Microeconomics for International Trade Theory

ECON0301

January 2011

Basic Trade Model

General equilibrium multiple agents (firms and consumers) all optimizing The aggregation problem

One shot long run income=expenditure => balance of trade

Market structure Constant returns to scale, perfect competition Variable returns to scale, market power (new trade

theory in 80s; new new trade theory in this past decade)

Basic Prerequisites

Consumption Production Perfectly Competitive Market General Equilibrium

I: Consumption Theory

Consumption Problem

Consumption choice problem

Results: Demand functions

Equalization of marginal utility per dollar

Marginal rate of substitution = relative price

,

max , subject to x yx yU x y I PY PY

, , and , ,x x y y x yx D P P I y D P P I

yx x x

x y y y

MUMU MU P

P P MU P

C

A

B

At point A, the budget line and some IC are tangential to each other

Omitting the negative signs,

The slope of the IC = MUx/MUy

The slope of the budget line = Px/Py

Marginal rate of substitution = relative price

Y

X

A

C

U1U0At point C, MUx/MUy=Px/Py. But obviously not an optimal bundle

Some properties on the shapes of ICs are required and will be assumed to hold.

Marginal rate of substitution = relative price

dyy

yxUdx

x

yxUdU

),(),(

Marginal rate of substitution

Change in utility

Along an IC

The slope of an IC

0

( , ) ( , )0

U x y U x ydU dx dy

x y

/0

/x

U Uy

dy U x MU

dx U y MU

Cobbs-Douglas Utility Function CD utility function: Marginal utilities:

MRS = relative price

Expenditure shares constant

x x x

y y y

MU P P xy

MU x P P y

and yxP yP x

I I

,U x y x y

1, ,

,

x

U x y U x yx yMU x y

x x xU x y

MUyy

Cobbs-Douglas Utility Function

Very nice demand functions

Without loss of generality, assume

If not, we can always represent the old CD utility function by a new CD function

, , and , ,x x y y x yx y

I ID p p I D p p I

P P

1

, where and a bU x y x y a b

U=10

U=20

U=30

V=100

V=200V=2001

An order-preserving re-labeling of ICs does not alter the preference ordering.

Measurability of Utility

11

22

66

222''''

'''

''

222'

yxUU

xyUU

xyUU

yxUU

xyU

Positive monotonic (order-preserving) transformation

They are called positive monotonic transformation

They all refer to the same preferences, leading to the same choice

Two properties of CD utility functions

Unit income elasticity 1% increase in income => 1% increase in

consumption The aggregation problem

Given total income, income distribution does not affect market demand

The aggregation problem Suppose there are N agents, each with the same CD utility

function Suppose their incomes are I1,I2, … , IN, summing up to I. The total market demand for x equals

Given the total income, the income distribution among the agents does not affect the market demand. As long as the latter is concerned, we do not need to know income distribution.

A property need not generally hold for other utility functions

1

1

N

x x

N

x x

II

P P

I I I

P P

An example where income distribution matters Two goods: necessity (x) and luxury (y) Two agents, where I1+I2 = I = 10

each will consume luxury only after x0 <5 units of necessity is consumed.

Suppose prices px=1, py=1.

Equal income, the market quantity demanded for x is 2x0; the market quantity demanded for y is 10 - 2x0

Unequal income; suppose I1 < I2 and I1 < x0. Then market quantity demanded for x is I1 + x0 < 2x0; market quantity demanded for y is 10-(x0 + I1).

Unequal income diverts resource to luxuries while basic necessities are not fully provided => income distribution matters

The aggregation problem

The Cobb-Douglas utility function is among the family of utility functions for which income distribution per se does not affect market demand

We can further define the concept of “the relative demand for x” by an individual

This relative demand is independent of the individual’s income

/ yx

y x y x

pD I I

D P P p

The aggregation problem

The relative demand for x in the whole economy (with individual incomes I1, I2, …, IN) is just the same as the relative demand for x by any individual

Take out: With CD utility function, we can talk about demand for x by the economy without knowing how

income is distributed among individuals Relative demand for x by the economy without

knowing the total income

1 2 1 2/ yx N N

y x y x

pD I I I I I I

D P P p

II. Production, Perfectly Competitive Market, Equilibrium

Production function

total product ( , )

average product of labor

marginal product of labor

average product of capital

marginal product of capital

L

L

K

K

Q f K L

QAP

LQ

MPLQ

APKQ

MPK

Suppose , some proportional change in inputs.

/ % change in output

/ % change in all input

1 i.r.t.s.

1 c.r.t.s.

1 d.r.t.s.

dL dK d

L K

dQ QR

d

Production Function

Output elasticity and returns to scale

Production function

When returns to scale do not change with scale, for any t>1, the technology exhibits

IRTS if , ,

CRTS if , ,

DRTS if , ,

f tK tL tf K L

f tK tL tf K L

f tK tL tf K L

Production function

For CRTS technology, there are two nice properties MPK and MPL depend on K/L only, but not on the

absolute scale E.g., MPK the same when you hire K=3 and L=5,

compared with when you hire K=6 and L=10. MPK*K+MPL*L = f(K,L)

When factors are hired up to r = p*MPK and w = p*MPL, the profit is just zero!

