Advanced Microeconomics - coin.wne.uw.edu.plcoin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/2011_micro...Advanced Microeconomics Small open economy Jan Hagemejer December 18, 2011

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.


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.


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.


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 .


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).


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.


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).


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.


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


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.


Pareto e�ciency


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.


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`


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.


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.


Equilibrium conditions - factor prices

z2

z1

q1=1/p1

q2=1/p2

1/w1

1/w2


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.


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).


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


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


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


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).


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).


Rybczynski Theorem


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).


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).


Documents

Advanced Microeconomics - coin.wne.uw.edu.plcoin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/2011_micro...Advanced Microeconomics Small open economy Jan Hagemejer December 18, 2011