Economics 266: International Trade — Lecture 4: Ricardian Theory (II)— Stanford Econ 266 (Donaldson)

Winter 2016 (lecture 4)

Stanford Econ 266 (Donaldson)

Ricardian Theory (I)

Winter 2016 (lecture 4)

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“Putting Ricardo to Work” (Title of nice survey by Eaton and Kortum (JEP, 2012))

Ricardian model has long been perceived as a useful pedagogic tool, with little empirical content... Great to explain undergrads why there are gains from trade But grad students should study richer models (e.g. Feenstra’s graduate textbook—edition 1, from 2003—had a total of 3 pages on the Ricardian model!)

Eaton and Kortum (Ecta, 2002) has lead to “Ricardian revival” Same basic idea as in Wilson (1980): Is it necessary to know about the pattern of trade to do certain types of counterfactual analysis? But more structure: Small number of parameters, so well-suited for quantitative work.

Goals of this lecture: 1 2 3

Present EK model Discuss estimation of its key parameters Introduce tools for welfare and counterfactual analysis

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Basic Assumptions N countries, i = 1, ..., N Continuum of goods u ∈ [0, 1] Preferences are CES with elasticity of substitution σ: Ui =

1

Z 0

qi (u )

(σ−1)/σ

σ/(σ−1) du

,

One factor of production (labor) There may also be intermediate goods (more on that later) ci ≡ unit cost of the “common input” used in production of all goods Without intermediate goods, ci is equal to “wage” wi in country i Stanford Econ 266 (Donaldson)

Ricardian Theory (I)

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Basic Assumptions (Cont.) Constant returns to scale: Zi (u ) denotes productivity of (any) firm producing u in country i Zi (u ) is drawn independently (across goods and countries) from a Fr´ echet distribution: −θ

Pr(Zi ≤ z ) = Fi (z ) = e −Ti z , with θ > 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 ≡ (Z1 , ..., ZN ).

Trade is subject to “iceberg” (Samuelson, 1954) costs dni ≥ 1 dni units need to be shipped from i so that 1 unit makes it to n; transport costs, but no real resources used in transport.

All markets are perfectly competitive Stanford Econ 266 (Donaldson)

Ricardian Theory (I)

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Four Key Results A - The Price Distribution

Let Pni (Zi ) ≡ ci dni /Zi be the unit cost at which country i can serve a good Z to country n and let Gni (p ) ≡ Pr(Pni (Zi ) ≤ p ). Then: Gni (p ) = Pr (Zi ≥ ci dni /p ) = 1 − Fi (ci dni /p )

Let Pn (Z ) ≡ min{Pn1 (Z ), ..., PnN (Z )} and let Gn (p ) ≡ Pr(Pn (Z ) ≤ p ) be the price distribution in country n. Then: Gn (p ) = 1 − exp[−Φn p θ ] where Φn ≡

N

∑ Ti (ci dni )−θ

i =1 Stanford Econ 266 (Donaldson)

Ricardian Theory (I)

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Four Key Results A - The Price Distribution (Cont.)

To show this, note that (suppressing notation Z from here onwards) Pr(Pn ≤ p ) = 1 − Πi Pr(Pni ≥ p )

= 1 − Πi [1 − Gni (p )] But using Gni (p ) = 1 − Fi (ci dni /p ) then 1 − Πi [1 − Gni (p )] = 1 − Πi Fi (ci dni /p )

= 1 − Πi e −Ti (ci dni ) θ = 1 − e − Φn p

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Ricardian Theory (I)

−θ θ p

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Four Key Results B - The Allocation of Purchases

Consider a particular good. Country n buys the good from country i if i = arg min{pn1 , ..., pnN }. The probability of this event is simply country i 0 s contribution to country n0 s price parameter Φn , πni =

Ti (ci dni )−θ Φn

To show this, note that 



πni = Pr Pni ≤ min Pns s 6 =i

If Pni = p, then the probability that country i is the least cost supplier to country n is equal to the probability that Pns ≥ p for all s 6= i

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Four Key Results B - The Allocation of Purchases (Cont.)

