Macro Lect 8 2015 Endogenous Growth - …-with-posive-externalies-from-capital- ... Macro Lect 8 2015 Endogenous Growth

Prof. George Alogoskoufis, Dynamic Macroeconomic Theory, 2015

Externali*es, Human Capital and Endogenous Growth

Externali*es from Capital Accumula*on, Investment in Human Capital and Research and Development


Economic Growth and Learning by Doing

• We now turn to a growth model which is based on the assump*on of posi*ve externali*es from aggregate capital accumula*on on labor efficiency. The main idea that drives this model is learning by doing, an idea introduced to growth models by Arrow (1962). This assump*on can, under certain condi*ons, lead to endogenous growth, as in Romer (1986).

• In the learning by doing model, labor efficiency is a func*on of both exogenous technical progress, as well as aggregate capital per worker. Thus, the efficiency of labor, which is the same for all firms, depends on capital per worker in the rest of the economy. Because of learning by doing, as suggested by Arrow (1962), the accumula*on of aggregate capital increases labor produc*vity both directly and indirectly, through “knowledge spillovers”, that have a direct effect on the efficiency of labor. It is assumed here that "knowledge" is like a public good, and that the accumula*on of knowledge depends on the accumula*on of aggregate capital.

• An important consequence of this approach is that diminishing returns from capital accumula*on set in more slowly, and that, under certain condi*ons, there may even be constant or increasing returns from capital accumula*on. In these laWer circumstances growth becomes endogenous and is determined by the rate of accumula*on of aggregate physical capital.

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Learning by Doing and Capital Accumula*on

“I advance the hypothesis here that technical change in general can be ascribed to experience, that it is the very ac*vity of produc*on which gives rise to problems for which favorable responses are selected over *me. … I therefore take … cumula*ve gross investment (cumula*ve produc*on of capital goods) as an index of experience. Each new machine produced and put into use is capable of changing the environment in which produc*on takes place, so that learning is taking place with con*nually new s*muli. This at least makes plausible the possibility of con*nued learning in the sense, here, of a steady rate of growth in produc*vity.” Arrow (1962).

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Produc*on and Externali*es from the Accumula*on of CapitalProduc*on of goods and services takes place through a large number of compe**ve firms. The produc*on func*on of firm i is given by,

4

Yi (t) = AKi (t)α (h(t)Li (t))

1−α

where 0<α<1

The efficiency of labor is a func*on of aggregate capital per worker (learning by doing) and exogenous technical progress. It is thus determined by,

h(t) = K(t)L(t)

⎛⎝⎜

⎞⎠⎟β

egt( )1−β

where 0≦β


The Aggregate Produc*on Func*on in the Learning by Doing Model

The aggregate produc*on func*on is given by,

5

Y (t) = A(K(t))α+β (1−α )(egtL(t))1−(α+β (1−α ))

Aggregate Output per Worker is given by,

y(t) = A(k(t))α+β (1−α ) egt( )1−α−β (1−α )


Output per Worker as a Func*on of Capital per Worker

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k

y

β=0

0<β<1

β=1β>1


Externali*es from the Accumula*on of Capital and the Aggregate Produc*on Func*on

• For β=0, there is no effect of the aggregate accumula*on of capital on labor efficiency, and we are back to an aggregate Cobb Douglas produc*on func*on without externali*es. The efficiency of labor depends only on the exogenous rate of technical progress g.

• For 0<β<1 the accumula*on of capital implies a posi*ve externality on the efficiency of labor, but the aggregate marginal product of capital tends to fall as the economy accumulates more capital per worker. Capital accumula*on leads to diminishing returns, although the produc*vity of capital declines at a slower rate than if there were no externali*es.

• For β=1, the aggregate marginal product of capital is constant, equal to total factor produc*vity A, and is not affected by the accumula*on of capital. There are no diminishing returns to capital accumula*on as the aggregate marginal product of capital is constant.

• Finally, for β>1, the marginal product of capital increases with capital accumula*on, but this assump*on violates the condi*on of constant returns to scale for the aggregate economy.

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A Learning by Doing Model of Endogenous Growth

We shall concentrate on the special case β=1, which implies endogenous growth without viola*ng CRS.