Production function

We show the second property:

, , ( CRTS)

Differentiating it w.r.t , we obtain

, ,

, ,, (chain rule)

, ,,

Now imposing

f tK tL tf K L

t

df tK tL f K L

dtf tK tL f tK tLtK tL

f K LtK t tL t

f tK tL f tK tLK L f K L

tK tL

the condition that 1, it becomes

, ,,

t

f K L f K LK L f K L

K L

Profit maximization problem

Each firm chooses

If an optimum exists, will hire K and L such that

,

max , - -K L

pf K L rK wL

,

,

K

L

f K LpMP p r

Kf K L

pMP p wL

Cost minimization problem

Sometimes it is easier to re-phase the problem as a cost minimization problem, following by the output choice problem

Given input prices, choose the input combination that minimizes cost

Cost function

,

min subject to ,K LrK wL f K L Q

, , , satisfying

, , , , for all 0

C r w Q

C tr tw Q tC r w Q t

Cost function

When RTS does not change with scale, for any t>1, the technology exhibits

, , , ,IRTS if

, , , ,CRTS if

, , , ,DRTS if

C r w tQ C r w Q

tQ Q

C r w tQ C r w Q

tQ Q

C r w tQ C r w Q

tQ Q

isoquant

Isoquant – the locus of K and L such that the output level is constant

Bending toward the origin

L

K

Q=10

Q=20

Iso-cost line

rK+wL=constant

Optimal input mix to produce Q=10

Equilibrium condition

Main assumptions: CRTS technology, perfect competition, all inputs variable (long run equilibrium).

The min. costs to produce $1 worth of a good are exactly $1.

Given output and input prices, we can determine the optimal mix of inputs

Or, given output price and K/L ratio, we can determine the relative input prices

L

K

Q=1/p

Q=1/p’

Iso-cost line

rK+wL=constant

Equilibrium condition

For CRTS technology and in LR equilibrium, we cannot tell the output level of a particular firm, because every output level will lead to the same profit (which is zero) given fixed input and output prices

1

1

(1 ) 1

1 1

( , )

When + =1,

and

Marginal products depend on the / ratio, not on the absolute scale

K

L

K

L

Q f K L K L

QMP K L

KQ

MP K LL

LMP K L

K

KMP K L

L

K L

Cobb-Douglas Production Function

Cobb-Douglas Production Function

What does α+β=1 mean? α+β>1; IRTS α+β=1; CRTS α+β<1; DRTS

The technology exhibits CRTS iff + =1.

2 ,2 2 2 2 2 ( , )f K L K L K L f K L

The aggregation problem

Consider an industry with all firms having the same CRTS technology: Q=f(K,L); output & input markets perfectly competitive; and firms maximizing profits. Collectively, the industry employs K* and L*.

What is the total output of the industry? f(K*,L*). The output produced would be the same as if there

were one single firm employing K* and L*. The “fine structure” of the industry (i.e., the number of firms, the sizes, etc.) is irrelevant.

The aggregation problem

Let K1, K2, …, KN be the amount employed in the N firms.

Let L1, L2, …, LN be the amount employed in the N firms.

Cost minimization requires that K1/L1=K2/L2=… = KN/LN=K*/L*=a.

Total output

1 1 2 2

1 1 2 2

1 2

* * *

* *

, , ,

, , ,

,1 ,1 ,1 ( CRTS)

,1 , ( CRTS)

,

N N

N N

N

f K L f K L f K L

f aL L f aL L f aL L

L f a L f a L f a

L f a f aL L

f K L

The aggregation problem

Question: What is the marginal product of capital (labor) in each firm?

Despite possibly different scales, each firm’s marginal product of capital is simply equal to ∂F(K*,L*)/∂K, i.e., marginal product of capital of a fictitious firm which employs just K* and L*.

Similarly, all firms have the same marginal product of labor = ∂F(K*,L*)/∂L.

The aggregation problem

Question: What is the rental rate of capital paid by each firm? What is the wage rate of labor paid by each firm?

Let P be the price of the good produced in the industry. The rental rate of capital is just P*∂F(K*,L*)/∂K, while the wage rate of labor is just P*∂F(K*,L*)/∂L.

The real rental rate (≡ w/p) and real wage rate (≡ r/p) should be ∂F(K*,L*)/∂K & ∂F(K*,L*)/∂L.

The aggregation problem

To know the output of an industry, as well the wage rate of labor and the rental rate of capital being paid by firms in the industry, there is no need to know the internal structure of the industry.

We can simply work out the problem by assuming all inputs (K* are L*) are hired by a single firm (which is nonetheless price taking in both input and output markets).

In this case, we can understand f(K,L) as an industry production function.

The aggregration problem

Takeout: given same CRTS technology, & LR competitive equil, the total K* and L* employed in the sector fully describe the output level as well as the real rental rate and real wage rate.