kiα ,
1 1+θ
(27)
αθ
ki1+θ .
(28)
The final issue is that in the first bracket terms are endogenous variables, not primitives. We can substitute them out in the following manner. First define N X
Ω=
Lk ykFT
1 ! 1+θ
1 1+θ
ki1+θ .
k=1
δi =
Ti Li
(29)
αθ
(30)
So that Ω is the non-country specific component and δi is the country specific component of real GDP. Now we will take logs and repeatedly substitute this expression into (28). Doing and walking through each of the steps gives. . . log yiF T
log yiF T
N X 1 = log Lk Ωδi 1+θ k=1
!
+ log δi
N X 1 1 log Ω + log Lk δi = 1+θ 1+θ k=1
!
+ log δi
N X
!
(32)
N X 1 Lk δi + log 1+θ k=1
1 1+θ
2
log Ω +
log yiF T =
1 1+θ
n
N X 1 n X Lk δi log Ω + log 1 + θ n=1 k=1
log yiF T
1 1+θ
n
N X 1 log Ω + log Lk δi 1+θ k=1
log yiF T =
1 1+θ
...
=
log yiF T =
log yiF T
1 log 1+θ
N X
Lk δi
k=1
N X 1 = log Lk δi θ k=1
!
!
2
(31)
log
Lk δi
k=1
1 1 + log δi 1 − 1+θ
+ log δi
!
!
+ log δi
X 1 n + log δi 1+θ n=0
!
+ log δi
(33)
(34)
(35)
(36)
(37) (38)
27
Then piecing everything together yields
yiFT =
N X
θ 1+θ
1 1+θ
Lk Tk
αθ 1+θ
kk
k=1
! θ1
Ti Li
1 1+θ
αθ
ki1+θ .
(39)
Algorithm for Calibrating the Model The strategy is we want to pick T s and one τ per country to exactly replicate the distribution of income per worker and home trade shares λii . Below we describe the procedure to calibrate the model in this way. Step 1. We first use our development accounting results to recover the T s. So Ti =
yi kiα
θ
λii .
(40)
This is straightforward. Step 2. With the assumption of one importing trade barrier per country, the home trade share can be expressed as ′
λii = 1 −
Φ (w, τ, T, k) τi−θ i Φi (w, τ, T, k)
(41)
′
where Φi is Φi minus the term for country i in the summation. This equation provides a map′ ping from observed trade, two endogenous objects Φi and Φ, into the unobserved trade friction. What this allows us to do is to set up a system of equations to find the τi that satisfies the relationship above subject to the equilibrium constraints, i.e. that wages are consistent with market clearing.
28