The previous probability is equal to −i p θ

Πs 6=i Pr(Pns ≥ p ) = Πs 6=i [1 − Gns (p )] = e −Φn where Φn−i =

∑ Ti (ci dni )−θ

s 6 =i

Now we integrate over this for all possible p 0 s times the density dGni (p ) to obtain Z ∞

−i p θ

e − Φn

0

=

Ti (ci dni )−θ Φn

= πni Stanford Econ 266 (Donaldson)

Ti (ci dni )−θ θp θ −1 e −Ti (ci dni ) !Z

Z ∞ 0

∞

0

−θ θ p

dp

θΦn e −Φn p p θ −1 dp θ

dGn (p )dp = πni Ricardian Theory (I)

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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 Gn (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 −i q θ

Πs 6=i Pr(Pni ≥ q ) = Πs 6=i [1 − Gns (q )] = e −Φn

Then 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 −i q θ

e − Φn Stanford Econ 266 (Donaldson)

dGni (q )

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Four Key Results C - The Conditional Price Distribution (Cont.) −i θ

Integrating this probability e −Φn q dGni (q ) over all prices q ≤ p and −θ θ using Gni (q ) = 1 − e −Ti (ci dni ) p then Z p Z0p

−i q θ

e − Φn

dGni (q )

−i q θ

e − Φn

θTi (ci dni )−θ q θ −1 e −Ti (ci dni ) 0  Z p θ Ti (ci dni )−θ = e −Φn q θΦn q θ −1 dq Φn 0 = πni Gn (p )

=

−θ p θ

dq

Given that πni ≡ probability that for any particular good country i is the least cost supplier in n, the conditional distribution of the price charged by i in n for the goods that i actually sells in n is Z p −i θ 1 e −Φn q dGni (q ) = Gn (p ) πni 0 Stanford Econ 266 (Donaldson)

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Four Key Results C - The Conditional Price Distribution (Cont.)

Hence, we see that in Eaton and Kortum (2002): 1

2

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. The share of spending by country n on goods from country i is the same as the probability πni calculated above.

We will establish (in Lectures 10-11) a similar property in models of monopolistic competition with Pareto distributions of firm-level productivity

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Four Key Results D - The Price Index

The exact price index for a CES utility with elasticity of substitution σ < 1 + θ, defined as pn ≡

1

Z 0

pn (u )1−σ du

1/(1−σ) ,

is given by pn = γΦn−1/θ where

  1/(1−σ) 1−σ γ= Γ +1 , θ R∞ where Γ is the Gamma function, i.e. Γ(a) ≡ 0 x a−1 e −x dx.

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Four Key Results D - The Price Index (Cont.)

To show this, note that pn1−σ =

=

Z ∞ 0

p 1−σ dGn (p ) =

Z 1 0

pn (u )1−σ du

Z ∞ 0

p 1−σ Φn θp θ −1 e −Φn p dp. θ

Defining x = Φn p θ , then dx = Φn θp θ −1 , p 1−σ = (x /Φn )(1−σ)/θ , and pn1−σ

=

Z ∞ 0

(x /Φn )(1−σ)/θ e −x dx

−(1−σ)/θ

Z ∞

x (1−σ)/θ e −x dx   1−σ −(1−σ)/θ +1 = Φn Γ θ

= Φn

This implies pn = γΦn−1/θ with function to be well defined Stanford Econ 266 (Donaldson)

0

1− σ θ

+ 1 > 0 or σ − 1 < θ for gamma

Ricardian Theory (I)

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Equilibrium Let Xni be total spending in country n on goods from country i Let Xn ≡ ∑i Xni be country n’s total spending We know that Xni /Xn = πni , so Xni =

Ti (ci dni )−θ Xn Φn

(*)

Suppose that there are no intermediate goods so that ci = wi . In equilibrium, total income in country i must be equal to total spending on goods from country i so wi Li =

∑ Xni n

Trade balance further requires Xn = wn Ln so that wi Li =

n

Stanford Econ 266 (Donaldson)

Ti (wi dni )−θ w L −θ n n j Tj (wj dnj )

∑∑

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Equilibrium (Cont.)

This provides system of N − 1 independent equations (Walras’ Law) that can be solved for wages (w1 , ..., wN ) up to a choice of numeraire We now know that this is unique: Scarf and Wilson (2003), Alvarez and Lucas (2007), Allen and Arkolakis (2015)

Everything is as if countries were exchanging labor (as we expected from Wilson, 1980). Fr´echet distributions just imply that labor demands are iso-elastic Begs the question: might there be other Ricardian microfoundations under which the global labor demand system is isoelastic? And what happens if it’s not isoelastic? See Lectures 14-15.

