Lecture 9: Growth and Trade - Princeton Universityerossi/Trade/Lecture9_552.pdf · 2018-02-01 ·...

Lecture 9: Growth and TradeEconomics 552

Esteban Rossi-Hansberg

Princeton University

ERH (Princeton University ) Lecture 9: Growth and Trade 1 / 62

Growth and Interdependence, Ventura (1997)

Two of the most interesting facts of the post-war international growthexperience are :

I Conditional convergenceI The East Asian Miracle

Standard explanations for conditional convergence rely on :I Establishing diminishing returns to investmentI Assuming low level of economic integration

Problems with standard explanations :I If diminishing returns are important, why have the growth rates of the EastAsian economies not been declining quickly enough as these economiesaccumulate capital?

I Inability of investors to exploit arbitrage opportunities


This Paper

Model of trade and growth that combines conditional version of FPE withthe neoclassical growth model

Technology exhibits diminishing returns, yet countries’ability to trade andeliminate price differentials implies that these diminishing returns areglobal but not localHolding constant differences in labor productivity, poor countries grow fasterthan rich ones if and only if factor prices do not change too fast as the worldeconomy grows

East Asian economies were able to beat the curse of diminishing returnsthrough their ability to trade


The Model : Countries and Goods

World economy with J countries

One final good used for consumption and investment

2 factors of production: Capital and Labor

πj is the share of world’s population located in country jI These shares are constant and population grows at the rate n

Final good is non-traded and is the numeraire

Costless trade in intermediate goods ⇒ Countries share the sameintermediate good price p1(t) and p2(t), where t ∈ [0,∞)No international factor movements

wj (t) is the wage rate while rj (t) is the rental rate in country j


The Model : Representative agent

Representative agent in each country consumes cj (t) and owns a capitalstock of kj (t)

Capital does not depreciate

The consumer maximizes ∫ ∞

0ln cj (t)e

−(ρ−n)tdt

The budget constraint of the consumer is

cj +.kj + n · kj = rj · kj + wj


The Model : Production

Final goods sectorI Many competitive firmsI (xb1j + x

b2j )

1/(1−b), where xij (t) denotes the purchases of good i by therepresentative firm of country j at date t

I σ = (1− b)−1 is the elasticity of substitution, b < 1Intermediate goods sector

I Aj is a measure of labor productivityI One worker produces Aj units of good 1I One unit of capital produces one unit of good 2


Competitive EquilibriumThe competitive equilibrium of the world economy consists of a sequence ofprices and quantities such that consumers and firms optimize and marketsclear

The first order conditions are :

rj = ρ+

.cjcj

.kj = (rj − n) · kj + wj − cj

limt→∞

kjcje−(ρ−n)t = 0

The demands of intermediates as a function of prices and the demand of finalgoods are given by

xij = p1/(1−b)i · (rj · kj + wj ) where

1 = pb/(1−b)1 + pb/(1−b)

2


Factor prices satisfy the following relationships :

wj = Aj · p1rj = p2

Rental rates are equalized across countries despite the absence ofinternational factor movements

Since x1j/x2j = A/k so market clearing in world commodity markets leads to

p1/p2 = (k/A)1−b

The dynamics of c and k (world averages) are

.cj/cj = (Ab + kb)(1−b)/b · kb−1 − ρ

.kj = (Ab + kb)1/b − n · k − c

Long run growth if aggregate technology is capable of sustaining the MPkabove ρ

I σ > 1⇒ MPk is bounded below at 1⇒endogenous growth modelI σ < 1⇒ MPk eventually equals ρ⇒exogenous growth model


Autarky

Diminishing returns ensure that additional investment is less productive ⇒poor countries grow faster than rich ones

Assuming that the world is a collection of autarkic economies allows us toextend the properties of time-series graph to cross-section graphs

However one needs to be aware that the convergence test is a joint test ofdiminishing returns and the view that international linkages do not matter forthe growth process


Free trade

In models of trading economies, the law of diminishing returns applies only toworld averages

Assume that countries are so interdependent that differences in rates ofreturn are arbitraged away

Define kRj = kj/k and ARj = Aj/A

The law of motion of kRj is given by

.

kRj = φ · (kRj − ARj ),

where φ is defined as follows

φ =Ak· [(ρ− n) ·

∫ ∞

tp1 · e−

∫ τt (p2−n)dv .dτ − p1 ]

