Mankiw_7e_Chapter 7.ppt

7/25/2019 Mankiw_7e_Chapter 7.ppt

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1ChapterSeven

CHAPTER 7Economic Growth I:Capital Accumulation and Population Growth

A PowerPointTutorial

To Accompany

MACROECONOMICS, 7th. EditionN. Gregory Mankiw

Tutorial written by:

Mannig J. SimidianB.A. in Economics with Distinction, Duke UniversityM.P.A., Harvard University Kennedy School of Government

M.B.A., Massachusetts Institute of Technolo (MIT) Sloan School of Manaement


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2ChapterSeven

The Solow Growth Model is designed to show how

growth in the capital stock, growth in the labor force,

and advances in technology interact in an economy,

and how they affect a nations total output of

goods and services.

Lets now examine how the

model treats the accumulationof capital.


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ChapterSeven


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!ChapterSeven

The production function represents the

transformation of inputs "labor "L#, capital "K#,production technology# into outputs"final goods

and services for a certain time period#.

The algebraic representation is$Y = F (K, L )

The %roduction &unction

The %roduction &unction

IncomeIncome

isissome function ofsome function of

our given inputsour given inputs

Lets analy'e the supply and demand for goods, and

see how much output is produced at any given time

and how this output is allocated among alternative uses.

Key Assumption: The Production unction h!s const!nt returns to sc!le"

# ##


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(ChapterSeven

This assumption lets us analy'e all )uantities relative to the si'e of

the labor force. Setz =$%L"

Y/L= F (K/L ,$ )

&utput&utput

Per wor'erPer wor'er

isis some function ofsome function of

the amount ofthe amount of

capital per workercapital per worker

*onstant returns to scale imply that the si'e of the economy asmeasured by the number of workers does not affect the relationship

between output per worker and capital per worker. +o, from now on,

lets denote all )uantities in per worker terms in lower case letters.

ere is our production function$ , wheref "k# = F "k,1#.y =f ( k )

This is a constantThis is a constant

that can bethat can be

ignored.ignored.


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-ChapterSeven

MPKf(k $) *f (k)

yy

kk

f"k#

The production function shows

how the amount of capital per

worker k determines the amountof output per workery =f"k#.

The slope of the production

function is the marginal product of

capital$ if kincreases by 1 unit,y

increases byMPKunits.

1%/


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0ChapterSeven

consumptionconsumption

per workerper worker

dependsdepends

onon savingssavings

raterate"between and 1#"between and 1#

utpututput


consumptionconsumption

per workerper worker investmentinvestment


y = c + iy = c + i1#

c =c =

"13"13ss

##yy

c =c ="13"13ss

##yy2#

yy4 "134 "13ss##yy5 i5 iyy4 "134 "13ss##yy5 i5 i#

!#i =i = ssyyi =i = ssyy

Investment s!vings. The rate of saving s

is the fraction of output devoted to investment.


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6ChapterSeven

ere are two forces that influence the capital stock$

7Investment:expenditure on plant and e)uipment.7+epreci!tion:wearing out of old capital8 causes capital stock to fall.

9ecall investment per worker i = s y.

Lets substitute the production function fory,we can express investment

per worker as a function of the capital stock per worker$

i = sf(k)

This e)uation relates the existing stock of capitalkto the accumulation

of new capital i.


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:ChapterSeven

;nvestment, sf(k)

utput,f (k)

c"per worker#

i"per worker#y"per worker#

The saving ratesdetermines the allocation of output between

consumption and investment. &or any level of k,output isf(k#,investment is sf"k#, and consumption isf"k#


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1ChapterSeven

;mpact of investment and depreciation on the capital stock$k= i k

*hange incapital stock

;nvestment =epreciation

9emember investment e)uals

savings so, it can be written$k = sf(k) k

k

kk

k

=epreciation is therefore proportional

to the capital stock.


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11ChapterSeven

Investment!nd depreci!tion

,!pit!l

per wor'er-k

i. k.

k.k$ k/

At k.- investment e0u!ls depreci!tion !nd

c!pit!l will not ch!nge over time" >elow k?,investment

exceedsdepreciation,

so the capital

stock grows.

>elow k?,investment

exceedsdepreciation,

so the capital

stock grows.

Investment- sf(k)

+epreci!tion- k

@bove k?, depreciationexceeds investment, so the

capital stock shrinks.

@bove k?, depreciationexceeds investment, so the

capital stock shrinks.


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12ChapterSeven

Investment!nd

depreci!tion

,!pit!l

per wor'er-k

i* = k.

k1. k/.

+epreci!tion- k

Investment- s1

f(Investment-

s/f(k

The +olow odel shows that if the saving rate is high, the economy

will have a large capital stock and high level of output. ;f the saving

rate is low, the economy will have a small capital stock and a

low level of output.

@n increase in

the saving rate

causes the capitalstock to grow to

a new steady state.

@n increase in

the saving rate

causes the capitalstock to grow to

a new steady state.


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1ChapterSeven

The steady3state value of kthat maximi'es consumption is called

the Golden Rule Level of Capial! To find the steady3state consumptionper worker, we begin with the national income accounts identity$

and rearrange it as$

c = y - i.

This e)uation holds that consumption is output minus investment.>ecause we want to find steady3state consumption, we substitute

steady3state values for output and investment. +teady3state output

per worker is f "k?# where k? is the steady3state capital stock per

worker. &urthermore, because the capital stock is not changing in thesteady state, investment is e)ual to depreciation k?. +ubstituting f"k?#foryandk? for i,we can write steady3state consumption per worker as$

c* = f (k*) - k*.

y - c + i


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1!ChapterSeven

c?4 f "k?# 3 k?.@ccording to this e)uation, steady3state consumption is whats left

of steady3state output after paying for steady3state depreciation. ;tfurther shows that an increase in steady3state capital has two opposing

effects on steady3state consumption. n the one hand, more capital

means more output. n the other hand, more capital also means that more

output must be used to replace capital that is wearing out.

