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Advanced MicroeconomicsSmall open economy
Jan Hagemejer
December 18, 2011
Jan Hagemejer Advanced Microeconomics
A small open economy
Consider an economy with:
J �rms, each producing a consumer good qj from primaryinputs/factors zj = (z1j , . . . , zLj) ≥ 0.
Production function fj(zj).
Total endowment of inputs z̄1, . . . , z̄L � 0.
Endowments are initially owned by consumers and consumerscannot consume them.
Suppose that the prices of the J produced commodities are�xed at p = (p1, . . . , pJ) - EXOGENEOUS world prices.
Factors can move freely between sectors. Immobileinternationally.
Jan Hagemejer Advanced Microeconomics
The �rms' problem.
The �rm solves the following problem:
Maxzj≥0
pj fj(z) − w · zj .
Denote optimal input demands: z(p,w) (vector).
Factor supply: (z̄1, . . . , z̄L) at positive input prices.
Jan Hagemejer Advanced Microeconomics
Factor market equilibrium
De�nition
A equilibrium in the factor market for this economy given theoutput prices p consists of an input price vectorw∗ = (w∗
1, . . . ,w∗
L ) � 0 and a factor allocation
(z∗1 , . . . , z∗J ) = ((z∗11, . . . , z
∗L1), . . . , (z∗
1J , . . . , z∗LJ)),
such that �rms receive their desired factor demands under prices(p,w∗) and all the factor markets clear, that is:
z∗j = zj(p,w) for all j = 1, . . . , J and∑j
z∗`j = z̄` for all ` = 1, . . . , L.
Jan Hagemejer Advanced Microeconomics
Factor market equilibrium
Given concavity of �rms' production function variablesw∗ = (w∗
1, . . . ,w∗
L ) and (z∗1, . . . , z∗J ) constitute an equilibrium if the
following hold:
pj
∂fj(z∗j )
∂z`j= w∗
` for all ` = 1, . . . , L and j = 1, . . . , J and∑j
z∗`j = z̄` for ` = 1, . . . , L.
Given the equilibrium in the factor market, the equilibrium outputlevels are q∗j = fj(z
∗j ) for every j .
Jan Hagemejer Advanced Microeconomics
Output market equilibrium
Alternative equilibrium speci�cation:
De�nition
Output levels (q∗1, . . . , q∗J) � 0 and factor prices w∗ � 0 constitute
an equilibrium if and only if the following conditions hold:
pj =∂cj(w
∗, q∗j )
∂qjfor j = 1, . . . , J,
(price equals marginal cost)
∑j
∂cj(w∗, q∗j )
∂w`= z̄` for ` = 1, . . . , L.
(sum of factor demands for each factor by all �rms equals demandfor that factor).
Jan Hagemejer Advanced Microeconomics
The social planner solution
Social planner wanting to maximize the country's revenue:
Maxzj≥0
pj fj(zj) s.t.∑j
zj = z̄ .
compare it to the joint pro�t maximization by j �rms subject to∑j z
∗j = z̄ (market clearing):
Maxzj≥0
∑j
(pj fj(zj) − w∗ · zj).
which in fact gives:
Maxzj≥0
∑j
pj fj(z) s.t.∑j
zj = z̄ .
So the two solutions generate same results.
Jan Hagemejer Advanced Microeconomics
The aggregate solution properties
The aggregate factor demands must then solve:
Maxz>0
(f (z) − w · z),
where f (z) is the aggregate production function �for dollars�f (z) = Max
z1,...,zJ
∑j pj fj(zj) subject to
∑j zj = z
The FOCs are:
w` =∂f (z)
∂z`for every `.
The equilibrium factor price must be:
w` =∂f (z̄)
∂z`for every `.
The price of factor ` must be equal to its aggregate marginalproductivity (in terms of total country revenue).
Jan Hagemejer Advanced Microeconomics
The model closure
Note that we did not talk about the consumer.
Consumers (in the same fashion as producers) take the worldprices as given.
Consumers maximize utility given the world prices.
However, we can solve the two completely independently: why- prices are exogenous (determined somewhere else in theworld).
If we did have consumers in our model, we would have somebudget constraints:
Balance of payments: the value of imports=value of exports
Equivalent to: value of sales of all produced goods = value of
purchases of all consumed goods.
Equivalent to: �rms pro�ts + factor earnings = value of all
consumed goods.
Jan Hagemejer Advanced Microeconomics
The 2x2 case
Assumptions:
J = L = 2
production function f1(z11, z21) and f2(z12, z22) areCRS/homothetic/homogeneous of degree 1.
factor 1 - labor, factor 2 - capital.
for every vector of factor prices w = (w1,w2),
cj(w) is the minimum cost of producing one unit of good j .(unit cost)
aj(w) = (a1j(w), a2j(w)) the (unique) input combination atwhich the min cost is reached.
aj(w) are sometimes called: unit factorrequirements/demands
Jan Hagemejer Advanced Microeconomics
Factor intensity
De�nition
The production of good 1 is relatively more intensive in factor 1than is the production of good 2 if:
a11(w)
a21(w)>
a12(w)
a22(w),
at all factor prices w = (w1,w2).NOW: ASSUME the above.
Jan Hagemejer Advanced Microeconomics
Pareto e�ciency
Jan Hagemejer Advanced Microeconomics
Important
Factor intensity (use of factor 1 relative to factor 2 inproduction) of one �rm exceeds the one of the other along allpoints of the Pareto set (except on diagonal).
