exogenous change and growth of knowledgechapter1

8/7/2019 Exogenous change and growth of knowledgechapter1

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Advanced Macroeconomics

Chapter 1: Exogenous Growth and accumulation of

Knowledge

Christian Ghiglino

October 8, 2009

Contents

1 Introduction 3

1.1 Fundamental economic issues . . . . . . . . . . . . . . . . . . 31.2 The Kaldor stylized facts and more . . . . . . . . . . . . . . . 31.3 What have been found (Salai-Martin (2002)) . . . . . . . . . 3

2 The Solow Model 4

2.1 Main assumptions . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 The dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Comparison with the facts . . . . . . . . . . . . . . . . . . . . 52.6 The Solow model with exogenous technological progress . . . 62.7 The Solow model in discrete time . . . . . . . . . . . . . . . . 6

2.7.1 Stability of the steady state . . . . . . . . . . . . . . . 72.7.2 Global properties . . . . . . . . . . . . . . . . . . . . . 7

3 The Ramsey Model 7

3.1 Main equations . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Neoclassical two-sector model 9

4.1 Main equations . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 The firms problem . . . . . . . . . . . . . . . . . . . . . . . . 94.3 The consumers problem . . . . . . . . . . . . . . . . . . . . . 104.4 Specific formulation of the model . . . . . . . . . . . . . . . . 11

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5 Diamond Model 12

5.1 The consumers problem . . . . . . . . . . . . . . . . . . . . . 125.2 The firms problem . . . . . . . . . . . . . . . . . . . . . . . . 135.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Knowledge accumulation without capital 14

6.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.3.1 Case 1: < 1 . . . . . . . . . . . . . . . . . . . . . . . 166.3.2 Case 2: = 1 . . . . . . . . . . . . . . . . . . . . . . . 176.3.3 Case 3: > 1 . . . . . . . . . . . . . . . . . . . . . . . 18

7 Knowledge accumulation with capital 19

7.1 Main equations . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2 decreasing returns: + < 1 . . . . . . . . . . . . . . . . . . 197.3 constant returns: + = 1 . . . . . . . . . . . . . . . . . . . 197.4 increasing returns: + > 1 . . . . . . . . . . . . . . . . . . 19

8 Learning by Doing 20

8.1 Main equations . . . . . . . . . . . . . . . . . . . . . . . . . . 208.2 Case 1: < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 Case 2: = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8.3.1 n=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.3.2 n>0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.4 Case 3: > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Human Capital 24

9.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.2 Main equations . . . . . . . . . . . . . . . . . . . . . . . . . . 249.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9.3.1 The steady state . . . . . . . . . . . . . . . . . . . . . 259.3.2 The phase diagram . . . . . . . . . . . . . . . . . . . . 26

9.3.3 Quantitative analysis . . . . . . . . . . . . . . . . . . . 27

9.3.4 An application . . . . . . . . . . . . . . . . . . . . . . 28

10 Malthus 30

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1 Introduction

1.1 Fundamental economic issues

What is the engine of long-run growth? Why is there a constant growthrate? How to stimulate growth?

Why is there structural change? Why the importance of services riseso sharply?

Why are there inequalities across countries? How to reduce them?

Why are there short-run fluctuations? Are these bad? How to reducethem?

Why is there unemployment? How to eliminate it?

1.2 The Kaldor stylized facts and more

Continuing growth in output per worker (balanced growth, constantgrowth rate)

Continuing growth in capital per worker

The capital-output ratio is constant

The rate of return to capital is constant

The shares of labor and capital in national income are constant

Systematic changes in the relative importance of various sectors (agri-culture, manufacturing, services)

Labor force in R & D follows a balanced growth path

Total number of Patents in the R & D sector is constant

Large differences in productivity growth rates across countries

1.3 What have been found (Salai-Martin (2002))

There is no good theory!!! There is no simple determinant for growth

The initial level of income is the most determinant and robust variable

The size of the government does not matter but its quality yes

Relation between human capital and growth is weak

Institutions are important for growth

More open economies growth faster

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2 The Solow Model

This is the standard model of capital accumulation. Without assuming exo-geneous technological progress, this model predicts convergence to a steadystate in per capita output. The driving force behind this result is the factthat there are diminsihung returns in capital, the only accumulable input.