The economys output is used for

consumption or investment. ;n the steady

state, investment e)uals depreciation.

Therefore, steady3state consumption is the

difference between output f "k?# anddepreciation k?. +teady3state consumptiois maximi'ed at the Aolden 9ulesteady

state. The Aolden 9ulecapital stock is

denotedk

*gold,and the Aolden 9uleconsum tion is c* .

k

kk

k

utput,f(k)

c ?gold

k?gold


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1(ChapterSeven

Lets now derive a simple condition that characteri'es the Aolden 9ule

level of capital. 9ecall that the slope of the production function is the

marginal product of capitalMPK.The slope of thek

? line is.>ecause these two slopes are e)ual at k?gold, the Aolden 9ulecan

be described by the e)uation$MPK = .

@t the Aolden 9ulelevel of capital, the marginal product of capital

e)uals the depreciation rate.

/eep in mind that the economy does not automatically gravitate toward

the Aolden 9ulesteady state. ;f we want a particular steady3state capital

stock, such as the Aolden 9ule, we need a particular saving rate tosupport it.


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The basic +olow model shows that capital accumulation, alone,cannot explain sustained economic growth. igh rates of saving

lead to high growth temporarily, but the economy eventually

approaches a steady state in which capital and output are constant.

To explain the sustained economic growth, we must expand the

+olow model to incorporate the other two sources of economic

growth.

+o, lets add population growth to the model. Bell assume that thepopulation and labor force grow at a constant rate n.


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10ChapterSeven

Like depreciation, population growth is one reason why the capital

stock per worker shrinks. ;f n is the rate of population growth and

is the rate of depreciation, then "5 n#kis 1re!'2eveninvestment- which is the amount necessary

to keep constant the capital stock

per worker k.

Investment-1re!'2even

investment

,!pit!l

per wor'er-k

k.

3re!'2even

investment-( n)

Investment- sf(k

&or the economy to be in a steady state,

investments f(k# must offset the effects of

depreciation and population growth "5 n#k.Thisis shown by the intersection of the two curves. @n

increase in the saving rate causes the capital stock

to grow to a new steady state.

&or the economy to be in a steady state,

investments f(k# must offset the effects of

depreciation and population growth "5 n#k.Thisis shown by the intersection of the two curves. @n

increase in the saving rate causes the capital stock

to grow to a new steady state.


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16ChapterSeven

Investment-1re!'2even

investment

,!pit!l

per wor'er-k

k.$

Investment- s f(k

( +

n$)k

@n increase in the rate of population growth shifts the line

representing population growth and depreciation upward. The new

steady state has a lower level of capital per worker than theinitial steady state. Thus, the +olow model

predicts that economies with higher rates

of population growth will have lower

levels of capital per worker andtherefore lower incomes.

k*/

( +

n/)k

@n increase in the rate

of population growth

from n1to n2 reduces the

steady3state capital stock

from k?1to k?2.

@n increase in the rate

of population growth

from n1to n2 reduces thesteady3state capital stock

from k?1to k?2.


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1:ChapterSeven

The ch!nge in the c!pit!l stoc' per wor'er is:k i * (+n)kThe ch!nge in the c!pit!l stoc' per wor'er is: k i * (+n)k

4ow- let5s su1stitute "f(k)for i: k (sfk) * (+n)k

This e)uation shows how new investment, depreciation, andpopulation growth influence the per3worker capital stock. Cew

investment increases k, whereas depreciation and populationgrowth decrease k. Bhen we did not include the DnE variable in our

simple versionFwe were assuming a special case in which the

population growth was .

4ow- let5s su1stitute "f(k)for i: k (sfk) * (+n)k

This e)uation shows how new investment, depreciation, andpopulation growth influence the per3worker capital stock. Cew

investment increases k, whereas depreciation and populationgrowth decrease k. Bhen we did not include the DnE variable in our

simple versionFwe were assuming a special case in which the

population growth was .

h d h i i ff f i h i l


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2ChapterSeven

;n the steady state, the positive effect of investment on the capital per

worker Gust balances the negative effects of depreciation and

population growth. nce the economy is in the steady state,

investment has two purposes$

1# +ome of it, "k?#, replaces the depreciated capital,

2# The rest, "nk?#, provides new workers with the steady state amountof capital.

,!pit!lper wor'er- k

k.k.6

The Ste!dy St!te

Investment-sf(k)

3re!'2even Investment- + n) k3re!'2even investment-

+

n6)k

An incre!se in the r!te

of growth of popul!tionwill lower the level of

output per wor'er"

sf(k)


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21ChapterSeven

7;n the long run, an economys saving determines the si'eof kand thusy.7The higher the rate of saving, the higher the stoc'of capital

and the higher the level ofy.7@n increase in the rate of saving causes a period of rapid growth,but eventually that growth slows as the new steady state is

reached.

,onclusion$ although a high saving rate yields a highsteady3state level of output, s!ving 1y itself c!nnot gener!te

persistent economic growth"

,onclusion$ although a high saving rate yields a high

steady3state level of output, s!ving 1y itself c!nnot gener!te

persistent economic growth"


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22ChapterSeven

+olow growth model

+teady stateAolden 9ule level of

capital

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