The above equation provides a full characterization of the cross section ofgrowth rates


Free trade

Ceteris paribus, if the growth of wages is low, φ is negative, and countriesthat have a low stock of capital relative to their labour productivityparameter, accumulate capital at a higher rate

The world is populated by countries that act as permanent income consumersI Countries exhibit identical rates of wealth accumulation, which in turn, is aweighted average of the growth rates of the stock of capital and the netpresent value of wages

I The growth rate of the latter is the same for all countriesI If this growth rate is low, countries must be accumulating capital at a ratethat exceeds that of total wealth, depending on how large the share of capitalis in consumer’s wealth

I The lower this share the higher is the growth rate

Symmetric argument for the case in which the net present value of wagesgrows at a high rate


Dynamics

Define z = c/k and note that φ =.z/z

The distribution of capital stocks approaches (moves away) from thedistribution of labour productivities if and only if the averageconsumption-capital ratio decreases (increases), φ < 0

.z/z = z − ρ+ n− (Ab + kb)(1−b)/b · Ab · k−1

.k = (Ab + kb)1/b − (n+ z) · k

In the steady state z is constant and the distribution of capital stocks doesnot change

During the transition, the behavior of z depends crucially on the assumedelasticity of substitution between capital and labor


Case 1 :σ > 1


but still high enoughσ < 1


and low enoughσ < 1


Cross Sections and Time Series

If the aggregate technology exhibits a high σ

I We would observe a downward-sloping cross section of growth ratesI We would also observe a (downward-sloping, but) almost flat time-series graphfor the growth rate

If the aggregate technology exhibits a low σ

I We would observe an upward-sloping cross section of growth ratesI We would also observe a (downward-sloping, but) steep time-series graph forthe growth rate

Since the existence of diminishing returns does not rule out any configurationfor a cross section of growth rates, one cannot use the conditionalconvergence finding as evidence of diminishing returns

However the conditional convergence finding can be used as evidence that σis high and that the world economy might be closer to a endogenous growthmodel


Making MiraclesGrowth in Miracle Economies

I Rental rate in a miracle economy is p2 · (1+ θj ), where θj is the discrepancybetween domestic and foreign rates, reflecting different taxation systems,investment subsidies, etc

I A small economy has the following dynamics for relative capital stock :.

kRj = (φ+ p2 · θj + ρ− ρj ) · kRj − φj · ARjwhere φj is defined as follows

φj =Ak· [(ρj − n) ·

∫ ∞

tp1 · e−

∫ τ

t (p2(1+θj )−nj )dv · dτ − p1 ]

I Miracle economies are those in which p2 · θj + ρ− ρj > 0I For the small economy to remain small asymptotically, population growthmust not be too large

I Even when σ ≤ 1, a small economy that combines a high savings rate with alow rate of population growth can beat diminishing returns by adopting anopen trade regime

I This economy behaves as if it had a linear technology

GDP(autarky ) = (kbj + Abj )1/b GDP(free − trade) = p1 · Aj + p2 · kj


Making Miracles

Structural Transformation versus Capital DeepeningI In autarkic models of economic growth, growth is basically equivalent tocapital deepening

I In this model, as the capital-labor ratio increases, instead of using morecapital-intensive techniques in each sector, the miracle economies absorb theextra capital by expanding capital-intensive sectors and contractinglabor-intensive ones - economic growth leads to structural transformation

I The share of manufacturing exports in GDP is

(1− µ) ·µ · (kRj − ARj )

µ · (kRj − ARj ) + ARj


The World Income Distribution, Acemoglu & Ventura(2002)

The picture of the world income distribution raises two questionsI Why are there such large income differences across countries?I Why has the world income distribution been relatively stable since 1960?

Existing explanations are built on two assumptions :I Diminishing returns to capitalI Technological spillovers


This paper

This paper offers an alternative framework for analyzing the world incomedistribution, based on international trade and specialization

Countries that accumulate capital faster than average experience decliningexport prices, reducing the value of the marginal product of capital

These terms-of-trade effects introduce de facto diminishing returns at thecountry level and ensure the stability of the world income distribution

Cross-country differences in economic policies, saving rates, and technologylead to differences in relative incomes, not in long-run growth rates

This result depends essentially on some degree of specialization


This paper

In the steady state, countries with lower rates of time preference and lowerprice of investment goods will have lower rental rates, hence higher relativecapital and income

Countries with better technologies will be richer because they have higherrental rates for a given level of relative capital and income

The speed of conditional convergence depends on the strength of theterms-of-trade effects, not on the capital share in output

Holding technology and other determinants of steady-state income constant,a 1 % faster growth is associated with a 0.6 % deterioration in the terms oftrade.