2.1 Main assumptions

There is one aggregate good which can be consumed or invested toaccrue capital.

There are two inputs, labor and capital.

There is constant returns to scale (possibility of replication).

Capital depreciates at the constant rate .

Total population and total workforce (assumed to be equal) have aconstant growth rate of n.

The saving rate is exogenous, s.

2.2 The equations

Let K(t) and L(t) be the total capital and labor and Y(t) be total outputin period t. Let C(t) be total consumption. Let F be the constant returnsproduction function. The main equations of the Solow model are:

Y(t) = F

K(t), L(t)

(1)

Y(t) = C(t) + K(t) + K(t) (2)

sF

K(t), L(t)

= K(t) + K(t) (3)

L(t) = nL(t) (4)

Equation (3) can be expressed in per capita terms:

k(t) = sf

k(t) ( + n)k(t) (5)

At a steady state k(t) = 0, so k is implicitly given by

sf(k) = (n + )k (6)

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2.3 The dynamics

To focus the analysis on a steady state is relevant only if this is dynamicallystable so that this equilibrium is robust to shocks. The diagrammatic anal-ysis of the dynamic equation shows that k is globally dynamically stable.The economy converges to k independently of the initial value k(0).

2.4 Results

1. y(t), k(t) and c(t) converge to their steady states values.

2. Y(t), K(t) and C(t) grow with the population growth rate n.

3. The speed of convergence is ....

2.5 Comparison with the facts

1. The Solow model do not explain well the differences across countries.Indeed, in order to explain the differences income level the differencesin capital per capita and in the interest rate across countries need tovery large. This is not compatible with the observed pattern.

2. The model do not allow for fluctuations without exogenous shocks tothe production function f(k)

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2.6 The Solow model with exogenous technological progress

Let A(t) be the state of knowledge. It is assumed to be labour augmenting,so Leff(t) = A(t)L(t). Intensive variables are defined through normalizingby A(t)L(t). Let F be the constant returns production function. The mainequations of the model are:

y(t) =Y(t)

A(t)L(t)= F

k(t), 1)

= f(k(t)) (7)

We assume

A(t) = gA(t) (8)

leading toA(t) = A(0)egt

The dynamics in intensive form is:

k(t) = sf

k(t) (g + + n)k(t) (9)

At a steady state k(t) = 0, so k is implicitly given by

sf(k) = (g + n + )k (10)

2.7 The Solow model in discrete time

Most economic models can be formulated in both continuous and discretetime. The choice is usually made considering the solvability of the model.Economically, discrete time makes more sense in real macro models becausemacro data is usually provided by discrete time series with large time peri-ods. On the other hand, financial data is usually so dense that a continuoustime formulation is more appropriated.

Let Kt and Lt be the total capital and labor and Yt be total outputin period t. Let Ct be total consumption. Let F be the constant returns

production function. The main equations of the Solow model are:

Yt = F

Kt, Lt

(11)

Yt = Ct + Kt+1 Kt + Kt (12)

sF

Kt, Lt

= Kt+1 Kt + Kt (13)

Lt+1 = (1 + n)Lt (14)

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Equation (3) can be expressed in per capita terms:

kt+1 = sf

k(t)

+ (1 n)kt (15)

At a steady state kt+1 = kt = k, so k is implicitly given by

sf(k) = (n + )k (16)

The value of the steady state (and the other properties) depends on theproperties of the function f. What do we know about f?

The function f inherits some properties of the production function F. Inparticular, F is assumed to be constant returns to scale, increasing in botharguments and with infinite marginal productivity at 0. A good example isF(K, L) = KL1 with 0 < < 1. The per capita form is f(k) = k whichis concave with infinite marginal productivity at zero and asymptoticallyzero.

