International Trade Lecture 3: Ricardian Theory (II)

Transcription

1 International Trade Lecture 3: Ricardian Theory (II) Week 2 Spring (Week 2) Ricardian Theory (I) Spring / 34

2 Putting Ricardo to Work Ricardian model has long been perceived has useful pedagogic tool, with little empirical content: Great to explain undergrads why there are gains from trade But grad students should study richer models (Feenstra s graduate textbook has a total of 3 pages on the Ricardian model!) Eaton and Kortum (2002) have lead to Ricardian revival Same basic idea as in Wilson (1980): Who cares about the pattern of trade for counterfactual analysis? But more structure: Small number of parameters, so well-suited for quantitative work Goals of this lecture: 1 Present EK model 2 Discuss estimation of its key parameter 3 Introduce tools for welfare and counterfactual analysis (Week 2) Ricardian Theory (I) Spring / 34

3 Basic Assumptions N countries, i = 1,..., N Continuum of goods u 2 [0, 1] Preferences are CES with elasticity of substitution σ: Z 1 σ/(σ 1) U i = q i (u) du (σ 1)/σ, 0 One factor of production (labor) There may also be intermediate goods (more on that later) c i unit cost of the common input used in production of all goods Without intermediate goods, c i is equal to wage w i in country i (Week 2) Ricardian Theory (I) Spring / 34

4 Basic Assumptions (Cont.) Constant returns to scale: Z i (u) denotes productivity of (any) rm producing u in country i Z i (u) is drawn independently (across goods and countries) from a Fréchet distribution: Pr(Z i z) = F i (z) = e T i z, with > σ 1 (important restriction, see below) Since goods are symmetric except for productivity, we can forget about index u and keep track of goods through Z (Z 1,..., Z N ). Trade is subject to iceberg costs d ni 1 d ni units need to be shipped from i so that 1 unit makes it to n All markets are perfectly competitive (Week 2) Ricardian Theory (I) Spring / 34

5 Four Key Results A - The Price Distribution Let P ni (Z) c i d ni /Z i be the unit cost at which country i can serve a good Z to country n and let G ni (p) Pr(P ni (Z) p). Then: G ni (p) = Pr (Z i c i d ni /p) = 1 F i (c i d ni /p) Let P n (Z) minfp n1 (Z),..., P nn (Z)g and let G n (p) Pr(P n (Z) p) be the price distribution in country n. Then: G n (p) = 1 exp[ Φ n p ] where Φ n N T i (c i d ni ) i= (Week 2) Ricardian Theory (I) Spring / 34

6 Four Key Results A - The Price Distribution (Cont.) To show this, note that (suppressing notation Z from here onwards) Pr(P n p) = 1 Π i Pr(P ni p) = 1 Π i [1 G ni (p)] Using then G ni (p) = 1 F i (c i d ni /p) 1 Π i [1 G ni (p)] = 1 Π i F i (c i d ni /p) = 1 Π i e T i (c i d ni ) p = 1 e Φnp (Week 2) Ricardian Theory (I) Spring / 34

7 Four Key Results B - The Allocation of Purchases Consider a particular good. Country n buys the good from country i if i = arg minfp n1,..., p nn g. The probability of this event is simply country i 0 s contribution to country n 0 s price parameter Φ n, π ni = T i (c i d ni ) Φ n To show this, note that π ni = Pr P ni min P ns s6=i If P ni = p, then the probability that country i is the least cost supplier to country n is equal to the probability that P ns p for all s 6= i (Week 2) Ricardian Theory (I) Spring / 34

8 Four Key Results B - The Allocation of Purchases (Cont.) The previous probability is equal to where Π s6=i Pr(P ns p) = Π s6=i [1 Φ i n = T i (c i d ni ) s6=i G ns (p)] = e Φ n i p Now we integrate over this for all possible p 0 s times the density dg ni (p) to obtain Z 0 e Φ n i p T i (c i d ni ) p 1 e T i (c i d ni ) p dp! = T i (c i d ni ) Z Φ n e Φ np p Φ n 0 1 dp Z = π ni dg n (p)dp = π ni (Week 2) Ricardian Theory (I) Spring / 34 0

