Slides growth

Growth Theory
PD Dr. M. Pasche
DFG Research Training Group “The Economics of Innovative Change”,
Friedrich Schiller University Jena
Creative Commons by 3.0 license – 2008/2013 (except for included graphics from other sources)
Work in progress. Bug Report to: markus@pasche.name
S.1

Outline:
1. The Empirical Picture of Growth
1.1 Some Stylized Facts
1.2 Convergence
1.3 Growth Accounting
1.4 Regressions on Growth Determinants
2. Some Preliminaries of Growth Theory
2.1 The Harrod-Domar Approach
2.2 The Basic Solow Model
2.3 Exogenous Technological Change
2.4 Intertemporal Optimization
2.5 Analyzing Growth Equilibria
3. Models of Endogenous Growth
3.1 Overview: Sources of Growth
3.2 AK model and Knowledge Spillovers
3.3 Models with Human Capital Accumulation
3.4 R&D based Growth with Increasing Product Variety
3.5 R&D based Growth with Increasing Product Quality
3.6 Technological Progress, Diﬀusion, and Human Capital
3.7 Further Issues S.2

4. Critique and an Evolutionary Perspective
4.1 Empirical Evidence
4.2 Methodological Objections
4.3 Evolutionary Approaches: Outline
4.4 Evolutionary Approaches: Example
Basic Literature:
* Barro, R.J., Sala-i-Martin, X. (1995), Economic Growth. New York:
McGraw-Hill.
* Aghion, P., Howitt, P. (2009), The Economics of Growth. MIT
Press.
◮ Acemoglu, D. (2008), Introduction to Modern Economic Growth.
Princeton University Press.
References to more speciﬁc literature can be found in the slide collection.
S.3

Literature:
◮ Barro, R.E., Sala-i-Martin, X. (1995), Economic Growth.
Chapter 1.1-1.2 (chapter 10-12 for a deep empirical analysis)
◮ Kaldor, N. (1963), Capital Accumulation and Economic Growth, in:
Lutz, F.A., Hague, D.C. (eds.), Proceedings of a Conference held by
the International Economics Association. London: Macmillan.
◮ Mankiw, N.G., Romer, D., Weil, D.N. (1992), A Contribution to the
Empirics of Economic Growth. Quarterly Journal of Economics
107(2), 407-437
◮ Temple, J. (1999), The New Growth Evidence. Journal of Economic
Literature 37(1), 112-156.
Symbols:
Y = A · F(K, N) = real output or income
K = capital stock
N = employed labor
A = total factor productivity
r = real interest rate S.4

◮ Income per capita y = Y /N is growing with a constant rate (but
declining growth rate in the 1970ies in most developped countries).
◮ The capital/output ratio (capital coeﬃcient) K/Y is stationary.
◮ The capital/labor ratio (capital intensity) K/N is increasing. This is
just an implication of a growing Y /N and a stationary K/Y .
◮ The rate of return to capital r = ∂Y /∂K is stationary (but has a
certain decline in developped countries).
◮ The income distribution is stationary (measured by V = rk/wN or
by wN/Y , rK/Y ).
◮ The rate of return to labor w = ∂Y /∂N is increasing. This is just
an implication of stationary distribution, stationary K/Y and
growing Y /N.
◮ The per capita growth rates diﬀer much across countries.
◮ The per capita growth rate cannot be explained solely by
accumulation of capital and growing labor force (→ technical
progress, human capital, knowledge etc.).
S.5

A note on growth rates:
Growth with a constant rate g means that the variable grows
exponentially:
y(t) = y(0)egt
Logarithm and differentiating with respect to time:
ln y(t) = ln y(0) + gt ⇒ gy ≡
d ln y(t)
dt
=
1
y
·
dy
dt
= g
For empirical data we use the first differences ∆ ln y(t) to
determine the growth rate.
Growth with a constant rate means that we have a linear trend of
ln y(t) in a figure with absolute scale, or, alternatively, a linear
trend of y(t) in a figure with a logarithmic scale.
S.6

Some illustrating empirical facts on growth dynamics:
◮ From 500 (roman imperium) to 1500: no signiﬁcant economic
growth!
◮ 1500-1800 about 0.1% growth rate.
◮ Moderate growth rates during the industrial revolution
1800-1900, increasing in the late 19th century.
◮ Massive acceleration of economic activity in the 20th century,
especially in the post war period.
◮ Decline of growth rates (in developped countries) starting
from the 1970ies.
Some illustrating empiricial facts on distribution (base = 2002):
◮ The richest country is Luxembourg with $ 49368 per capita,
the poorest country is Kongo with $ 344 (= factor 143!)
◮ If Bangladesh grows with its average post war growth rate of
1.1% then it approaches the 2002 level of per capita income
of the USA in 200 years. S.7

S.8

S.9

1.2 Convergence
Are less developped countries growing faster
(“catching-up”)?
Measuring convergence:
◮ β-convergence: Negative relationship between per capita
income y = Y /N and growth rate gy .
◮ σ-convergence: Decline of a dispersion measure (like
standard deviation of (logarithmic) per capita income, Gini
coeﬃcient etc.)
S.10

1.2 Convergence
Problems:
◮ To be comparable, per capita income has to measured with
the same unit (e.g. Dollar). Hence we have to multiply the
values with the exchange rate.
◮ The exchange rates are fluctuating and are determined by
variables which are not related to real income (i.e.
non-fundamental expectations). Thus, the per capita income
measured in a foreign currency may change even if ther real
output remains the same: distortion of the measure.
◮ Moreover, we have eventually different inflation rates in the
countries. Since we can measure the nominal income and the
inflation rate, we have to account for the different purchasing
power when expressing the income in a foreign currency.
Solution: Construcing “purchasing power parity” exchange rates
(PPP) to express all values in Dollar (e.g. Penn World tables)
S.11

1.2 Convergence
General result: There is no overall β-convergence!
(Penn World Tables, x-axis = y1960, y-axis = gy as φ 1960-1992)
S.12

1.2 Convergence
Average growth rate of per capita income in 1960-1985 vs. ln(y) in 1960; 117
countries.
S.13

1.2 Convergence
Speciﬁc results: There is β-convergence within a group of
countries which are “similar” regarding properties like high human
capital endowment, stable political institutions etc.
⇒ conditional β-convergence
⇒ “convergence clubs”
⇒ the gap between “rich” and “poor” countries is growing.
S.14

1.2 Convergence
Frequency of per capita income classes; 117 countries.
In 1960: E[ln(y)] = 7.296, V [ln(y)] = 0.81275, V /E = 0.1114.
S.15

1.2 Convergence
Frequency of per capita income classes; 117 countries.
In 1985: E[ln(y)] = 7.7959, V [ln(y)] = 1.2126, V /E = 0.1555.
S.16

Literature:
◮ Solow, R.M. (1957), Technical Change and the Aggregate
Production Function. Review of Economics and Statistics 39,
312-320.
We start from a stylized production function Y = A · F(K, N), where
A = A(t) is a time-dependend function for the total factor productivity
(e.g. A = exp(ηt)).
Y (t) = A(t) · F(K(t), N(t))
ln Y (t) = ln A(t) + ln F(K(t), N(t))
Diﬀerentiating with respect to time:
gY = gA +
FK
˙K
F
+
FN
˙N
F
= gA +
AFK
Y
˙K +
AFN
Y
˙N
S.17

gY = gA +
AFK
Y
˙K +
AFN
Y
˙N
with AFK = r and AFN = w we have
= gA +
rK
Y
˙K
K
+
wN
Y
˙N
N
and with a linear homogenous production function
= gA + α(t)gK + (1 − α(t))gN
This can be transformed into an estimation equation for (non-observable)
gA in discrete time.
Measuring Y , K, N the growth contributions of the physical inputs K and
N can be estimated. The part of output growth which cannot be
explained by K and N is the “residual” which is interpreted as technical
progress = increase in the total factor productivity (Solow residual).
S.18