2.7.1 Stability of the steady state

A steady state is economically relevant only if it is robust to perturbations,i.e. locally stable (The Samuelson principle). There are several ways toanalyze the local stability properties of a steady state: using the phasediagram, computing the eigenvalues of the Jacobian matrix at the steady

state, and by hand kt+1 < kt).

2.7.2 Global properties

Globally stable.

3 The Ramsey Model

The structure of the Ramsey model is as for the Solow model but the savingrate is endogenous. This means that the behavior of the consumer should

be explicitly modeled as an optimization.

3.1 Main equations

Yt = F(Kt, Lt) (17)

It = sYt = Kt+1 Kt + Kt (18)

Yt = It + Ct (19)

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Substitute (??) into (??), divide by Lt, use the fact that kt+1 =Kt+1Lt+1

=

Kt+1Lt

LtLt+1 =

Kt+1Lt

11+ , rearrange and you get capital per capita:

kt+1 =sf(kt) + (1 )kt

1 + =

f(kt) ct + (1 )kt1 +

(20)

Rearrange (??) and you get consumption per capita:

ct = f(kt) (1 + )kt+1 + (1 )kt (21)

The maximization problem of the representative consumer is given by:

M ax

t=0

t

U(ct) s.t. (??) (22)

Substitute (??) into the utility function of (??), take the derivations withrespect to kt and you get the Euler equations:

U(ct)

U(ct1)=

1 +

f(kt) + (1 ) t 1 (23)

3.2 Phase diagram

The dynamics of capital per capita (kt) and consumption per capita (ct) isdescribed by equations (??) and (??)

k=0: If kt = kt+1 := kt, then we have with (??):

ct = f(kt) ( + )kt (24)

c=0: If ct = ct+1, then we have with (??):

f(kt) =1 +

(1 ) (25)

Equation (25) determines the level of capital per capita, where consumptionis constant. We denote this level by kc=0

ct > f(kt) ( + )kt: ct > f(kt) ( + )kt. Substitute (??) for ct andyou get: f(kt) (1 + )kt+1 + (1 )kt > f(kt) ( + )kt. Rearrange andyou finally get: kt+1 kt < 0.We therefore have the following dynamics:

ct > f(kt) ( + )kt kt+1 kt < 0 (26)

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kt > kc=0: If kt > kc=0, then we have with (25) and because f(kt) < 0:

f

(kt) kc=0 f(kt) 0 so that gA(t) increases. Equation (87) de-scribes a one-dimension, first-order, linear and homogeneous dynamicallysystem. In other words, the equation

A(t)

A(t) = g

A =n

1

has as the solution:A(t) = A(0)e

n

1t (88)

Consequently, both A(t) and Y(t)/L(t) grow at the rate n1

n=0 From equation (88), knowledge A(t) is constant and the growth rateof knowledge is 0:

A(t) = A(0) (89)

From (89), (78) and (75) output is constant and the growth rate of output

is 0:Y(t) = (1 aL)L(0)A(0) (90)

1. Conclusion: The growth rates of output, output per capita and knowl-edge are all 0

n>0 With (88), (75) and (78), output is given by:

Y(t) = A(0)L(0)(1 aL)et

n

1+n

(91)

With (91) and (78), output per capita is given by:

y(t) = Y(t)L(t)

= A(0)(1 aL)et n1 (92)

From (91), the growth rate of output is given by:

Y(t)

Y(t)=

n

1 + n (93)

From (92), the growth rate of output per capita is given by:

y(t)

y(t)=

n

1 (94)

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Observations:

1. The growth rates of output and output per capita depend on popula-tion growth n. This is called the scale effect. It is not observed inthe empirical comparisons across countries.

2. The growth rates of output and output per capita do not depend onaL: the size of the Research & Development sector is irrelevant forgrowth.