9 Four Key Results C - The Conditional Price Distribution The price of a good that country n actually buys from any country i also has the distribution G n (p). To show this, note that if country n buys a good from country i it means that i is the least cost supplier. If the price at which country i sells this good in country n is q, then the probability that i is the least cost supplier is Π s6=i Pr(P ni q) = Π s6=i [1 G ns (q)] = e Φ n i q The joint probability that country i has a unit cost q of delivering the good to country n and is the the least cost supplier of that good in country n is then e Φ n i q dg ni (q) (Week 2) Ricardian Theory (I) Spring / 34

10 Four Key Results C - The Conditional Price Distribution (Cont.) Integrating this probability e Φ n i q dg ni (q) over all prices q p and using G ni (q) = 1 e T i (c i d ni ) p then Z p 0 e Φ n i q dg ni (q) = Z p 0 = e Φ n i q T i (c i d ni ) q 1 e T i (c i d ni ) p dq Ti (c i d ni ) Z p e Φ nq Φ n q 1 dq Φ n 0 = π ni G n (p) Given that π ni probability that for any particular good country i is the least cost supplier in n, then conditional distribution of the price charged by i in n for the goods that i actually sells in n is 1 π ni Z p 0 e Φ n i q dg ni (q) = G n (p) (Week 2) Ricardian Theory (I) Spring / 34

11 Four Key Results C - The Conditional Price Distribution (Cont.) In Eaton and Kortum (2002): 1 All the adjustment is at the extensive margin: countries that are more distant, have higher costs, or lower T 0 s, simply sell a smaller range of goods, but the average price charged is the same. 2 The share of spending by country n on goods from country i is the same as the probability π ni calculated above. We will establish a similar property in models of monopolistic competition with Pareto distributions of rm-level productivity (Week 2) Ricardian Theory (I) Spring / 34

12 Four Key Results D - The Price Index The exact price index for a CES utility with elasticity of substitution σ < 1 +, de ned as is given by where Z 1 1/(1 σ) p n p n (u) du 1 σ, γ = 0 1 Γ p n = γφ 1/ n σ 1/(1 σ) + 1, where Γ is the Gamma function, i.e. Γ(a) R 0 x a 1 e x dx (Week 2) Ricardian Theory (I) Spring / 34

13 Four Key Results D - The Price Index (Cont.) To show this, note that Z 0 pn 1 σ = p 1 σ dg n (p) = Z 1 0 Z 0 p n (u) 1 σ du = p 1 σ Φ n p 1 e Φ np dp. De ning x = Φ n p, then dx = Φ n p 1, p 1 σ = (x/φ n ) (1 σ)/, and Z pn 1 σ = 0 (x/φ n ) (1 = Φ σ)/ e x dx Z (1 σ)/ n x (1 0 σ)/ e x dx = Φ (1 σ)/ n 1 Γ σ + 1 This implies p n = γφ 1/ n with 1 σ + 1 > 0 or σ 1 < for gamma function to be well de ned (Week 2) Ricardian Theory (I) Spring / 34

14 Equilibrium Let X ni be total spending in country n on goods from country i Let X n i X ni be country n s total spending We know that X ni /X n = π ni, so X ni = T i (c i d ni ) Φ n X n (*) Suppose that there are no intermediate goods so that c i = w i. In equilibrium, total income in country i must be equal to total spending on goods from country i so w i L i = X ni n Trade balance further requires X n = w n L n so that w i L i = n T i (w i d ni ) j T j (w j d nj ) w nl n (Week 2) Ricardian Theory (I) Spring / 34