◮ Measuring Y : usually real GDP (from national statistics
agency)
◮ Measuring N: number of employed and self-employed people,
or: time measure (work hours)
◮ Measuring K: This is non-trivial since the accounting systems
measure gross investment and depreciation.
◮ Depreciation depends on legal regulation and is only a rough
proxy for physical depreciation.
◮ In balance sheets the “capital” is evaluated according to
diﬀerent and changing legislation rules.
◮ Perpetual Inventory Method:
Kt = Kt−1 + Igross
t − δKt−1
with δ ∈ (0, 1) as the constant depreciation rate.
S.19

Results:
S.20

Some problems:
◮ Measuring the capital stock (see above), in addition we need
estimates about the utilization of the present capital stock.
Generally, the estimation results are often not robust for changes in
the measurement concept.
◮ All qualitative changes in capital as well as in labor are captured
indirectly in the TFP. However, much progress is embodied in the
physical inputs. It is reasonable to disaggregate the inputs to
account for these effects, e.g. including human capital or
distinguishing groups of different skilled worker (with different
average wages), or distinguishing capital vintages.
◮ The empirical validity of constant returns of scale and competitive
factor markets is questionable.
◮ Growth is also affected by non-technical determinants like stability
of political institutions, tax system, integration into global markets,
protection of intellectual property rights etc. Hence, institutional
change is captured as “technological” change.
S.21

Literature:
◮ Mankiw, N.G., Romer, D., Weil, D.N. (1992), A Contribution
to the Empirics of Economic Growth. Quiarterly Journal of
Economics 107(2), 407-437
◮ Barro, R.E., Sala-i-Martin, X. (1995), Economic Growth.
Chapter 1.1-1.2 (and chapter 12)
◮ Starting point is not a certain production function.
◮ Instead: looking for resonable determinants/regressors
S.22

Example from Mankiw/Romer/Weil:
gyi
= 3.04
(3.66)
−0.289
(4.66)
ln yi,1960+0.524
(6.02)
ln si −0.505
(1.75)
ln(ni +g+δ)+0.233
(3.88)
SCHOOLi +ui
gyi per capita GDP in country i in 1960-1990
yi,1960 per capita GDP in country i in 1960
si saving rate (average 1960-1985)
ni population growth rate
SCHOOLi schooling rate (secondary school, average 1960-1985)
g rate of technical progress
δ depreciation rate
ui error term (iid)
Sample: 98 countries, t-values in brackets
Problems:
◮ Endogenous regressors/multicollinearity
◮ Model uncertainty
S.23

Some “stylized” facts from growth regressions:
* Signiﬁcant positive impact of human capital
(Barro, R.J. (1991), Economic Growth in a Cross Section of Countries.
Quarterly Journal of Economics 106(2), 407-443)
* Knowledge as a public good: positive impact
(Caballero, R.J., Jaﬀe, A.B. (1993), How High are the Giants’ Shoulders:
An Empirical Assessment of Knowledge Spillovers and Creative
Destruction .... NBER Working Paper No. 4370)
◮ Life expectancy, health: positive
◮ Governmental consumption: negative
◮ Political instability: negative;
quality of political institutions: positive
S.24

◮ Financial development (financial institutions): positive
◮ Market distortions (like tariffs): negative
* Integration in global markets: positive
(Balassa, B. (1986), Policy Responses to Exogenous Shocks in
Developping Countries. American Economic Review 76(2), 75-78.
◮ etc. etc.
There are also a lot of ambigous/insignificant results.
S.25

Are high growth rates always “good”?
◮ no information about income distribution and welfare
◮ no information about welfare improving governemntal
acrivities (health care, social insurance etc.) which may damp
the growth rates
◮ environmental degradation and ressource exploitation
◮ increasing “defensive expenditures”: a growing part of the
income is needed to compensate the negative impact of
growth on welfare.
S.26

Role of Growth Theory:
◮ Explanation of the stylized facts = explaining the economic
mechanisms driving the economic activities, depending on
exogenous variables.
◮ Giving advice for growth policy (if there is any); not
neccessarily in order to accelerate growth rates but to realize a
pareto-eﬃcient growth path.
S.27

An economic theory cannot include all reasonable determinants
and eﬀects: Some variables (like Y and K) are endogenously
determined, others (like N) are exogenous, others are not taken
into considration (like human capital in the standard Solow model).
The question is whether the primary source of growth (“growth
engine”) is an endogenous part of the model or not:
◮ “Old” growth theory, where technological progress as a
primary source of growth is exogenous.
◮ “New” growth theory, where diﬀerent types of technical
progress are endogenously explained.
S.28

Remarks:
◮ The “old” growth theory is sometimes called “neoclassical” as
opposed to the “new” endogenous growth theory. This is
misleading since the “new” models follow the neoclassical
paradigm in a more rigorous fashion (intertemporally
optimizing representative agents, perfect (future) markets,
Walrasian equilibrium).
◮ “New” is not always superior (for a critical assessment see the
last section).
◮ Non-mainstream theorizing like evolutionary or Post-Keynesian
growth theory does not ﬁt in the scheme of “old” and “new”.
S.29

Literature:
◮ Harrod, R.F. (1939), An Essay in Dynamic Theory. Economic
Journal 49, 14-33.
◮ Domar, E. (1946), Capital Expansion, Rate of Growth, and
Employment. Econometrica 14, 137-250.
Common Features:
◮ Tradition of Keynesian Macroeconomics; studying the income
and capacity eﬀects of investments
◮ Linear-limitational production function:
Y = min{σK, αL}
with constant σ = 1/ν (σ = capital productivity, ν = capital
coeﬃcient) and a natural growth rate ∆L/L = n = gn
S.30

The Domar Growth Model:
Domar considers the income and the capacity effect of investment:
◮ Income effect: Investments are part of the realized output
(income) Yt.
◮ Capacity effect: Investment augments the capital stock and
therefore enhance the production capacity Y p
t .
S.31

Capacity effect:
◮ Realized investment have an effect on the potential output
according to the constant capital coefficient:
Kt = νY p
t
∆Kt = Kt+1 − Kt = It = ν(Y p
t+1 − Y p
t )
∆Y p
t = Y p
t+1 − Y p
t =
1
ν
It (1)
Income effect:
◮ Constant saving ratio: St = sYt
◮ Goods market equilibrium: It = St. It follows:
Yt =
1
s
It
∆Yt = Yt+1 − Yt =
1
s
(It+1 − It) (2)
ˆYt = Ît (3)
S.32

Assume that additional capacity is utiized:
Then from (1) and (2) we have
∆Yt = Yt+1 − Yt = Y p
t+1 − Y p
t = ∆Y p
t
1
s
(It+1 − It) =
1
ν
It
It+1 − It
It
= ˆIt =
s
ν
and from (3) we have
ˆYt =
s
ν
= σs = gw
This could be called a “balanced growth rate”.
S.33