8

Y (t) = AK(t)Aggregate output per worker is given by,

y(t) = Ak(t)Because of the linearity of the aggregate produc*on func*on, the rate of growth of output per worker, or per capita income g is equal to the rate of growth of capital per worker. The accumula*on of capital per worker does not lead diminishing returns.

y•(t)y(t)

= k•(t)k(t)

= g


Determina*on of the Real Interest Rate and Real Wages

Firms operate under perfect compe**on and they maximize profits by taking the prices of inputs as given. Profit maximiza*on implies that the real interest rate will be equal to the marginal product of capital for individual firms, and the real wage to the marginal product of labor for individual firms.

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r(t) = aAki (t)α−1k(t)1−α −δ =αA −δ = r

w(t) = (1−α )Aki (t)α k(t)1−α = (1−α )Ak(t) = (1−α )y(t)

The real interest rate is constant and equal to the private marginal product of capital, as calculated by each individual firm. The real wage is a constant share of output per worker. If output per worker is growing at a rate g, then the real wage per worker will also be growing at a rate g.


Endogenous Growth and the Savings Rate in the Learning by Doing ModelLet us first assume that, as in the Solow model, consumer behavior is described by a constant savings rate. Per capita consump*on is thus given by,

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c(t) = (1− s)y(t)

k•(t) = sy(t)− (n +δ )k(t) = (sA − n −δ )k(t)

The accumula*on of per capita capital is given by,

As a result,

g = k•(t)k(t)

= y•(t)y(t)

= sA − n −δ


• The higher the savings rate s, and total factor produc*vity A, the higher the growth rate of per capita income. On the other hand, popula*on growth (employment) and the deprecia*on rate have a nega*ve impact on the endogenous growth rate.

• In this model the savings rate plays a similar role to its role in the exogenous growth Solow model. In the exogenous growth Solow model, the savings rate has a posi*ve impact on steady state capital per effec*ve unit of labor k*, and the growth rate during the convergence process towards k*, but does not affect the long-‐term growth rate, which is equal to the exogenous rate of technical progress g. In this endogenous growth Solow model, with posi*ve externali*es from capital accumula*on, the saving rate s determines the gross investment rate, and, through the gross investment rate the endogenous growth rate.

• The accumula*on of capital does not imply diminishing returns for the marginal product of capital in this endogenous growth model, and growth con*nuous for ever.

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Endogenous Growth and the Savings Rate in the Learning by Doing Model


Convergence in the Endogenous Growth Learning by Doing Model• In the endogenous growth learning by doing model, "poor" and "rich" economies in terms of ini*al capital, will not converge to the same per capita income, even if they have the same savings rate, the same total factor produc*vity, the same rate of popula*on growth and the same deprecia*on rate.

• They will simply have the same steady state growth rate, without converging to the same per capita income.

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Endogenous Growth in a Representa*ve Household Learning by Doing Model

From the Euler equa*on for consump*on, which describes the op*mal savings behavior of the representa*ve household, the rate of change of per capita consump*on is given by,

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c•(t)c(t)

= 1θr(t)− ρ( ) = 1

θαA −δ − ρ( )

As all per capita variables grow at the same rate on the balanced growth path, the growth rate will be determined by,

g = c•(t)c(t)

= k•(t)k(t)

= y•(t)y(t)

= 1θaA −δ − ρ( )


The Determinants of Endogenous Growth in the Representa*ve Household Learning by Doing

Model• A higher pure rate of *me preference of households ρ, rela*ve to the private marginal product of capital to firms (the real interest rate), results in a lower endogenous growth rate of per capita income and consump*on. This is because a higher pure rate of *me preference of households implies lower savings and a lower rate of accumula*on of capital.

• On the other hand, a higher total factor produc*vity A, or a higher private contribu*on of capital to output α, lead to a higher endogenous growth rate, as both result in a higher equilibrium real interest rate, and higher savings and capital accumula*on rates.

• For the opposite reason, the deprecia*on rate δ has a nega*ve impact on the endogenous growth rate.

