New Final Growth

8/7/2019 New Final Growth

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Annual Average Growth Rate, g, for Y from

1960 to 1997 comes from:

( Y1997/Y1960 ) = (1+g)1997-1960

Thus, g = ( Y1997/Y1960 )(1/37) - 1

Or: g = exp{(1/37)( ln(Y1997) - ln(Y1960 ) )} - 1

Rule of 70:If Y grows at g percent per year, then:

The level of Y Doubles every 70/g years


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t


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Table 1.1


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The world income distribution is highly

skewed About 2/3 of the world population has at most

20% of the GDP per worker of the US

About 1/10 of the world population at least

80% of the GDP per worker of the US


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The world income distribution is changing

over time

The bulk of the worlds population issubstantially richer today than it was in 1960.

The fraction of people living in poverty has

fallen since 1960.

A major reason for these changes is therecent economic growth in China and India

which together account for 40 percent of the

world population.


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In addition to the level of income rising

over time, the growth rate of world incomehas also been rising over time

Note the ratio or logarithmic scale in thefollowing graph


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In contrast to world income growth rates

that are accelerating, the growth rate ofincome for the US has not been growing at

a faster and faster rate over time

But the US growth rate has tended to be

positive over time and has been roughly

constant over the long run


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Fig. 1.4


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Growth Theory


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Chapter 2

The Solow Model

Norton Media Library

Charles I. Jones


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The Production Function


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A production function describes the way we canuse inputs to produce outputs technically speaking, a production function tells us the

maximum amount of output we can produce for everycombination of inputs

An equation for the production functionY=AF(K,L)

where :

F is some function of A, K and L,

K=stock of capital,L=labor

and A is productivity, which accounts for all

other factors that may be involved


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For a production function

Y=AF(K,L)

we typically assume:

Positive marginal products

when A rises Y rises

when K rises Y rises when N rises Y rises

Diminishing marginal products for K and N

As K gets larger, an additional unit of K causes Y toincrease by a smaller amount

As N gets larger, an additional unit of N causes Y toincrease by a smaller amount

But as A rises, Y rises proportionally

no diminishing marginal product here


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These conditions on marginal productstranslate directly into the derivatives on the

production function Positive marginal products

First derivative of F with respect to K is positive

First derivative of F with respect to L is positive

Diminishing marginal products Second derivative of F with respect to K is negative

Second derivative of F with respect to L is negative

It is easy to show that any percentage change in

A has the same percentage effect on Y in thisproduction function

The elasticity of Y with respect to A is 1


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We like to compare countries in terms of

output per capita, so divide both sides of the

equation by L

Assume L is population, not number of

workers, and for now dont worry about the

distinction (Y/L) = (1/L)F(K,L) = F(K,L)/L


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A common assumption in economics is that

the production function has constant returns

to scale (CRS)

That means if we double the inputs we will

double the amount of output that we can

produce CRS has a certain intuitive appeal

If we own one factory and build a second factory

that has precisely the same facilities and we employ

identical workers in the second factory, we shouldbe able to produce twice as much output


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More generally, CRS means that a givenpercentage change in every input will change the

amount of output we can produce by precisely thatsame percentage

Assume F(K,L) is a CRS production function

Suppose z is some percentage change in the factors

CRS means that if Y=F(K,L) for particular values of K,L and Y, then zY=F(zK,zL)

This makes it easy to turn our production functioninto a per capita measure

Let z=(1/L):

(Y/L) = F( (K/L), 1 )


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It is convenient to redefine variables in percapita terms

y=Y/L output per capita

k=K/L capital per capita

And to use f to represent the new function

y = F(k,1) = f(k)

A graphical version of the production

function


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Properties of the f(k) production function Positive marginal product with respect to k

Increased k raises the maximum amount of output that can be

produced Upward sloping

Diminishing marginal product of k Increased k raises the amount of y that can be produced, but

does so at a diminishing rate Concave shape

These properties derive from our initial productionfunction, F(K,N), which has positive anddiminishing marginal products for K and for Nseparately

Recall that F(K,N) has constant returns to scale (CRS)when K and N each change proportionately this feature was used to derive the production function in terms

of output per person and capital per person


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A popular example of a production functionis the Cobb-Douglas production function

Y=AKL1- with 0 < < 1where Y = output, K = capital, L = labor,

and A is productivity.

We need this productivity term in a productionfunction since there are times when twoeconomies use essentially the same amount ofthe factors of production, K and L, and yetproduced substantially different amounts of Y

The assumption that is bounded between zero andone comes from the marginal product being positiveand diminishing


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HomeworkShow that the Cobb-Douglas production

function has:

A. Positive marginal product of capital

B. Positive marginal product of labor

C. Diminishing marginal product of capital

D. Diminishing marginal product of laborE. Constant returns to scale.


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This production function can be converted

into y as a function of k by dividing it by L

y = Y/L = (1/L) AK

L1-

= L-1AKL1-

= AKL1- L-1

= AKL1--1

= AKL-

= AK(1/L)

= A(K/L)

Therefore, we get y = Ak


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For the Cobb-Douglas function The marginal product of y with respect to k is equal to

the derivative of y with respect to k:

Ak-1 Assuming A and k are positive, in order for marginal product

of y with respect to k to be positive, must be positive number

Using our rules for taking derivatives, the derivative ofthis marginal product with respect to k is:

(-1)Ak-1-1

OR (-1)Ak-2

Since A, k and are positive numbers, the only way formarginal product to be diminishing is if

(-1) < 0Which means < 1


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Graphing the Cobb-Douglas function


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The Basic Solow Model Solow won the Noble Prize primarily for his

contributions to Growth Theory

His model is the foundation of all modern

growth theories


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The simplified Solow model

Accumulation of physical capital, K, is animportant feature in this model Capital is built by investing in capital goods.