Domar paradoxon:
◮ Assume that real investment growth ˆIt > gw . Then the
demand Yt grows faster than the capacities Y p
t . That means
that too large investment implies underutilization of
capacities.
◮ Assume that real investment growth ˆIt < gw . Then the
demand Yt grows slower than the capacities Y p
t . That means
that too low investment implies overutilization of capacities.
S.34

Equilibrium and natural growth rate:
◮ Recall, that we have a linear-limitational production function.
Then the growth rate of Y p
t is determined by the growth of
the limiting factor!
◮ The growth rate of labor is gn = n. It is very unlikely that
gn = gw . Note, that n, ν, s are exogenously given parameter.
◮ If gw > gn then we have growing capacities that could not be
utilized due to a scarcity of labor.
◮ If gw < gn then the capacity grows slower than population.
We have increasing unemployment.
S.35

Assume a utilization factor
θ =
Yt
Y p
t
, θ ∈ [0, 1]
ˆθ = ˆYt − ˆY p
t
From the capacity effect we have
Y p
t = σKt
ˆY p
t = ˆKt =
It
Kt
=
sθY p
t
Kt
= sθσ
and hence
⇒ ˆθ = Î − sσθ
This growth rate of capacity utilization depends linearly on the degree of
capacity utilization. A steady state solution ˆθ = 0 leads to
θ∗
=
Ît
sσ
< 1
in case of Ît = ˆYt < sσ = gw . S.36

If growth is lower than the “balanced growth rate” then the
economy evolves into a stable steady state with underutilization of
production capacity which is not desirable.
θ
ˆθ
θ∗
S.37

The Harrod Growth Model:
◮ Harrod considers only the income eﬀect of investment.
◮ Assumption of linear-limitational production function is not
neccessary.
◮ The constant capital coeﬃcient plays a role in the
determination of investment behavior, i.e. ν is a behavioral
parameter of the investment function (“accelerator”).
I = ν(Y e
− Y )
with Y e as the expected demand. With the saving function as
given above and I = S we have
I = S = sY = ν(Y e
− Y )
⇒
Y e − Y
Y
=
s
ν
= ge (4)
with ge as the expected (“warranted”) growth rate.
S.38

◮ If the realized and the expected (constant) growth rate are
equal (¯g = ge) then we have equilibrium growth: The realized
growth of Y leads to a growth of S = I which conforms the
expectations of the investors.
◮ Problem: What happens if realized and expected/warranted
growth rate diﬀers?
S.39

◮ If the realized growth rate is larger, ¯g > ge, then the investors
correct their expectations Y e upwards and invest more. Due
to the income eﬀect this fosters the growth rate: The
economy diverges from the balanced growth path.
◮ If the realized growth rate is lower, ¯g < ge, then the
expectations are corrected downwards, this lowers the realized
growth rate: The economy also diverges from the balanced
growth path.
◮ The equilibrium growth path is dynamically unstable!
(“growth on a knife edge”)
S.40

t
log Yt
S.41

Analytical description:
Deﬁne:
gt ≡
Yt − Yt−1
Yt
(5)
ge
t ≡
Y e
t − Yt−1
Y e
t
(6)
Solving (6) to Yt−1 and employing into (5) yields
gt =
Yt − (Y e
t − ge
t Y e
t )
Yt
= 1 − (1 − ge
t )
Y e
t
Yt
(7)
Recall that from (4) and (6) we have
Y e
t − Yt
s/ν
= Yt,
Y e
t − Yt
ge
t
= Y e
t
S.42

Enployng these expressions into (7) we have
gt = 1 −
(1 − ge
t )
ge
t
s
ν
(8)
Now assume adaptive expectations:
ge
t+1 = ge
t + α(gt − ge
t ), α ∈ (0, 1) (9)
Employing (8) for gt we have
ge
t+1 − ge
t = α
1 − ge
t
ge
t
ge
t −
s
ν
Obviously, we are on a balanced growth path, when gt = ge
t = s/ν.
S.43

◮ With ge
t < s/ν we have ge
t−1 − ge
t < 0,
i.e. growth expectations becomes more and more pessimistic,
inducing a growing (negative) deviation from the balanced
growth path.
◮ With ge
t > s/ν we have ge
t−1 − ge
t > 0,
i.e. growth expectations becomes more and more optimistic,
inducing a growing (positive) deviation from the balanced
growth path.
S.44

ge
t
s
ν
ge
t+1 − ge
t
S.45

Some problems:
◮ The empirical ﬁndings contradict Harrod’s result of a “knife
edge” growth path.
◮ The stable growth with underutilization of capacities
according to Domar does not take into account that in the
long run labor and physical capital should be regarded as
substitutional rather than complementary factors.
⇒ From Keynesian to Neoclassical Growth Theory: Solow Model.
S.46

Literature:
◮ Solow, R.M. (1956), A Contribution to the Theory of Economic
Growth. Quarterly Journal of Economics 70, 65–94.
◮ Swan, T.W. (1956), Economic Growth and Capital Accumulation.
Economic Record 32, 334-361.
Assumptions:
◮ Closed economy without government.
◮ Identical profit-maximizing firms are producing a homogenous good
Y which can either be consumed or invested Y = C + Igross
.
◮ Perfect competition on goods and factor markets, full-employment,
flexible factor prices according to their marginal return, the goods
price index is normalized to one.
◮ Labor supply A (and due to full employment also the demand for
labor N) is growing with the rate n:
gA =
˙A
A
= gN = n
S.47

◮ There is no investment function. Since we have goods market
equilibrium, it is always I = S. By deﬁnition we have
˙K = I = Igross
− δK, δ ∈ (0, 1) depreciation rate
◮ There is a production technology Y = F(K, N) with the
following properties:
◮ FK , FN > 0, FKK , FNN < 0, FKN > 0
◮ Linear homogeneity: λY = F(λK, λN).
Then the output per capita can be expressed by
y =
Y
N
= F
K
N
, 1 ≡ f (k)
with k = K/N, fk > 0, fkk < 0.
◮ Inada conditions: limk→0 f (k) = 0, limk→∞ f (k) = ∞,
limk→0 fk (k) = ∞, limk→∞ fk (k) = 0
◮ Constant savings: S = Y − C = sY , s ∈ (0, 1)
S.48

Derivation of the dynamic equation:
From derivation of k with respect to time we have (quotient rule)
˙k =
˙K
N
− nk
From Y = C + Igross = C + I + δK = C + ˙K + δK we have
˙K = Y − C − δK
Inserting ˙K into ˙k (with y = Y /N = f (k)) we have
˙k =
Y − C − δK
N
− nk
⇒ ˙k = sf (k) − (n + δ)k (10)
For the per capita income we have
y = f (k(t))
˙y = fk
˙k = fk(sf (k) − (n + δ)k) (11)
S.49

k
kk∗
˙k
f (k)
sf (k)
(n + δ)k
S.50

The steady state k∗ is deﬁned as an equilibrium where all values
are growing with a constant rate (and all per capita values are
constant).
Steady state condition ˙k = 0 leads to
sf (k∗
) = (n + δ)k∗
(12)
Since k = K
N doesn’t change in time, we have gK = gN = n and
due to linear homogeneity we have also gY = n. Hence the per
capita output y = Y /N is constant in steady state (as it can also
seen directly in (11)).
S.51

