Advance Macroeconomic-Knowledge Accumulation

8/7/2019 Advance Macroeconomic-Knowledge Accumulation

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Advanced Macroeconomics

Lecture 2a

Endogenous growth: Knowledge accumulation

Christian Ghiglino
http://find/http://goback/


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In the previous lecture the Solow model was extendedintroducing exogenous technological progress. The processwas spontaneous without the need of any investment anddidnt use any resources.

It is now time to model technological progress. A firstpossibility is to assume that knowledge is produced in theeconomy and specify the required technology to produceideas. In this Section it is assumed that only labour is neededbut the model can be extended to one in which both capital

and labour are used as input to produce knowledge.


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Knowledge accumulation without capital

Modeling knowledge rises two issues:

What are the properties of knowledge?Knowledge is a non-rival good, i.e. its use by a person does notmake its use by another person more difficult. Knowledge, may beexcludable (encoded TV program) or non-excludable (thePythagoras theorem).

Is the production function for knowledge constant returns inthe physical (rival) inputs?

At least CRS is assumed in physical goods production because ofthe possibility to replicate individual firms even when these have anoptimal size, i.e. returns become eventually decreasing. On theother hand, increasing returns to scale are possible wheneverexternalities exist between firms. Increasing returns within a firmare usually excluded. In the case of knowledge, constant returns

cannot really be assumed.


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The structure of the model

One physical output which is fully consumed

One physical input, labour. No need of capital.

A non-rival good, A, interpreted as knowledge.

The production function for output is CRS in labour. Theproduction function for knowledge is arbitrary.


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The main equations of the model are:

Y(t) = FAY(t),KY(t), LY(t) = (1 aL)L(t)A(t) (1)A(t) = B

aLL(t)

A(t) (2)

L(t) = nL(t) (3)

(3) is a first-order, linear and homogenous dynamical system. Thesolution is given by:

L(t) = L(0)ent (4)

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

knowledge with equation (2):

gA(t) =A(t)

A(t)= B

aLL(t)

A(t)1 (5)


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Take the derivation ofgA(t) with respect to time and you get

gA =

A(t)

A(t)

= B[aLL(t)]

1

L(t)A(t)1 + (6)

+( 1)B[aLL(t)]A(t)2

A(t) (7)

so that

gA =

A(t)

A(t) = B[aLL(t)]1L(t)

[L(t)/L(t)]A(t)1 + (8)

+( 1)B[aLL(t)]A(t)1

[A(t)/A(t)] (9)


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and finally

gA(t) = ngA(t) + g2A

(t)( 1) (10)

The steady state is obtained for

0 = ngA(t) + g2A(t)(

1) (11)

giving

gA =n

1 = gA (12)


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Analysis

Case 1: < 1

If < 1 it follows from (10) that gA(t) converges to g

Awhere

gA =n

1 (13)

Indeed, for gA(t) > g

Awe have

gA(t) < 0 so that gA(t) decreases

while gA(t) < g

A we have

gA(t) > 0 so that gA(t) increases.Equation (13) describes a one-dimension, first-order, linear andhomogeneous dynamically system. In other words, the equation

A(t)

A(t) = g

A =

n

1

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

n1

t (14)

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

n

1


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Case n=0

From equation (14), knowledge A(t) is constant and the growthrate of knowledge is 0:

A(t) = A(0) (15)

From (15), (4) and (1) output is constant and the growth rate ofoutput is 0:

Y(t) = (1

aL)L(0)A(0) (16)

1. Conclusion: The growth rates of output, output per capitaand knowledge are all 0

Case n>0

With (14), (1) and (4), output is given by:

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

n1

+n

(17)


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With (17) and (4), output per capita is given by:

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

= A(0)(1 aL)et n1 (18)

From (17), the growth rate of output is given by:

Y(t)Y(t)

= n1

+ n (19)

From (18), the growth rate of output per capita is given by:

y(t)y(t)

= n1

(20)


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Observations:

1. The growth rates of output and output per capita depend onpopulation growth n. This is called the scale effect. It is not

observed in the empirical comparisons across countries.2. The growth rates of output and output per capita do not

depend on aL: the size of the Research & Development sectoris irrelevant for growth.


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Case 2: = 1

From (2), we have:

A(t) = BaLL(t)

A(t) (21)

Substitute (4) into (21), rearrange and you get the growth rate of

knowledge:

gA(t) =A(t)

A(t)= Ba

LL(t) (22)

Furthermore, from (10) you get:

gA(t) = ngA(t) (23)


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Case n=0

From (22) we get:

A(t) = BaLL(0)

A(t) (24)

and

gA(t) = 0 (25)

(24) is a first-order, linear, homogeneous dynamical system. Thesolution is given by:

A(t) = A(0)eBa

LL(0)t

(26)


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With (26), (1) and (4), output is given by:

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

LL(0)

t (27)

With (27) and (4), output per capita is given by:

y(t) =Y(t)

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

BaLL(0)t (28)

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

y(t)y(t) =

BaLL(0)

(29)


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Observations:

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

2. The growth rates of output, output per capita and knowledgedepend on aL: The amount of labor in the Research &Development sector matters for the long run behavior.


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Case n>0

From (22) we have the growth rate of knowledge

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

BentaLL(0)

(30)

Furthermore, from (23) we have the evolution of the growth rate:

gA(t) = ngA(t) (31)

Observations:

1. Because of (30), the growth rate of knowledge and thereforethe growth rates of output and output per capita are > 0

2. Because of (31), the growth rate of knowledge and thereforethe growth rates of output and output per capita are evenincreasing. This is not consistent with empirical data thatshows nearly constant growth rates.


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Case 3: > 1

From (10), we have the evolution of the growth rate of knowledge

gA(t) = ngA(t) + g2A(t)( 1) (32)

Observations:

1. Because of (32) we have gA(t) > 0 even ifn = 0 the growthrate of knowledge and therefore the growth rates of outputand output per capita are increasing forever.

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Advance Macroeconomic-Knowledge Accumulation