This explains a significant fraction of cross-country income differences


The Model

Continuum of countries with mass 1

Capital is the only factor of production

Continuum of intermediate goods z ∈ [0,M ]2 final goods - consumption good and investment good

Free trade in intermediate goods, no trade in final goods

Each country is defined by a triplet (µ, ρ, φ)I µ - technologyI ρ - rate of time preferenceI φ - effect of policies and institutions on the incentives to invest

The joint distribution of these characteristics is given by G (µ, ρ, φ)


The Model

Each country has a representative consumer with utility function :

u =∫ ∞

0ln c(t) · e−ρ·t · dt

The budget constraint is given by

pI ·.k + pc · c = y ≡ r · k

Products are differentiated by origin

µ is the measure of intermediates produced by the (µ, ρ, φ) country with∫µ · dG = M

Intermediates are produced by competitive firms. One unit of capital is usedto produce one intermediate good


The Model continued

Each country also contains many competitive firms in the consumption andinvestment goods sectors with unit cost functions :

Bc (r , p(z)) = r1−τ · (∫ M0p(z)1−ε · dz)τ/(1−ε)

BI (r , p(z)) = φ−1 · r1−τ · (∫ M0p(z)1−ε · dz)τ/(1−ε)

where p(z) is the price of intermediate with index z

τ is the share of intermediates in production and will turn out to be the ratioof exports to income. It is usually interpreted as a measure of openness

ε > 1 is the elasticity of substitution and as well as the price elasticity offoreign demand for the country’s products. The inverse of this elasticity isoften interpreted as a measure of the degree of specialization


World EquilibriumThe competitive equilibrium of the world economy consists of a sequence ofprices and quantities such that consumers and firms optimize and marketsclearConsumer optimization leads to the following first-order conditions:

r +.pI

pI−

.pcpc

= ρ+

.cc

limt→∞

pI · kpc · c

· e−ρ·t = 0

The optimal rule is given by

pc · c = ρ · pI · k

Firm maximization in the intermediate sector leads to

p = r

Price index for intermediates is the numeraire∫ M0p(z)1−ε · dz =

∫µ · p(z)1−ε · dG = 1


World Equilibrium

p is also the terms of trade

Firms in the consumption and investment sectors take prices as given andchoose factor inputs to maximize profits

pc = r1−τ

pI = φ−1 · r1−τ

Market clearing for capital, by Walras’Law, is equivalent to imposing tradebalance

y = µ · p1−ε · Ywhere Y =

∫y · dG is world income


World Dynamics

The state of the world economy is fully described by a distribution of capitalstocks. The law of motion for the world economy is given by

.k/k = φ · rτ − ρ

r · k = µ · r1−ε ·∫r · k · dG

Define x ≡.Y /Y and yR ≡ y/Y

Setting the same growth rate for all countries,.

we obtain the steady-state cross section of rental rates as

r∗ = (ρ+ x∗

φ)1/τ

The world distribution is stable because countries that accumulate more facelower terms of trade, reducing the interest rate and the incentives for furtheraccumulation


World distribution of income

y∗R = µ · ( φ

ρ+ x∗)(ε−1)/τ

∫µ · ( φ

ρ+ x∗)(ε−1)/τ · dG = 1

terms of trade = p = (µ

yR)1/(ε−1)

rate of return =r +

.pI

pI−

.pcpc= φ · pτ

For countries with greater patience and better economic policies, lower termsof trade are suffi cient to ensure accumulationLower terms of trade and better technology is translated into a greaterrelative income level yRCountries with low values for ρ and high values for φ and µ will be richerThe strength of diminishing returns depends on the volume of trade τ andthe extent of specialization ε


Empirical Implications

Relation between steady-state income and savings rate ( x ∗ρ+x ∗ ) is given

y∗R = µ · φ(ε−1)/τ · ( sx∗)(ε−1)/τ

τ = 0.3 (share of exports) and ε = 2.6 (changes in terms of trade)

