Download Factor price equalization

Chapter 4: Increasing returns to scale
4.1 Empirical observations
4.2 Homogeneous products
4.2.1 External economies of scale
4.2.2 Contestable markets
4.2.3 Oligopolies
4.3 Differentiated products
4.3.1 Love of variety
4.3.2 Ideal variety
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4.1 Empirical observations
Volume of trade largest between „similar“
countries within OECD (more than 50% of world
trade).
Substantial intra-industrial trade (more than 25%)
Trade grows faster than world income
Growing importance of transactions within multinational corporations
Trade liberalization and free trade associations
should have led to a restructuring of production and
changes in factor income – not observed, all factor
incomes grow at the same pace.
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4.2 Homogeneous products
4.2.1 External economies of scale
Production function of representative firm:
xi = f(vi,)
(1)
where  denotes the positive external effects which
are beyond the control of firm i (e.g. domestic output
of the commodity, world output, total domestic
income etc.)
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Justifications:
Cheaper inputs for larger industries (however,
begs the question of where cheaper inputs come
from)
Approximation of model with price equal to
average cost in equilibrium (however, compare to
contestable markets)
Technical progress cannot be appropriated
(however, begs the question where it comes from).
Learning by doing – larger well trained labor force.
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Autarky equilibrium
Unit cost function: bj = bj(w,), bj < 0, bjw = aj.
aj =
(a1j,…,aij,…aMj) = vector of inputs per unit of output of
good j.
Equilibrium conditions:
pa = b(wa,a)
(2)
v = Axa
(3)
Trade equilibrium
p  b(w,)
(4)
v = Ax
(5)
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Welfare gains from trade:
Graham (1923): Free trade may reduce welfare through
reallocation of resources from industries with scale economies to
those without.
Kemp and Negishi (1970): Reallocation in opposite direction
ensures gains from trade.
Helpman and Krugman (1985): Sufficient condition for gains from
trade:
pjfj(vja,)  pjfj(vja,a),
(6)
i.e. at world market prices „on average“ factor productivity is
greater than in autarky, given the factor allocation of autarky and
 instead of a.
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Proof:
Idea: Show that (6) implies feasibility of ca = xa in the
free trade equilibrium, then apply WARP:
pxa  pjfj(vja,)  G(p,v,) = GDP-function.
Alternative sufficient condition for gains from trade:
bj(w,a)xja  bj(w,)xja
(7)
i.e. at world market prices costs of production of
autarky outputs are smaller in the free trade
equilibrium than in the autarky equilibrium.
The proof is similar to the one above:
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Proof:
(4) and (7) imply
pxa  bj(w,)xja  bj(w,a)xja ,
and by definition
bj(w,a) = waj(w,a)  waj(wa,a), hence
pxa  waj(wa,)xja = waj(wa,)xja = vw = G(p,v,),
Thus xa = ca is feasible in the free trade equilibrium
which cannot be inferior to the autarky equilibrium.
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Trade structure
In addition to different technologies (Ricardo) and/or
different factor endowments (Heckscher-Ohlin)
different utilization of economies of scale as cause
of comparative advantage: Predictions about trade
structure become even more difficult. Sufficient
condition for traditional results to hold: external
effects are the same for all countries in the free
trade equilibrium.
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Examples:
a) Ricardo:
Production functions:
fj(vj,) = j()vj/aj,
Fj(Vj,) = j()Vj/Aj
(8)
The domestic country will export good one if
a1/a2 < A1/A2.
This does not rule out the possibility that in autarky
we get
[2(a)a1]/[1(a)a2] > [2(A)A1]/[1(A)A2], hence
predictions about trade patterns may not be possible
by using only information about autarky.
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b) Heckscher-Ohlin
All countries have identical production functions: xj =
fj(vj,)
(9)
i.e. external effects are the same and have the same
effects in all countries in the free trade equilibrium –
predictions about trade patterns using information
after trade takes place are possible.
c) Counterexample: sector- and country-specific
externalities:
xj = xjavj/aj
(10)
Firm perceives infinite costs without domestic
production.
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Factor price equalization:
Approach: determination of FPE-set (allocations of
factor endowments between countries which allow
replication of integrated equilibrium through free
trade equilibria).
Focus on country- and sector specific external scale
economies – assumed to exist for subset IE of all
goods. Their production functions are
xi = fi(vi,xi), i  IE
(11)
Subset of goods produced with constant returns to
scale: IC.