6.3.2 Case 2: = 1

From (76), we have:

A(t) = BaLL(t)A(t) (95)Substitute (78) into (95), rearrange and you get the growth rate of knowl-edge:

gA(t) =A(t)

A(t)= BaLL(t)

(96)

Furthermore, from (84) you get:

gA(t) = ngA(t) (97)

n=0 From (96) we get:

A(t) = B

aLL(0)

A(t) (98)

andgA(t) = 0 (99)

(98) is a first-order, linear, homogeneous dynamical system. The solution isgiven by:

A(t) = A(0)eBaLL(0)t (100)

With (100), (75) and (78), output is given by:

Y(t) = (1 aL)L(0)A(0)eBa

L

L(0)t

(101)

With (101) and (78), output per capita is given by:

y(t) =Y(t)

L(t)= (1 aL)A(0)e

BaLL(0)t (102)

Therefore, the growth rate of output and the growth rate of output percapita are given by:

y(t)

y(t)= B

aLL(0)

(103)

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Observations:

1. The growth rates of output, output per capita and knowledge areconstant and the same: B(aLL(0))

2. The growth rates of output, output per capita and knowledge dependon aL: The amount of labor in the Research & Development sectormatters for the long run behavior.

n>0 From (96) we have the growth rate of knowledge

gA(t) =A(t)

A(t)= Bent

aLL(0)

(104)

Furthermore, from (97) we have the evolution of the growth rate:

gA(t) = ngA(t) (105)

Observations:

1. Because of (104), the growth rate of knowledge and therefore thegrowth rates of output and output per capita are > 0

2. Because of (105), the growth rate of knowledge and therefore thegrowth rates of output and output per capita are even increasing.

This is not consistent with empirical data that shows nearly constantgrowth rates.

6.3.3 Case 3: > 1

From (84), we have the evolution of the growth rate of knowledge

gA(t) = ngA(t) + g2A(t)( 1) (106)

Observations:

1. Because of (106) we have gA(t) > 0 even if n = 0 the growth rate ofknowledge and therefore the growth rates of output and output percapita are increasing forever.

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7 Knowledge accumulation with capital

7.1 Main equations

The main equations of this model are:

K(t) = sY(t) = (1 K)(1 L)

1K(t)

A(t)L(t)1

(107)

A(t) = B

KK(t)

LL(t)

A(t) (108)

7.2 decreasing returns: + < 1

The main results are the same as in the model of section (6.3.1):

1. The growth rate of knowledge converges: limt gA(t) = gA(t)

2. n=0: the growth rates of knowledge, output and output per capita areall 0

3. n>0: the growth rates of knowledge, output and output per capita areconstant and depend on the growth rate of population

4. L and K have no impact on the growth rates of knowledge, outputand output per capita

7.3 constant returns: + = 1


1. n=0: the growth rates of knowledge, output and output per capita areconstant

2. n>0: the growth rates of knowledge, output and output per capita areincreasing

3. L, K matter

7.4 increasing returns: + > 1


1. The growth rates of knowledge, output and output per capita arepositive and ever increasing even if n=0

2. n, K, L matter

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8 Learning by Doing

Another way to obtain growth is to assume that the process of productionbecomes more effective as production occurs. This is different than assumingexogeneous technological progress because progress here depends on howmuch it is produced. However, no specific investment is required.

8.1 Main equations

The main equations of this model are:

Y(t) = K(t)

A(t)L(t)

1

(109)

A(t) = BK(t) (110)

sY(t) = K(t) (111)

L(t) = L(0)ent (112)

Substitute (110) and (109) into (111) and you get:

K(t) = sB1L(t)1K(t)+(1) (113)

In order to analyze the dynamics, define the growth rate of capital:

gK(t) =

K(t)K(t) = sB1L(t)1K(t)+(1)1 (114)

Take the derivation of gK(t) with respect to time and you get:

gK(t) = n(1 )gK(t) + (1 )( 1)gk(t)2 (115)

8.2 Case 1: < 1

If < 1, then it follows from (115) that the growth rate of capital gK(t)converges to gK, where gK(t) = 0. The growth rate of capital is given by:

gK =

K(t)K(t)

= n1

(116)

