Economics 266: International Trade | Lecture 4: Ricardian ...dave-donaldson.com/wp-content/uploads/2016/10/Lecture-4_Ricardo... · Economics 266: International Trade | Lecture 4

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) 1 / 35

“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 tradeBut grad students should study richer models (e.g. Feenstra’s graduatetextbook—edition 1, from 2003—had a total of 3 pages on theRicardian model!)

Eaton and Kortum (Ecta, 2002) has lead to “Ricardian revival”

Same basic idea as in Wilson (1980): Is it necessary to know about thepattern of trade to do certain types of counterfactual analysis?But more structure: Small number of parameters, so well-suited forquantitative work.

Goals of this lecture:

1 Present EK model2 Discuss estimation of its key parameters3 Introduce tools for welfare and counterfactual analysis


Basic Assumptions

N countries, i = 1, ...,N

Continuum of goods u ∈ [0, 1]

Preferences are CES with elasticity of substitution σ:

Ui =

(∫ 1

0qi (u)

(σ−1)/σdu

)σ/(σ−1),

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


Basic Assumptions (Cont.)

Constant returns to scale:

Zi (u) denotes productivity of (any) firm producing u in country iZi (u) is drawn independently (across goods and countries) from aFrechet distribution:

Pr(Zi ≤ z) = Fi (z) = e−Ti z−θ

,

with θ > σ− 1 (important restriction, see below)Since goods are symmetric except for productivity, we can forget aboutindex 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


Four Key ResultsA - The Price Distribution

Let Pni (Zi ) ≡ cidni/Zi be the unit cost at which country i can servea good Z to country n and let Gni (p) ≡ Pr(Pni (Zi ) ≤ p). Then:

Gni (p) = Pr (Zi ≥ cidni/p) = 1− Fi (cidni/p)

Let Pn(Z ) ≡ minPn1(Z ), ...,PnN(Z ) and let Gn(p) ≡Pr(Pn(Z ) ≤ p) be the price distribution in country n. Then:

Gn(p) = 1− exp[−Φnpθ ]

where

Φn ≡N

∑i=1

Ti (cidni )−θ


Four Key ResultsA - 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 usingGni (p) = 1− Fi (cidni/p)

then

1−Πi [1− Gni (p)] = 1−ΠiFi (cidni/p)

= 1−Πie−Ti (cidni )

−θpθ

= 1− e−Φnpθ


Four Key ResultsB - The Allocation of Purchases

Consider a particular good. Country n buys the good from country iif i = arg minpn1, ..., pnN. The probability of this event is simplycountry i ′s contribution to country n′s price parameter Φn,

πni =Ti (cidni )−θ

Φn

To show this, note that

πni = Pr

(Pni ≤ min

s 6=iPns

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


Four Key ResultsB - The Allocation of Purchases (Cont.)

The previous probability is equal to

Πs 6=i Pr(Pns ≥ p) = Πs 6=i [1− Gns(p)] = e−Φ−in pθ

whereΦ−in = ∑

s 6=i

Ti (cidni )−θ

Now we integrate over this for all possible p′s times the densitydGni (p) to obtain∫ ∞

0e−Φ−in pθ

Ti (cidni )−θ θpθ−1e−Ti (cidni )

−θpθdp

=

(Ti (cidni )

−θ

Φn

) ∫ ∞

0θΦne

−Φnpθpθ−1dp

= πni

∫ ∞

0dGn(p)dp = πni


Four Key ResultsC - The Conditional Price Distribution

The price of a good that country n actually buys from any country ialso has the distribution Gn(p).

To show this, note that if country n buys a good from country i itmeans that i is the least cost supplier. If the price at which country isells this good in country n is q, then the probability that i is theleast cost supplier is

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

Then the joint probability that country i has a unit cost q ofdelivering the good to country n and is the the least cost supplier ofthat good in country n is then

e−Φ−in qθdGni (q)


Four Key ResultsC - The Conditional Price Distribution (Cont.)

Integrating this probability e−Φ−in qθdGni (q) over all prices q ≤ p and

using Gni (q) = 1− e−Ti (cidni )−θpθ

then∫ p

0e−Φ−in qθ

dGni (q)

=∫ p

0e−Φ−in qθ

θTi (cidni )−θqθ−1e−Ti (cidni )

−θpθdq

=

(Ti (cidni )−θ

Φn

) ∫ p

0e−Φnq

θθΦnq

θ−1dq

= πniGn(p)

Given that πni ≡ probability that for any particular good country i isthe least cost supplier in n, the conditional distribution of the pricecharged by i in n for the goods that i actually sells in n is

1

πni

∫ p

0e−Φ−in qθ

dGni (q) = Gn(p)


Four Key ResultsC - The Conditional Price Distribution (Cont.)