Each dollar of investment, I, builds an additional dollarof capital. This means the capital stock will increase.This suggests

K=I

Where K is the change in the capital stock Capital increases if we invest

This simple equation implies the capital stock willnever fall since I can never be negative. But thatimplication is not true empirically.

What is missing is the fact that capital depreciates andthat a lot of investment is done to replenish depreciatedcapital. Let D=depreciated capital, and the equationbecomes:

K=I-D


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Capital depreciates by

wearing out from use or age

becoming technologically obsolete (i.e. out of date)

Since we model variables in per capita terms to

compare across countries, divide all variables in

the equation by L

Redefine variables so that lower case letters denote

per capita values:k=i-d

For now assume that L is fixed not growing

I

L L L

(!


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Assume that investment is a fixed share of

income: =i/y noting that I/Y=i/y and

that 0 < < 1

Assume that capital depreciates at a constant rate:

d=k

(from D= K and then dividing by L ) Assume a general form for a production function:

y=Af(k)

K

K

K


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Putting all of these assumptions into the equation for

capital accumulation ( k=i-d )

We obtain:

This equation tells us how the capital stock k is

determined given specific values of , A, and

parameters in the production function (e.g. in a

Cobb-Douglas production function).

It is convenient to analyze this model graphically (To

make my equation line up with the following graph let

A in the equation be 1, but this is only temporary itwill be very important to allow for changes in A)

k A f (k) k ( ! K H

K


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The graph plots three equations

Investment per capita:

Depreciation per capita:

Output per capita:

The gap between the investment per capita line and

depreciation per capita tells us how much the capital

per capita will change

(k)K

kH

(k)


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If the levels of , A, and the parameters in theproduction function ( if its a Cobb-Douglas productionfunction) are constant, these 3 curves will be fixed that is

they will never moveAn important implication of these assumptions is that

the economy eventually settles down to a steady state levelof k - and consequently to a steady state level of y as well.

Why?

If we start out with k>kss depreciation exceeds investment and thecapital stock falls

If we start out with k


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Output is determined by the production function

and the level of capital

Output follows movements in the capital stock

At this point, nothing else is changing in the production

function

When capital settles down to its steady state, output

reaches its steady state, yss, which is determined usingthe production function:

yss = Af(kss)


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Homework Solve for the steady state values of y and k

in our Solow model, using the Cobb-

Douglas production function.


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The parameters that we have assumed are constantneed not always be constant.

For example, may change because:

Policymakers implement a policy to stimulate more investmentOR

More savings becomes available to a country, which induces moreinvestment in that country

Why? Investment is financed by savings (I=S-CA)

A or also may change for various reasons (we willconsider this later)

What happens when the investment rate (investmentshare of income) increases in our model?

That is to say, when rises from to curves move and what happens as a result?

K

K

2

KK1

K


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When considering how the economy responds to

some change in a parameter or an exogenous

variable, it is useful to assume the economy startsoff in an equilibrium position

Then we can analyze

If a change in some parameter (or exogenous variable) pushes the

economy away from equilibrium

And if so, how this works in the economy and in what way are

variables affected

If the economy eventually returns to equilibrium

And if so, how it does this and how is the equilibrium position of

the economy affected

In the picture the initial position of the economy is

ss1 ss1k k and y y! !


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An increase in the investment rate causes

the steady state value of k to increase

In the graph the new steady state k is kss2

kss2 is greater than kss1

So k must rise from kss1 and eventually reaches

kss2

Once k reaches kss2 it will stop rising and stay

there (until, and unless, something else

happens)


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Similar behavior is found for y

The initial steady state is yss1.

As k rises so does y

This comes from the production function and the fact that k has

a positive marginal product

k stops rising when it reaches its steady state

at that point y also reaches its new steady state labeled yss2 inthe graph

An important implication: A one time increase in the

investment rate will not make the output per capita

grow faster in the long-run since it settles down to asteady state value in this model


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The previous analysis explained what happens if theinvestment rate rises for a country

But it can just as well describe what happens if two countries have

different investment rates Suppose Country 1 has an investment rate of

Country 2 has an investment rate of and

Our graphical model predicts that a higher investment shareyields a higher steady state level of output per capita

What does the cross-country evidence tell us?