(116) describes a first-order, linear, homogeneous dynamical system whosesolution is given by:

K(t) = etn1 K(0) (117)

With (112), capital per capita is given by:

k(t) =K(t)

L(t)= k(0)et

n1 (118)

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Take the derivation of k(t) with respect to t, and you get the growth rate

of capital per capita:k(t)

k(t)=

n

1 (119)

Substitute (110), (112) and (117) into (109), and you get the output:

Y(t) = K(0)+(1)L(0)1B1ent1 (120)

From (120) you get the growth rate of output:

Y(t)

Y(t)=

n

1 (121)

Divide (120) by L(t) and use (112) to get output per capita:

y(t) =Y(t)

L(t)= k(0)K(0)(1)et

n1 (122)

From equation (122), the growth rate of output per capita is straightforward:

y(t)

y(t)=

n

1 (123)

Observations:

1. The growth rates depend on the population growth rate and are inde-

pendent of the saving rate s

2. If n=0, then all the growth rates are equal to 0

3. If n>0, then the growth rates are positive and constant. They dependon the value of n.

8.3 Case 2: = 1

If = 1, we have from (115):

gK(t) =K(t)

K(t)= n(1 )gK(t) (124)

Equation (124) describes a first-order, linear, homogeneous dynamical sys-tem. The solution is given by:

gK(t) =K(t)

K(t)= etn(1)

K(0)

K(0)= etn(1)gK(0) (125)

If = 1, then gK(0) is given from (114):

gK(0) =K(0)

K(0)= sB1L(t)1 = sB1L(0)1ent(1) (126)

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8.3.1 n=0

If n=0, we have from (125) and (126):

K(t) = K(t)gK(0) = sB1L(0)1K(t) (127)

The solution to (127) is given by:

K(t) = K(0)etsB1L(0)1 (128)

Therefore, the growth rate of capital is given by:

K(t)

K(t)

= sB1L(0)1 (129)

From (128) you get immediately capital per capita and the growth rate ofcapital per capita:

k(t) =K(t)

L(t)= k(0)etsB

1L(0)1 (130)

k(t)

k(t)= sB1L(0)1 (131)

Substitute (110), (112) and (128) into (109) and you get output and thereforethe growth rate of output:

Y(t) = K(0)L(0)1B1etsB1L(0)1 (132)

Y(t)

Y(t)= sB1L(0)1 (133)

Output per capita and the growth rate of output per capita are given by:

y(t) = k(0)L(0)1B1etsB1L(0)1 (134)

y(t)

y(t)= sB1L(0)1 (135)

Observations:

1. Even if n=0, the growth rates are positive and constant

2. All the growth rates are the same

3. The growth rates depend on the saving rate s

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8.3.2 n>0

Substitute (126) into (125) and you get the growth rate of capital:

gK(t) =K(t)

K(t)= sB1L(0)1e2nt(1) (136)

Observations:

1. Because of (136), the growth rate of capital is positive

2. Because of (124), the growth rate of capital and therefore all the othergrowth rates are ever increasing

3. The growth rates depend on the populations growth n and the savingrate s

8.4 Case 3: > 1

Observations:

1. Because of (114), the growth rate of capital and therefore all the othergrowth rates are positive

2. Because of (115), the growth rates are ever increasing

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9 Human Capital

Theories based on knowledge accumulation cannot explain cross-countrydifferences in incomes because knowledge is not rival and most of the timenon-excludable. In fact even when it is excludable it is likely that all coun-tries could buy the use of the best technology.

However, the use of knowledge requires workers with appropriated skillsor abilities. This specific knowledge is called human capital.

Human capital is excludable and rival as usual physical capital. In-deed, human capital is attached to a worker. When a worker is used toproduce output his human capital cannot be used by another worker.

The accumulation of human capital is similar to the accumulation ofphysical capital. Devoting more resources to its accumulation increasesthe output.

Some workers earning reflect these acquired skills. The estimation ofthe part of national income devoted to accumulable inputs is not onlyreflected by the capital share. Indeed, the labour share is the sum ofthe pure raw labour share and the rewards to human capital.

