1

UPF Macroeconomics I

LECTURE SLIDES SET 6

Professor Antonio Ciccone

2

Ideas and Economic Growth

3

Producing output versus ideas

• Ideas: non-rival, accumulable input

• Ideas: may be excludable (patents, secrecy) or not

• Ideas: producing them lowers current, but increases future output

4

Producing output versus ideas

• Question 1: what is the growth process for a given allocation of inputs between producing output and producing ideas?

Characterize the join evolution of ideas and output in the “spirit” of Solow

5

Producing output versus ideas• Question 2: how much of inputs is

allocated to producing ideas in decentralized equilibrium?

Difficulties:-In these models there are at least 3 inputs: capital, labor, and ideas-Holding ideas constant, our “reproduction argument” implies that there are at least constant returns to capital (K) and labor (L)-HENCE, there are increasing returns to K,L, and ideas! It took quite a while to develop a set of models (“toolbox”) where the decentralized dynamic general equilibrium could be characterized

6

1. A FRAMEWORK FOR ANALYZING GROWTH WITH RESEARCH AND

DEVELOPMENT

Quantity of output produced

fraction of total capital stock used in production

fraction of total labor force used in production

Y (t) (1 ak )K (t)

(1 al )A(t)L(t) 1

0 1

1 ak

1 al

taken to be exogenous; in the “spirit” of Solow

7

= level of technology = stock of ideas

ideas: non-rival inputs

= new ideas, which are created by using capital, labor, and old ideas in the RESEARCH AND DEVELOPMENT (R&D) process

A(t)

A(t)

8

Production of new ideas• Research and Development (R&D) technology

fraction of total capital stock used in R&D fraction of total labor force used in R&D

A(t) G ak K (t) , al L(t) , A(t)

A(t) B ak K (t)

al L(t)

A(t)

0 B,,,

ak

al

taken to be exogenous; this is in the “spirit” of Solow

9

Returns to scale to K and L in production of IDEAS could be increasing or decreasing:

• DECREASING: replicating inputs could lead to same discoveries being made twice

• INCREASING: doubling inputs could lead to more than twice the discoveries because of interactions among researchers (“the whole is more than sum of its parts”)

10

Also, what is the link between stock of ideas and new ideas?

• presumably : OLD ideas are useful for developing new ideas

• : doubling stock, doubles discoveries holding inputs L and K constant

• : effect of stock of ideas on creation less than proportional

• : effect of stock of ideas on creation more than proportional

0

1

1

1

11

: ideas keep growing at same rate even if resources allocated to R&D constant

growth of ideas accelerates when resources allocated to R&D constant

to keep growth of ideas constant, more and more resources must be allocated to R&D

1( )( ) ( ) ( )

( ) k lA t

B a K t a L t A tA t

12

2. GROWTH WITH RESEARCH AND DEVELOPMENT: THE CASE

WITHOUT CAPITAL

Quantity of output produced

Production of new ideas

Population growth (exogenous):

(1 )t l t tY a A L

t l t t l t tA B a L A Ba L A

0 B,,

/t tL L n

13

Growth of ideas ( ) gA(t)

gA(t)

A(t)A(t)

Bal L(t) A(t) 1

( )1 ( )

( )A

AA

g tn g t

g t

( ) 1 ( ) ( )A A Ag t n g t g t

14

CASE 1:

Balanced (constant) growth path

(0 ) 1

gA(t) 0 n 1 gA(t) gA(t)

gA *(t) gA*

n1

15

Is the BGP stable?

• Graph on the vertical axis against on the horizontal axis

• Check that is increasing when below and decreasing when above

gA(t) gA(t)

gA(t)

gA*

n1

16

( )Ag t

( )Ag t

*Ag0

STABILITY OF BGP

( ) 1 ( ) ( )A A Ag t n g t g t

17

Note that implies that a faster population growth n translates into faster growth of ideas in the balanced growth path.

Is there empirical support for the positive relationship between n and the long run growth rate?

Hard to test as we need long time series for that; but Michael Kremer 1993, QJE used population growth data going back to 1 Million B.C.

gA*

n1

18

Why does an increase in not raise the long run growth rate?