1K2K

2 1K " K


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At this point in the course, we dont knowhow to measure steady state output per

capita But we do know how to measure output per

capita in a given year

Also, we can measure the averageinvestment share over a period of time

When we plot actual y in a given yearagainst the average investment rate

(investment share of GDP) we find there isa significant positive relationship


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This positive relationship between output-per-

capita and the investment share across countries is

predicted by the Solow model But we also saw that a permanent increase in the

investment share WILL NOT lead to persistent

growth in y or k

Eventually k and y settle down to a steady state andgrowth discontinues

The model still needs something to make y and k

both grow over a long period of time as is

observed in most countries In order to get y to grow in the Solow model,

productivity (A) must grow over time


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Some may think y and k could grow forever if theinvestment share were to continually increase over time

This is NOT reasonable for at least 2 reasons: First, many of the worlds economies have had output per capita growconsistently over long periods of time even while their investmentshares have stayed roughly constant

Second, the investment share, I/Y, can not grow forever

If it kept rising eventually all a countrys income would go toward

investment leaving nothing for consumption, This would make peoplevery unhappy, in fact dead, since we all need a minimum amount ofconsumption just to survive

And once again, after this share stops rising a country will eventuallyarrive at a steady state and stop growing. There could be no persistentgrowth.


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STOPPED HERE FOR

MIDTERM #1


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Homework: Suppose that falls from 1 to

2

in our model (1

>2

). Hold all other

parameters as exogenous. Explain what

happens to y and k over time. Do y and k

grow forever or will they eventually arrive

at a steady state? How does the new steadystate compare to the initial steady state?

How would have to change in order for y

to grow forever? Is it reasonable to thinkthat the depreciation rate can continue to

change like this over time?


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Homework: Suppose that A rises from A1 to

A2

in our model (A1


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The conclusion from class and the two most recenthomework problems is: Rising investment share can not cause continuing growth of y

and k for reasons cited a few pages back The only way that changes in may cause y and k to grow

indefinitely is if the depreciation rate continues to fall. But theproblem with this hypothesis is that

Depreciation rates do not exhibit any tendency to fall over time

The depreciate rate can not fall below zero. This means that eventuallythere would come a time when depreciation would stop falling - thus kand y would eventually stop rising

The only plausible way for y and k to grow indefinitely is if Acontinues to rise


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Unfortunately, Solow assumed productivity

grew at some exogenous rate Exogenous means outside of or external to thesystem. Hence, Solows growth model doesntexplain what determines growth in productivity orthe economy

While Solows model does not explain whyproductivity grows it has been used byeconomists to try to explain cross-countrydifferences in income per capita. We willalso examine some of these cross-countrydifferences.


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An advantage of the Cobb-Douglas production

function is that it allows us to get an analytical

solution for all variables of interest in the model

Putting Cobb-Douglas in to our capital accumulationequation yields

If there is no growth in A, then k settles down to a

steady state: k=0

k A k k E( ! K H

An Analytical Solution to Solows Model


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When k=0:

Solve for the kss:

ss ssk k 0EK H !

ss ssk kEK ! H

1/(1 )

ssk

EK ! H

ss

ss

k

kE

K !

H1

sskE K !

H


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Then we can solve for yss: : ss ssy kE!

/(1 )

ssyE EK ! H

1 / ( 1 )

s sy

E E K ! H

/(1 ) /(1 )(1 )/(1 ) /(1 ) 1/(1 )

ssyE E E E

E E E E EK K ! ! H H


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Our final result is:

Suppose we have two countries, Country i and

Country j. Each will have its own steady state, yi

and yj and the ratio of the steady states is:

/(1 )

1/(1 )

ssy A

E E

E K ! H

i i

i

j j

j

/(1 )

1 /(1 ) ii

ii

/(1 )j 1 /(1 ) j

j

j

Ay

yA

E E

E

E E E

K H

! K H


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The previous equation allow each country to

potentially have a unique value , , A or

Suppose the investment share is the only thing that

differs across countries.

Then Ai=Aj, i=j and i=j,

And our general result simplifies as follows:

i i

i

j j

j

/ (1 )

1 /(1 ) i/ (1 )

i

ii i

/ (1 )

j j1 /(1 ) j

j

j

Ay

yA

E E

EE E

E E

E

K H K

! ! K K H

K


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Numerical examples

If

then

If

then

The ratio of income between richest and poorest countriesis more that 50 today

This evidence suggests that cross-country variation in investment

rates will not be sufficient to make the Solow model explain cross-

country differences in income per capita

i

.2 (20 ) and .05 (5 )K ! K !

1 / 2iy 4 2y

! !

1 / 2iy 1 6 4y

! !

i .32 (32 ) and .02 (2 )K ! K !


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Lets look at a broad selection of countries to seehow well or poorly the model performs

We use a graph and a large cross section ofcountries that plots

actual data yi/yj along with

predicted yi/yj which equals:

(this prediction comes from assuming countries onlydiffer in terms of the investment share and =1/3 for allcountries)

1/ 2

i

j

K K


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If this simple version of Solows model is correct,

all the data points will be on a straight line with a

slope of 1 that goes through the origin The model is perfectly correct if the only thing thatdiffers across countries is the investment share

We dont expect any macroeconomic theory to be perfectly

correct because macroeconomies are rather complicated social

systems How far the data points are from this line yields some

indication of how well or how badly the simple version of

Solows model actually works

So putting that specific line on the previous graph


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Th US i b t ti th li


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The US is, by construction, on the line Output per capita in US relative to output per capita in the US is always

equal to 1 Nothing is learned here, but this helps us draw the line

What do we learn from this picture? First, there is a positive relationship between the investment

rate and income per capita, as is predicted by the model Countries with higher investment rates have higher GDP per capita

Of course, we already saw this in a previous graph

Second, the Solow model with only the investment ratediffering across countries does NOT do a good job ofexplaining the observed differences in output-per-capita

