micro for trade.pdf

7/29/2019 micro for trade.pdf

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ECON0301/ECON2252

Sept 2013

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General equilibrium multiple agents (firms and consumers) all optimizing The aggregation problem: no inconsistency between

aggregate and individual level decisions

One shot environment a static model income=expenditure => balance of trade

Market structure Constant returns to scale, perfect competition

Variable returns to scale, market power (new tradetheory in 80s; new new trade theory in this pastdecade)

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Preference: Consumer Theory

Technology: Producer Theory

Market Structure: Perfectly Competitive Market

General Equilibrium

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Consumption choice problem

max,

, subject to + .

Equalization of marginal utility per dollar

Marginal rate of substitution = relative price

yx x x

x y y y

MUMU MU P

P P MU P

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A

At point A, the budget lineBLand some ICaretangential to each other

Omitting the negative signs,

Slope of the IC=

Slope of the BL=

6

y

xBudget line

Indifferencecurve


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U=10

U=20

U=30

V=100

V=200V=2001

An order-preserving re-labeling ofICs does not alter thepreference ordering.

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x

y


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11

22

66

222''''

'''

''

222'

yxUU

xyUU

xyUU

yxUU

xyU

They are called positive monotonic transformation

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

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CD utility function:

Marginal utilities:

1, ,

,

x

U x y U x yx yMU x y

x x x

U x yMUy

y

,U x y x y

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MRS = relative price

Expenditure shares constant

Very nice demand functions

x x x

y y y

MU P P xy

MU x P P y

andyx

P yP x

I I

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

I ID p p I D p p I

P P

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Without loss of generality, we assume

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

The preference ordering is still preservedafter such positive monotonic transformation.

The demand functions found out are just thesame as before

1

, where anda bU x y x y a b

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

consumption

The aggregation problem Given total income, income distribution does not

affect market demand

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Suppose there are Nagents, each with the same CD utilityfunction

Suppose their incomes are I1,I2, , IN, summing up to I. The total market demand for xequals

Given the total income, the income distribution itself doesnot affect the market demand. 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

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Two goods: necessity (x) and luxury (y): px=1, py=1.

Two agents, where I1+I2= I= 10 Suppose each will consume luxury only after x0 < 5 units of

necessity is consumed.

Equal income: the market quantity demanded for xis 2x0;the market quantity demanded for yis 10 - 2x0

Unequal income: suppose I1 < I2 and I1 < x0. Then marketquantity demanded for x is I1 + x0 < 2x0; market quantitydemanded for yis 10-(x0 + I1).

Unequal income diverts resource to luxuries while basicnecessities are not fully provided => income distributionmatters

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The CD utility function is in the family ofutility functions, for which the ratios of goodsdemanded depend only on relative prices, noton income

can define the relative demand for x by anindividual For example, with CD utility function,

the relative demand is independent of the individualsincome

/yx

y x y x

pD I I

D P P p

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The relative demand for xin the whole economy isjust the same as the relative demand for xby anyindividual

Take out: With CD utility function, we can talk about The demand for xby the economy without

knowing income distribution, and Relative demand for xby 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

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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

L

QMP

L

QAP

K

QMP

K

Definitions:

Assumptions:

Diminishing marginal productivity

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Output elasticity and returns to scale

Suppose

>1 Increasing return to scale (IRTS)

=1 Constant return to scale (CRTS)


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

IRTS if , ,CRTS if , ,

DRTS if , ,

f tK tL tf K Lf tK tL tf K L

f tK tL tf K L

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For CRTS technology, there are two niceproperties:

MPKand MPLdepend on capital-labor ratio only,

but not on the absolute scale e.g., MPKthe same when you hire K=3 and L=5,

compared with when you hire K=6 and L=10.

Eulers equation: + = ,

When factors are hired up to = and = , the profit is just zero!

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, , ( 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

dt

f tK tL f tK tLtK tLf K L

tK t tL t

f tK tL f tK tL

K L f K LtK tL

the condition that 1, it becomes

, ,,

t

f K L f K LK L f K L

K L

We show the second property:

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Firms problem:

Assume each firm is too small to affect inputprices (w, r) and output price (p). If anoptimum exists, will hire Kand Lsuch that

,

max , - -K L

pf K L rK wL

,

,

K

L

f K LpMP p r

Kf K L

pMP p wL

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Treat the firms problem into a two-stage problem: costminimization, following by output choice.

First, given output quantity, the firm chooses the inputquantities to minimize costs,

Then, chooses quantity

Perfect Competition: uses p = MC.

Monopoly Competition: uses MR = MC.

Imperfect Competition: strategic consideration

,

min subject to ,K L

rK wL f K L Q

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Doubling input prices simply doubles the total cost:

When RTS does not change with scale, for anyt> 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

, , , satisfying

, , , , for all 0

C r w Q

C tr tw Q tC r w Q t

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Isoquant the locusofK and Lsuch thatthe output level isconstant

Bending toward theorigin

L

K

Q =10

Q =20

Iso-cost line

rK+wL= constant

Optimal input

mix to produce

Q=10

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

input and output prices

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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

K

QMP K LL

LMP K L

K

KMP K LL

K L

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The technology exhibits CRTS iff + =1.

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

What does +=1 mean?

+>1; IRTS

+=1; CRTS

+


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Consider an industry with all firms having thesame CRTS technology: Q= f(K, L); output &input markets perfectly competitive; and firmsmaximizing profits. As a whole, the industry

employs K* and L*.

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Q1: What is the industrial output? f(K*,L*)

Let K1, K2, , KNbe the amount employed inthe Nfirms; L1, L2, , LNbe the amount employed.

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

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

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Q2: What is the marginal product of capital (labor)in each firm?

Despite possibly different scales, each firmsmarginal product of capital is simply equal to

f(K*,L*)/K. Similarly, all firms have the same marginal product

of labor = f(K*,L*)/L .

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Q3: What is the rental rate of capital paid byeach firm? the wage rate of labor paid byeach firm?

Let pbe the price of the good produced in

the industry.

The rental rate of capital is just ,

.

the wage rate of labor is just ,

.

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What do we know about industrial output? simply work out the problem by assuming all

inputs (K* and L*) are hired by a single firm(which we assume is price taking in both inputand output markets).

we can understand f(K, L) as an industryproduction function.

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Takeout: given same CRTS technology & LRcompetitive equilibrium, total K* and L* inthe sector fully describe the output level,as well as the real rental rate and real wage

rate.

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micro for trade.pdf