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Equilibrium (Cont.) Under frictionless trade (dni = 1 for all n, i) previous system implies wi1+θ =

Ti ∑n wn Ln Li ∑j Tj wj−θ

and hence (for any 2 countries, i and j), writing this as “relative labor demand = relative labor supply”   1+ θ Tj wi L = i wj Ti Lj Compare with similar equation in DFS (1977), in Lecture 3, but with symmetric Cobb-Douglas prefs (i.e. θ (ze) = ze): w 1 − ze = w ∗ ze   −1 w 1 − A−1 ( ww∗ ) ⇒ = w ∗ A−1 ( ww∗ ) Stanford Econ 266 (Donaldson)

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The Gravity Equation Letting Yi = ∑n Xni be country i 0 s total sales, then Yi =

Ti (ci dni )−θ Xn = Ti ci−θ Ωi−θ ∑ Φ n n

where Ωi−θ ≡ Solving

Ti ci−θ

from Yi =

∑

n −θ −θ Ti ci Ωi

Xni =

dni−θ Xn Φn and plugging into (*) we get

Xn Yi dni −θ Ωiθ Φn

Using pn = γΦn−1/θ we can then get Xni = γ−θ Xn Yi dni −θ (pn Ωi )θ This is the Gravity Equation, with bilateral resistance dni and multilateral resistance terms pn (inward) and Ωi (outward). Stanford Econ 266 (Donaldson)

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The Gravity Equation A Primer on Trade Costs

From (*) we also get that country i’s share in country n’s expenditures normalized by its own share is   Φi − θ pi dni −θ Xni /Xn = d = Sni ≡ Xii /Xi Φn ni pn 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 Sni = 1.  1/2 in Letting Bni ≡ XXniii · XXnn then  1/2 Bni = (Sni Sin )1/2 = dni−θ din−θ Under symmetric trade costs (i.e., dni = din ) then Bni−1/θ = dni can be used as a measure of trade costs. (Later we will call this the Head and Ries (AER, 2001) index of trade costs.) Stanford Econ 266 (Donaldson)

Ricardian Theory (I)

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The Gravity Equation A Primer on Trade Costs

The Gravity Equation A Primer on Trade Costs We can also see how Bni varies with physical distance—perhaps a plausible also see how ni varies proxy for We dni can —between n Band i: with physical distance between n and i:

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Stanford Econ 266 (Donaldson)

Ricardian Theory (I)

Ricardian Theory (I)

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How to Estimate θ, “The Trade Elasticity”?

As we will see, θ, often referred to as “the trade elasticity,” is the key structural parameter for welfare and counterfactual analysis in EK model In order to estimate θ directly from Bni = dni−θ we need a measure of dni , not just a proxy (like distance). Negative relationship in Figure 1 could come from strong effect of proxy variable (distance) on dni or from mild CA (high θ), so θ not identified.

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How to Estimate the Trade Elasticity? EK use price data to measure pi dni /pn , and then use fact that  −θ Sni = pipdnni . 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 pi (j ) of individual goods j in the model. They note that for goods that n imports from i we should have pn (j )/pi (j ) = dni , whereas goods that n doesn’t import from i can have pn (j )/pi (j ) ≤ dni . Since every country in the sample does import manufactured goods from every other, then maxj {pn (j )/pi (j )} should be equal to dni . To deal with measurement error, they actually use the second highest pn (j )/pi (j ) as a measure of dni . Stanford Econ 266 (Donaldson)

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HowtotoEstimate Estimatethe the Trade Trade Elasticity? Elasticity? How

Letrnirni(j()j )≡ lnlnppn n(j()j )− lnlnppi i((jj)).. EK They calculateln(lnp(np/p themean mean Let calculate n /p i )i )asasthe acrossj jofofrnirni(j()j .).Then Thenthey they measure measure ln(pii dni ni /p across /pnn) by by max 2j frni (j )g = max 2j {rni (j )} DDnini = /50 ∑j rrni((jj))/50 ∑ j ni θ Given Sni = pipdipnidnni−θ they estimate θ from ln(Sni ) = θDni . Given Sni = pn they estimate θ from ln(Sni ) = −θDni . Method of moments: θ = 8.28. OLS with zero intercept: θ = 8.03. Method of moments: θ = 8.28. OLS with zero intercept: θ = 8.03. Stanford Econ 266 () Ricardian Theory (I) Winter 2015 (lecture 4) 21 / 34 Stanford Econ 266 (Donaldson)

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Alternative Strategies Simonovska and Waugh (2011) argue that EK’s procedure suffers from upward bias: Since EK are only considering 50 goods (real world has more), 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 θ closer to 4.