Ireland is the only other country on the line (the US is also, but it is byconstruction - incomes are measured relative to the US level)

Different investment rates seem to explain the difference in output-per-

capita between Ireland and the US Except for Luxemburg which is well above the line, all other countries

are below the line, and most of them are well below the line

Interpretation: For nearly all of the countries, the model predicts thatoutput-per-capita should be much higher in countries than is actuallyseen when we examine the cross-country data


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So the model in its current form does not lookvery promising

It doesnt explain why countries grow for an extendedperiod of time

It doesnt explain most of the variation in income percapita that we observed across countries

But dont give up totally on Solow We can reexamine the model under alternative

assumptions

The key conclusion from the last graph: Cross-

country variation in investment rates, by itself, isunable to explain some of the most importantfeatures in the data


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Our current simplistic version of Solows model

does have an interesting property: The farther a country operates below its steady state,

the faster capital-per-worker and output-per-worker will

grow

And similarly, the farther a country operates above itssteady state, the slower capital-per-worker and output-

per-worker will grow

that property turns out to be a general property of

the Solow model under all kinds of conditions It is also a feature of many other growth theories


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To see this point, return to the capital accumulation

equation

Divide both side of the equation by k

Define the growth rate of k as:

Then

1k

kk

E(

! K Hkk

k

(!

k k k

E

( ! K H

1k kE! K H


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We can plot the function that determines growth in k. It isconstructed from two terms

1. The function of k:

Which is

A declining function of k

(negative first derivative with respect to k,

which stems from diminishing marginal product of k) A convex function of k

(positive second derivative with respect to k)

2. the depreciation rate

(a straight line since it is not a function of k) The difference between the two terms gives us the growth

rate of k

1A kE

K


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First notice that the graph provides another way of

seeing why kss is a steady state

k rises when kkss

This graph also shows that the farther k is below kss,

the larger is the growth rate of k

And similarly, the farther k is above kss the slower

is the growth rate of k

in this case, growth of k is negative when k is above kss


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Output per capita is determined by the production function,

and so it behaves like capital per capita

Recall the Cobb Douglas production function

It turns out that if y is determined by this production function, one can

show that the growth rate of y is equal to the growth rate of A plus

time the growth rate of k:

The farther y is below yss, the larger is the growth rate of y

The farther y is above yss the slower is the growth rate of y in this case, when productivity growth is zero, the growth of y is negative

when y is above yss

y AkE!

y A k! E


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We can derive 3 important conclusions from thisanalysis (these can also be are found in a more general

Solow model that would allow more than just theinvestment share to vary)

1. If two countries have the same levels of productivity,

depreciation rate and investment rate and they both usethe same production function, but one country starts outat a lower level of k and y, that country will grow at afaster rate (until both countries get to the steady state)

This result is a direct implication of the last graph

This illustrates a concept known as convergence holding allparameters the same across economies, poorer economies willgrow faster than richer economies


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2. If two countries start at the same level of y, but oneof the countries has a higher investment rate, thenthat country will grow at a faster rate The country with higher has higher steady state levels

for k and y.

If both countries start at the same levels of k and y, the one thathas farther to go (higher steady state levels of k and y) must growat a faster rate to make it to these higher levels of k and y (untilboth countries get to their steady states)

In the following graph, country 2 has a higher investmentrate than country 1:

and so country 2 is growing faster (vertical gap betweenpoints b and a) than country 1 (which is in steady state)

K

2 1K " K


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1

1 A k

E

K

1

2 A kEK

2 1

K K

c

a

b

kak ck

Growth rate of k


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3. When a country raises it will begin to grow a

faster rate The higher investment rate causes k to grow faster whichof course causes y to also grow at a faster rate

This is seen in the previous graph, by assuming a countrys

growth rate rises from to

Hence, the growth rate for k is initially equal to the

vertical gap between points b and a in the previous graph

However, as k rises the growth rate begins to slow (the

vertical gap decreases) until the economy reaches the new

steady state (kc) at which point growth stops

Note: We still are assuming no productivity growth

2K

K

1K


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Savings and Investment Savings and investment are intimately connected

by an important national income accountingidentity

I = S - CA

(I/Y) = (S/Y) - (CA/Y)

If CA (the current account) is equal to zero (mustbe zero for a closed economy) then savings isequal to investment. Furthermore,

(I/Y) = (S/Y)

investment rate = savings rate

B t i l CA i t ll d d th


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But in general CA is not small and so we need the moregeneral result:

(I/Y) = (S/Y) - (CA/Y)

Feldstein and Horioka (1980) showed a very closecross-country relationship between the investment rate(I/Y) and the savings rate (S/Y)

This is called the Feldstein and Horioka puzzle because:

International capital markets should funnel savings toward the highestrate of return, adjusted for risk

The fact that people do not invest that much in these high yieldingforeign assets implies a level of risk aversion that is unbelievably high

Hence, the positive relationship between investment

shares and savings rate implies a positive relationshipbetween savings rates and income-per-capita, which isvery evident in aggregate data


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Using our definition of national savings

S=Spvt+Sgovt

In the equation from before yields:

OR

Thus for the domestic investment rate to increase, at least one ofthe following must occur:

An increase in the private savings rate;

An increase in the government savings rate;

An increase in the rate at which net foreign savings flows in (the netinflow of foreign savings is equal to CA)

pvt govtS +SI CA= -

Y Y Y

pvt govtS SI CA= -

Y Y Y Y


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The empirical evidence relating savings

rates to income per capita provides somesupport for Solows model But other

interpretations are possible

Frequently, alternative structural explanations

for a set of empirical evidence exist, and many

times an alternative works in the opposite way

to the first theory

H d l ti th ff t


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How does population growth affect

variables in the Solow model?

