Econ 191: Measuring the Gains from Trade · 2012-11-13 · Introduction The Armington Model Next...

AnnouncementsMeasuring the Gains of Trade

Summary

Econ 191: Measuring the Gains from Trade

Preparatory lecture for Professor Andrés Rodríguez-Clare

Dawn Powers

U. C. Berkeley

Nov. 6, 2012

Dawn Powers Background for Nov. 13: “Measuring Gains from Trade”


Summary

Outline

1 Announcements

2 Measuring the Gains of TradeIntroductionThe Armington Model



Summary

Announcements

Assignment due:Data (empirical papers)Model setup (theoretical papers)

General feedback handouts on the course webpageIf you change your topic, submit a new 2-page researchproposal ASAP



SummaryIntroductionThe Armington Model

Next Week’s Topic“Measuring the Gains from Trade”

Next week, Prof. Rodriguez-Clare will discuss:Andres Rodriguez-Clare (with Costas Arkolakis, SvetlanaDemidova and Pete Klenow), "Endogenous Variety and theGains from Trade," American Economic Review Papers andProceedings, May 2008.Andres Rodriguez-Clare (with Costas Arkolakis and ArnaudCostinot), "New Trade Models, Same Old Gains?" AmericanEconomic Review, February 2012.




Measuring the Gains from Trade




Outline

1 Announcements





Introduction

Samuelson (1939): There are gains in international trade.




Introduction

Samuelson (1939): There are gains in international trade.Under perfect competition, opening up to trade expands thePPF and leads to Pareto superior outcomes.




Introduction

Samuelson (1939): There are gains in international trade.Recall Econ 1: Assume that the production technologies incountries A and B produce the following yields per acre forwheat and cotton:

A BWheat 2 bushels 6 bushelsCotton 6 bales 2 bales

Who has the absolute advantage in producing wheat? Thecomparative advantage?




Introduction

Assuming that each country has 100 acres of farmland, and botheconomies are closed, their PPFs look as the following:




Introduction

Side note: Without trade, where will these countries produce onthe PPF?

Hint: Recall last week’s lecture...




Introduction

A: Depends on the preferences of each country’s consumers forcotton in relation to wheat. If both countries’ consumers get equalutility from cotton and wheat, these preferences could berepresented in the indifference curves below (assuming preferencescan be aggregated here).




Introduction

Okay, back to trade: Because each country has an absoluteadvantage in producing one product, specialization and tradewill benefit both.Consider the total production of each country before and afterspecialization and trade, assuming each country’s consumers getequal utility from wheat and cotton:

Before trade:

A B

Wheat (75 acres)*(2 bushels/acre) (25 acres)*(6 bushels/acre)

Cotton (25 acres)*(6 bales/acre) (75 acres)*(2 bales/acre)

In this scenario, each country produces and consumes 150 bushels of wheat and

150 bales of cotton.




Introduction

Okay, back to trade: Because each country has an absoluteadvantage in producing one product, specialization and tradewill benefit both.Consider the total production of each country before and afterspecialization and trade, assuming each country’s consumers getequal utility from wheat and cotton:

After trade:

A B

Wheat 0 acres = 0 bushels (100 acres)*(6 bushels/acre)

Cotton (100 acres)*(6 bales/acre) 0 acres = 0 bales

In this scenario, each country produces 600 units of the commodity for which it

has the absolute advantage, and trades half for half of the other country’s

commodity.




Introduction

Where does this new production level fall on each country’s PPF?




Outline

1 Announcements





Tools for the Armington Model

CES utility functions

We’ve already looked at utility functions in general in previousweeks, so let’s focus on building up the concept of constantelasticity of substitution (CES).






Marginal rate of substitution (MRS): The rate at which aconsumer is ready to give up one good in exchange for anothergood while maintaining the same level of utility.

E.g. If goods A and B are continuously divisible, then at levels(a, b) of goods A and B, a consumer’s MRS of A for B (i.e.willingness to give up units of good A for units of good B) is:

MRSa,b = MUaMUb

= dln(UaUb

)

where MUa is the marginal utility of consuming one unitbeyond a of good A,and where the last result is derived from the properties ofnatural logs.