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

1 All the adjustment is at the extensive margin: countries that are moredistant, have higher costs, or lower T ′s, simply sell a smaller range ofgoods, but the average price charged is the same.

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

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


Four Key ResultsD - The Price Index

The exact price index for a CES utility with elasticity of substitutionσ < 1 + θ, defined as

pn ≡(∫ 1

0pn(u)

1−σdu

)1/(1−σ)

,

is given bypn = γΦ−1/θ

n

where

γ =

[Γ(

1− σ

θ+ 1

)]1/(1−σ)

,

where Γ is the Gamma function, i .e. Γ(a) ≡∫ ∞0 xa−1e−xdx .


Four Key ResultsD - The Price Index (Cont.)

To show this, note that

p1−σn =

∫ 1

0pn(u)

1−σdu

=∫ ∞

0p1−σdGn(p) =

∫ ∞

0p1−σΦnθpθ−1e−Φnp

θdp.

Defining x = Φnpθ, then dx = Φnθpθ−1, p1−σ = (x/Φn)(1−σ)/θ, and

p1−σn =

∫ ∞

0(x/Φn)

(1−σ)/θe−xdx

= Φ−(1−σ)/θn

∫ ∞

0x (1−σ)/θe−xdx

= Φ−(1−σ)/θn Γ

(1− σ

θ+ 1

)This implies pn = γΦ−1/θ

n with 1−σθ + 1 > 0 or σ− 1 < θ for gamma

function to be well definedStanford Econ 266 (Donaldson) Ricardian Theory (I) Winter 2016 (lecture 4) 13 / 35

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 (cidni )−θ

ΦnXn (*)

Suppose that there are no intermediate goods so that ci = wi .

In equilibrium, total income in country i must be equal to totalspending on goods from country i so

wiLi = ∑n

Xni

Trade balance further requires Xn = wnLn so that

wiLi = ∑n

Ti (widni )−θ

∑j Tj (wjdnj )−θwnLn


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), Alvarezand Lucas (2007), Allen and Arkolakis (2015)

Everything is as if countries were exchanging labor (as we expectedfrom Wilson, 1980).

Frechet distributions just imply that labor demands are iso-elasticBegs the question: might there be other Ricardian microfoundationsunder which the global labor demand system is isoelastic? And whathappens if it’s not isoelastic? See Lectures 14-15.


Equilibrium (Cont.)

Under frictionless trade (dni = 1 for all n, i) previous system implies

w1+θi =

Ti

Li

∑n wnLn

∑j Tjw−θj

and hence (for any 2 countries, i and j), writing this as “relative labordemand = relative labor supply”(

wi

wj

)1+θ Tj

Ti=

LiLj

Compare with similar equation in DFS (1977), in Lecture 3, but withsymmetric Cobb-Douglas prefs (i.e. θ(z) = z):

w

w ∗1− z

z=

L∗

L

⇒(

w

w ∗1− A−1( w

w ∗ )

A−1( ww ∗ )

)−1=

L

L∗


The Gravity Equation

Letting Yi = ∑n Xni be country i ′s total sales, then

Yi = ∑n

Ti (cidni )−θ Xn

Φn= Tic

−θi Ω−θ

i

where

Ω−θi ≡∑

n

d−θni Xn

Φn

Solving Tic−θi from Yi = Tic

−θi Ω−θ

i and plugging into (*) we get

Xni =XnYidni

−θΩθi

Φn

Using pn = γΦ−1/θn we can then get

Xni = γ−θXnYidni−θ(pnΩi )

θ

This is the Gravity Equation, with bilateral resistance dni andmultilateral resistance terms pn (inward) and Ωi (outward).


The Gravity EquationA Primer on Trade Costs

From (*) we also get that country i ’s share in country n’sexpenditures normalized by its own share is

Sni ≡Xni/Xn

Xii/Xi=

Φi

Φnd−θni =

(pidnipn

)−θ

This shows the importance of trade costs and comparative advantagein determining trade volumes. Note that if there are no trade barriers(i.e, frictionless trade), then Sni = 1.

Letting Bni ≡(XniXii· XinXnn

)1/2then

Bni = (SniSin)1/2 =

(d−θni d

−θin

)1/2

Under symmetric trade costs (i.e., dni = din) then B−1/θni = dni can

be used as a measure of trade costs. (Later we will call this the Headand Ries (AER, 2001) index of trade costs.)