For a fixed level of capital (K), a higher population

(L) reduces capital per person (k) and so output per

person (y) would also be smaller

This effect of population on capital is known as capitaldilution (As L gets larger a given stock of capital must

be shared by more workers)

for output per capita to reach a steady state, capital

per person must also reach a steady state Thus if population grows, capital must grow at the same

rate as population growth to maintain a steady state

capital per person


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We need to modify the capital accumlation

equation to allow for population growthsince we derived the earlier version

assuming there was no population growth

First, we will show that the earlier capital

acummulation equation doesnt work

properly and then we will repair it


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Recall our capital per person accumulation equation

(assuming fixed population)

We know that by dividing both sides of this

equation by k we get an expression for the growth

rate of k

k k f (k) / k

k

(! ! K H

k f (k) k ( ! K H

S h h i d h d i i


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Suppose that the investment rate and the depreciation rate are

equal to zero. In that case the last equation says that k grows at

the rate zero. If k growth is zero then k is at some constant

value.

Recall an even earlier equation: K=I-D

If the investment rate = 0, investment = 0

If the depreciation rate = 0, depreciation = 0

Thus K=0, which means K grows at a rate of zero (divide both side ofthe equation by K, to get K/K

K


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Heres the problem: k=K/L

If K is NOT growing while L is growing then k must be shrinking ---- itcan NOT reach a positive steady state level

We need to fix our k/k equation

The fix is simple and intuitive: If and are zero then K mu

also be zero. And, if L grows at a constant rate, n, k will be

shrinking at that rate, that is k will grow at the rate n

This suggests that the fix obtains by subtracting the population

growth rate from the right side of the previous growth rate of k

equation

K


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We change the capital accumulation

equation in specific ways It is convenient to change capital accumulation from a

difference equation to a differential equation (this

means changing from discrete time to continuous timemodel)

From calculus we know that:

Dividing by some finite number doesnt change that, so:

K dK

as t 0, lim t dt

(

( p !(

1 K 1 dK as t 0, lim

K t K dt

(( p !

(

Thus our original capital accumulation equation


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Thus our original capital accumulation equation

becomes:

We, also, will modify our Cobb Douglas production

function to have labor augmenting productivity

Labor augmenting means that in the production

function A multiples L. AL is sometimes called effective labor input.

It adjusts labor effort by the productivity of labor

d (A , , L )d t ! K H

1(A, , L) (AL)

E E!


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Putting this all together yields our

new capital accumulation equation

But we want to put things in terms of output per capita and

given constant returns to K and L that means when the

production function is divided by L that will make of output

per capita a function of productivity and capital per capita.

Thus we would like to put our capital accumulation equation

in terms of capital per capita.

1dK K (A L ) K dt

E E! K H

We will assume for convenience:


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We will assume for convenience:

Population (or the work force) grows

at a constant rate

Where n is a constant equal to the growth rate

This is a differential equation that is easy to solve (if

you know something about differential equations) The solution is: L(t)=Loent

This is the equation for the level of L if it grows at aconstant rate

1 dLn

L dt

!


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How are capital growth and capital

per capita growth related? First, recall that k = K/L

We can take the logarithm of the last equation

Then take time derivative of this equation

And finally obtain

log(k) log(K) log(L)!

d d dlog(k) log(K) log(L)

dt dt dt!

1 d k 1 d K 1 d L

k d t K d t L d t

!


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Thus, we can take the production capital accumulation equation from beforedivide by K, and use the last equation plus the assumption that labor grows ata constant rate to get:

Or simplifying we obtain:

Note that the resulting equation:

Is the same as our previous equation for growth rate of k, except that we havean additional term, -n. We need that sort of adjustment for population growth,to make the analysis right in terms of k.

1 1 1 11 dk

K (AL) ( n) k A ( n)k dt

E E E E ! K H ! K H

1 11 dk

k A ( n)k dt

E E

! K H

11 d k 1 K ( A L ) K nk d t K

E E ! K H


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Thus we will see that there is a steady state for this

model too, and the condition is the same as before:

It comes essentially from the same graph, assuming all

the parameters, including productivity, stay fixed. In this case, the steady state is the solution for kss to:

Which has the following solution:

1 1

ss0 k ( n)E E! K H

dk0

dt!

1/(1 )

ssk

n

EK !

H


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To prove that the model reaches a steady state, first

multiply the growth of k equation by k:

And note that from our previous Cobb-Douglas

production function for Y, it is easy to show that:

Thus we can graph the dynamic equation in k, assumingall parameters (including A) are fixed

(note that =d in the graph, both refer to the

depreciation rate)

1dk k A ( n)k dt

E E! K H

1y k AE E!