Elasticity of substitution (in a utility function): For ageneric utility function U(c1, c2) where c1 and c2 represent theamounts consumed of goods 1 and 2 respectively,the elasticity of substitution of c1 for c2 is the amount bywhich the consumer’s willingness to give up c1 for units of c2changes when the relative MUs of consuming goods 1 and 2change:

ε =dln( c2

c1)

dln(MUc1MUc2

)=

dln( c2c1

)

dln(MRS1,2)= −

dln( c2c1

)

dln(MRS2,1)= −

dln( c2c1

)

dln(Uc2Uc1

)= −

d (c2/c1)c2/c1

(d (Uc2/Uc1 )

Uc2/Uc1)

where Uc1 is the derivative of U WRT c1,and where we again exploit the properties of natural logs tosimplify the expression.






Different consumers may display (theoretically) different elasticitiesof substitution:

Constant elasticity of substitution (CES): For every valueof consumption ratio c2

c1, a particular percentage change in

MRS will cause c2c1

to change at a constant rate (i.e. toexperience a constant percentage change).Increasing elasticity of substitution (IES): As consumptionratio c2

c1increases, a particular percentage change in MRS will

cause c2c1

to change at an increasing rate (i.e. to experienceincreasing percentage changes).






Different consumers may display different types of elasticity ofsubstitution:

Decreasing elasticity of substitution (DES): Asconsumption ratio c2

c1increases, a particular percentage change

in MRS will cause c2c1

to change at a decreasing rate (i.e. toexperience decreasing percentage changes).






We have now built our way to describe a CES utility function:Constant elasticity of substitution (CES) utility function:A utility function that displays a constant elasticity ofsubstitution.

An individual with CES utility choosing levels of each goodi = 1, 2, ..., n would have the following utility function:

C = (�n

i=1 c(σ−1)/σi )σ/(σ−1)

where ci is the individual’s demand for good i ,and where σ > 1 is the (constant) elasticity of substitutionbetween goods.





Gravity equation

Gravity equation (Tinbergen 1962): Countries with larger GDPs, orwho are closer to each other, have more trade between them.

Underlying idea: Models of monopolistic competition predictthat countries with large GDP should trade the most with eachother because:

1 Larger countries export more because they produce moreproduct varieties

2 Larger countries import more because their demand is higher.





Gravity equation

Newton’s Universal Law of Gravitation: Suppose two objectshave mass M1 and M2 and are located distance d apart. The forceof gravity Fg between these two objects is

Fg = G ∗ M1∗M2d2

where G is a constant describing the magnitude of the relationshipbetween mass and distance and force.

Punchline: M ↑ or d ↓ =⇒ F ↑.





Gravity equation

The gravity equation in trade: Suppose two countries haveGDP1 and GDP2, respectively, and are located distance d apart.The amount of trade predicted to occur between the countries is:

Trade = B ∗ GDP1∗GDP2dn

Note here that n is not necesarily set equal to 2 because we do nothave complete information on the precise relationship betweendistance and trade.

Punchline: GDP1,2 ↑ or d ↓ =⇒ Trade ↑.





Gravity equation

Trade = B ∗ GDP1∗GDP2dn

Also note one of the differences between theoretical andeconometric models here:

In theoretical models, we can assume causality because we areproposing an idea about a system works in absence ofconfounding variables.In both lab and field research, however, confounding variableswill always be present: The only way to convincingly arguethat they don’t affect your results is to achieve conditionalindependence through research design.





Gravity equation: Criticism and evolution

Krugman (1995), Leamer and Levinsohn (1995): Gravity modelsare strictly descriptive, and lack a theoretical underpinning.

However, the last 10 years has seen an ’explosion’ of alternativemicrotheoretical foundations underlying gravity equations, whichgive:

The same macro-level prediction regarding the structure ofbilateral trade flows as a function of costAdditional micro-level predictions that vary with the marketstructure assumed.




The Armington Model

Gravity equation: Use in international trade

Trade economists use multi-country gravity models forcounterfactual analysis.

Counterfactual analysis here involves the use of gravitymodels to quantify the gains from international trade thatwould be associated with moving one country from thecurrent, observed trade equilibrium to a counterfactualequilibrium with no trade.

Side note: See Sekhon (2004): “Quality Meets Quantity: Case Studies, ConditionalProbability, and Counterfactuals” for a discussion on the weaknesses of counterfactualanalysis.




The Armington Model

Brief description: The Armington Model is the simplest gravityequation used in the study of international trade.

Assumptions and Setup

Consider a world economy comprising j = 1, ..., n countries.Each country is endowed with Qi units of a distinct goodi = 1, ..., n.

Note that, because each country produces one good distinctfrom all other goods, the country and good notations areinterchangeable to some extent.




