MalthusandSolow(中级宏观经济学,香港中文大学)

Chapter

6Economic

Growth:Malthus

and

SolowThe

Malthusian

Model

of

Economic

GrowthMalthus

argued:advancesin

the

technology

for

producing

foodincreased

population

growthno

increase

in

the

standard

of

living

unless

there

were

somelimits

on

population

growthA

dynamic

model

with

many

periodsConfine

attention

to

whathappens

in

the

current

period

and

thefuture

periodAggregateproduction

function

with

constant

returns

to

scale:(1)whereis

current

aggregate

outputis

current

fixed

supply

of

landis

current

laborThere

is

no

investment

and

no

government

spendingAssume

each

person

is

willing

to

work

at

any

wage

and

has

oneunit

of

labor

to

supply

(a

normalization),

so

that

in

Equation

(1),is

both

the

population

and

the

labor

input

Suppose

population

growth

depends

on

the

quantity

ofconsumption

per

worker(2)whereis

the

population

in

the

future

(next)

periodis

an

increasing

functionis

aggregate

consumptionso

thatis

current

consumption

per

worker•,mainly

due

to

the

fact

that

higher

foodconsumption

per

worker

reduces

death

rates

through

betternutritionAll

goods

produced

are

consumed,

so

C

=

Y.Hence,

(3)Then

use

Equation

(3)

to

substitute

for

C

in

Equation

(2):(4)

The

constant-returns-to-scale

property

of

the

production

functioimplies

thatAfter

multiplying

each

side

by

N,

Equation

(4)

can

be

rewritten

as(5)

Population

growth

dependson

consumption

per

workerin

the

Malthusian

model•is

the

steady

state

forthe

populationIfthenpopulation

increasesIfthenpopulationdecreasesSteady

State

Analysis

of

the

Malthusian

ModelRecallLetting,andThen

from

Equation(2):soIn

the

steady

state,then

can

be

determinedIn

panel

(b),is

determinedis

determined

from

the

per-worker

production

In

panel

(a),functionSteady

state

population

is

given

byTheSteady

State

Effects

of

an

Increase

in

z,

Suppose

the

economy

is

initially

in

a

steady

state,

withwhich

then

increases

once

and

for

all

time

tounchangedfallsSteady

state

population

increases

Population

increases

overtime

to

its

steady

state

valueincreases

at

time

Tconsumption

per

workerincreasesthen

decline

to

its

steadystate

value

Another

example:

populationcontrol

both

c*

and

l*increase.How

Useful

is

the

Malthusian

Model

of

EconomicGrowth?

Before

the

Industrial

Revolution

in

about1800:economic

growthconsistent

with

the

Malthusian

Model

From

the

perspective

of

the

early

21st

century:Malthus

waswrongWhy?Did

not

allow

for

the

effect

of

increases

in

the

capital

stock

onproduction,

andDid

not

account

for

all

the

effects

of

economic

forces

onpopulation

growthThe

Solow

Model:

Exogenous

GrowthConsumers:The

population

grows

over

time:whereis

the

population

in

the

future

periodis

the

rate

of

growth

in

the

population

Assume

that

consumers

consume

a

constant

fraction

of

income

ineach

period:whereis

current

consumptionis

the

aggregate

savings

rateThe

Representative

Firm:Constant

returns

production

function

with

capital

and

labor

inputwhereis

output

per

workeris

capital

per

worker•whereis

constant

depreciation

rate

andCompetitive

Equilibrium:Equilibrium

condition:Steady

State

Analysis

Quantity

of

capital

perworker

converges

to

aconstant,quantity

of

output

perworker

converges

to

aconstant,

If

s,

n

and

z

are

constant,real

income

per

workercannot

growno

betterment

in

livingstandards

In

the

long

run,

when

the

economy

converges

to

the

steady

statequantity

of

capital

per

worker,

k*,

all

aggregate

quantities

(K,and

C)

grow

at

the

rate

n.Analysis

of

the

Steady

StateIn

the

steady

state,by

rearranging,Comparative

Statics:

An

Increase

in

the

Savings

Rate,

shifts

The

curve,uplevels

of

capital

perworker

and

output

perworker

are

higherBUT

there

is

no

effect

on

thegrowth

rates

of

aggregatevariables

Before

time

T,

aggregateoutput

is

growing

at

theconstant

rate,

Savings

rate

increases

attime

T

After

time

T,

output

thenconverges

in

the

long

run

toa

new

higher

steady

stategrowth

pathConsumption

per

Worker

and

Golden

Rule

CapitalAccumulation

Consumption

per

worker

inthe

steady

state

is•(golden

rule

quantityof

capital

per

worker)

givesthe

maximum

consumptionper

worker,Property

of

the

goldenrule:Comparative

Statics:

An

Increase

in

Labor

Force

Growth

An

increase

in

the

laborforce

growth

rate

(n)

causesa

decrease

in

the

quantity

ofcapital

per

worker

andoutput

per

worker Growth

rates

in

aggregateoutput,

aggregateconsumption

and

aggregateinvestment

increaseComparative

Statics:

An

Increase

in

Total

FactorProductivity The

Solow

Model

predictsthat

a

country’s

standardof

living

can

continue

toincrease

in

the

long

run

onlyif

there

are

continuingincreases

in

total

factorproductivity

(z)Mathematical

Solution

(Appendix

p.636)In

the

steady

state,Hence,

The

sign

is

ambiguous,

so

that

consumption

per

worker

couldincrease

or

decrease

with

an

increase

in

the

savings

rate

The

golden

rule

steady

state

quantity

of

capital

per

worker

solvesthe

problem•solvesorAs,Recall

in

the

steady

state,Totally

differentiating

Equation

(A.1),

we

getHence,

solving

for

the

appropriate

derivatives,Growth

Accounting

If

aggregate

real

output

is

to

grow

over

time,

it

is

necessary

forfactor

or

factors

of

production

to

be

increasing

over

time,

or

forthere

to

be

increases

in

total

factor

productivity.

Growth

Accounting:

anexercise

to

measure

how

much

of

thegrowth

in

aggregate

output

overa

period

of

time

is

accounted

forby

growth

in

each

of

the

inputs

to

production

and

by

increases

intotal

factor

productivityGrowth

Accounting

Growth

accounting

starts

by

considering

the

aggregate

productionfunction

from

the

Solow

growth

model,

Cobb-Douglas

productionfunction

is

a

good

analytical

tool

forgrowth

accounting.

The

production

function

takes

the

formwhereGrowth

AccountingSuppose

we

set=

0.36,

then

the

production

function

isIf

wehave

measures

of

aggregate

output,,

then

total

factthe

capital

inputproductivity,

and

the

labor

inputcan

be

measured

as

a

residualAGrowth

Accounting

Exercise

The

table

shows

how

the

growth

in

the

capital

stock,

inemployment,

and

in

total

factor

productivity

contribute

to

thegrowth

in

real

output

Increases

in

measured

total

factor

productivity

could

be

the

resuof:new

inventoriesgood

weathernew

management

techniquesfavorable

changes

in

government

regulationsdecreases

in

the

relative

price

ofenergy(any

other

factor

that

causes

more

aggregate

output

to

beproduced

given

the

same

quantities

of

aggregate

factor

inputs)Solow

Residuals

and

the

Productivity

Slowdo