• Reason analogous to why increase in savings rate s in the Solow model does not increase long run growth: “Decreasing returns”

• Note that yielded

– Increase in al increase the short-run growth rate of ideas– But when <1 we get that maintaining the same growth rate of

ideas becomes harder and harder as the stock of idea increases (“fishing out the pond effect”)

– In the long-run we get a level effect only

The fraction of resources allocated to R&D is IRRELEVANT for long-run growth rate !!

al

A(t) Bal

L(t) A(t)

A(t)A(t)

Bal L(t) A(t) 1

19

IMPORTANT TO NOTE:

Balanced growth path growth rate:

• there can only be long run growth of ideas and output if:

n>0

• if n=0, there is NO long run growth

gA*

n1

20

R&D and endognous growth

• Hence, there can be long run growth even without exogenous technological progress

• BUT the growth rate is linked to population growth, which we don’t usually think of as a “policy parameter”

21

CASE 2:

Hence implies ever accelerating growth

1

gA(t) n 1 gA(t) gA(t)

1

22

In this case, a small increase in ends up having a very large effect on the stock of ideas in the long run

An increase in implies• short term increase in growth of ideas (as before)• these additional ideas further increase the growth of

ideas when

for any future time t, the growth rate will be higher after the increase in

al

al

al

A(t)A(t)

Bal L(t) A(t) 1

1

23

CASE 3: 1

gA(t)

A(t)A(t)

Bal L(t) A(t) 1

gA(t) Bal L(t)

• NOW, there is long run growth even if n=0!!!

gA(t) gA* Bal L

24

3. GROWTH WITH RESEARCH AND DEVELOPMENT: THE CASE WITH

CAPITAL

Quantity of output produced

Production of new ideas

Y (t) (1 ak )K (t)

(1 al )A(t)L(t) 1

0 1

A(t) B ak K (t)

al L(t)

A(t)

0 B,,,

25

+

standard assumptions of Solow model:

• constant savings rate s

• constant population growth rate n

• no depreciation of capital

26

The idea and capital growth equations

gA(t)

A(t)A(t)

Bak K (t) al

L(t) A(t) 1

gA(t)

gA(t)

K (t)K (t)

n 1 gA(t)

gA(t)

K (t)K (t)

n 1 gA(t)

gA(t)

gA(t) gK (t) n 1 gA(t) gA(t)

27

gK (t) K (t)

K (t)s

Y (t)K (t)

s(1 ak ) (1 al )

1 K (t) L(t)1 A(t)1

K (t)

gK (t)

gK (t)

( 1)gK (t) (1 )n (1 )gA(t)

28

Evolution of growth rate of ideas and capital (analyzing the two dimensional dynamic system)

gA(t) gK (t) n 1 gA(t) gA(t)

gK (t)

( 1)gK (t) (1 )n (1 )gA(t) gK (t)

29

-ISOCLINE ( )

• Above this line: falls• Below this line: increases

gA gA(t) 0

gK (t) n 1 gA(t) 0

gK (t)

1

gA(t)

n

gA

gA

CASE 1: (i.e. or ) 1 1 1 (1 ) /

30

-ISOCLINE ( )

gK gK (t) 0

( 1)gK (t) (1 )n (1 )gA(t) 0

gK (t) n gA(t)

• Above this line: falls• Below this line: increases

gK

gK

31

( )Kg t

( )Ag t

Ag ISOCLINE

0

Kg ISOCLINE

ISOCLINES in GROWTH RATES space

32

( )Ag t

( )Kg t

Ag ISOCLINE

0

Kg ISOCLINE

EVOLUTION OF GROWTH RATES

33

( )Ag t

( )Kg t

Ag ISOCLINE

0

gK ISOCLINE

DYNAMICS

34

( )Ag t

( )Kg t

Ag ISOCLINE

0

gK ISOCLINE

DYNAMICS plus INITIAL CONDITION

STARTING POINT

35

Starting point of dynamical system is GIVEN by INITIAL capital, technology, and labor force

A(t)A(t)

Bak K (t) al

L(t) A(t) 1

K (t)K (t)

sY (t)K (t)

