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1

A

C

B

Dirty good

zA

zC

zB

zS

z=φ1(θ)x

z=φ2(θ)x

Cleangood

pT

pW

Source: Adapted fro mAntweiler e t al. 2001

Composition

Scale Technique

Pollution

UT

UW

YT YW

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2

Tariff

C- utility fromenvironmentalquality

R – value of outputWelfare

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3

Spot test!

Q. In the previous example, what is the optimal tariff, and how is it calculated?

A. Where absolute values of slopes of R and C are equal (marginal environmental benefit=marginal cost in terms of consumption)

B. Q. What is the optimal tariff on imports of a dirty good?C. A. t = 0.

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4

II. General equilibrium approaches—theory

A. Analytical tools: producers, consumers, markets and trade

B. Geometric models of trade and environment

- What are we measuring? Environmental and welfare outcomes

C. Comparative static results and standard theorems

D. Simple models of trade and environmental policy

- Environmental and welfare costs of trade policies.

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5

Trade expenditure function

For the model just described, the aggregate budget constraint,

e(p, u) = r(p, v)

contains all the information necessary to describe an equilibrium.

We can also use it for comparative static analysis of the effectsof price, endowment and technology shocks.

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6

Comp. statics with TEF

Method: take total differential of TEF.

e(p, u) = r(p, v)

eudu + epdp = rpdp + rvdv

Rearrange, noting that ep – rp = net imports:

eudu = –(ep – rp)dp + rvdv

-- LHS is a money-metric of welfare

-- RHS captures effects of price and endowment changes

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7

Details

• Welfare measure: eu = ∂e/∂u is the reciprocal of ∂u/∂y, the marginal utility

of income. So eudu = dy, a money-metric of welfare change.

• Welfare effects of terms-of-trade shocks:• Sign depends on whether goods are net imports or

exports.

• Welfare effects of endowment growth:• Recall that ∂r/∂v = w, the shadow factor price.

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8

Extensions

• Policies, e.g. trade policy

• Externalities

• Non-traded goods

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9

Trade policy distortion (tariff)

• Suppose 2 goods, exports (x) and imports (m).

• Let px = 1 and q = pm + t (= tariff)

• Adding tariff revenues to income:• e(1, q, u) = r(1,q) + t(em – rm)

• Then by differentiation (using dq = dt), dy = t(emm - rmm)dt < 0

• where = (1- tem) > 0.

• A tariff increase reduces welfare.

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10

Spot test!

In this model, a tariff clearly reduces welfare. What effect does it have on the sectoral structure of production?

How do we know?

The tariff raises output in the protected sector, and reduces it in the other sector.

Check 2nd derivatives of revenue function, using homogeneity property.

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11

Externalities

• E.g. env. externality in production• TEF is now:

• e(p, u) = r(p, v) - z'y• where z is qty of pollution per unit of y

produced.

• Env. externality in consumption: • u = u(c, z) ==> e(p, z, u)

• NB assumption of separability.

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Non-traded goods

• Goods may be non-traded (or effectively so) for intrinsic and policy reasons.

• If one good is non-traded, for this, mn = 0. Equilibrium now requires additional equation:

e(p, u) = r(p, v)

en(p, u) = rn(p, v)

and solves for pn as well as agg. welfare. • With endogenous prices, preferences play a role

in economic structure.

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13

Salter-Swann diagram

(yT, yN) = (cT, cN)

RER = pN/pT

N

T

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14

Effects of growth with non-traded good

N

T

Income exp. path

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Two fundamental GE results

• Distributional effects of a price change: the Stolper-Samuelson theorem

• Production effects of a factor endowment change: the Rybczinski theorem

• Assume:– Two factors of production, two products, so

yj = yj(x1, x2), for j = 1,2

– Complete and competitive markets, CRTS.

– Prices are ‘given’ in world markets.