Existence and uniqueness of the equilibrium:
◮ The linear function (n + δ)k is starting in the origin and has a
positive ﬁnite slope (n + δ).
◮ Due to the Inada condition the saving function sf (k) also
starts in the origin but has an inﬁnite slope near to the origin.
With k → ∞ the slope of the saving function decreases to
zero. Both functions are monotonously increasing.
◮ Hence there must exist a unique intersection point with the
linear function (n + δ)k.
S.52

Stability of the equilibrium:
The equilibrium is stable if d ˙k(k∗)/dk < 0:
d ˙k(k∗)
dk
= sfk − (n + δ)
Inserting the steady state condition (12)
= sfk − s
sf (k)
k
< 0
⇒ fk <
f (k)
k
This is ensured by the concavity of the function (see assumption
fk > 0, fkk < 0) [gradient inequality condition].
S.53

Compatible with stylized facts?
◮ Growing y = Y /N cannot be explained without technical
progress!
◮ Growing capital/labor ratio k = K/N cannot be explained.
◮ Constant ratio K/Y is compatible with the model.
◮ Constant income distribution is compatible with the model.
◮ In a transient phase (before approaching the steady state) we
should observe growing per capita income, growing K/N, and
β-convergence, but a changing income distribution.
S.54

Convergence:
For an economy which has not yet reached the steady state equilibrium
we can calculate the per capita growth rate from (11):
gy =
˙y
y
=
fk
y
(sf (k) − (n + δk))
˙k
> 0
This is positive as long k < k∗
⇐⇒ ˙k > 0 (before reaching the steady
state). The dependency of gy from k is negative:
dgy
dk
=
fkk
f (y)2
((sf (k) − (n + δ)k)
˙k
f (k) − fk (n + δ) (f (k) − kfk ))
>0
< 0
This inequality holds true since ˙k > 0 because ykk < 0. Furthermore fk is
the return to capital and hence kfk is the capital income per capita. Thus
f (k) − kfk is the (positive) labor income.
As a result the growth rate gy is high for a low k and vice versa. This
implies unconditional β-convergence!
S.55

◮ In a widely used form technical progress enters the production
function by enhancing the total factor productivity A:
Y = A · F(K, N)
◮ In the “old” growth theory the sources and economic
mechanisms driving the technical progress are not part of the
model.
◮ Technical progress (TP) is modeled as an exogenously
determined process A(t) = A(0)eγt.
S.56

TP – Hicks concept:
◮ TP affects the productivity of both, capital and labor. The
productivity growth has the same impact on the output like
an augmentation of both input factors.
◮ As the growth of (marginal) productivity affects both factors
uniformly, the TP does not affect the relation between factor
prices (wages, interest rate)!
◮ TP is called Hicks-neutral, if the income distribution
V = rK/wN remains unchanged. Since TP does not change
the ratio r/w this implies that capital intensity K/N does also
not change.
◮ TP is called Hicks-labor augmenting if K/N and V increase,
and it is called capital-augmenting if K/N and V decrease.
S.57

N
K
¯Yt
¯Y TP
t
tan α = K/N
V = tan α
tan β = rK
wN
tan β = w/r
S.58

Growth rates in case of Hick-neutral TP and a linear
homogenous Cobb-Douglas production function:
TP is measured by an eﬃciency factor η(t) = η(0)eγt (with
η(0) = 1) which is multiplied with capital and labor
Y = F(ηK, ηN) = (ηK)α
(ηN)1−α
= ηKα
N1−α
= eγt
Kα
N1−α
ln Y = γt + α ln K + (1 − α) ln N
gY = γ + αgK + (1 − α)gN
Since Hicks-neutrality implies gK = gN
gY = γ + gN
S.59

Compatibility with stylized facts?
◮ Per capita income grows with the positive rate
gy = gY − gN = γ.
◮ Income distrbution is constant.
◮ The constant capital intensity K/N does not conform stylized
facts!
◮ With gY > gN = gK the capital coefficient K/Y declines.
This does not conform the stylized facts!
An increasing capital intensity K/N would require Hicks labor
augmenting TP. Unfortunately, then we would have a trend in
income distribution which contradicts the stylzed fact.
Furthermore, the decline of the capital coefficient would be still
conflict with the stylized facts.
S.60

TP – Harrod concept:
◮ TP affects the productivity of labor. The (marginal)
productivity of labor increases and hence the ratio of factor
prices r/w decreases due to TP.
◮ TP is called Harrod-neutral if the income distribution
V = rK/wN remains unchanged. Since r/w decreases, K/N
must increase with the same rate. Furthermore,
Harrod-neutrality implies a constant capital coefficient K/Y .
◮ Harrod-capital or labor augmenting TP could also be defined
but are of minor interest in this context.
S.61

N
K
¯Yt
¯Y TP
t
V = tan α
tan β = tan α′
tan β′ = rK
wN
tan βtan α tan β′
tan α′
S.62

Growth rates in case of Harrod-neutral TP and a linear
homogenous Cobb-Douglas production function:
TP is measured by an eﬃciency factor η(t) = η(0)eγt (and
η(0) = 1) which is multiplied with labor
Y = F(K, ηN) = Kα
(ηN)1−α
= η1−α
Kα
N1−α
= e(1−α)γt
Kα
N1−α
ln Y = (1 − α)γt + α ln K + (1 − α) ln N
gY = (1 − α)γ + αgK + (1 − α)gN
Since Harrod-neutrality implies gK = gY
gY = (1 − α)γ + αgY + (1 − α)gN
(1 − α)gY = (1 − α)γ + (1 − α)gN
gY = γ + gN
(the same result as in case of Hicks-neutral TP).
S.63

Compatibility with a steady state:
◮ Obviously, a steady state cannot be deﬁned as an equilibrium
where all per capita values are constant. It is more generally
deﬁned as an equilibrium, where all per capita values grow
with a constant rate (in case of the standard Solow model:
zero).
◮ From the Solow model we have the steady state condition:
˙k
k
= s
f (k, η)
k
− (n + δ) = const (= γ)
Since s, n, δ are constant, this condition holds true only if
f (k, η)/k = Y /K is also constant which requires
Harrod-neutral technical progress.
◮ As we have seen, it is gY = n + γ. From Harrod-neutrality it
follows gY = gK = n + γ and hence gk = ˙k/k = γ.
S.64

Compatibility with stylized facts?
◮ Per capita income grows with the positive rate
gy = gY − gN = γ.
◮ Income distrbution is constant.
◮ Increasing capital intensity K/N since gK = gY > gN.
◮ Constant capital coeﬃcient K/Y .
⇒ most stylized facts are compatible with Harrod-neutral TP.
S.65

Remarks:
◮ In practice it is not possible to discriminate which part of
output growth is due to capital or due to labor augmenting
TP.
◮ If we interpret growing output as a result of inreased labor
productivity and therefore increase real wages and hence w/r
(e.g. as a result of “productivity-oriented wage policy”) then
we treat TP as if it is Harrod-neutral.
◮ It is unsatisfactory that the TP itself is not explained, i.e. TP
is not generated by economic activity which requires some
ressource input.
S.66

◮ In the standard Solow model, the saving rate s is assumed to
be exogenously given, i.e. the households do not maximize
their utility (problem of missing “microfoundation”).
◮ In a ﬁrst step, we determine the optimal saving rate in a
simple comparative-static framework:
Households maximize their utility from per capita
consumption in the steady state. Since the utility function is
unique up to positive-aﬃn transformation, we could maximize
the per capita consumption in steady state, instead.
S.67