An alternative approach is to use tariffs (Caliendo and Parro, 2011). If dni = tni τni where tni is one plus the ad-valorem tariff (they actually do this for each 2 digit industry) and τni is assumed to be symmetric, then:     dni dij djn −θ tni tij tjn −θ Xni Xij Xjn = = . Xnj Xji Xin dnj dji din tnj tji tin They can then run an OLS regression and recover θ. (In practice, this is done over time and Their preferred specification leads to an estimate of 8.22. Stanford Econ 266 (Donaldson)

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Gains from Trade Consider again the case where ci = wi From (*), we know that πnn =

Tn wn−θ Xnn = Xn Φn

We also know that pn = γΦn−1/θ , so −1/θ ωn ≡ wn /pn = γ−1 Tn1/θ πnn .

Under autarky we have ωnA = γ−1 Tn1/θ , hence the gains from trade are given by −1/θ GTn ≡ ωn /ωnA = πnn Trade elasticity θ and share of expenditure on domestic goods πnn are sufficient statistics to compute GT Stanford Econ 266 (Donaldson)

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Gains from Trade (Cont.)

A typical value for πnn (manufacturing) is 0.7. With θ = 5 this implies GTn = 0.7−1/5 = 1. 074 or 7.4% gains. Belgium has πnn = 0.2, so its gains are GTn = 0.2−1/5 = 1. 38 or 38%. One can also use the previous approach to measure the welfare gains associated with any foreign shock, not just moving to autarky: 0 ωn0 /ωn = πnn /πnn


−1/θ

For more general counterfactual scenarios, however, one needs to 0 and π . (In autarky we knew that π know both πnn nn nn = 1.)

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Adding an Input-Output Loop

Now 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 β 1− β then write ci = wi pi .

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Adding an Input-Output Loop (Cont.) The analysis above implies πnn = γ

−θ

 Tn

cn pn

−θ

and hence −1/θ cn = γ−1 Tn−1/θ πnn pn β 1− β

Using cn = wn pn

this implies β 1− β

wn pn

−1/θ = γ−1 Tn−1/θ πnn pn

so

−1/θβ

wn /pn = γ−1/β Tn

−1/θβ

πnn

The gains from trade are now −1/θβ

ωn /ωnA = πnn

Standard value for β is 1/2 (Alvarez and Lucas, 2007). For πnn = 0.7 and θ = 5 this implies GTn = 0.7−2/5 = 1. 15 or 15% gains. Stanford Econ 266 (Donaldson)

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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.

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Adding Non-Tradables (Cont.) The price index computed above is now pgn , but we care about ωn ≡ wn /pfn , where 1− α pfn = wnα pgn This implies that ωn =

wn 1− α α wn pgn

= (wn /pgn )1−α

Thus, the gains from trade are now −η/θ

ωn /ωnA = πnn where η≡

1−α β

Alvarez and Lucas argue that α = 0.75 (share of labor in services). Thus, for πnn = 0.7, θ = 5 and β = 0.5, this implies GTn = 0.7−1/10 = 1. 036 or 3.6% gains Stanford Econ 266 (Donaldson)

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Comparative statics (Dekle, Eaton and Kortum, 2008) See also Rutherford (1994), “Lecture Notes on Constant Elasticity Functions”

Go back to the simple EK model above (α = 0, β = 1). We have

=

Xni

∑ Xni

Ti (wi dni )−θ Xn −θ ∑N i =1 Ti (wi dni )

= wi Li

n

As we have already established, this leads to a system of non-linear equations to solve for wages, wi Li =

∑ n

Stanford Econ 266 (Donaldson)

Ti (wi dni )−θ ∑k Tk (wk dnk )

Ricardian Theory (I)

−θ

wn Ln .