The Armington Model


Each country is populated by a representative agent whosepreferences are represented by a CES utility function:

Cj = (�n

i=1 C (σ−1)/σij )σ/(σ−1)

where Cij is the demand for good i in country j , and σ is theelasticity of substitution between goods i = 1, ..., n.




The Armington Model


The associated price index in country j for goods goodsi = 1, ..., n is given by

Pj = (�n

i=1 P1−σij )1/(1−σ)

where Pij is the price of good i in country j .




The Armington Model


Each country is subject to iceberg transport costs ininternational trade:

Iceberg trade costs (Samuelson 1954): A cost oftransporting a good that uses up only some fraction of thegood itself, rather than using any other resources.

One way to formally model iceberg trade costs is to introducethe following assumption:

In order to sell one unit of good i in country k, firms fromcountry j must ship τi,jk ≥ 1 units, where country j ’s domestictransport cost for good i τi,jj equals 1.




The Armington Model


Assume no opportunities for arbitrage exist.Arbitrage: A trading strategy that requires the investment ofno capital, cannot lose money, and has a positive probability ofmaking money.E.g. If a commodity is listed at different prices simultaneouslyon different stock exchanges, a trader could purchase thecommodity at the cheaper price and sell it on the other marketat the more expensive price.

For this no-arbitrage condition to hold, the price of countryj ’s good i in country k must only be a function of the good’sdomestic price and transport cost:

Pi ,jk = τi ,jkPi ,jj .




The Armington Model


The domestic price Pi ,jj of good i , then, is a function ofcountry j ’s total income Yj , and j ’s endowment of its distinctgood i , Qi ,j :

Pi ,jj =YjQi,j

.

Combining this expression with the no-arbitrage condition, weget the price of country j ’s good i in country k :

Pi ,jk = τi ,jkYjQi,j

.




The Armington Model


Let Xjk denote the total value of country j ’s imports fromcountry k .

Given CES utility functions, bilateral trade flows satisfy

Xjk = (Pi,jkPj

)1−σEj

where Ej =�n

i=1�n

k=1 Xjk is country j ’s total expenditure ongoods from all countries.




The Armington Model

Solution

Combining the expressions for Pi ,jj , Pi ,jk , and Xjk , we obtainthe following gravity equation:

Xjk =χi,j (Ejτi,jk )

−ε�n

l=1 χl,j (Elτl,jk )−ε Ej

where

trade elasticity ε ≡ (σ − 1) denotes how the demand forimports relative to domestic demand (

XjkXjj

) changes as bilateraltrade costs τi,jk change

exogenous import demand shifter χi,j ≡ Qσ−1i,j is a function of

country j ’s endowment and its trade elasticity.




The Armington Model


Assume market-clearing conditions for goods:

Yj =�n

i=1�n

k=1 Xi ,kj .

Assume that each country spends its entire income:

Ej = Yj .




The Armington Model

Solution

Finally, we combine the market-clearing and budget constraintequality conditions to construct the trade-balance condition:

�ni=1

�nk=1 Xi ,jk =

�ni=1

�nk=1 Xi ,kj .




The Armington Model

Solution:

A trade equilibrium in the Armington model is thus composed ofthe matrix of bilateral trade flows X ≡ {Xi ,jk} and the vector ofcountry expenditure levels E ≡ {Ej} such that the above gravityequation and trade balance condition hold simultaneously.#




The Armington Model

Strengths and Weaknesses of the Armington Model

Strengths:Simple.

=⇒ Makes the counterfactual equilibrium with no trade (inthe next section) easy to compute.

Weaknesses:Makes the ad-hoc assumption that each country isexogenously endowed with a distinct good.




The Armington ModelCounterfactual Analysis

Recall that international trade economists use the gravity equationfor trade to quantify the gains from international trade viacounterfactual analysis.

In this context, the gains from international trade equalsthe (absolute value of) the percentage change in real incomeassociated with moving one country from the current,observed trade equilibrium to a counterfactual equilibrium.





Rodríguez-Clare and Costinot (2012; Hereafter RCC) illustrate howthe gravity equation can be used in 2 steps to quantify the welfareconsequences of globalization, which they model as an exogenousshock to transport costs τ and exogenous import demand shifters χ:

1 They show how changes in real consumption (i.e. gains fromtrade) can be observed from changes in two observable macrovariables: bilateral trade X and income expenditures E.

2 They compute the effects of exogenous shocks to τ and χ onX and E.






Define country j ’s real (i.e. price-adjusted) consumption as

Cj ≡EjPj

.





Step 1: Show how changes in Cj can be inferred fromchanges in bilateral trade and income expenditures alone.