From the steady state condition (k = k∗(s)) we have
sf (k) = (n + δ)k (13)
⇒ f (k) − c = (n + δ)k
max
s
c = f (k) − (n + δ)k
⇒
dc
ds
=
dk
ds
(fk − (n + δ)) = 0 (FOC)
Dividing by dk/ds and inserting the condition (13) yields
fk =
sf (k)
k
⇒ s = fk ·
k
f (k)
(14)
which is known as the “golden rule” of optimal growth.
S.68

k
f (k)
(n + δ)k
k∗
sf (k)
C/Y
⇒ fk = (n + δ)
S.69

Assumptions for intertemporal maximization:
◮ Arrow-Debreu economy:
◮ There exist complete (future) markets for all goods.
◮ The representative agents (household, ﬁrm) are perfectly
informed about all present and future prices.
◮ In each t it is possible to arbitrage goods between all present
and future markets.
◮ Perfect competition on all present and future goods and factor
markets (implying compensation by marginal product).
◮ There are no externalities or other market imperfections
(otherwise intertemporal optimization is possible but yields
pareto-inferior outomes).
◮ Households maximize the net present value of the utility ﬂow
from consumption according to an intertemporal budget
constraint.
S.70

◮ Firms are maximizing their proﬁts, they are price-takers on
goods and factor markets. They produce the homogenous
good Y with constant returns to scale.
◮ Since all present and future markets are in equilibrium, we
have an equilibrium path of goods price, wages and interest
rates.
◮ It is suﬃcient in case of perfect foresight that all optimal
plans are contracted in t = 0. Afterwards there is no need to
revise any decision (markets are open in t = 0, afterwards the
contracts are executed for all t).
◮ If there are stochastic elemets (like technical progress or
uncertainty about the outcome of an R&D process) then we
have no perfect foresight, and the model has to operate with
rational expectations. Agents will immediately adapt their
plans to the stochastic shocks.
S.71

Introduction into Intertemporal Optimization
◮ The representative agent has a control variable c(t). The
decision about consumption implies a decision about savings
an hence capital accumulation.
◮ The state of the economy is represented by a state variable
k(t).
◮ In each time the present value of the utility (objective) is
given by v(c(t), k(t), t).
A typical example is v(c(t), k(t), t) = e−ρtu(c(t))
with ρ > 0 as the time preference.
S.72

◮ The agent’s goal in t = 0 is to maximize the present value:
Finite time horizon:
T
0
v(c(t), k(t), t)dt
Infinite time horizon:
∞
0
v(c(t), k(t), t)dt
which requires that utility is additive-separable in time.
◮ Maximization under the constraint that the state variable
develops according to a differential equation (“law of motion”,
transition equation):
˙k = g(k(t), c(t), t)
A typical example is ˙k = f (k(t)) − c(t) − δk(t).
◮ Of course, for the state variable we have to define the initial
value: k(0) = k0 > 0.
S.73

◮ We need a condition about the value of k at the end of the
time horizon: Typically,
Finite time horizon: k(T)e−¯r(T)T
≥ 0
Inﬁnite time horizon: lim
t→∞
k(t)e−¯r(t)t
≥ 0
where ¯r(t) ∈ (0, 1) is the average discount rate, deﬁned as
¯r(t) =
1
t
t
0
r(v)dv
◮ This means that the present value of the state variable should
be non-negative at the end of the planning horizon. Usually,
the discount rate is the net interest rate = fk(k(t)) − δ.
S.74

The complete problem:
max
c(t)
T
0
v(c(t), k(t), t)dt
subject to ˙k(t) = g(k(t), c(t), t)
k(0) = k0 > 0 given
k(T)e−¯r(T)T
≥ 0
or for an inﬁnite time horizon:
max
c(t)
∞
0
v(c(t), k(t), t)dt
subject to ˙k(t) = g(k(t), c(t), t)
k(0) = k0 > 0 given
lim
t→∞
k(t)e−¯r(t)t
≥ 0
S.75

For solving this problem we build the Hamiltonian function:
H(c(t), k(t), t, µ(t)) = v(c(t), k(t), t) + µ(t)g(c(t), k(t), t)
where µ(t) is a Lagrangian multiplier for each t.
[This expression could be derived from principles of optimization
theory which is not part of the course.]
S.76

Economic interpretation of the multiplier:
◮ In each t the agent consumes c(t) and owns k(t).
◮ Both affects the utility:
◮ Choice of consumption (and eventually k(t)) enters directly
the utility function
◮ Choice of consumption affects the savings and hence the
development of k(t) according to the law of motion. This
affects the future output/income and hence future
consumption and therefore the present value of utility.
◮ The multiplier µ(t) is therefore a shadow price (or
opportunity cost) of a unit of capital in t expressed in units of
utility at time t = 0.
◮ For a given value of µ(t) the Hamiltonian expresses the total
contribution of the choice of c(t) to present utility.
S.77

Solution of the problem:
Let c∗(t) a solution (time path) of the optimization problem, and
k∗(t) is the associated time path of the state variable. Then there
exists a function µ∗(t) (so-called costate variable) so that for all t
following statements hold true:
a) First order condition (FOC):
∂H
∂c(t)
= 0
b) Canonical equations (CE):
∂H
∂µ(t)
= g(c(t), k(t), t) = ˙k(t)
−
∂H
∂k(t)
= ˙µ(t)
The latter is the law of motion for the shadow price.
S.78

c) Transversality condition (TC):
µ(T)k(T) = 0
This means that if the inequality restriction of the problem is
not binding = the final state variable k(T) has a positive
value, then its shadow price must be zero. Otherwise the
agent would leave a positive capital stock unused which could
contribute positively to the present utility. Hence, the TC is
an dynamic efficiency condition!
In case of an infinite time horizon the transversality condition
reads
lim
t→∞
µ(t)k(t) = 0
[We do not discuss the case of non-discounting which requires
another type of TC; see Barro/Sala-i-Martin, appendix 1.3 for
details,]
S.79

How to proceed (this will be demonstrated by an example):
◮ From the FOC and the CE we obtain differential equations for
state variable k and the costate variable µ.
◮ Since the FOC relates c to µ it is possible to eliminate µ and
to derive a differential equation for c instead (the
“Keynes-Ramsey rule”).
◮ Both differential equations ˙c and ˙k have steady state (c∗, k∗)
where ˙k = ˙c = 0.
◮ Depending on the initial conditions, it is usually not clear
whether the system converges to the steady state
(“saddle-point equilibrium”). Since the initial conditions are
chosen by the optimizing agents, they will choose c(0) (for a
given k(0)) which is consistent with the FOC, CE and the
transversality condition. This ensures that the system will be
on a stable path to the steady state.
S.80