The model’s predictions for cross-country income differences are identical tothose of the Solow model with α = 0.85

Yields a large elasticity of output to savings and generates cross-countryincome differences even larger than those observed in the data

Furthermore, the speed of convergence is 1.1 percent a year, which is slowerthan observed in the data


Empirical Evidence

Equation to be tested

πt = (gt − xt )/(ε− 1) + ∆ ln µt


Empirical Evidence

Since changes in technology, as captured by ∆ ln µt are directly correlatedwith changes in income, the above relationship will be biased

A possible source of variation in growth would come from countries growingat different rates because they have started in different positions relative totheir steady state income level and are therefore accumulating at differentrates to approach their steady state. This should be orthogonal totechnological change

The estimating equation is

πt = δ · gt + Z ′t ·ω+ υt

where gt is instrumented for using

gt = −β · ln yt−1 + Z ′t · θ + ut


Extensions

Add labour as a factor of productionI Production of consumption good requires laborI The determinants of the steady state distribution of income and world growthrate are still the same

I Wages tend to be higher in rich countries (those with high µ and high φ).Follows from a greater demand for consumption by richer countries, increasingthe demand for labor and wages

I The cost of living tend to be higher in rich countries since consumption goodsare more expensive there

Endogenous determination of the number of varietiesI Firms in all countries know how to produce an infinite mass of intermediates

µ∗ = τ · ρ+ x∗ · [1− (1− γ) · (1− τ)]

ρ · [τ + ε · (1− γ) · (1− τ)] + τ · x∗

I Countries with low ρ endogenously specialize in the production of more goods


Product Development and International Trade, Grossmanand Helpman (1989)

This paper develops a multicountry, dynamic, general equilibrium model ofproduct innovation and international trade to study the creation ofcomparative advantage through research and development (R&D) and theevolution of world trade over timeWorld economy with 3 activities :

I A "traditional" commodity produced under competitive conditionsI A continuum of varieties of "modern" industrial product (horizontallydifferentiated

I R&D that leads to the acquisition of the know-how needed to produce newbrands of the industrial good

Forward-looking potential producers conduct R&D and enter the productmarket whenever profit opportunities existNew products substitute imperfectly for the old, and prices, interest rates andthe pattern of trade evolve over time as more commodities become availableTrade has both intra-industry and inter-industry componentsInternational capital flows take place to finance R&D and in somecircumstances, multinational corporations may emerge


Main Results

If both R&D and the production of differentiated goods are more humancapital-intensive activities than the production of a traditional good and if allactivities bear Leontief production technologies, then the human capital-richcountry will be a net exporter of differentiated products and an importer ofthe traditional good at every moment in time

The model predicts a rising share of trade in world GNP, at least when R&Dis the most human capital intensive of the three activities

The human capital-rich country has both a greater incentive to invest and agreater incentive to save (per capita)

For certain compositions of factor endowments, the extent ofmultinationality, as measured by output, employment of subsidiaries, or thenumber of multinational firms, expands over time, at least initially and as theworld economy approaches the steady state


A Dynamic Model of R&DConsumers

U =∫ ∞

0e−ρt log u(.)dt

u = [∫ n0cx (i)αdi ]

sx/αc1−sxy , α, sx ∈ (0, 1)

The consumer’s maximization problem can be solved in two stages

First stage : Consumer chooses {cx (i)} and cy to maximize u(.)

cx (i) = sxEp(i)−σ∫ n(t)

0 p(j)−σdj

cy = (1− sx )E/py

The budget constraint is given by∫ ∞

te−[R (τ)−R (t)]E (τ)dτ =

∫ ∞

te−[R (τ)−R (t)]I (τ)dτ + A(t)

Second stage : The F.O.C. for maximizing U s.t. the B.C. is.EE =

.R − ρ


A Dynamic Model of R&DProducers

φx (wf )→ unit cost of productionφn(wf )→ cost of developing a brandThe number of potential products is infiniteEntrepreneur takes as given the aggregate level E (t) and the pricing policy ofcompetitors (of measure n(t))and sets the price of his brand so as tomaximize profitsIn a symmetric equilibrium, output per variety x(i) = xand prices p(i) = p∀i ∈ [0, n(t)] satisfy

x = sxEpn

andαp = φx (wf )