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A useful tool: ‘hat’ calculusRules. For all variables x, a and b,

Levels Proportional changesx = ab

ˆ x = ˆ a +ˆ b

x = a + bˆ x =

a

x

ˆ a +

b

x

ˆ b

x = a – b ˆ x = ˆ a −ˆ b

x = kab ˆ x = ˆ a +ˆ b

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Effects of a price changeBy CRTS, pjyj

= w1x1j + w2x2j for all j = 1,2

Divide both sides by yj to obtainpj

= w1a1j + w2a2j,

where aij = xij/yj is unit input requirement.

Totally differentiate this expression to obtaindpj

= a1j dw1 + a2jdw2 for all j = 1,2

Convert to proportional changes:

ˆ p 1

= θ11

ˆ w1

+ θ21

ˆ w2

ˆ p2

= θ12

ˆ w1

+ θ22

ˆ w2

where θij

= wi

xij

/ pj

yj

, θ1 j

+ θ2 j

= 1 .

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18

Choose p2 as numeraire price, so ˆ p 2

= 0 . Now solve as a syste m ofequations to obtain:

ˆ w1

=

θ22

θ

ˆ p1

and ˆ w2

=

− θ21

θ

ˆ p1

where θ = θ11θ22 - θ12θ21 = θ22 – θ21.

If commodit 1y i s intensi vein factor 1, then:

ˆ w1

ˆ p1

= θ22

θ > 0 and ˆ w2

ˆ p1

< 0

Moreove,r since θ1j + θ2j = 1, we have ˆ w1

ˆ p1

> 1 .

Thus: ˆ w1

> ˆ p1

> ˆ p2

= 0 > ˆ w2

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Stolper-Samuelson theorem

A rise in one commodity price raises the real return to the factor used intensively in producing that commodity, and reduces the real return to the other factor.

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20

Applications of S-S

• Effects of trade shocks or trade policy reforms on the returns to factors– For environmental analysis: changes in factor

returns indicate incentives for exploitation or investment

• Ex.1: If forests are open-access, a ‘shock’ that raises returns to timber may increase harvests

• Ex. 2: raising returns to agriculture may promote soil-conserving investments

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21

Effects of endowment growthEach factor endowment vi is fully employed in the two sectors, so:

vi

= y1

ai 1

+ y2

ai 2

, where aij

= xij

/ yj

Fi ndthe effect of a change i nvi on yj.

- Define proportional c hangesi n variable :s ˆ vi

= dvi/vi, etc.- T henfor each inpu t vi:

ˆ vi

= λi 1

ˆ y1

+ λi 2

ˆ y2

where λij are employment shares.

Equilibrium sectora l output cha nges ar e given bysoluti onto:

λ11

λ12

λ21

λ22

⎡

⎣

⎢

⎤

⎦

⎥

ˆ y1

ˆ y2

⎡

⎣

⎢

⎤

⎦

⎥=

ˆ v1

ˆ v2

⎡

⎣

⎢

⎤

⎦

⎥

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22

For simplicity let ˆ v 2

= 0 . Then:

ˆ y1

=

λ22

λ

ˆ v1

and ˆ y2

=

− λ21

λ

ˆ v1

where λ = λ11λ22 − λ21λ12.

If λ > 0 (sector 1 is intensive in use of v1) then ˆ y1

ˆ v1

> 1 , and

ˆ y1

> ˆ v1

> ˆ v2

> ˆ y2

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Rybczinski Theorem

At constant prices, expansion of one factor endowment raises output of the good that uses that factor intensively, and reduces that of the other good. If 2 factors expand, one good's output grows more slowly than the rate of growth of either factor.

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Implications of Ryb. result

• Unequal rates of factor accumulation alter structure of production– Capital deepening causes labor-intensive or

NR-intensive sectors to decline, other things equal

• Thus investment in non-agricultural sectors may diminish pressures exerted on the natural resource base by primary industries.

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25

Discussion of fundamental results

• The S-S and R theorems provide ‘core’ insights for any GE analysis.

• They hold for 2X2 models; similar, but weaker predictions apply for higher-dimension models.

• Both theorems break down in presence of externalities.

• In practice, however, can use these predictions to check the credibility of predictions obtained from larger models.