• A higher elas*city of inter-‐temporal subs*tu*on of consump*on 1/θ results in a higher growth rate, as it facilitates savings.

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The Savings Rate in the Endogenous Growth Ramsey Model

From the capital accumula*on equa*on in the representa*ve household endogenous growth model, the change in per capita capital is given by,

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k•(t) = Ak(t)− c(t)− (n +δ )k(t)

As a result,

k•(t)k(t)

= A − n −δ − c(t)k(t)

= g

The savings rate is thus determined by,

s = 1− c(t)Ak(t)

= n + g +δA

= 1A

n +δ + 1θ(αA −δ − ρ)⎛

⎝⎜⎞⎠⎟


The Inefficiency of Compe**ve Equilibrium in the Endogenous Growth Ramsey Model

Let us assume there is a social planner who maximizes the inter temporal u*lity func*on of the representa*ve household, under the aggregate and not the private capital accumula*on constraint. The first order condi*on for an op*mum would we given by,

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c•(t)c(t)

= 1θA −δ − ρ( ) = g*> g = 1

θαA −δ − ρ( )

The endogenous growth rate in the compe**ve economy is lower than the socially efficient growth rate. This is because the compe**ve real interest rate (αΑ-‐δ) underes*mates the social net marginal product of capital (Α-‐δ), which, because of the posi*ve externality from capital accumula*on, is higher than the private net marginal product of capital. Since the posi*ve externality from capital accumula*on is not reflected in the real interest rate, households have a smaller incen*ve to save and accumulate capital, and, as a result, the investment rate and the growth rate of the compe**ve economy are lower than what would be socially op*mal.


Endogenous Growth in an Overlapping Genera*ons Model: The Behavior of Consumers

The change in per capita consump*on in the Blanchard Weil overlapping genera*ons model is given by,

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c•(t) = r(t)− ρ( )c(t)− nρk(t) = αA −δ − ρ( )c(t)− nρk(t)

Dividing by per capita output, and assuming that per capita consump*on and output grow at the same rate, we get,

c(t)y(t)

= nρA αA −δ − ρ − g( )


Endogenous Growth in an Overlapping Genera*ons Model: The Accumula*on of Capital

From the capital accumula*on equa*on, ager we divide through by the per capita capital stock, we get,

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g = 1− c(t)y(t)

⎛⎝⎜

⎞⎠⎟A − n −δ

The aggregate rate of economic growth g+n is determined by the difference between the savings (investement) rate *mes total factor (and capital) produc*vity A, minus the deprecia*on rate.


Endogenous Growth in an Overlapping Genera*ons Model

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g

c/y

(c/y)=0

(c/y)E

gE

E

ER

gR=aA-δ-ρ


Endogenous and Exogenous Growth and Real Convergence

• In endogenous growth models of the learning by doing model, there is no convergence process.

• In exogenous growth models, any two economies characterized by the same parameters describing the technology of produc*on, household preferences and economic policy, will converge to the same balanced growth path, even if they start from different ini*al condi*ons.

• In endogenous growth models they will have the same endogenous growth rate, but they will not converge to the same per capita income. Their ini*al differences will remain for ever.

• The available empirical evidence from post war interna*onal experience indicates that convergence cannot be dismissed easily.

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Inter-‐temporal Path of Per Capita Income and Real Convergence in Endogenous and Exogenous

Growth Models

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Models of Human Capital Accumula*on and Economic Growth• In the learning-‐by-‐doing endogenous growth model, endogenous growth is essen*ally a by-‐product of the accumula*on of physical capital, because of the assump*on that labor efficiency is a func*on of the aggregate physical capital stock per worker.

• An alterna*ve class of growth models (Lucas (1988), Mankiw, Romer and Weil (1992), Jones (2002)) emphasizes the educa*on and training of workers and the accumula*on of human capital that it implies. The accumula*on of human capital brings about an increase in the efficiency of labor.

• Under some condi*ons, this class of models can also lead to endogenous growth.

• Endogenous growth in such models is not a by-‐product of physical capital accumula*on, as in the Arrow-‐Romer model, but also depends on the factors that determine the accumula*on of human capital.