9.1 Assumptions

There is one physical output. There are four inputs: Physical capital, Raw labour, Human capital

and non-rival Knowledge.

Physical and Human capital accumulate according to the correspond-ing investment in the physical good, noted sK and sH. These couldbe endogeneised but are here exogenous.

The accumulation of Knowledge could be endogeneised as in the pastlectures. It is here assumed to grow at the rate g.

There are CRS in respect to both sorts of capital and labour.

9.2 Main equations

The main equations of the model are the following:

Y(t) = K(t)H(t)

A(t)L(t)1

(137)

K(t) = sKY(t) (138)

H(t) = sHY(t) (139)

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L(t) = L(0)ent L(t) = nL(t) (140)

A(t) = A(0)egt A(t) = gA(t) (141)

From (137) and (138) we get:

K(t) = sKK(t)H(t)

A(t)L(t)

1(142)

From (137) and (139) we get:

H(t) = sHK(t)H(t)

A(t)L(t)

1(143)

(142) and (143) define a two-dimensional, first-order differential equation.

9.3 Analysis

Define capital per effective labor and human capital per effective labor:

k(t) =K(t)

A(t)L(t)(144)

h(t) =H(t)

A(t)L(t)(145)

Take the derivations of k(t) and h(t) with respect to time and use (142) and(143) to get:

k(t) =

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

=

KALK(A

L +

AL)

A2L2(146)

=

K

AL

K

AL

AA

+

L

L

(147)=

sKKH(AL)1

AL

K

AL(g + n) (148)

= sKk(t)h(t) (n + g)k(t) (149)

Similarlyh(t) = sHk(t)

h(t) (n + g)h(t) (150)

9.3.1 The steady state

(146) and (??) define a two-dimensional, first-order differential equation thatcan be studied within a phase diagram. At the steady state, k(t) = 0 andh(t) = 0. Use this together with (146) and (??) to calculate the steadystate:

k =

s1K sH

1

n + g

11

(151)

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h = sKs1H1

n + g1

1(152)

An important relation is that

sKsH

=k

h

At the steady state (k*, h*), we have:

0 =k(t)

k(t)=

ln

k(t)

=

ln K(t)

A(t)L(t)

(153)

This means that

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

= constant

Manipulate this equation and rearrange to get:

K(t)

K(t)= g + n (154)

Analogously, the growth rate of human capital is given by:

H(t)

H(t)= g + n (155)

Observations: The growth rate of output do not depend on the savingrates sK and sH.It only depends on g and n. The per capita output onlydepends on g. As the steady state will be shown to be stable, the long rungrowth is independent of the initial conditions. However, the level of outputper capital and capital per capita do depend on the saving rates.

9.3.2 The phase diagram

Situation

k(t) 0

From

k = sKkh (n + g)k 0

we getk(n + g) sKk

h

k1 sKh(n + g)1

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k sKn + g

1

1

h

1

with

1

1

1 = 1

So the curve is convex. To the left of

h(t) = 0 the derivative is positive so h

is increasing with time. The arrow goes to the right. To the right of

h(t) = 0the situation is opposite. The phase diagram shows that the steady state isstable.

9.3.3 Quantitative analysis

In order to calculate the elasticity of output per effective labor with respectto sK and to sH at the steady state, first take the logarithm on both side ofequations (151) and (152) and manipulate them to get:

ln(k) =1

1

(1 )ln(sK) + ln(sH) ln(n + g)

(156)

ln(h) =1

1

ln(sK) + (1 )ln(sH) ln(n + g)

(157)

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Divide (137) by A(t)L(t) to get output per effective labor:

y(t) = k(t)h(t) (158)

Take the logarithm to both side of (158), manipulate and evaluate at thesteady state to get:

ln(y) = ln(k) + ln(h) (159)

Substitute (156) and (157) into (159) and rearrange:

ln(y) =

1 ln(sK) +

1 ln(sH)