Step 1a: Show that, for small shocks, changes in Cj can beinferred from 2 statistics alone:

1 λjj ≡Xi,jj�

i�

k Xi,jk, changes in country j ’s share of

expenditures on domestic goods2 ε, the trade elasticity from the gravity equation

Xjk =χi,j (Ejτi,jk )

−ε�n

l=1 χl,j (Elτl,jk )−ε Ej .

Recall that trade elasticity here is change in imports relative to domesticdemand (

XjkXjj

) caused by changes in bilateral trade costs τi,jk .





Step 1a: Show that, for small shocks, λjj and ε are sufficientto infer changes in Cj .

Note that CES utility functions for each country j allow us todecompose changes in Pj into changes in domestic and importprices:

dlnPj = λjjdlnPjj + (1 − λjj)dlnPMj

where PMj ≡ (

�j �=k P1−σ

i,jk )1

1−σ is the component of the price indexassociated with imports.






Also note that by differentiating bilateral trade flow condition

Xjk = (Pi,jkPj

)1−σEj ,

we can show that

dln(1 − λjj)− dlnλjj = (1 − σ)(dlnPMj − dlnPjj).






Combining this result with the decomposition of Pj , and byapplying the arithmetic natural log identity

λjjdlnλjj = −(1 − λjj)dln(1 − λjj),

we get that

dlnPj = dlnPjj −dlnλjj1−σ .






Plugging this result into real consumption definition Cj ≡EjPj

and taking natural logs and derivatives, we can expresschanges in real consumption resulting from changes in λjj as:

dlnCj = (dlnEj − dlnPjj) +dlnλjj1−σ .

Moreover, we can have ε enter this expression by recalling thatε ≡ 1 − σ:

dlnCj = (dlnEj − dlnPjj)−dlnλjj

ε .






dlnCj = (dlnEj − dlnPjj)−dlnλjj

ε .

Finally, we can simplifying the expression by noting that theexogeneity of each country j ’s endowment Qi in theArmington model causes (dlnEj − dlnPjj) to equal 0 by thedefinitions above, implying:

dlnCj = −dlnλjjε .






dlnCj = −dlnλjjε

In sum, whatever the origins of the exogenous shock to X and Emay be, two statistics–trade elasticity ε and changes in the shareof expenditure on domestic goods λjj–are sufficient to infer welfarechanges.





Step 1: Show how changes in Cj can be inferred fromchanges in bilateral trade and income expenditures alone.

Step 1b: Show that, for large shocks, integration of theresult derived in Step 1a is sufficient to infer changes in Cj .





Step 1b: Show that, for large shocks, integration of theresult derived in Step 1a is sufficient to infer changes in Cj .

Since the previous expression

dlnCj = −dlnλjjε

holds for any small shock to τ and χ, the welfare (i.e.consumption) consequences of large shocks to τ can becalculated by integrating the result from Step 1a over allvalues of λjj :

Cj = λ−1/εjj ,

where the “hat”-valueˆof any variable v denotes the change in thevariable between the initial and counterfactual equilibria.





Step 2: Compute the effects of exogenous shocks to τ and χon X and E.

Dekle, Eaton and Kortum (2008):

Let λi ,jk ≡ Xi,jk�l Xlk

denote the share of country k ’s totalexpenditure on goods from country j .

Because the gravity equation holds in both the initial andcounterfactual equilibria, the difference in λi ,jk between theinitial and counterfactual equilibria can be estimated from theobserved differences in equilibrium values of E , τ , and χ:

λi ,jk =χi,j (Ej τi,jk )

−ε

�nl=1 λlj χl,j (El τl,jk )−ε

.







The trade balance condition�

k=1 Xjk =�

k=1 Xkj of theArmington model implies that each country’s expendituresequals the sum of every other country’s expenditures oncountry j ’s good i :

E �j =

�ni=1

�nk=1 λ

�i ,kjE

�k .

Note that this condition holds in both the initial and counterfactual

equilibria, but we will only exploit it in the counterfactual case.






Dekle, Eaton and Kortum (2008):Combining the above trade balance condition with theexpression for λi ,jk above, we get

EjEj =�n

i=1�n

k=1λi,kj χi,j (Ej τi,kj )

−εEkEk�nl=1 λi,lj χi,l (El τlj )−ε

.

This expression allows us to compute Ej , the effect of exogenous

shocks to τ and χ on country j ’s expenditure levels, as long as we

also know ε’s value.