36

IMPORTANT TO NOTE:

• there can only be long run growth of ideas, capital, and output if:

n>0

• if n=0, there is NO long run growth

37

CASE 2:

• we are interested in whether IN THIS CASE there will be long run growth even if n=0

• hence ALSO assume n=0

1

38

gA

gK (t)

1

gA(t)

n

gK (t) gA(t)

n

gK (t) gA(t)

-ISOCLINE

39

gK

gK (t) n gA(t)

gK (t) gA(t)

HENCE:

• The two isoclines lie on top of each other• NOW, there is long run growth even if n=0!!!

-ISOCLINE

40

( )Ag t

( )Kg tAg ISOCLINE

0

Kg ISOCLINE

beta+theta=1 CASE WITHOUT POPULATION GROWTH

41

Michael Kremer's model

42

Michael Kremer's model

"Population Growth and Technological Change One Million B.C. to

1990", Q.J.E. 1993

• Michael Kremer's intuition was that in a Malthusian world, i.e. a world in which population is just big enough to survive, there is a link between the state and technology and the amount of population: If everyone consumes just a "subsistence" amount, societies with more advanced technology (say, better agriculture) will be able to support larger populations

• Hence, we could infer from the level of A L

43

The framework

Assume the following production function:

where:• indicates the level of technological progress• is population• is land

At least for a pre-industrial society, it may make sense to have only labour and land as production inputs. Note that the production function has constant returns to scale: the replication argument is valid! (ie, double the amounts of input, and you double output)

Y ALT 1

ALT

44

Malthusian case

Now express the production function in per-capita terms:

and assume that population increases when is above some subsistence level .

This will reduce output per capita, so that it is reasonable to assume - if population growth reacts fast enough - that population will constantly adjust such that always holds.

y Y

LAL 1T 1

y

y y

y

y

45

Malthusian case

We can solve for the population level that corresponds to

What does it mean?

• In the absence of changes in , population will be constant

• Ceteris paribus, population will be proportional to land area

• If separate regions have different levels of technology , population or population density will be increasing in

L A

y

1

1 T

T

L

A

AAL / T

46

Technological progress

Now: enter technological progress. Assume that What does this imply for population growth?

Take logs and derivatives of

and obtain

• population will grow at a constant rate. True?

L A

y

1

1 T

AA

g

LL

1

1

AA

g

1

47

48

49

50

More than exponential growth

So population growth appears to be increasing in population. This implies faster than exponential growth, which is what you would achieve with

• Why?

Key insight: Each person has a constant probability of inventing a new technology. But because "ideas" (insights, designs. . . ) are nonrival, the whole society should profit from it.

LL

const.

51

Ideas as public goods

• "As for the Arts of Delight and Ornament, they are best promoted by the greatest number of emulators. And it is more likely that one ingenious curious man may rather be found among 4 million than among 400 persons."

William Petty

• "If I could redo the history of the world, halving population size each year from the beginning of time on some random basis, I would not do it for fear of losing Mozart in the process."

Edmund Phelps

52

Modelling growth and ideas

So, because every individual has the same probability of inventing

something new, should be proportional to population size:

Insert this into:

Result: • population growth is itself proportional to population. • Aside: Results don't change substantially if we assume ,

i.e.

AA

gL

LL

1

1

AA

gL

1

A gLA

AA

1

53

A natural experiment

Could we somehow test the model?

Kremer suggests to consider a "natural experiment": the end of the last ice age around 10'000 B.C., when previously connected land masses (Eurasia+Africa, the Americas, Australia, Tasmania) were separated and technological diffusion wasn't possible any more.

Assumptions:

• These 4 regions had shared the same basic technology up to that point (same ).

• Hence, their populations must have been proportional to the land areas

A

54

4 (or 5) separated regions

55

A natural experiment

Prediction:

• 11500 years of separation (until 1500) should have lead technology levels to diverge

• Growth rates of technology will be proportional to initial

population:

• Higher growth rates of translate into larger populations

• or, given constant area of land masses, into higher densities

AA

gL

A

56

The natural experiment: results