An example: Cass-Koopman-Ramsey Model
◮ Cass, D. (1965), Optimum Growth in an Aggregate Model of
Capital Accumulation. Review of Economic Studies 32 (3),
233–240.
◮ Koopmans, T.C. (1965), On the Concept of Optimal Growth.
In: The Econometric Approach to Development Planning,
225–287, North–Holland, Amsterdam.
The basic idea is to provide a microfoundadtion for the neoclassical
Solow model by assuming an intertemporal maximizing household.
It is assumed that the assumptions of the standard Solow model
hold true (with except for the constant consumption/saving rate
which will be replaced by c(t)).
S.81

a) The household:
◮ The household has a time-separable utility function u(c) with
uc > 0, ucc < 0 (1. Gossen Law). He maximizes
max
c
U(0) =
∞
0
u(c)e−ρt
ent
dt =
∞
0
u(c)e−(ρ−n)t
dt
subject to ˙k = w + rk − (n + δ)k − c
k(0) > 0
where w is the wage, r the interest rate. Therefore w + rk is the per
capita income from labor and holding an individual capital stock.
Subtracting consumption, w − rk − c is the (gross) saving per
capita which increases the capital stock. However, depreciation δ
and the growth of the population diminishes the capital per capita.
◮ In the objective function, ρ is the time-preference rate. The
representative household has to take into account that the
“members” of the household grow with the rate n. We must assume
ρ > n, otherwise the integral diverges.
S.82

Solution:
The Hamiltonian is
H(c, k, t, µ) = u(c)e−(ρ−n)t
+ µ · (w + rk − (n + δ)k − c)
The conditions for an optimum are
∂H
∂c
= uc(c)e−(ρ−n)t
− µ = 0 (15)
−
∂H
∂k
= −(r − n − δ)µ = ˙µ (16)
∂H
∂µ
= w + rk − (n + δ)k − c = ˙k (17)
S.83

Diﬀerentiating (15) with respect to time
ucc (c)˙ce−(ρ−n)t
− (ρ − n)uc (c)e−(ρ−n)t
= ˙µ
Substituting ˙µ (r.h.s.) by condition (16):
− (ρ − n)uc (c)e−(ρ−n)t
= −(r − n − δ)µ
Substituting µ by condition (15) ﬁnally eliminates µ:
− (ρ − n)uc (c)e−(ρ−n)t
= −(r − n − δ)uc (c)e−(ρ−n)t
Dividing by e−(ρ−n)t
and rearranging leads to
ucc (c)˙c = uc (c)(r − (ρ + δ))
Dividing by ucc (c)c yields the Keynes-Ramsey rule:
gc =
˙c
c
= −
uc (c)
ucc (c) · c
σ
(r − ρ − δ)
S.84

◮ The expression −uc/(ucc · c) = σ is the intertemporal
elasticity of substitution of the utility function.
◮ In many growth models it is assumed that the utilitiy function
is isoelastic (constant σ). Examples:
u(c) =
c1−θ − 1
1 − θ
, θ > 0, σ = 1/θ
u(c) = log(c) (σ = 1)
◮ The Keynes-Ramsey rule implies that we have a positive
growth rate for the per capita consumption as long as the net
return to capital r − δ exceeds the timepreference rate ρ.
Since there are decreasing returns to capital and hence a
decreasing r the growth rates will also decrease until the path
approaches the steady state.
S.85

b) The Firm:
◮ The representative firm is a price taker (price level is
normalized to 1) and maximizes its period profit:
max
K,N
π(t) = N(t) · [f (k(t)) − r(t)k(t) − w(t)]
◮ From the first order conditions we have
r(t) = fk(k(t))
w(t) = f (k(t)) − fk(k(t))k(t)
In the optimum there are zero profits and the factors are
compensated by their marginal product.
◮ Alternatively, the firm’s objective could also be seen in
maximizing the firm’s present value (net present value of the
proft flow).
S.86

c) Market equilibrium:
In equilibrium all produced goods are demanded either as
consumption or as investment goods:
y = f (k) = ˙k + (n + δ)k + c
Summing up, the optimization behavior of households and ﬁrms
leads to a two-dimensional system of diﬀerential equations
(Keynes-Ramsey rule, intertemporal budget restriction):
˙c = −
uc(c)
ucc(c)
(fk(k) − (ρ + δ)) (18)
˙k = f (k) − (n + δ)k − c (19)
S.87

All time paths {c(t)}∞
t=0 and {k(t)}∞
t=0 generated by this system must
additionally obey the transversality condition
lim
t→∞
µ(t)k(t) = 0
From (16) we have (note that r = r(t))
˙µ
µ
= −(r − n − δ)
⇒ µ(t) = µ(0)e−(¯r(t)−n−δ)t
and from (15) we have for t = 0
µ(0) = uc (c)e−(ρ−n)0
= uc (c)
hence the transversality condition reads
lim
t→∞
uc (c)k(t)e−(¯r(t)−n−δ)t
= 0
Obviously, this requires that average net return of capital exceeds the
growth rate of population: ¯r(t) − δ > n.
S.88

◮ Each growth model with intertemporal optimization yields a
system of diﬀerential equations – e.g. the law of motion for
the per capita capital stock (˙k) and the Keynes-Ramsey rule
for the development of the per capita consumption (˙c).
Furthermore, the transversality condition must hold true.
◮ We are interested in the steady state = ﬁxpoint of the
dynamic system
◮ existence of a (non-trivial) steady state
◮ stability of the steady state
◮ The analysis is demonstrated by the example of the
Cass-Koopman-Ramsey model.
S.89

The Cass-Koopman-Ramsey model has three ﬁxpoints:
(a) c∗ = k∗ = 0. This is the trivial solution will not be discussed
(b) c∗ = 0, k∗ = ¯k with f (¯k) = (n + δ)¯k. In this case the output
is used only to maintain the capital stock, there is no
consumption. This contradicts the TVC.
(c) c∗, k∗ as the solution of ˙c = ˙k = 0.
Equalizing (18) and (19) with zero yields the steady state
fk(k∗
) = ρ + δ (20)
c∗
= f (k∗
) − (n + δ)k∗
(21)
In equilibrium the net return to capital equals the time prefernce
rate, and the per capita savings maintain the equilibrium capital
stock.
S.90

Graphical reresentation:
◮ Phase diagramm: (k, c)-space, each point (vector) is a
certain state of the model. The dynamic equations determine
how this state evolves in time. For a marginal time step this
could be represented by a vectorﬁeld in the (k, c)-space.
◮ Trajectory: Time path of {(k(t), c(t))} starting from any
initial value.
◮ Isocline: The implicit function of all (k, c)-combinations
where ˙c = 0 or ˙k = 0. The intersection point of both isoclines
is the steady state.
S.91

k
˙c = 0
˙k = 0
k∗ ¯k
c∗
c
S.92

◮ The isoclines ˙k = 0 separates the regions with ˙k > 0 and
˙k < 0 (and analogous for ˙c).
◮ We have
∂ ˙c
∂k
= −
uc(c)
ucc(c)
fkk < 0
∂ ˙k
∂c
= −1 < 0
Hence, we obtain the arrow directions of the vector ﬁeld for
the development of an arbitrary trajectory.
◮ We see the trivial solution c∗ = 0, k∗ = 0 as well as the
TVC-violating solution c∗, k∗ = ¯k in the diagramm.
◮ Since the isoclines have a unique intersection point (steady
state) which is a “saddle point”.
S.93

Since we assumed ρ > n the steady state consumption is lower than in
the golden rule due to time preference.
k
˙c = 0
˙k = 0
k∗ ¯k
c∗
c
k
f (k)
(n + δ)k
S.94

Stability of the steady state:
[A detailed introduction into the analysis of dynamical systems is
provided by the course “Economic Dynamics” by Prof. Lorenz!]
The standard analysis of stability is based on linear systems.
Therefore, we linearize the nonlinear Cass-Koopman-Ramsey model
around the steady state. This is a Taylor approximation (1. degree)
of the original system at (c∗, k∗).
˙k
˙c
=
∂ ˙k/∂k ∂ ˙k/∂c
∂ ˙c/∂k ∂ ˙c/∂c
·
k − k∗
c − c∗
S.95