+

1 ln(n + g) (160)

Take the derivative with of (160) with respect to sK, sH respectively and

you get:y

y= [ln(y)] =

1

1

sK(161)

y

y= [ln(y)] =

1

1

sH(162)

The elasticities of output per effective labor with respect to sK and sH arethen given by:

esK =

yy

sKsK

=

1 (163)

esH =yy

sHsH

=

1 (164)

9.3.4 An application

Some part of the workers earning is due to human capital. To find outhow much we assume that the minimum wage reflects the inherent skills.For example if we assume then that 14 the average wage w is associated tonon-accumulated skills while 34w is due to human capital. As usual let

be the physical capital share of national income. Then (1 ) is the shareremaining for both acquired and non-aquired labour abilities. Assumingthat 34 comes from human capital we get =

34(1 ).

In agreement with this order of magnitude let = 0.35 and = 0.4 sothat 1 = 0.25. With these values we get

ln(y) =0.35

0.25ln(sK) +

0.4

0.25ln(sH)

0.75

0.25ln(n + g) (165)

orln(y) = 1.4ln(sK) + 1.6ln(sH) 3ln(n + g) (166)

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while with Solow we would get

ln(y) = 0.54ln(sK) 0.54ln(n + g) (167)

The difference between the two model;s is large. Indeed, assume that thesaving rates are twice in country B compared to country A and that (n + g)is 20% smaller in country B. Then

log yA log yB = 1.4(log sAK log s

AK) + 1.6(log s

AH log s

AH) 3log(0.8)

giving

log yA log yB = 1.4 log 2 + 1.6log2 3log(0.8) = 2.75

so thatyByA

= e2.75 = 15.6

while with Solow we would have

yByA

= e0.4 = 1.6

Because of large elasticities of output respect the underlying determinantsof the model, the model has the potential to explain cross country incomedifferences.

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10 Malthus

Historical data shows that before the industrial revolution, increase in to-tal output was used to increase population rather than income per capita.This is because the minimal subsistance level constraint was binding. Malthu-sian models try to explain the observed rise in population during this pe-riod. They endogeneise n. The models of knowledge accumulation with andwithout capital can we used as they predict a relationship between growthin total output and population growth n.The main assumptions of the model we consider here are the following:

1. There is an input in fixed supply: Land (:=R). Capital is ignored. We

assume Y(t) = RA(t)L(t)1 (168)2. Population adjusts so that the output per capita is fixed at the sub-

sistance level yY(t)

L(t)= y Y(t) = yL(t) (169)

3. Growth rate of knowledge is proportional to L(t)

gA(t) =A(t)

A(t)

= BL(t) (170)

The model can be solved. Indeed, first put (eq: 64) and (169) together,multiply both side of this equation by L(t)1

L(t)1R

A(t)L(t)1

= yL(t)L(t)1 (171)

giving

R

A(t)1

= yL(t) (172)

Solving for L(t) you get:

L(t) = R A(t)

1

y1

(173)

The growth rate of population is then given by:

n(t) =L(t)

L(t)=1

A(t)A(t)

=1

BL(t) (174)

Equation (174) can be tested empirically by an econometric regression. Theresult is

n(t) = 0.0023 + 0.524L(t)

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and the fit is very good.

A second test of the theory is to consider economies that were isolatedfor long periods of time. They were four blocs that were separated at theend of the ice age with no communication until 1500. As we can assumethay the density was identical when they were separated, the populationdensity in 1500 reflects the poplutation growth rate since the end of the iceage. Larger blocs had a larger initial population so that they are expectedto grow faster with a resulting higher population density in 1500. The dataagree with the prediction

Eurasia-Africa: Surface=84 millions km2. Density in 1500: 4.9 h/km

Americas: Surface=38 millions km2. Density in 1500: 0.45 h/km

Australia: Surface=8 millions km2. Density in 1500: 0.03 h/km

Tasmania: Surface=0.1 millions km2. Density in 1500: 0.03 h/km

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exogenous change and growth of knowledgechapter1

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