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The Produc*on Func*on in Models with Human Capital Accumula*on

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Y (t) = AK(t)α a(t)L(t)( )1−α

a(t) = (h(t))γ (egt )1−γ

where 0<α<1, 0≦γ≦1

h=H/L is human capital per worker


The Generalized Solow Model (Mankiw, Romer and Weil)

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K•(t) = sKY (t)−δK(t)

H•(t) = sHY (t)−δH (t)

Y (t) = C(t)+ K•(t)+ H

•(t)+δ (K(t)+ H (t))

C(t) = (1− sK − sH )Y (t)


The Generalized Solow Model in Exogenous Efficiency Units of Labor

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h~•

(t) = sH y~(t)− (n + g +δ )h

~(t)

y~(t) = c

~(t)+ k

~•

(t)+ h~•

(t)+ (n + g +δ )(k~(t)+ h

~(t))

c~(t) = (1− sK − sH )y

~(t)

k~•

(t) = sK y~(t)− (n + g +δ )k

~(t)

where, k~(t) = K(t)

egtL(t)h~(t) = H (t)

egtL(t)y~(t) = Y (t)

egtL(t)c~(t) = C(t)

egtL(t)


The Balanced Growth Path in the Exogenous Growth Generalized Solow Model

(Mankiw, Romer and Weil)

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k~*= sK

sHh~* h

~*= A(sK

α sH1−α )

n + g +δ⎛⎝⎜

⎞⎠⎟

1(1−γ )(1−α )

y~*= A(sK

α sHγ (1−α ) )

n + g +δ( )α+γ (1−α )

⎛

⎝⎜⎞

⎠⎟

1(1−γ )(1−α )

0<γ<1, implies exogenous growth at a rate g

c~*= (1− sK − sH )y

~*


The Balanced Growth Path in the Exogenous Growth Generalized Solow Model

• The level of per capita output on the balanced growth path depends posi*vely on total factor produc*vity A and the shares of output invested in physical and human capital (sK και sH).

• As in the original Solow model it depends nega*vely on the popula*on growth rate n, the rate of exogenous technical progress g, and the deprecia*on rate δ.

• The rate of growth of per capita output on the balanced growth path is equal to the rate of exogenous technical progress g.

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Endogenous Growth in the Generalized Solow Model

(Mankiw, Romer and Weil)

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k *(t) = sKsHh*(t)

γ=1, implies endogenous growth at a rate,

g = y•*(t)y*(t)

= Askα sh

1−α − (n +δ )


Endogenous Growth in the Generalized Solow Model

• The Generalized Solow Model becomes an endogenous growth model if γ=1.

• On the balanced growth path, the ra*o of physical to human capital is stabilised at the ra*o of the investment rates in physical and human capital. Because both physical and human capital are growing at the same rate, we have endogenous growth. The accumula*on of physical capital causes an increase in income, that in turn causes an increase in human capital through expenditure on educa*on and training. This in turn leads to a further rise in output, which causes new investment in physical capital. The parallel accumula*on of physical and human capital leads to endogenous growth.

• The endogenous growth rate depends posi*vely on total factor produc*vity A, and a weighted average of the income ra*os invested in physical and human capital sK and sH.

• The rate of growth of popula*on n, and the deprecia*on rate δ, have a nega*ve impact on the endogenous growth rate.

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The Produc*on Func*on of the Generalized Solow Model in Discrete Time

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Yt = AKtα atLt( )1−α

at = (ht )γ (1+ g)(1−γ )t

Lt = (1+ n)t

where, 0<α<1, 0≦γ≦1.

h=H/L is human capital per worker and L the number of workers.