15 Equilibrium (Cont.) This provides system of N 1 independent equations (Walras Law) that can be solved for wages (w 1,..., w N ) up to a choice of numeraire Everything is as if countries were exchanging labor Frechet distributions imply that labor demands are iso-elastic Armington model leads to similar eq. conditions under assumption that each country is exogenously specialized in a di erentiated good In the Armington model, the labor demand elasticity simply coincides with elasticity of substitution σ Under frictionless trade (d ni = 1 for all n, i) previous system implies and hence w 1+ i = T i L i n w n L n j T j w j w i Ti /L 1/(1+) i = w j T j /L j (Week 2) Ricardian Theory (I) Spring / 34

16 The Gravity Equation Letting Y i = n X ni be country i 0 s total sales, then where Solving T i c i Y i = n T i (c i d ni ) X n Φ n Ω i n d ni X n Φ n = T i ci Ωi from Y i = T i ci Ωi and plugging into (*) we get Using p n = γφ 1/ n X ni = X ny i d ni we can then get Φ n X ni = γ X n Y i d ni Ω i (p n Ω i ) This is the Gravity Equation, with bilateral resistance d ni and multilateral resistance terms p n (inward) and Ω i (outward) (Week 2) Ricardian Theory (I) Spring / 34

17 The Gravity Equation A Primer on Trade Costs From (*) we also get that country i 0 s share in country n 0 s expenditures normalized by its own share is S ni X ni /X n = Φ i d pi d ni ni = X ii /X i Φ n p n This shows the importance of trade costs and comparative advantage in determining trade volumes. Note that if there are no trade barriers (i.e, frictionless trade), then S ni = 1. Letting B ni Xni X ii X in X nn 1/2 then B ni = (S ni S in ) 1/2 = 1/2 dni din Under symmetric trade costs (i.e., d ni = d in ) then B 1/ ni be used as a measure of trade costs. = d ni can (Week 2) Ricardian Theory (I) Spring / 34

18 The Gravity Equation A Primer on Trade Costs We can also see how B ni varies with physical distance between n and i: 1 Trade and Geography Normalized import share: (xnl / xn) / (xll / xl) Distance (in miles) between countries n and i Image by MIT OpenCourseWare (Week 2) Ricardian Theory (I) Spring / 34

19 How to Estimate the Trade Elasticity? As we will see the trade elasticity is the key structural parameter for welfare and counterfactual analysis in EK model Cannot estimate directly from B ni = dni because distance is not an empirical counterpart of d ni in the model Negative relationship in Figure 1 could come from strong e ect of distance on d ni or from mild CA (high ) Consider again the equation pi d ni S ni = p n If we had data on d ni, we could run a regression of ln S ni on ln d ni with importer and exporter dummies to recover But how do we get d ni? (Week 2) Ricardian Theory (I) Spring / 34

20 How to Estimate the Trade Elasticity? EK use price data to measure p i d ni /p n : They use retail prices in 19 OECD countries for 50 manufactured products from the UNICP 1990 benchmark study. They interpret these data as a sample of the prices p i (j) of individual goods in the model. They note that for goods that n imports from i we should have p n (j)/p i (j) = d ni, whereas goods that n doesn t import from i can have p n (j)/p i (j) d ni. Since every country in the sample does import manufactured goods from every other, then max j fp n (j)/p i (j)g should be equal to d ni. To deal with measurement error, they actually use the second highest p n (j)/p i (j) as a measure of d ni (Week 2) Ricardian Theory (I) Spring / 34

21 How to Estimate the Trade Elasticity? Econometric Society. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Let r ni (j) ln p n (j) ln p i (j). They calculate ln(p n /p i ) as the mean across j of r ni (j). Then they measure ln(p i d ni /p n ) by D ni = max 2 j fr ni (j)g j r ni (j)/50 Given S ni = pi d ni p n they estimate from ln(sni ) = D ni. Method of moments: = OLS with zero intercept: = (Week 2) Ricardian Theory (I) Spring / 34