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Integrated equilibrium conditions:
pi = bi(w,xi)
(12)
vi = aji(w,xi)xi
(13)
where vi denotes the vector of factor inputs in industry
i and aji is the input of factor j per unit of output of
industry i.
The set of all factor allocations compatible with FPE
(denoted as ) is defined analogously to the standard
case with the additional requirement that the
integrated equilibrium output of goods with increasing
returns to scale can be produced in one country:
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= {v1,...,vJ|ij  0, jij =1iI, ij{0,1} iIE}
vj = iijvi iJ
(14)
where ij denotes the share of country j of the vector
of factor inputs in industry i in the integrated
equilibrium, and vj denotes the vector of factor
endowments of country j.
For two countries and three industries, one of which
has external economies of scale, the set  is shown
in figure 4.1.
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Figure 4.1 FPE-set for industry 1 being capital intensive and
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enjoying
increasing returns to scale.
Remarks:
 The diagonal need not belong to .
 The equilibrium is not unique with respect to the
structure of production.
 Increasing returns to scale will generate division
of labor and trade even between identical
countries.
 It need not be the relatively capital rich country
that produces the capital intensive good, absolute
size also matters.
 Net factor import is uniquely determined.
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Figure 4.2: Indeterminacy with respect to the location of industry 1
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Non-uniqueness
Integrated economy, two goods, one factor,
production function x1 = x1v1/a1. Let v = 1,v1 < 1/2.
The following figure (4.3) shows  for this case:
0___________Q__________Q‘___________0*
vh measured from left to right. Let 0Q = Q‘0* = v1.
vh between Q and Q‘: integrated equilibrium is
reproduced, location of industry 1 indeterminate.
vh < v1: integrated equilibrium reproduced if industry
1 is in foreign country.
Second equ.: Industry 1 in home country, foreign
country firms perceive infinite production costs.
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Non-uniqueness cont’d
Let v1 > 1/2.  is shown in figure (4.3):
0___________Q’_ _ _ _ _ _Q___________0*
 consists of the solid lines 0Q‘ and Q0*.
 vh between Q‘Q: integrated equilibrium cannot be
reproduced.
 vh in 0Q‘: integrated equilibrium reproduced,
industry 1 in foreign country.
 Industry 1 in home country: no FPE.
 Both countries produce good 1: FPE, but no
reproduction of integrated world equilibrium.
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4.2.2 Contestable markets
Extension of Bertrand model to industries with subadditive cost-functions:
c(xi) < c(xi)
Reasons: Economies of scale, fixed costs.
Crucial assumptions: no sunk costs, no entry or exit
costs.
Dasgupta-Stiglitz: Theory well funded, but not well
founded.
Main difference to previous model: scale economies
are internal for firms and known to them.
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Integrated world equilibrium: additional condition:
ci(w,xi(pi))  pi  pi < pi*
rules out „inefficient“ equilibria (see Figure 4.5 below)
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Sufficient condition for gains from trade:
ci(w,xi)xia   ci(w,xia)xia
Non-existence of free trade equilibrium: Integrated
economy, two goods, one factor, production function
x1 = x1v1/a1. Let v = 1,v1 < 1/2. The following figure
(4.6) shows  for this case:
0___________Q__________Q‘___________0*
vh measured from left to right. Let 0Q = Q‘0* = v1.
vh between Q and Q‘: w greater in country that
produces good 1  provokes entry from other
country, but two firms cannot co-exist in a
contestable market.
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4.2.3 Oligopolies
One-shot Cournot-model
 Market concentration and trade: Even if two
countries are identical, free trade increases
competitive pressure and lowers prices as
compared to autarky
 Oligopoly and transport costs: may lead to
asymmetric oligopoly in each country, but above
effect still possible.
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4.2.3.1 Concentration in partial equilibrium
k identical consumers, n firms, market demand: X(p)
= kx(p).
Firm i maximizes
i(xi) = xip([jixj + xi]/k) – c(w,xi),
(15)
FOCs:
p + p‘xi/k = cx(w,xi)
(16)
Identical firms  xi = kx(p)/n
Elasticity of demand of representative consumer:
 = - x‘(p)p/x
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Substituting this into (16) yields
p[1 – 1/n(p)] = cx(w,kx(p)/n) = pR-1
(17)
Suppose there is free trade between two perfectly
identical countries (same number of identical
consumers and firms). Instead of (17) we get
p[1 – 1/2n(p)] = cx(w,kx(p)/n)
(18)
i.e. the price goes down even though no commodity
flows will be observed.