The Pareto set lies all above or all below the diagonal. If itcuts the diagonal, it has to lie completely on the diagonal(CRS).
Any ray from the origin cuts the Pareto set at most once. Thefactor intensities move monotonically along the Pareto set.
Jan Hagemejer Advanced Microeconomics
Solution to the problem (1): factor demands
Factor demands are obtained through Shephard's lemma.
The total cost of production is:
Cj(w1,w2, q) = cj(w1,w2)qj
Therefore:
z`j(w1,w2, qj) =∂cj(w1,w2)qj
∂w`
or:
a`j(w) =∂cj(w1,w2)
∂w`
Jan Hagemejer Advanced Microeconomics
Solution to the problem (2): factor wages
In equilibrium:
c1(w1,w2) = p1 and c2(w1,w2) = p2.
Goods prices have two equal to the unit costs production of goods:
with CRS: MC = AC =unit cost and this follows from pro�tmaximization.
Jan Hagemejer Advanced Microeconomics
Equilibrium conditions - factor prices
Lerner-Pearce diagram (or how too graphically determine wages):
We can think of required production levels of good 1 and 2that give us the same unit (i.e. 1 dollar) revenue
we can draw unit isoquants for both goods - the unit isovalue
lines:
1 = p1f1(z11, z21) and 1 = p2f2(z12, z22)
We also can �nd all the input combinations that cost exactly 1dollar
the so called unit isocost line
z1w1 + z2w2 = 1
The isocost that that is tangent to both isoquants pins downthe factor wages.
Jan Hagemejer Advanced Microeconomics
Equilibrium conditions - factor prices
z2
z1
q1=1/p1
q2=1/p2
1/w1
1/w2
Jan Hagemejer Advanced Microeconomics
Solution to the problem (3): factor allocation
Once the factor wages are known, thefactor demands satisfy the following:
z∗11
z∗21
=a11(w∗)
a21(w∗)and
z∗12
z∗22
=a12(w∗)
a22(w∗).
The above expression, together withthe resource constraint
z̄` = z∗`1 + z∗`2, ` = 1, 2.
gives us the equilibrium factor usesand outputs.
Jan Hagemejer Advanced Microeconomics
Factor price equalization
Theorem
The equilibrium factor prices depend only on the technologies of
the two �rms and on the output prices p.
NOTE: NOT on the factor endowments (but only in the smallopen economy, i.e. with EXOGENEOUS prices of �nal good).
Jan Hagemejer Advanced Microeconomics
Comparative statics
How does a change in the price of one of the outputs (eg. p1)a�ect the equilibrium factor prices and factor allocations?p1 goes up. We need to produce less q1 to get one dollar of output.w1 goes up and w2 goes down.
z2
z1
q1=1/p1
q2=1/p2
1/w1
1/w2
1/w'1
1/w'2
Jan Hagemejer Advanced Microeconomics
Comparative statics
How does a change in the price of one of the outputs (eg. p1)a�ect the equilibrium factor prices and factor allocations?p1 goes up. The green line shows proportional change in w1.Therefore w1 goes up more proportionately than p1.
z2
z1
q1=1/p1
q2=1/p2
1/w1
1/w2
1/w'1
1/w'2
Jan Hagemejer Advanced Microeconomics
Comparative statics
How does a change in the price of one of the outputs (eg. p1)a�ect the equilibrium factor prices and factor allocations?p1 goes up. As factor 1 becomes more expensive, both �rms startusing more of factor 2.
z2
z1
q1=1/p1
q2=1/p2
1/w1
1/w2
1/w'1
1/w'2
Jan Hagemejer Advanced Microeconomics
Stolper-Samuelson Theorem
Theorem
In the 2x2 production model with the factor intensity assumption, if
pj increases, then the equilibrium price of the factor more
intensively used in the production of good j increases, while the
price of the other factor decreases (assuming interior equilibria both
before and after the price change).
Jan Hagemejer Advanced Microeconomics
Rybczynski Theorem
Theorem
In the 2x2 production model with the factor intensity assumption, if
the endowment of a factor increases, then the production of the
good that uses this factor relatively more intensively increases and
the production of the other good decreases (assuming interior
equilibria, both before and after the change of endowment).
Jan Hagemejer Advanced Microeconomics
Rybczynski Theorem
Jan Hagemejer Advanced Microeconomics
Factor price equalization
It happens because:
c1(w1,w2) = p1 and c2(w1,w2) = p2.
Or: p1∂f (z11,z21)
∂z11= w1 and p2
∂f (z12,z22)∂z12
= w2 . So wages areonly dependent on the value of marginal productivities.
Therefore a change in endowment does not a�ect the level ofwages (because the ratios factor use stay constant).
However, the economy will specialize more in labour (capital)intensive goods if it is more abundant in labour (capital).
Jan Hagemejer Advanced Microeconomics
2x2 production economy recap
Distinct consumer and producer problems given world prices(we so far focus only on the producer problem).
Factor price equalization: factor prices are only a function oftechnology and world prices. Not endowments. If all countrieshave the same technology, the wages are the same everywhere.
Stolper-Samuelson - p1 up, w1 up, w2 down (good 1 moreintensive in factor 1).
Rybczynski theorem - ω1 up, y1 up, y2 down (good 1 moreintensive in factor 1).
Jan Hagemejer Advanced Microeconomics