We can also see how Bni varies with physical distance—perhaps a plausibleproxy for dni—between n and i :


We can also see how Bni varies with physical distance between n and i :

Stanford Econ 266 () Ricardian Theory (I) Winter 2015 (lecture 4) 18 / 34


How to Estimate θ, “The Trade Elasticity”?

As we will see, θ, often referred to as “the trade elasticity,” is the keystructural parameter for welfare and counterfactual analysis in EKmodel

In order to estimate θ directly from Bni = d−θni we need a measure of

dni , not just a proxy (like distance).

Negative relationship in Figure 1 could come from strong effect of proxyvariable (distance) on dni or from mild CA (high θ), so θ not identified.


How to Estimate the Trade Elasticity?

EK use price data to measure pidni/pn, and then use fact that

Sni =(pidnipn

)−θ.

They use retail prices in 19 OECD countries for 50 manufacturedproducts from the UNICP 1990 benchmark study.

They interpret these data as a sample of the prices pi (j) of individualgoods j in the model.

They note that for goods that n imports from i we should havepn(j)/pi (j) = dni , whereas goods that n doesn’t import from i canhave pn(j)/pi (j) ≤ dni .

Since every country in the sample does import manufactured goodsfrom every other, then maxjpn(j)/pi (j) should be equal to dni .

To deal with measurement error, they actually use the second highestpn(j)/pi (j) as a measure of dni .


How to Estimate the Trade Elasticity?How to Estimate the Trade Elasticity?

Let rni (j) ≡ ln pn(j)− ln pi (j). They calculate ln(pn/pi ) as the meanacross j of rni (j). Then they measure ln(pidni/pn) by

Dni =max 2jrni (j)∑j rni (j)/50

Given Sni =(pidnipn

)−θthey estimate θ from ln(Sni ) = −θDni .

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

Let rni (j) ≡ ln pn(j)− ln pi (j). EK calculate ln(pn/pi ) as the meanacross j of rni (j). Then they measure ln(pidni/pn) by

Dni =max 2jrni (j)∑j rni (j)/50

Given Sni =(pidnipn

)−θthey estimate θ from ln(Sni ) = −θDni .

Method of moments: θ = 8.28. OLS with zero intercept: θ = 8.03.


Alternative Strategies

Simonovska and Waugh (2011) argue that EK’s procedure suffersfrom 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 elasticitySimulation based method of moments leads to a θ closer to 4.

An alternative approach is to use tariffs (Caliendo and Parro, 2011).If dni = tniτni where tni is one plus the ad-valorem tariff (theyactually do this for each 2 digit industry) and τni is assumed to besymmetric, then:

XniXijXjn

XnjXjiXin=

(dnidijdjndnjdjidin

)−θ

=

(tni tij tjntnj tji tin

)−θ

.

They can then run an OLS regression and recover θ. (In practice, thisis done over time and Their preferred specification leads to anestimate of 8.22.


Gains from Trade

Consider again the case where ci = wi

From (*), we know that

πnn =Xnn

Xn=

Tnw−θn

Φn

We also know that pn = γΦ−1/θn , so

ωn ≡ wn/pn = γ−1T 1/θn π−1/θ

nn .

Under autarky we have ωAn = γ−1T 1/θ

n , hence the gains from tradeare given by

GTn ≡ ωn/ωAn = π−1/θ

nn

Trade elasticity θ and share of expenditure on domestic goods πnn aresufficient statistics to compute GT


Gains from Trade (Cont.)

A typical value for πnn (manufacturing) is 0.7. With θ = 5 thisimplies GTn = 0.7−1/5 = 1. 074 or 7.4% gains. Belgium hasπnn = 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 gainsassociated with any foreign shock, not just moving to autarky:

ω′n/ωn =(π′nn/πnn

)−1/θ

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


Adding an Input-Output Loop

Now imagine that intermediate goods are used to produce acomposite good with a CES production function with elasticity σ > 1.

This composite good can be either consumed or used to produceintermediate goods (input-output loop).

Each intermediate good is produced from labor and the compositegood with a Cobb-Douglas technology with labor share β. We can

then write ci = wβi p

1−βi .


Adding an Input-Output Loop (Cont.)

The analysis above implies

πnn = γ−θTn

(cnpn

)−θ

and hencecn = γ−1T−1/θ

n π−1/θnn pn

Using cn = wβn p

1−βn this implies

wβn p

1−βn = γ−1T−1/θ

n π−1/θnn pn

sown/pn = γ−1/βT

−1/θβn π

−1/θβnn

The gains from trade are now

ωn/ωAn = π

−1/θβnn

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


Adding Non-Tradables

Assume now that the composite good cannot be consumed directly.