Consump*on and Accumula*on Equa*ons of the Generalized Solow Model in Discrete Time

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Kt+1 = sKYt + (1−δ )Kt

Ht+1 = sHYt + (1−δ )Ht

Yt = Ct + Kt+1 + Ht+1 − (1−δ )(Kt + Ht )

Ct = (1− sK − sH )Yt


The Generalized Solow Model per Exogenous Efficiency Unit of Labor

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k~t+1 =

1(1+ n)(1+ g)

sK y~

t+ (1−δ )k~t

⎛⎝

⎞⎠

h~t+1 =

1(1+ n)(1+ g)

sH y~

t+ (1−δ )h~t

⎛⎝

⎞⎠

y~

t = Ak~t

α

h~t

γ (1−α )c~t = (1− sK − sH )y

~

t

k~t =

Kt

(1+ g)t Lth~t =

Ht

(1+ g)t Lty~

t =Yt

(1+ g)t Ltc~t =

Ct

(1+ g)t Ltwhere,


The Real Interest Rate and the Real Wage per Exogenous Efficiency Unit of Labor

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rt =αAk~t

α−1

h~t

γ (1−α )

−δ

w~t = (1−α )Ak

~t

α

h~t

γ (1−α )


Steady State Capital, Human Capital, Output and Consump*on in the Generalized Solow Model

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k~*= sK

sHh~* h

~*= A(sK

α sH1−α )

(1+ n)(1+ g)− (1−δ )⎛⎝⎜

⎞⎠⎟

1(1−γ )(1−α )

y~*= A A(sK

α sHγ (1−α ) )

(1+ n)(1+ g)− (1−δ )⎛⎝⎜

⎞⎠⎟

1(1−γ )(1−α )

c~*= (1− sK − sH )y

~*

0<γ<1, implies exogenous growth at a rate g


Steady State Real Interest Rate and Real Wage in the Generalized Solow Model

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r*=α (1+ n)(1+ g)− (1−δ )sK

−δ

w~*= (1−α )A A(sK

α sHγ (1−α ) )

(1+ n)(1+ g)− (1−δ )⎛⎝⎜

⎞⎠⎟

1(1−γ )(1−α )


A Dynamic Simula*on of the Generalized Solow Model

Assump*ons about parameters:

Α=1, α=0.333, γ=0.333, sK=0.25, sH=0.05, n=0.01, g=0.02, δ=0.03

We inves*gate two alterna*ve scenarios

1. An increase of the investment rate in human capital by 0.005, from 0.05 to 0.055).

2. A correspondin increase of the investment rate in physical capital by 0.005, from 0.25 to 0.255.

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Values on the Balanced Growth Path

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Αρχική Αύξηση sH Αύξηση sK

k 10.99 11.52 11.38

h 2.20 2.54 2.23

y 2.65 2.77 2.69

c 1.85 1.93 1.87

r 0.050 0.050 0.049

w 1.76 1.85 1.79




The Exogenous Growth Model of Jones

The efficiency of labor is determined by,

40

a(t) = h(v,ψ )egt

where the func*on h(v,ψ) determines human capital per worker. This depends posi*vely on the *me each worker devotes to educa*on and training v, and the rate of return to inves*ng in human capital (rate of return to educa*on) ψ. Jones assumes this func*on is exponen*al.

h(v,ψ ) = eψ v


Per Capita Income in the Jones exogenous growth model

Per Capita Output and Income on the Balanced growth path evolves as,

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y*(t) = A sAn + g +δ

⎛⎝⎜

⎞⎠⎟

α1−α

eψ vegt

Per capita output and income on the balanced growth path is a posi*ve func*on of the amount of *me spent in educa*on and training v, and the rate of return on investment in human capital ψ. However, in other respects, this model is an exogenous growth model, similar to the Solow model.


The Lucas Endogenous Growth Model

According to Lucas (1988), the produc*on of human capital per worker is determined by,

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a(t) = h(t) = eζ (1−u*−(n+δ ))t = e1θ(ζ −δ −ρ )t

h•(t) = ζ (1− u(t))h(t)− (n +δ )h(t)

where h(t) denotes human capital. The produc*on of human capital depends posi*vely on the the propor*on of *me that workers devote to educa*on and training 1-‐u(t), and the efficiency of produc*on of human capital ζ.

In steady state, labor efficiency per worker evolves according to,


The Determinants of Endogenous Growth in the Lucas Model

• Lucas assumed that u is chosen by a representa*ve household in order to maximize its inter-‐temporal u*lity of consump*on. As a result, the choice of u depends both on the preferences of the representa*ve household, and on the technological parameters characterizing the produc*on of goods and services and human capital.