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Comparative statics (Dekle, Eaton and Kortum, 2008) Consider any shock to labor endowments (Li → Li0 ), trade costs (dni → dni0 ), or productivity (Ti → Ti0 ). Two methods for solving for the change in endogenous variables: 1

2

0 , T and T 0 . As in EK (2002): estimate or calibrate θ and Li , Li0 , dni , dni i i Then solve for the endogenous variables at old and new equilibrium values. As in DEK (2008): If the initial equilibrium corresponds to a setting where all endogenous variables (matrix of Xni values) are observed (and no measurement error), and have estimate of θ, then can instead solve for changes in endogenous variables. Advantages:

Often simpler to solve for one set of changes than two sets of levels. Often simpler to explain to audience. Model fits data in observed pre-shock year exactly.

NB: DEK procedure creates impression that hard and often controversial task of estimating many (L, d, T ) parameters has been avoided, but that’s not true. Stanford Econ 266 (Donaldson)

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Comparative statics (Dekle, Eaton and Kortum, 2008) To see how it works, note that trade shares are (from *) πni =

Ti (wi dni )−θ ∑k Tk (wk dnk )

−θ

and

0 πni

=

Ti0 (wi0 dni0 )

−θ

0 ) ∑k Tk0 (wk0 dnk

−θ

.

Letting xˆ ≡ x 0 /x, then we have πˆ ni

= =

=

Tˆi wˆ i dˆni 0 ) ∑k Tk0 (wk0 dnk

−θ


−θ

/ ∑j Tj (wj dnj )−θ
−θ Tˆi wˆ i dˆni


−θ Tk (wk dnk )−θ / ∑j Tj (wj dnj )−θ ∑k Tˆk wˆ k dˆnk
−θ Tˆi wˆ i dˆni
−θ . ∑k πnk Tˆk wˆ k dˆnk

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Comparative statics (Dekle, Eaton and Kortum, 2008) On the other hand, for equilibrium we have wi0 Li0 =

∑ πni0 wn0 Ln0 = ∑ πˆ ni πni wn0 Ln0 n

n

Letting Yn ≡ wn Ln and using the result above for πˆ ni we get wˆ i Lˆ i Yi =

∑ n

πni Tˆi wˆ i dˆni


−θ

∑k πnk Tˆk wˆ k dˆnk


−θ wˆ n Lˆ n Yn

This forms a system of N equations in N unknowns, wˆ i , from which we can get wˆ i as a function of shocks and initial observables (establishing some numeraire). Here πni and Yi are data (obtainable from Xni matrix) and we know dˆni , Tˆi , Lˆ i , as well as θ.

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Comparative statics (Dekle, Eaton and Kortum, 2008) To compute the implications for welfare of a foreign shock, simply impose that Lˆ n = Tˆn = 1, solve the system above to get wˆ i and get the implied πˆ nn through Tˆi wˆ i dˆni

πˆ ni =


−θ

∑k πnk Tˆk wˆ k dˆnk


−θ .

and use the formula to get −1/θ ωˆ n = πˆ nn

Of course, if it is not the case that Lˆ n = Tˆn = 1 (i.e. there is some “domestic” component to the shock too), then one can still use this approach, since in general we have: ωˆ n = Tˆn Stanford Econ 266 (Donaldson)


1/θ

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−1/θ πˆ nn Winter 2016 (lecture 4)

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Some Examples of Extensions of EK (2002) But there are many others...

Bertrand Competition: Bernard, Eaton, Jensen, and Kortum (2003) The same (Extreme Value Theory) tricks that EK (2002) show work for characterizing the lowest price work for finding the second-lowest, etc. Bertrand competition ⇒ variable markups at the firm-level Measured productivity varies across firms ⇒ one can use firm-level data to calibrate model

Multiple Sectors: Costinot, Donaldson, and Komunjer (2012) Tik ≡ fundamental productivity in country i and sector k One can use EK’s machinery to study pattern of trade, not just volumes More in next lecture (on empirics of Ricardian models)

Non-homothetic preferences: Fieler (2011) Rich and poor countries have different expenditure shares Combined with differences in θ k across sectors k, one can explain pattern of North-North, North-South, and South-South trade Stanford Econ 266 (Donaldson)

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