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We can add the production function toanswer questions about what happens to

output per capita and consumption percapita

Output per capita comes directly from theproduction function

Consumption per capita comes from: y i,which is the gap between the productionfunction and the investment line


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We can use this model (or any

model) to do experiments The first experiment examines the effects of

an increased investment rate


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The model predicts that countries with

higher investment rates will tend to have

higher levels of output per capita (andhigher levels of capital per capita)

There is some empirical support for the

hypothesis that countries with higherinvestment rates have higher output per

capita


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A second experiment examines the

relationship between population growth rate

and output per capita


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The model predicts that countries with higherpopulation growth rates will tend to have lowervalues for output per capita (and lower capital per

capita) We could also examine the effects of an increase in thedepreciation rate, using the previous graph. The resultsare the same, both qualitatively and quantitatively. Butwe dont expect there to be great differences indepreciation rate across countries.

There is some empirical support for the hypothesisthat countries with higher population growth rateshave lower output per capita


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We can take the previous capital accumulation equation

and divide by k:

This is the growth equation for k, similar to one we

derived before when there was no population growth Compared to the earlier graph, the only differences are that

we used a differential equation and we allowed for population

growth

It looks very similar and has similar implications

1 11 dk k A ( n)k dt

E E! K H


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This model maintains the key conclusions we mentioned before


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y

in a model without population growth

- Convergence

- Transition dynamics

If two countries have the same levels of productivity,

depreciation rate, investment rate and NOW POPULATION

GROWTH, they both use the same production function, but onecountry starts out at a lower level of k and y, that country will

grow at a faster rate (until both countries get to the steady state)

Another way of looking at this graph: The farther below (above)its steady state the faster (slower) an economy will grow


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You can also show that: If two countries start at the same level of y, but one of

the countries has a higher investment rate, then that

country will grow at a faster rate initially

When a country raises it will begin to grow at a fasterate (though not forever because it will again reach a

steady state in the long run)K


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This version of Solows model maintains a keylimitation: There is no growth in k or y in the long-run

That is the implication of these two variables settling downto a steady state

You might conjecture that a continual decline in thepopulation growth rate could be a plausible explanation forpersistent increases in output per capita. But it is not.

Eventually n would have to become negative for this decliningpopulation growth rate story to explain continual growth in y. And

once population growth becomes negative the level of population isheaded to zero

Also, many countries experience growth in output per capita with nochange in population growth rate

Now we will allow for productivity growth


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If productivity grows at a constant rate, g, then, we can write this

as the following differential equation

As before, we can solve this sort of differential equation for the

level of A: A(t)=Aoegt

Productivity growth occurs whenever there is a new idea about a

product innovation or a new idea about how to produce a product

more efficiently (e,g, producing the same goods at a lower cost or

producing more goods at no additional cost)

We will see that productivity growth is the only thing that yields

a plausible explanation for growth in y, k or any other per capita

variables

1 dA gA dt

!

It would be nice if we could put the model in


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It would be nice if we could put the model in

terms of variables that do not grow. Then we

might be able to obtain variables that achievesteady states.

We saw previously that by dividing by labor

achieved steady state when that was the only

thing growing. Since we now have labor andproductivity growing, we can try to divide by

both to see if that will give us a transformation of

variables that achieves steady state.

Recall our newest production function


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p

If we divide by AL we get:

Define

Then the production function becomes y kE! %%

1Y K (AL) K K

AL AL (AL) AL

EE E E

E

! ! !

1Y F(A, K, L) K (AL)E E! !

K

k AL!%Y

y AL!%

these transformed variables are defined as:


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these transformed variables are defined as:

output per effective labor unit:

capital per effective labor unit:

where AL measures effective labor units, being the

quantity of labor adjusted for productivity

Jones likes to call the capital technology ratiok

y

k

Now, return to our capital accumulation equation


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, p q

First lets use a new notation that is common for

growth modeling and common in mathematics.

For any variable Z, define:

That means:d

d t

!&

d ZZd t

!&

d

Yd t ! K H

Using this definition in the capital accumulation


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equation

Then dividing by K yields:

Note that

And so we obtain:

1

YY yA L k

KK k

A L

E ! ! !% %%

K Y

K K! K H

K Y K! K H

1K k

K

E ! K H%

We need an expression for the growth rate of capital in

terms of capital per effective labor unit.


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p p

Given that:

We know we can take logarithms:

Then take time derivative of this equation

And finally obtain

log(k) log(K) log(L) log(A)! %

d d d dlog(k) log(K) log(L) log(A)

dt dt dt dt

! %

1 dk 1 dK 1 dL 1 dA

dt K dt L dt A dtk

!

%

%

Kk

AL

!%

And the last equation becomes:

k K

%


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And plugging this result into:

Yields:

Which can be written as:

1K

kK

E

! K H%

1k k (g n )

k

E ! K H%

%%

k Kn g

Kk

!

%

%

k k (g n )k E! K H% % %


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Homework Solve for the steady state values of capital and output per

effective labor unit in our model with population growthand productivity growth, assuming a steady state occurs.(This means solve for the values of endogenous variables

in terms of parameters and exogenous variables) Calculate the steady state ratio of K to Y (called theaggregate capital to output ratio). Is it the same as theratio of k to y or is it different?

On a ratio scale, graph steady state values of K and Y

(hint the ratio scale and the logarithm scale are thesame). How fast are these variables each growing?

On the same ratio scale, graph steady state values of kand y. How fast are each of these variables growing?

Does this model achieve a steady

state?


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state?