22 Alternative Strategies Simonovska and Waugh (2011) argue that EK s procedure su ers from upward bias: Since EK are only considering 50 goods, maximum price gap may still be strictly lower than trade cost If we underestimate trade costs, we overestimate trade elasticity Simulation based method of moments leads to a closer to 4. An alternative approach is to use tari s (Caliendo and Parro, 2011). If d ni = t ni τ ni where t ni is one plus the ad-valorem tari (they actually do this for each 2 digit industry) and τ ni is assumed to be symmetric, then X ni X ij X jn X nj X ji X in = dni d ij d jn tni t ij t jn = d nj d ji d in t nj t ji t in They can then run an OLS regression and recover. Their preferred speci cation leads to an estimate of (Week 2) Ricardian Theory (I) Spring / 34

23 Gains from Trade Consider again the case where c i = w i From (*), we know that π nn = X nn X n = T nwn Φ n We also know that p n = γφn 1/, so ω n w n /p n = γ 1 T 1/ n π 1/ nn. Under autarky we have ω A n = γ 1 Tn 1/, hence the gains from trade are given by GT n ω n /ω A n = πnn 1/ Trade elasticity and share of expenditure on domestic goods π nn are su cient statistics to compute GT (Week 2) Ricardian Theory (I) Spring / 34

24 Gains from Trade (Cont.) A typical value for π nn (manufacturing) is 0.7. With = 5 this implies GT n = 0.7 1/5 = or 7.4% gains. Belgium has π nn = 0.2, so its gains are GT n = 0.2 1/5 = or 38%. One can also use the previous approach to measure the welfare gains associated with any foreign shock, not just moving to autarky: ω 0 n/ω n = π 0 nn/π nn 1/ For more general counterfactual scenarios, however, one needs to know both π 0 nn and π nn (Week 2) Ricardian Theory (I) Spring / 34

25 Adding an Input-Output Loop Imagine that intermediate goods are used to produce a composite good with a CES production function with elasticity σ > 1. This composite good can be either consumed or used to produce intermediate goods (input-output loop). Each intermediate good is produced from labor and the composite good with a Cobb-Douglas technology with labor share β. We can then write c i = w β β i p 1 i (Week 2) Ricardian Theory (I) Spring / 34

26 Adding an Input-Output Loop (Cont.) The analysis above implies and hence Using c n = w β n p 1 so n β π nn = γ T n cn p n c n = γ 1 T 1/ this implies The gains from trade are now n πnn 1/ p n wn β pn 1 β = γ 1 T 1/ n πnn 1/ p n w n /p n = γ 1/β Tn 1/β πnn 1/β ω n /ω A n = π 1/β nn Standard value for β is 1/2 (Alvarez and Lucas, 2007). For π nn = 0.7 and = 5 this implies GT n = 0.7 2/5 = or 15% gains (Week 2) Ricardian Theory (I) Spring / 34

27 Adding Non-Tradables Assume now that the composite good cannot be consumed directly. Instead, it can either be used to produce intermediates (as above) or to produce a consumption good (together with labor). The production function for the consumption good is Cobb-Douglas with labor share α. This consumption good is assumed to be non-tradable (Week 2) Ricardian Theory (I) Spring / 34

28 Adding Non-Tradables (Cont.) The price index computed above is now p gn, but we care about ω n w n /p fn, where p fn = wn α p 1 α This implies that w n gn α ω n = wn α p 1 Thus, the gains from trade are now where gn = (w n /p gn ) 1 α ω n /ω A n = π η/ nn η 1 α β Alvarez and Lucas argue that α = 0.75 (share of labor in services). Thus, for π nn = 0.7, = 5 and β = 0.5, this implies GT n = 0.7 1/10 = or 3.6% gains (Week 2) Ricardian Theory (I) Spring / 34

29 Comparative statics (Dekle, Eaton and Kortum, 2008) Go back to the simple EK model above (α = 0, β = 1). We have n X ni = γ T i (w i d ni ) p nx n pn = γ N T i (w i d ni ) i=1 X ni = w i L i As we have already established, this leads to a system of non-linear equations to solve for wages, w i L i = n T i (w i d ni ) k T k (w k d nk ) w nl n (Week 2) Ricardian Theory (I) Spring / 34