From (19) and (20) we have
∂ ˙k
∂k
= fk(k∗
) − (n + δ) = (ρ + δ) − (n + δ) = ρ − n > 0
Furthermore,
∂ ˙k
∂c
= −1 < 0
∂ ˙c
∂k
= −
uc(c∗)
ucc(c∗)
· fkk(k∗
) < 0
∂ ˙c
∂c
=
[ucc(c∗)]2 − uccc(c∗) · uc(c)
[ucc(c∗)]2
· [fk(k∗
) − (ρ + δ)]
=0, see (20)
= 0
Thus we have
˙k
˙c
=
ρ − n −1
− uc (c∗)
ucc (c∗) · fkk(k∗) 0
J
·
k − k∗
c − c∗
S.96

The determinant of the Jacobian matrix J is
det J = −
uc(c∗)
ucc(c∗)
· fkk(k∗
) < 0
The characteristic polynom is
λ2
− (ρ − n)λ + det J
with the roots (eigenvalues)
λ1,2 =
ρ − n
2
±
1
2
(ρ − n)2 − 4 det J
Since the determinant det J is negative the square-root is taken
from a positive term (real valued ⇒ non-cyclical behavior) and we
have two diﬀerent real-valued roots.
S.97

Cases:
◮ λ1, λ2 < 0: steady state globally stable
◮ λ1, λ2 > 0: steady state globally unstable
◮ λ1 and λ2 have diﬀerent signs: saddle point equilibrium
The last case can be proven to hold true:
λ1λ2 = det J < 0
With the eigenvalues it is now possible to provide a solution
k(t), c(t) for the linearized model (will not be treated in this
course).
S.98

Consequence of saddle-point stability:
◮ In the intertemporal maximization problem we have an initial value
k(0) > 0. To determine a starting point we need a value c(0). As
the vector ﬁeld shows, an in-appropriate choice of c(0) will let the
trajectory diverge from equilibrium!
◮ From the solution of the linearized model it can be seen that for
every given k(0) there exists one speciﬁc c(0)∗
which leads the
trajectory along the saddle path to the steady state.
◮ The transitory dynamic in case of c(0) = c(0)∗
are depicted in the
following graphic by the thin dashed lines (example). The transitory
danmic for c(0) = c(0)∗
is depicted by the bold dashed line.
◮ A choice of the initial c(0) = c(0)∗
either contradicts the
Keynes-Ramsey rule or it contradicts the transversality condition.
By rationality assumption, the representative agent will hence
properly choose c(0)∗
and therefore the saddle-point stability of the
steady state is ensured.
S.99

stable saddle−path
c=0
k=0
k
c
(with f (k) = k0
.6, u(c) = log(c), (n + δ) = (ρ + δ) = 0.2)
S.100

Further properties of the Cass-Koopman-Ramsey model
(details see Barro/Sala-i-Martin, chapter 2)
◮ Pareto-Optimality: Sinde the markets are perfect and there are no
externalities, the intertemporal decisions and hence the growth path of
the model is pareto-optimal. Due to the time preference rate the saving
ration in the steady state is below the “golden rule” in the Standard
Solow model.
◮ Transitory dynamics: The saddle point stability of the steady state
implies a certain policy function c(k), i.e. for each k the policy function
ensures that the economy is on the saddle path to the steady state. It
describes the transitory dynamnics on the saddle path. c(k(t)) could be
computed numerically by approximation technologies.
◮ Convergence:Compared to the Solow model the saving rate is now
endogenously determined but we have to additional stratctural
parameters: intertemporal elasticity of substitution σ and time preference
rate ρ. These parameters shape the rate of convergence but the Solow
results for β- and σ-convergence also hold true for the CKR- model.
◮ Policy implications: Policy may change preference parameters (taxing
household income andb governmental expenditures = changing the saving
ratio). This aﬀects only the per capita income level, not the growth rate! S.101

◮ In the “neoclassical” growth theory (Solow,
Cass-Koopman-Ramsey) we have no steady state growth
neither of per capita income nor of labor productivity.
◮ Extending these models with Harrod-neutral technological
progress lacks an explanation of such a progress. Progress
takes place without any economic activities and without
spending ressources (opportunity cost) to promote this
progress.
◮ Technically spoken, the absence of steady state of per capita
growth is a result from decreasing returns of capital. In a
transitory phase we have an incentive to accumulate capital
but with decreasing r = fk(k) (Inada conditions) the per
capita growth rates diminish and fall to zero in the steady
state (see Keynes-Ramsey rule).
S.102

Solution: Y = K · N1−α?
◮ Increasing returns of scale: not compatible with perfect
competition, no factor compensation according marginal
productivity, Euler theorem not valid!
⇒ No solution!
Looking for models...
◮ with non-diminishing returns of capital
◮ which are compatible with perfect competition (or
monopolistic competition)
◮ with endogenous explanation for technological progress
◮ with policy advice
S.103

Some sources of endogenous growth
a) (Technical) Knowledge:
◮ may be embodied in humans (→ human capital) or
disembodied (“blue prints”, knowledge stock)
◮ in case of disembodied knowledge: non-rival in use,
(non-) disclosure regulated by
◮ intellectual property rights (patents)
◮ high firm specifity
◮ limited absorbability
◮ to the extent where we have disclosure and free access to
knowledge there are positive spillover effects (externalities)
◮ externalities imply that the price system is incomplete and
market based allocation is pareto-inferior
◮ to the extent of non-disclosure there is a private return from
producing knowledge and hence an incentive for R&D
◮ increasing knowledge regarding
◮ new products (variety approaches)
◮ higher product quality (quality approaches)
◮ production efficiency S.104

b) Human Capital:
◮ skills and speciﬁc knowledge of human beings
◮ rival in use, excludability ⇒ private good with a positive return
⇒ incenive to invest into HC.
◮ Accumulation of HC by
◮ learning by doing
◮ by schooling (investment)
◮ Not all eﬀects of HC may be appropriatable, positive
externalities possible
◮ one-sector versus two-sector models
S.105

Literature:
◮ King, R.G., Rebelo, S. (1990), Public Policy and Economic
Growth: Developing Neoclassical Implications. Journal of
Political Economy 98 (5), S126–S150.
◮ Rebelo, S. (1991), Long–Run Policy Analysis and Long–Run
Growth. Journal of Political Economy 99, 500–521.
◮ Barro/Sala-i-Martin (chapter 4.1)
In all models of endogenous growth we assume n = 0, i.e. there is
no population growth!
S.106

a) Households maximize:
max
c
U(0) =
∞
0
u(c(t))e−ρt
dt (22)
conditional to ˙k = f (k) − δk − c
k(0) > 0
and furthermore the TVC holds true:
lim
t→∞
[µ(t)k(t)] = 0
The solution leads to the Keynes-Ramsey rule
gc = σ(r(t) − (ρ + δ))
where σ is assumed to be constant.
S.107

b) Firms produce the output only with capital (constant labor
force is neglected here). Capital includes physical as well as human
capital (“broad measure of capital”, Romer (1989))
y = Ak, A > 0
Hence we have r = fk(k) = A for all t (non-diminishing retuirns of
capital).
The Keynes-Ramsey rule thus reads
gc = σ(A − ρ − δ)
and gc > 0 if net return to capital A − δ exceeds the time
preference rate ρ.
S.108