Sufficient condition for gains from trade: output in
oligopolistic industries grows (pro-competitive effect
of trade).
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The direction of trade
Depends not only on costs and on pre-trade prices,
but also on the number of firms and consumers:
Identical countries except for the number of firms:
country with larger number of firms exports (but
also had lower pre-trade price).
Identical costs, but different number of firms and
consumers: net exporter is country with larger
firms/consumers ratio.
Identical countries except for costs: country with
smaller costs is net exporter.
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In general no clear prediction on the basis of only
one variable possible: cost advantages may be overcompensated by market size, etc.
Reason: Less efficient firms may still be active in a
Cournot-equilibrium.
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4.2.3.2 General equilibrium trade patterns
Equilibrium condition for a Cournot-oligopoly:
p[1 – 1/n(p)] = cx(w,kx(p)/n) = pR-1
(17)
implying
R = [1 – 1/n(p)]-1
(19)
“Wedge” between price and marginal cost: 1 − R-1
Let Ri denote the „mark-up” perceived by firm i.
Constant returns to scale imply that resources are
allocated s.t. Ri-1pixi is maximized (Helpman 1984).
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Factor abundancy:
If all firms in a country perceive the same Ri then
differences in output will depend solely on
differences in supply functions, hence with identical
production functions on differences in factor
endowments.
Ri‘s most likely to be identical across countries with
factor price equality.
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Factor price equalization
Integrated world economy:
Two types of industries: Io oligopolies, Ic competitive
industries.
Equilibrium conditions:
:= cxx/c….elasticity of costs, measure for increasing
returns to scale.
:= 1/ = c/cx, c = average cost. Consequently,
pR-1 = cx can be written as
pR-1 = c/.
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Collecting terms, the equilibrium conditions are
Ri  p, ni 
pi

 i w, xi  ci w, xi 
pi  ci w
i  I O
i  I C
A(w)xc + A(w)xO = v
Factor price equalization (and reproduction of integrated
world equilibrium) requires that factor allocation is
compatible with number of firms of the oligopoly in each
country.
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s :
j
ni
n ji
ni

...share of country j's firms in industry i
  v1,..., v J ij  0, ij  1i  I , ij  snij i  I 0 , v j  ij vˆ i  j  J
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
Figure 4.7: FPE-set, industry 1: oligopoly, industries 2 and 3:
competitive.
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Net import of all factors‘s services: possible for
a country with a large share in oligopolistic
industries.
Suppose sj (share of country j in world GDP)
satisfies
sj > v1j/v1 = ...= vNj/vN
i.e. share in world GDP is greater than share in
factor endowments: possible due to profits.
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Figure 4.8: capital rich country exports embodied capital services
and imports embodied labor services. C’: share in factor income, C”:
share in world GDP.
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Figure 4.9: capital rich country imports embodied capital services
and embodied labor services. C’: share in factor income, C”: share
in world
GDP.
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4.2.3.3 Market segmentation
Introduction of transport costs t:
Example: two identical countries, each with one firm.
Home market: 2 firms, asymmetric Cournot model:
domestic firm has marginal costs c, foreign firm has
marginal costs c + t. Equilibrium conditions:
p(xd + xf) + xdp‘(xd + xf) = c,
p(xd + xf) + xfp‘(xd + xf) = c + t.
Equilibrium with xd > xf > 0 exists for t < pm – c.
pm = monopoly price.
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Welfare effects:
w = u(xd + xf) – c(xd + xf) – txf
dw = u’(xd + xf)(dxd + dxf) – c(dxd + dxf) – tdxf – dtxf
Normalize u(x) s.t. u‘(x) = p:
dw = (p – c – t)(dxd + dxf) + tdxd – dtxf
Note that dt < 0  dxd < 0 < dxf
1. xfdt...cost reduction  w, but xf = 0 if
t = pm – c.
2. (p – c – t)(dxd +dxf)  w (export increase), but 0 if t = pm – c.
3. tdxd...  w (replacement of domestic shipments through
exports).
dw/dt > 0 at t  p – c.
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Figure 4.10: Gains and losses due to changes of t.
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