Yes The two lines are fixed that is they stay in one

position for fixed values of parameters of n, g, ,

and

There is only one positive steady state value for

And starting from any positive level ofthe dynamics will push the economy in the direction

of this steady state.

K

K

K

k

k


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Now we can examine an important question

that weve asked before with these models:

What happens to the economy when theinvestment rate increases?


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An increase in the investment rate shifts the

investment per effective labor unit line up

Capital per effective labor unit begins to rise sinceinvestment exceeds depreciation, with both of

these now measured per effective labor unit

Finally, the economy settles down to a new steady

state level of capital per effective labor unit

We can also analyze this model using the growth rate


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We can also analyze this model using the growth rate

of capital per effective labor unit form of the

equation

Graphing this equation yields the following whichillustrates how an increase in the investment rate

causes fast growth of capital per effective labor unit

initially,

But, as the economy gets closer and closer to the

steady state, the growth rate of capital per effective

labor unit slows down

1k k (g n )k

E ! K H&


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And using the last equation it is easy to solve for


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And using the last equation it is easy to solve for

the steady state value of capital:

And steady state output per capita comes from

putting the previous solution into the production

function:

1 /(1 )

sskg n

E K

! H %

/(1 )

ssyg n

E

E

K! H %


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This model yields the same predictions as

before:

The farther an economy is below its steady statethe faster it grows

The farther an economy is above its steady state

the slower it grows

We now examine the growth of


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We now examine the growth of

output per capita

We can get output per capita from the definition of


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output per effective labor unit. Since

Multiplying both sides by A gives: If we divide by

AL we get:

This means that:

YyAL

!%

Yy yA

L! ! %

y y A

y y A!

%

%

And from our production function, in per effective

l b i


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labor units:

We know that:

So finally,

y kE!

y A k

y A k! E

&&&

y k

y k! E

&&

Using the constant growth assumption for A, we get:


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This equation tells us that:

In the steady state y grows at the rate of g

When the economy is below its steady state, y grows at a rate greater

than g

And when the economy is operating above its steady state y is grows

at a rate less than g

Thus, when the investment rate went up, y grows faster than g

for a while, but eventually the economy returns to steady state

and y growth rate returns to g

y kgy k! E

%

%


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Recall that:t t ty y A! %


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Using our expression for A, we obtain: :

This equations tells us how y behaves over time inthis model:

Changes in output per effective labor unit shift the level of

y, but dont affect its long-run trend which is given by the

trend in A

Changes in A0 will also shift the level of y, but not its

trend line

This idea inspired what became known as Real Business Cycle

models (the first generation of those models)

gt

t t 0y y A e! %

t t ty y

Thus we see that an increase in the investment rate

ill th l l f t i b t t ff t it


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will cause the level of y to increase, but not affect its

long-run growth rate The following graph illustrates precisely what would

happen in this model, plotting how the log of y

changes over time:

t t 0log y log(y ) gt!


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Once again the model predicts


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Once again the model predicts

convergence and transition dynamics The economy converges to a steady state for

fixed values of the parameters

In the steady state real per capita variables growat the rate g

The farther below/above steady state the

faster/slower the economy will grow.


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Suppose there are two economies named

InitiallyBehind (IB)

InitiallyAhead (IA) We know that IB will grow faster in this

model

Transitional dynamics are the driving forcebehind convergence


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How well do the data support the


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How well do the data support the

convergence hypothesis?

We have convergence when we look

h l f i f hi h


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at the sample of countries for which

we have long time series

For some countries we can obtainreasonably reliable estimates of GDP and

population dating back at least as far as the

late 1800s


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We also find convergence when we

l k t l ti l ll d l d


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look at relatively well-developed

countries in the postwar period

This is a broader selection of countries, as well

as a more recent and more reliable sample ofdata for the countries


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However, when we broaden the set

of countries to include all countries


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in the postwar period, there isabsence of convergence

Many of the countries that are much poorerthan the developed countries of the world are

growing slower than those developed countries


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Adding Human Capital to the model


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g p

It seems plausible that human capitaldifferences across countries may provide areasonable explanation for why countrieshave different performance levels

Specifically, a low level of human capital mayexplain why many poor countries show noevidence of converging

We also want to allow for differences in

population growth and investment rate acrosscountries


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Assume that a persons human capital augments


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their labor input (labor measured in terms of hours

or people) In that case, human capital enters the production

function like productivity

We multiply labor input, L, by human capital perperson, h

This change will lead us to redefine output pereffective labor unit as:

But, after that redefinition our analysis remainsessentially the same

Yy

h L A

!%


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So we can plot the predicted ratio of GDP

per capita versus the actual GDP per capita,

holding A to be the same for each country


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This is a standard problem with Solow type

models: They tend to over-predict how much

income per capita countries will have, relative to

US income per capita when we fix productivity inall countries to be the same

So what if we allow productivity to differ across

countries? We can use the equation to measure

productivity in each country


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When you allow productivity to be differentacross countries, and use the Solow modelssteady state to calculate it, you find

extremely wide variation in productivity in theworld

Productivity is highly correlated with incomeper capita

richer countries are more productive thanpoorer countries


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In fact, if you were to decompose cross

country output per capita differences into

productivity differences and differencesexplained by all the other factors in the

Solow model, productivity differences

account for most of the cross country

differences in y

Conditional Convergence


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Conditional Convergence

Our model allows each country to have its own

steady state depending on its own investment rate,

population growth rate and human capital.