• The endogenous growth rate g=ζ(1-‐u*)-‐(n+δ) in the Lucas model works like the exogenous rate of technical progress in exogenous growth models.

• The higher is the steady state propor*on of non-‐leisure *me devoted to educa*on and training 1-‐u*, the higher the endogenous growth rate. In the Lucas model, 1-‐u* is chosen endogenously, and, in conjunc*on with the exogenous ζ, δ and n, determines the steady state growth rate of per capita output.

• The steady state endogenous growth rate is equal to (1/θ)(ζ-‐ρ-‐δ).

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Models of the Produc*on of Ideas and Innova*ons

• A final category of growth models, model technical progress as the result of ideas and innova*ons, that lead to higher total factor produc*vity or labor efficiency.

• These models emphasize the externali*es involved in genera*ng new ideas and innova*ons that increase the efficiency of produc*on. As the learning by doing model of Arrow emphasizes the externali*es from capital accumula*on, so the ideas and innova*ons models emphasize the externali*es of the produc*on of ideas and innova*ons.

• Although models in this category, some*mes called research and development (R&D) models, date from the late 1960s, the microeconomic founda*ons of these models and their implica*ons for the func*oning of markets were developed in the early 1990s, inspired by the work of Romer (1990). These models, under certain condi*ons, can lead to endogenous growth as well.

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Key Features of Ideas and Innova*ons

• A crucial assump*on of such models is that ideas and innova*ons can improve the efficiency of produc*on, either through total factor produc*vity or through labor efficiency.

• These models recognize that, unlike most other goods and services, the use of an idea and/or innova*on by a par*cular firm, or employee, does not prevent to use of the same idea and innova*on from other firms or workers. The use of a par*cular idea or innova*on is non rivalrous, in contrast to the use of a par*cular machine or a specific employee. From the *me an idea or innova*on has been produced, anyone with knowledge of this idea can use it, independently of how many others use it simultaneously. If a firm uses a specific machine or a par*cular employee, it automa*cally excludes any other firm from simultaneously using this same machine or this same employee. The property of non-‐rivalry gives ideas and innova*ons a character of a quasi public good.

• On the other hand, in contrast to purely public goods, the use of an idea can be parYally excludable by law. This allows the producer of an idea to charge for the use of his idea. For example, if an idea or innova*on is legally covered by a patent, then a firm or an employee who wants to use this idea or innova*on will have to pay a fee to the holder of the patent, for their copyright.

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Externali*es and the Produc*on of Ideas and Innova*ons

• Purely public goods and services are both non rivalrous and non excludable. Other goods and services, such as ideas that can be covered by copyright laws, may be non rivalrous, but may be excludable. Consequently, the producers of goods and services that use an idea or innova*on, may be charged for the benefits arising from their use.

• The produc*on of non rivalrous and non excludable goods and services implies externali*es, which are not reflected in the remunera*on of producers of these goods and services. Goods and services that result in posi*ve externali*es, such as ideas and innova*ons, will be produced in smaller quan**es than would be socially desirable, and goods and services that result in nega*ve externali*es, such as pollu*on of the environment, will be produced in larger quan**es than would be socially desirable.

• If the use of ideas and innova*ons is both non rivalrous and non excludable, then the market will produce fewer ideas and innova*ons than would be socially desirable. However, if the use of ideas and innova*ons can be made excludable by the protec*on of the law on copyright or a patent, then the produc*on of ideas and innova*ons can rise, as producers of ideas and innova*ons will be paid for the value of their product.

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The Key Elements of an Ideas and Innova*ons Growth Model

We assume a Cobb Douglas produc*on func*on of the form,

47

Y (t) = AK(t)α H (t)Ly(t)( )1−α

where H is the exis*ng aggregate stock of ideas and innova*ons, affec*ng the efficiency of labor in the goods and services sector, and Ly is the number of workers who are employed in the goods and services sector.

Apart from goods and services, the economy produces new ideas and innova*ons. The new ideas and innova*ons produced per instant depend on the exis*ng stock of ideas and innova*ons H , and the number of research workers, i.e those employed in the produc*on of ideas and innova*ons. Lh.