However, we still need to compute the effect of exogenous shocks to

τ and χ on bilateral trade X . We will represent X by λi,jk , the share

of country k’s total expenditures spent on goods from country j .Dawn Powers Background for Nov. 13: “Measuring Gains from Trade”






Recall the formula for λi ,jk , the change in country k ’s totalexpenditure on goods from country j :

λi ,jk =χi,j (Ej τi,jk )

−ε

�nl=1 λlj χl,j (El τl,jk )−ε

.

Because changes Ej were calculated from τ and χ in theprevious slide, we can now use this formula to compute λi ,jk .

We now have Ej and λi ,jk , the effects of exogenousshocks to τ and χ on X and E.





Summary

In order to calculate the welfare gains from international trade,RCC:

1 Show how changes in real consumption (i.e. gains from trade)can be observed from changes in two observable macrovariables: bilateral trade X and income expenditures E.

2 Compute the effects of exogenous shocks to τ and χ on X andE.




The Armington ModelCounterfactual Analysis: Practical Example

In order to demonstrate the usefulness of the simple Armingtonmodel, RCC examine a counterfactual exercise in which they moveto autarky (a closed economy).





Assumptions

Assume that the iceberg trade costs in the counterfactualequilibrium are now infinite, i.e. τ �i ,jk = +∞ for any pair ofcountries j �= k , so that no international trade occurs.

All other structural parameters are the same as in the initialequilibrium.





Setup

Here, RCC define country j ’s gains from international trade Gjas the absolute value of the percentage change in real incomeassociated with moving to the counterfactual equilibrium.

Note that, because λ�jj , the share of expenditure on domestic

goods in autarky, equals 1, it will be the case that λjj =1λjj

.

Combining this fact with the general counterfactual analysismodel’s formula for welfare consequences of a foreign shock

Cj = λ−1/εjj ,

we get the following formula for gains of trade:Gj = 1 − λ1/ε

jj .





Setup

Gj = 1 − λ1/εjj

In order to compute Gj , we still need measures of tradeelasticity ε and λjj , country j ’s share of expenditure ondomestic goods λjj in the current equilibrium.





Setup

Computing trade elasticity ε using data, however, is problematiceconometrically (Hummels and Hillberry 2012; Head and Mayer2012).

RCC ID a common practice in the literature: Estimating ε byrunning a cross-sectional regression analogous to the log formof the Armington model’s gravity equation:

lnXi ,jk = δXj + δM

k − εlnτi ,jk + δi ,jk ,

whereδXj ≡ lnχi,j − εlnYj is treated as an exporter FEδMj ≡ lnYk − ln[

�nl=1 χi,l(Ylτlk)−ε] is treated as an importer FE

δi,jk is treated as ME in trade flows orthogonal to lnτi,jk .





Solution

lnXi ,jk = δXj + δM

k − εlnτi ,jk + δi ,jk .

Because RCC is a theoretical paper, rather than estimating thecoefficient ε with data, they set trade elasticity ε = 5, a valuefound commonly in the empirical literature (Anderson and VanWincoop 2004).





Setup

However, RCC are able to calculate λjj from year-2000OECD-STAN Input-Output tables by algebraically exploitingthe definition of λjj :

λjj ≡XjjEj

= 1 −�

j �=k Xjk�nj=1 Xjk





Results

When RCC calculate each country j ’s gains from trade

Gj = 1 − λ1/εjj

using trade elasticity ε = 5 recommended in the literature andcountry-level share of expenditure on domestic goods λjjcalculated from OECD data, they conclude:

There is a considerable amount of variation in themagnitude of the gains from trade predicted by thesimple Armington model: Gj ranges from 0.6 percent(Japan) to 4.6 percent (Slovakia).




The Armington ModelResources

Resources used for Armington model tools, setup, solution, andapplications:

Rodríguez-Clare and Costinot, 2012: “Trade Theory withNumbers: Quantifying the Consequences of Globalization.”Rodríguez-Clare, Graduate International Trade 280A, UCBerkeley, Lecture 1 notes.Feenstra and Taylor, International Trade.



Summary

Summary

Trade economists use multi-country gravity models forcounterfactual analysis.

The Armington model is the simplest gravity model used ininternational trade.

Counterfactual analysis in the context of international trademodels the gains from international trade defined as thepercentage change in real income associated with moving onecountry from the current, observed trade equilibrium to acounterfactual equilibrium with no trade.



Summary

10-minute Break

Stay tuned for a skills lecture: “How to Obtain and PresentResults”...


Econ 191: Measuring the Gains from Trade · 2012-11-13 · Introduction The Armington Model Next...

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