30 Comparative statics (Dekle, Eaton and Kortum, 2008) Consider a shock to labor endowments, trade costs, or productivity. One could compute the original equilibrium, the new equilibrium and compute the changes in endogenous variables. But there is a simpler way that uses only information for observables in the initial equilibrium, trade shares and GDP; the trade elasticity, ; and the exogenous shocks. First solve for changes in wages by solving ŵ i ˆL i Y i = n π ni ˆT i k π nk ˆT k ŵ i ˆd ni and then get changes in trade shares from ŵ k ˆd nk ŵ n ˆL n Y n ˆT i ŵ i ˆd ni ˆπ ni =. k π nk ˆT k ŵ k ˆd nk From here, one can compute welfare changes by using the formula above, namely ˆω n = ( ˆπ nn ) 1/ (Week 2) Ricardian Theory (I) Spring / 34

31 Comparative statics (Dekle, Eaton and Kortum, 2008) To show this, note that trade shares are π ni = T i (w i d ni ) k T k (w k d nk ) and π0 ni = T 0 i (wi 0d ni 0 ) k Tk 0 (w k 0 d nk 0 ). Letting ˆx x 0 /x, then we have ˆT i ŵ i ˆd ni ˆπ ni = k Tk 0 (w k 0 d nk 0 ) / j T j (w j d nj ) ˆT i ŵ i ˆd ni = k ˆT k ŵ k ˆd nk Tk (w k d nk ) / j T j (w j d nj ) ˆT i ŵ i ˆd ni =. k π nk ˆT k ŵ k ˆd nk (Week 2) Ricardian Theory (I) Spring / 34

32 Comparative statics (Dekle, Eaton and Kortum, 2008) On the other hand, for equilibrium we have w 0 i L 0 i = n π 0 ni w 0 nl 0 n = n ˆπ ni π ni w 0 nl 0 n Letting Y n w n L n and using the result above for ˆπ ni we get ŵ i ˆL i Y i = n π ni ˆT i k π nk ˆT k ŵ i ˆd ni ŵ k ˆd nk ŵ n ˆL n Y n This forms a system of N equations in N unknowns, ŵ i, from which we can get ŵ i as a function of shocks and initial observables (establishing some numeraire). Here π ni and Y i are data and we know ˆd ni, ˆT i, ˆL i, as well as (Week 2) Ricardian Theory (I) Spring / 34

33 Comparative statics (Dekle, Eaton and Kortum, 2008) To compute the implications for welfare of a foreign shock, simply impose that ˆL n = ˆT n = 1, solve the system above to get ŵ i and get the implied ˆπ nn through ˆπ ni = ˆT i ŵ i ˆd ni. k π nk ˆT k ŵ k ˆd nk and use the formula to get ˆω n = ˆπ 1/ nn Of course, if it is not the case that ˆL n = ˆT n = 1, then one can still use this approach, since it is easy to show that in autarky one has w n /p n = γ 1 Tn 1/, hence in general ˆω n = ˆT n 1/ ˆπ 1/ nn (Week 2) Ricardian Theory (I) Spring / 34

34 Extensions of EK Bertrand Competition: Bernard, Eaton, Jensen, and Kortum (2003) Bertrand competition ) variable markups at the rm-level Measured productivity varies across rms ) one can use rm-level data to calibrate model Multiple Sectors: Costinot, Donaldson, and Komunjer (2012) Ti k fundamental productivity in country i and sector k One can use EK s machinery to study pattern of trade, not just volumes Non-homothetic preferences: Fieler (2011) Rich and poor countries have di erent expenditure shares Combined with di erences in k across sectors k, one can explain pattern of North-North, North-South, and South-South trade (Week 2) Ricardian Theory (I) Spring / 34

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