II-C
Clean
good
UW
UT
A
pT
C
B
pW
YT
YW
Dirt y good
zA
zC
z=2()x
Composition
zB
Scale
zS
Techn ique
z=1()x
Poll ution
Sour ce: Adapted from Antweil er et al. 2001
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II-C
Welf are
R – value of output
C- utilit y from
env ir onmental
qua lit y
Tariff
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II-C
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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II-C
II. General equilibrium
approaches—theory
A. Ana lytical tools : produce rs, consumers, markets and trade
B. Geo metric models of trade and env ironment
- What are we measuring? Environmental and welfare outcomes
C. Compara tive static results and standard theorems
D. Sim ple models of trade and env ir onmental poli cy
- Environmental and welfare costs of trade poli cies.
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II-C
Trade expendi ture function
For the mod el just described, the agg rega te budge t cons traint,
e(p, u) = r(p, v)
contains all the information nece ssary to describe an equ ili brium.
We can a lso use it for comparative static ana lysis of the effects
of price, endow ment and techno logy shocks .
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II-C
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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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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Extensions
• Policies, e.g. trade policy
• Externalities
• Non-traded goods
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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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II-C
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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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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Salter-Swann diagram
T
RER = pN/pT
(yT, yN) = (cT, cN)
N
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Effects of growth with nontraded good
T
Income exp. path
N
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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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II-C
A useful tool: ‘hat’ calculus
Rules. For all variables x, a and b,
Leve ls
x = ab
Proportiona l change s
xˆ  aˆ  bˆ
x=a +b
xˆ 
x=a –b
xˆ  aˆ  bˆ
x = kab
xˆ  aˆ  bˆ
a
b
ˆa  bˆ
x
x
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II-C
Effects of a price change
By CRTS, pjyj = w1x1j + w2x2j fo r all j = 1,2
Divide both sides by yj to obtain
pj = w1a1j + w2a2j,
whe re aij = xij/yj is unit i nput requireme nt.
Totally differentiate this expre ssion to obtain
dpj = a1j dw1 + a2jdw2 for all j = 1,2
Convert to proportiona l change s:
pˆ 1  11wˆ 1  21wˆ 2
pˆ 2  12 wˆ1  22 wˆ 2
whe re  ij  wi xij / pj y j , 1 j   2 j  1.
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II-C
Choose p2 as numeraire price, so pˆ 2  0 . Now solve as a system o f
equa tions to obtain:

21
ˆw1  22 ˆp1
ˆ
and
w2 
pˆ1


whe re
 = 1122 - 1221 = 22 – 21.
If comm odit y 1 is intensive in factor 1, then :
ˆ 1 ˆp1   22   0
w
Moreove r, since 1j + 2j = 1, we have
Thus :
and
ˆ 2 ˆp1  0
w
ˆ 1 ˆp1 1 .
w
ˆ 1  pˆ1  ˆp2  0  w
ˆ2
w
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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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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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Effects of endowment growth
Each factor endow ment vi is full y e mployed in the two sectors, so:
vi  y1ai1  y2ai 2, where aij  x ij / yj
Find the effect of a change in vi on yj.
- Define proportiona l changes in va riables: vˆi = dvi/vi, etc.
- Then for each input vi:
whe re ij are employment shares.
vˆi   i1 ˆy1  i 2 yˆ2
Equi li brium sectoral ou tput changes a re given by solution to:
11  12ˆy1  vˆ1 

yˆ   ˆv 

 21
22 2 
 2 
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For sim pli cit y le t vˆ2  0. Then :
 22
ˆy1 
vˆ1

and
21
ˆy2 
ˆv1

whe re = 1122  2112.
If  > 0 (sector 1 is intensive in use of v1) then ˆy1 vˆ1 1 , and
ˆy1  ˆv1  ˆv2  ˆy2
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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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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.
25