The resulting operating profits per variety are

π = (1− α)sxEn


A Dynamic Model of R&DProducers continued

An entrepreneur has perfect foresight regarding the evolution of spending Eand the number of firms nIn equilibrium, we must have∫ ∞

te−[R (τ)−R (t)]π(τ)dτ = φn [wf (t)]

Prices are normalized so that

1 = φn [wf (t)]

This choice of numeraire implies that

π(t) =.R(t)

The traditional good is produced subject to CRS. Therefore,

py = φy (wf )


A Dynamic Model of R&DIntegrated Equilibrium

Differential equation for the rate of change of spending.EE= (1− α)sx

En− ρ

Differential equation for the change in the number of brands.n = v(E )


A Dynamic Model of R&DSpecial Case

Two factors of production - skilled and unskilled labor

The traditional sector is the least human capital intensive and the overallhuman capital to labor ratio satisfies aHy/aLy < H/L < aHj/aLj for j = x , nProposition

I (a).w/w >

.py/py >

.p/p >

.r/r

I (b).py > 0

I (c).r < 0

I (d).p > 0 if and only if aHn/aLn > aHx/aLx

I (e) ∂.n/∂t < 0

I (f).X > 0

I (g).Y < 0 if and only if aHn/aLn > aHx/aLx

I (h).E > 0


Pattern of trade in a two-country world

Two countries a (human capital rich) and b with common tastes andtechnologiesFactor services and blueprints aren’t tradeable

A suffi cient condition for factor price equalization to obtain along the path ofthe trade equilibrium is that the human capital to labor ratios of the twocountries be bounded by the less human capital-intensive among R&D andproduction of differentiated productsERH (Princeton University ) Lecture 9: Growth and Trade 41 / 62

At an arbitrary point in time, full employment conditions are given by

Li = aLyLi + aLxsxEnpni + aLn

.ni

Hi = aHyLi + aHxsxEnpni + aHn

.ni

for i=a,b

Combining the above two equations :

.ni + b(t)ni = ki

Solving the above equation :

ni (t) = ki

∫ t0e−

∫ tz b(τ)dτdz

.na(t)/

.nb(t) is a constant and equal to ka/kb , as is the ratio of the total

outputs of modern goods Xa/Xb


Ratio of outputs of traditional goods

Ya(t)Yb(t)

=LaLb

[hc (t)− (Ha/La)][hc (t)− (Hb/Lb)]

where hc (t) is the human capital to labor ratio in the composite activity

Can be shown that Ya(t)Yb (t)< La

LbI hc (t) is bounded by hn ≡aHn/aLn and hx ≡aHx/aLxI Under conditions for FPE, each of these exceed Hi/Li

Ratio of demands for the traditional goods

Ea(t)Eb(t)

=Ea(0)Eb(0)

=LaLb

∫ ∞0 e

−R (t)[w(t) + r(t)ha ]dt∫ ∞0 e

−R (t)[w(t) + r(t)hb ]dt

Can be shown that cya(t) > Ya(t)

I ha/hb > 1 implies thatEa(0)Eb (0)

> LaLb

I cya(t)cyb (t)

> LaLb

I Follows immediately that cya(t)cyb (t)> Ya(t)

Yb (t)I Use market clearing condition cya(t) + cyb(t) = Ya(t) + Yb(t)


Each differentiated product is manufactured in only one country butconsumed worldwide⇒ The direction of trade in individual products isclear-cut

Aggregate pattern of trade → cxacxb< Xa

Xb

I Have already shown that XaXb is constant over timeI So is cxacxbI Can’t have cxa

cxb> Xa

Xb⇒ Country A

The volume of trade as a fraction of world spending

VTE= (1− sx )(

YbY− sb) + sx (sa

XbX+ sb

XaX)

If product development is human capital intensive relative to the productionof differentiated products, the volume of world trade grows faster than worldspending and GDP


Multinationals

Headquarter services are produced with human capital only and theproduction plants use these services and unskilled labor only

Headquarter services must be provided in the country in which the brand wasdeveloped


Learning by Doing and the Dynamic Effects ofInternational Trade, Young (1991)

Develops a model in which endogenous growth is generated bylearning-by-doing

Learning by doing incorporates two important characteristics :I Substantial spillovers in the development of knowledge across industriesI Existence of strong diminishing returns in the learning-by-doing process