Conditional convergence is the idea that countries

converge to their own steady state, and how far

they are away from their own steady state is what

determines how fast they are growing

We are assuming that g is the same for all countries

Conditional convergence holds up

fairly well in the data


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fairly well in the data

Growth rate since 1960 is plotted against1960y

y *


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Growth Accounting


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is is a technique for decomposing economicgrowth into contributions from various sources

More specifically, growth accounting decomposes

output per capita growth into contributions from

the various factors (e.g. capital per capita growth)and also from productivity growth

This last component is sometimes called

the Solow residual or

the contribution from multifactor productivity

And is sometimes also called our measure of ignorance since

it is the part of growth that we cant explain with measurable

changes in factors


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A simplistic example of how to do a

decomposition calculation


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In practice economists have tried to furtherdecompose growth into additionalcomponents such as

Labor composition

This is how labor may be shifting from productionto R D, or vice versa

Also, the capital per worker contribution can be

separated into IT (information technology)capital and other components


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Table 2.1

Growth accounting teaches us that the big changesin the growth rate of output per capita (or per


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worker) are largely the result of changes in the

growth rate of productivity From this accounting we observe the productivity

growth slow-down that plagued all countriesstarting in the early 1970s

This was particularly problematic for most of thedeveloped countries

And we can also see that strong productivitygrowth returned starting in the mid 1990s for the

US. The US benefited more from this productivity growth

than most other developed countries

Why did productivity growth slow


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down in the early 1970s? Historically, the slow-down has been attributed to

various sources

One that looked particularly promising was the

rapid rise in oil price shocks in the 1970s

In out model this amounts to a reduction in A0, or better

yet, a persistent decline in A0 which would lead to a

persistent slow down in the growth rate of output per

capita

The problem: In the 1980s there was no resurgence of

rapid productivity growth as oil prices fell dramatically

A variety of other explanations have also beenoffered . For example:


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Running out of ideas

Natural resource depletion

Excessive regulatory burden on firms

Reduced labor ability and/or reduced labor force

work ethic No single explanation seems a totally

satisfactory by itself

Each one may provide a partial explanation

But none of them, or all taken together, seem tobe a good candidate for explaining whyproductivity growth picked up in the mid-1990s

An explanation for both why the productivity growth slowed andwhy it then increased in 1990s: Technological revolution

The revolution is thought to be the expanding use of computersand information technology. The basic idea:


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Computers (and later, the Web) represent a general purpose technology

that can broadly affect how we produce and the types of goods we make Any new technology requires time to learn how to utilize it properly

This takes labor time away from production causes output to fall relativeto the amount of labor input

A larger share of labor is being devoted to learning rather than production

Economic historians, such as Paul David, have examined how

the introduction of new general purpose technologies (e.g. steamengines, electricity, etc.) in earlier periods coincided with adecline in output per capita that lasted for some period of time,but eventually in all those cases more rapid growth returned

This story can explain the resurgence of productivity growth in

the 90s, By then, people understood the new technologies wellenough to make significant and frequent improvements inproductivity. Thus caused output per capita to start growing at afaster rate

Endogenous Growth Models

E d th ll k th


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Endogenous growth usually make the

growth rate of productivity endogenous

However, some models generate growth by

having constant returns to scale for capital

in the production function This can be generated from learning by doing,

for example.

However, CRS for capital yields implicationsthat do not appear valid in the data

So Im going to ignore these models from now on

Romers model of Endogenous


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Growth


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The effectiveness of workers depends on

how many ideas there are in the economy

Assumption: The more ideas there are, the moreways ideas can be combined to create new ideas

OR the more ideas can be refined/extended to

create new ideas


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World population has been increasing since

the dawn of man (with some notable

exceptions like the Dark Ages)


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p g )

But while the population was rising thepopulation growth was at a very small rate,

until relatively recently in human history

Around 300 years ago the rate of output

growth started to increase and that coincides

with the increasing world population growth

rate

There is some evidence that the number of

new ideas has been growing rapidly,

Suppose new ideas are measured by the number


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Suppose new ideas are measured by the number

of patents US patents have increased dramatically over

time

But the growth rate of ideas has not been rising

Trust me it may not be evident from this graph


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However, the number of scientists has risen

dramatically over time

This implies that if the model is correct, the


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This implies that if the model is correct, the

growth rates for productivity and output percapita should have been rising during this time

period


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Romers model predicts that this rise in the

number of scientists and engineers will lead

to faster productivity growth


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There is no clear relationship between the

number of scientists and either the pace of

output growth in the developed world or the


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p g p

pace at which ideas are created. Since 1970 the dramatic increase in

scientists and engineers in the developed

world has not cause these countries to growat noticeably faster rates

This has suggested to Jones and others thatwe modify our growth model along certain

plausible lines


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Therefore, this version of the endogenous growth

model predicts that higher growth rates of

productivity and output will coincide with a higher

population growth rate


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population growth rate.

This is what the evidence shows, in contrast to the

first endogenous growth model which predicted

that higher level of population would lead to

higher growth rates in productivity and output. This implication of the model will be tested in

your (expected) lifetime.

World population growth will slow down substantially

within about 50 years maybe go to 0%. This model predicts productivity growth will slow

dramatically, maybe even stop growing

Schumpeterian models of endogenous

growth


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