H•(t) = hH (t)β Lh (t)

γ h > 0,0 < β <1,0 < γ <1


Diminishing Returns in the Produc*on of New Ideas and Innova*ons

• The produc*on func*on implies that the exis*ng stock of ideas and innova*ons has diminishing returns in the produc*on of new ideas and innova*ons, because the higher the exis*ng stock of ideas and innova*ons, the more difficult it will be to discover new ones.

• It also implies that the number of research workers is also associated with diminishing returns, as the likelihood of duplica*on of effort in the produc*on of new ideas and innova*ons is growing with the number of research workers.

48


A Generalized Solow Model based on the Produc*on of Ideas and Innova*ons

49

L(t) = Ly(t)+ Lh (t)

L•(t) = nL(t)

K•(t) = sYY (t)−δK(t)

Lh (t) = sH L(t)

Ly(t) = (1− sH )L(t)


The Determina*on of the Rate of Technical Progress

50

H•(t)

H (t)= hsH

γ( )H (t)β−1L(t)γ

g = γ n1− β

Under these assump*ons, the produc*on func*on of new ideas and innova*ons can be wriWen as,

On the balanced growth path, the rate of technical progress will be constant. Taking logarithms and then taking first deriva*ves with respect to *me, it follows that the steady state rate of technical progress is given by,


The Determinants of Endogenous Technical Change in the Ideas and Innova*ons Solow

Model

51

g = γ n1− β

The rate of technical progress on the balanced growth path, denoted by g, will be propor*onal to the popula*on growth rate n. The reason is that the popula*on growth rate determines the growth rate of research workers, who contribute to the genera*on of new ideas and innova*ons.

The only other parameters that determine the endogenous rate of technical progress are the parameters of the produc*on func*on of new ideas and innova*ons, β and γ. Both have a posi*ve impact on the endogenous rate of technical progress. The higher the elas*city of produc*on of new ideas and innova*ons with respect to the exis*ng stock of ideas and innova*ons (β) and with respect to the number of research workers (γ), the higher the rate of technical progress on the balanced growth path.

Thus, this model aWributes technical progress to popula*on growth, and the parameters of the produc*on func*on of new ideas and innova*ons.


The Balanced Growth Path in the Ideas and Innova*ons Solow ModelOn the balanced growth path, all per capita variables grow at the endogenous rate of technical progress g. Variables per efficiency unit of labor are determined in a way similar to the corresponding Solow model. One can show that, on the balanced growth path, we shall have,

52

k*= sY A(1− sH )1−α

n + g +δ⎛⎝⎜

⎞⎠⎟

11−α

y*= A(1− sH )1−α k *( )α c*= (1− sY )y*

where k = KHL

, y = YHL

,c = CHL

and H•(t)

H (t)= γ n1− β

= g


Conclusions from the Generalized Growth Models

• We have analyzed more general growth models, which, instead of relying only on the assump*on of exogenous technical progress, are based on the assump*ons of either external effects from the accumula*on of physical capital, or investment in human capital, or even the endogenous genera*on of ideas and innova*ons.

• Endogenous growth models do not necessarily provide for convergence, as the corresponding exogenous growth models. However, the available empirical evidence from post war interna*onal experience (see for example Mankiw, Romer and Weil (1992) and Barro (1997)) indicates that the issue of convergence of per capita incomes of the various economies cannot be dismissed easily. This convergence can be explained by generalized models in which there is learning by doing, accumula*on of human capital and endogenous technical progress, but not to a degree that would completely neutralize the diminishing returns from the accumula*on of physical capital.

• Consequently, generalized exogenous growth models, in which there are externali*es from capital accumula*on, and/or accumula*on of human capital and endogenous technical progress, could, in principle, explain most of the aspects of the process of economic growth that cannot be explained by the original Solow model, or the corresponding representa*ve household or overlapping genera*ons models.

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Macro Lect 8 2015 Endogenous Growth - …-with-posi*ve-externali*es-from-capital- ... Macro Lect 8 2015 Endogenous Growth

Macro Lect 8 2015 Endogenous Growth - …-with-posive-externalies-from-capital- ... Macro Lect 8 2015 Endogenous Growth