Learning by doing is bounded in each good

The knowledge generated by learning-by-doing is in the public domain

The unit output labor requirement of each good evolves over time

At any given time, learning-by-doing will have been exhausted in a subset ofgoods, but will continue in the remainder


Learning by Doing and the Dynamic Effects ofInternational Trade, Young (1991)

Over time, growth will involve the production of a changing basket of goods,with both the variety and quantity of goods consumed increasing

Although production takes place under conditions of perfect competition,there are gains from increasing variety

In the absence of international diffusion of knowledge, the effect of trade ontechnical progress and growth will depend on which goods the economyspecializes in based on its static comparative advantage

Trade improves the intertemporal welfare of consumers in economies in whichit accelerates technical progress and growth

However even in a country in which technical progress slows down, consumersmight experience an improvement in welfare


The Model

Large number of consumers

Perfectly competitive firms

Goods indexed by s along [B,∞) ordered according to the degree ofsophistication

Labor is the sole factor of production

Unit labor requirements are a(s, t) where a(s, t) is continuous andlimt→∞ a(s, t) = ∞Lower bound on potential input requirements, a(s) which is non-increasing ins

For each t∃ s∗t s.t. a(s, t) ≥ a(s∗t , t)


Learning-by-doing

Bounded learning-by-doing :I If a(s , t) = a(s), then

∂a(s , t)/∂t = 0

I If a(s , t) > a(s), then

∂a(s , t)/∂ta(s , t)

= −∫ ∞

BB(s , v , a(v , t)/a(v ))L(v , t)dv

where L(v , t) is the amount of labor devoted to the production of good v attime t and B(s, v , a(v , t)/a(v)) are the learning-by-doing coeffi cientsB(s, v , a(v , t)/a(v)) ≥ 0 for all vB(s, v , 1) = 0

B(s, v , a(v , t)/a(v)) > 0 for all v ∈ (s − αs , s + αs ) such thata(v , t) > a(v)

sups supv B(s, v , a(v , t)/a(v)) < ∞B(s, v , a(v , t)/a(v)) is continuous in s


Consumer optimizationInstantaneous utility function :

V (t) =∫ ∞

BU(C (s, t))ds with U ′(0) < ∞,U(0) = 0

U(·) is strictly concave and continuously differentiable ⇒ A strong, but notunbounded, preference for variety

No storage technology

Consumers inelastically supply one unit of labor for wage W

Budget constraint is given by

1 ≥∫ ∞

Ba(s, t)C (s, t)ds

where we use the relation P(s, t) = Wa(s, t)

The first order condition is given by

U ′(C (s, t))a(s, t)

=U ′(C (v , t))a(v , t)

∀ s, v such that C (s, t) > 0, C (v , t) > 0


Since all goods enter symmetrically into the utility function, the consumershould consume the cheapest of all goods, i.e. good s∗tFor finite quantities of good s∗t consumed, there exists some good M suchthat as the price of goods approached a(M, t) from below, consumption goesto zero

For all goods x such that a(x , t) ≥ a(M, t), consumption is zeroFor all goods s whose price is less than a(M, t), consumption quantities aredetermined by

U ′(C (s, t)) = a(s, t)U ′(0)/a(M, t)


Growth RateGrowth rate is measured by

g(t) =

∫ ∞B a(s, t)∂X (s, t)/∂tds∫ ∞B a(s, t)X (s, t)ds

− dL(t)/dtL(t)

where L(t) equals the population at time t and X (s, t) equals aggregateoutput of good s at time t

Using∫ ∞B a(s, t)X (s, t)ds = L(t), it follows that

g(t) =

∫ ∞B −∂a(s, t)/∂tX (s, t)ds

L(t)

Allowing SL to denote the set of goods s such that C (s, t) > 0 anda(s, t) > a(s) and using X (s, t) = C (s, t)L(t) under autarky

g(t) =∫s∈SL

a(s, t)C (s, t)(∫ ∞

BB(s, v , a(v , t)/a(v))L(v , t)dv)ds

It can be shown that if lims→∞ a(s) = 0, then lims→∞V (t) = ∞


Dynamic Effects of International TradeAutarky : Preferences and Technology

Intertemporal utility function :

P =∫ ∞

tV (x)e−ρ(x−t)dx

where V (x) denotes instantaneous utility at time x , which is given by

V (x) =∫ ∞

Blog (C (s, x) + 1)ds

where C (s, x) denotes consumption of good s at time x and B is a largenegative numberThe lower bound on potential unit labor requirements is given by a(s) = ae−s