Instead, it can either be used to produce intermediates (as above) orto produce a consumption good (together with labor).

The production function for the consumption good is Cobb-Douglaswith labor share α.

This consumption good is assumed to be non-tradable.


Adding Non-Tradables (Cont.)

The price index computed above is now pgn, but we care aboutωn ≡ wn/pfn, where

pfn = w αn p

1−αgn

This implies that

ωn =wn

w αn p

1−αgn

= (wn/pgn)1−α

Thus, the gains from trade are now

ωn/ωAn = π

−η/θnn

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 impliesGTn = 0.7−1/10 = 1. 036 or 3.6% gains


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 =Ti (widni )−θXn

∑Ni=1 Ti (widni )−θ

∑n

Xni = wiLi

As we have already established, this leads to a system of non-linearequations to solve for wages,

wiLi = ∑n

Ti (widni )−θ

∑k Tk (wkdnk)−θ

wnLn.


Comparative statics (Dekle, Eaton and Kortum, 2008)

Consider any shock to labor endowments (Li → L′i ), trade costs(dni → d ′ni ), or productivity (Ti → T ′i ).

Two methods for solving for the change in endogenous variables:1 As in EK (2002): estimate or calibrate θ and Li , L

′i , dni , d

′ni ,Ti and T ′i .

Then solve for the endogenous variables at old and new equilibriumvalues.

2 As in DEK (2008): If the initial equilibrium corresponds to a settingwhere all endogenous variables (matrix of Xni values) are observed(and no measurement error), and have estimate of θ, then can insteadsolve 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 oftencontroversial task of estimating many (L, d ,T ) parameters has beenavoided, but that’s not true.



To see how it works, note that trade shares are (from *)

πni =Ti (widni )

−θ

∑k Tk (wkdnk)−θ

and π′ni =T ′i (w

′i d′ni )−θ

∑k T′k (w

′kd′nk)−θ

.

Letting x ≡ x ′/x , then we have

πni =Ti

(wi dni

)−θ

∑k T′k (w

′kd′nk)−θ / ∑j Tj (wjdnj )

−θ

=Ti

(wi dni

)−θ

∑k Tk

(wk dnk

)−θTk (wkdnk)

−θ / ∑j Tj (wjdnj )−θ

=Ti

(wi dni

)−θ

∑k πnk Tk

(wk dnk

)−θ.



On the other hand, for equilibrium we have

w ′i L′i = ∑

n

π′niw′nL′n = ∑

n

πniπniw′nL′n

Letting Yn ≡ wnLn and using the result above for πni we get

wi LiYi = ∑n

πni Ti

(wi dni

)−θ

∑k πnk Tk

(wk dnk

)−θwnLnYn

This forms a system of N equations in N unknowns, wi , from whichwe can get wi as a function of shocks and initial observables(establishing some numeraire). Here πni and Yi are data (obtainablefrom Xni matrix) and we know dni , Ti , Li , as well as θ.



To compute the implications for welfare of a foreign shock, simplyimpose that Ln = Tn = 1, solve the system above to get wi and getthe implied πnn through

πni =Ti

(wi dni

)−θ

∑k πnk Tk

(wk dnk

)−θ.

and use the formula to get

ωn = π−1/θnn

Of course, if it is not the case that Ln = Tn = 1 (i.e. there is some“domestic” component to the shock too), then one can still use thisapproach, since in general we have:

ωn =(Tn

)1/θπ−1/θnn


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 forcharacterizing the lowest price work for finding the second-lowest, etc.Bertrand competition ⇒ variable markups at the firm-levelMeasured productivity varies across firms ⇒ one can use firm-leveldata to calibrate model

Multiple Sectors: Costinot, Donaldson, and Komunjer (2012)

T ki ≡ fundamental productivity in country i and sector k

One can use EK’s machinery to study pattern of trade, not just volumesMore in next lecture (on empirics of Ricardian models)

Non-homothetic preferences: Fieler (2011)

Rich and poor countries have different expenditure sharesCombined with differences in θk across sectors k, one can explainpattern of North-North, North-South, and South-South trade


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Economics 266: International Trade | Lecture 4: Ricardian ...dave-donaldson.com/wp-content/uploads/2016/10/Lecture-4_Ricardo... · Economics 266: International Trade | Lecture 4