B(s, v , a(v , t)/a(v)) = 2 for all s and v such that a(v , t)/a(v) > 1Unit labor requirements are given by

a(s, t) =ae−s ∀ s≤ T (t); a(s, t) =ae−T (t)es−T (t) ∀ s≥ T (t)with T (t) evolving according to the learning-by-doing equation

dT (t)dt

=∫ ∞

T (t)L(s, t)ds


Dynamic Effects of International TradeAutarky : General Equilibrium

For all goods s which are consumed in positive quantities :

a(s)C (s) = a(M)− a(s)Since a(s) is symmetric around T , to each M there corresponds a N suchthat a(M) = a(N)Denoting T −M = N − T = τ, it can be shown that

1 > dτ/dT > 0


Dynamic Effects of International TradeAutarky Growth Rate and Rate of Technical Progress

Rate of technical progress :

dT (t)dt

=∫ ∞

T (t)L(s, t)ds =

∫ N (t)T (t)

a(s, t)X (s, t)ds =L(t)2

Growth rate of GDP per capita :

g(t) =

∫ ∞B −∂a(s, t)/∂tX (s, t)ds

L(t)

=L(t)2


Dynamic Effects of International TradeFree trade : Preferences and Technology

Two economies - LDC with population L and DC with population L∗

Identical preferences

T ∗ > T

No spillovers of knowledge between the two economies and there’s nointernational borrowing or lending

Let X = T ∗ − TLet W be the numeraire and

ω = W ∗/W , p(s) = P(s)/W

Under perfect competition, if the good is produced in the LDC, thenp(s) = a(s), whereas if it is produced in the DC, thenp(s) = ωa∗(s)

Each good s is produced by the least cost producer


Dynamic Effects of International TradeFree trade : General Equilibrium

At each time t, consumers in both the LDC and the DC only consume goodspriced at less than p(M) and p(M∗) respectively

p(s)C (s) = p(M)− p(s) p(s)C ∗(s) = p(M∗)− p(s)

Denote the set of goods produced in the DC and the LDC by GDC and GLDC

M is determined by the B.C. of individuals in the LDC :

1 =∫s∈GDC

p(s)C (s)ds +∫s∈GLDC

p(s)C (s)ds

M∗ is determined by the B.C. of individuals in the DC :

1 =∫s∈GDC

p(s)C ∗(s)ds +∫s∈GLDC

p(s)C ∗(s)ds

The relative wage is determined by the trade balance condition :∫s∈GDC

Lp(s)C (s)ds =∫s∈GLDC

L∗p(s)C ∗(s)ds


ω = 1

ω = e2X


1<ω < e2X


Technical Progress and Equilibrium Dynamics

Equilibrium A :

dTdt(A) = 0

dT ∗

dt(A) =

L∗ + L2

Equilibrium B :

dTdt(B) =

L−ωL∗

2dT ∗

dt(B) = L∗

Equilibrium C,D :

L2>dTdt(C ,D) > 0

dT ∗

dt(C ,D) >

L∗

2

Equilibrium E :dTdt(E ) =

L2

dT ∗

dt(E ) =

L∗

2

To summarize, we have L2 ≥dTdt ≥ 0 and

dT ∗dt ≥

L∗2


Growth of GDP per capita and Intertemporal Welfare

The growth rates are given by

g(t) =2(dT/dt)2

L≤ L2

g∗(t) =2(dT ∗/dt)2

L∗≥ L∗

2

Effect of free trade on welfareI Unambiguously improve the intertemporal utility (i.u.) of DC consumers if theDC is never overtaken by the LDC

I Improve the i.u. of DC consumers even if the DC is overtaken by the LDC,provided that the time spent along the catch-up path is suffi ciently large

I Unambiguously improve the i.u. of LDC consumers if the LDC’s population issuffi ciently small relative to that of the DC

I Unambiguously reduce the i.u. of LDC consumers if the LDC’s population issuffi ciently large relative to that of the DC and it is unable to overtake the DC


Lecture 9: Growth and Trade - Princeton Universityerossi/Trade/Lecture9_552.pdf · 2018-02-01 ·...

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Transcript of Lecture 9: Growth and Trade - Princeton Universityerossi/Trade/Lecture9_552.pdf · 2018-02-01 ·...