All values are growing with a constant steady state rate
gy = gc = gk = σ(A − δ − ρ)
Observe that the Keynes-Ramsey rule implies a time-independent
growth rate for c(t) (and henceforth for k(t)). Therefore there is
no transitory dynamic! If TVC holds true, the model starts in
t = 0 in the steady state, i.e. for a given k(0) the initial c(0) is
determined.
S.109

Convergence:
◮ Since there is no transitory dynamic, there is no “catching
up”.
◮ Similar countries (technology, time preference, intertemporal
elasticity of substitution) grow with the same rate.
◮ Growth rate diﬀerences have to be explained by diﬀerent
structural parameters.
S.110

How to justifiy such an AK technology?
◮ Arrow, Kenneth J. (1962), The Economic Implications of Learning
by Doing. Review of Economic Studies 29, 155–173.
◮ Romer, Paul M. (1986), Increasing Returns and Long–Run Growth.
Journal of Political Economy 94, 1002–1037.
◮ Basic idea: There is no explicit “investment” into HC and no
explicit income share for this production factor. HC is
modelled as an external effect or as a by-product of physical
investment. Operating with physical capital goods leads to
“learning by doing” effects which increase human capital ¯K.
S.111

◮ Here HC/knowledge is non-rival in use and there is no
excludability (public good). Each investor also contribute to a
public good.
◮ As for a small firm the influence on the human capital stock is
marginal, it takes ¯K as given.
◮ Profit maximizing implies that the capital cost equals the
privately appropriatbale marginal returns of capital (ignoring
the external effect). Social return exceeds private return of
capital.
S.112

Production function (Cobb-Douglas technology):
Y = f (K, ¯K, L)
In case of Arrow (1962):
y = f (k, ¯K) = ¯Kη
kα
= Nη¯kη
kα
(where η + α = 1 yields the standard AK model)
In case of Romer (1986):
Y = f (K, ¯K · N) = Kα
( ¯KN)1−α
⇒ y = kα ¯K1−α
= N1−α
kα¯k1−α
S.113

a) Households maximize (22) and we have the Keynes-Ramsey
rule
gc = σ(r(t) − (ρ + δ))
where σ is assumed to be constant.
b) Firms maximize
max
K,N
π(k) = N · [kα ¯K1−α
− rk − w] (23)
From the first order conditions we have (with ¯K = Nk)
r = αkα−1 ¯K1−α
= αN1−α
(24)
w = (1 − α)kα ¯K1−α
= (1 − α)kN1−α
(25)
The marginal returns depend on firm specific k as well as on the
given human capital stock ¯K.
S.114

c) Decentral planning (market solution):
◮ With a given labor force N the return to capital r in (24) is
constant.
◮ The Keynes-Ramsey rule with decentralized planning reads
gc = σ(αN1−α
− (ρ + δ))
which is also the steady state growth rate for k.
◮ Since there are positive externalities = the social returns of
capital by inducing growing human capital are neglected in
the factor price r. Hence, the 1. theorem of welfare economics
does not hold true, and the growth path is pareto-ineﬃcient.
S.115

d) Social planner:
◮ A social planner is aware of the externalities, she does not
take ¯K as given. Hence the proﬁts according to (23) reads
max
K,N
π(k) = N · [
Kα
Nα
K1−α
− rk − w] = N · [
K
Nα
− rk − w]
◮ She calculates the FOC as
r = N1−α
and hence the Keymes-Ramsey rule is
g∗
c = σ(N1−α
− (ρ + δ)) > gc
S.116

Policy implications:
◮ Since the decentralied planning leads to pareto-ineﬃcient
steady state growth rates, there is room for welfare increasing
policy.
◮ Generally, incentives for economic activities with positive
spillovers must be increased (e.g. by subsidies), the incentives
for activities with negative spillovers have to be reduced (e.g.
by taxes).
◮ In each case it has to be taken into account that subsidies
have to be ﬁnanced and taxes generate expenditures. Both
has an economic impact on welfare.
S.117

Since physical investment have positive spillovers by creating
human capital, there should be subsidies θ to increase the incentive
to invest. The marginal return is then:
r = α(1 + θ)N1−α
and the Keynes-Ramsey rule is
g∗∗
c = σ(α(1 + θ)N1−α
− (ρ + δ))
By the “method of eyeballing” it is obvious that the optimal rate
of subsidies is
θ∗
=
1 − α
α
because then g∗∗
c = g∗
c .
S.118

How to finance this subsidy?
◮ Income tax: In most democratic systems such a tax is perceived as
“fair”. However, it lowers the marginal returns of the production
factors. As a response, an intertemporally maximizing agent would
then shift his consumption expenditures from the future to the
presence = lower saving = lower capital accumulation = lower
steady state growth rate!
◮ Per capita tax: This tax is perceived as “unfair” because it doesn’t
regard the agent’s ability to pay taxes. However, such a tax does
not affect allocation and has no negative impact on the steady state
growth rate.
◮ Consumption tax: This would not affect the intertemporal decision
between consumption and saving, but it would affect the decision
between working and leisure time. In our model (unelastic labor
supply) this doesn’t play a role.
◮ A subsidy θ∗
combined with a per capita tax is therefore the
optimal tax-transfer system in this model.
S.119

Convergence:
◮ There is no transitory dynamic.
◮ Countries with similar characteristics grow with the same
growth rate.
◮ Countries with different scale of labor force N grow with
different rates: Large countries are growing faster than small
countries (see Keynes-Ramsey rule!). There is no (or only
weak) empirical support for this effect.
◮ This scale effect could be avoided by assuming that the
external effect depends on the average human capital ¯K/N.
S.120

Literature:
◮ Lucas, R.E. (1988), On the Mechanics of Economic Development.
Journal of Monetary Economics 22, 3–42.
Basic idea:
◮ In the models of Romer and Arrow knowledge or human capital has
been represented as a positive externality of physical investment.
Lucas suggests that HC is a speciﬁc producable factor. It is
produced in a separate education sector (2-sector model).
◮ Producing HC requires ressources (opportunity costs) ⇒ allocation
between physical production and human capital accumulation.
◮ HC is treated as a private good. Investments into HC yield a
positive marginal return. In an extension of the model there are also
positive externalities.
S.121

◮ The representative household decides about
◮ intertemporal consumption/saving
◮ allocation of human capital to both sectors
education
production
k
h
y
c
mh
(1 − m)h
S.122

Simplifying assumptions:
◮ To avoid too much notation, we assume no population growth
and no depreciation of physical and human capital (which is
assumed to be identical in the original Lucas-model).
◮ The constant labor force is normalized to one (N = 1).
◮ Accumulation of human capital (schooling) only leads to
opportunity costs since the houshold could either spend time
in the schooling sector or in the production sector. There is no
market price for schooling.
◮ Human capital H is a private good. Hence it is possible to
deﬁne the per capita human capital (individual skill level) as
h(t) = H(t)/N.
S.123

The two sectors:
◮ Human capital (schooling) sector:
˙h(t) = A(1 − m(t))h(t), A > 0, m(t) ∈ [0, 1] (26)
where A is the productivity of the sector, and m(t) is the fraction of
human capital which is allocated to physical production. HC
(output) is produced only with the factor HC (input).
Therefore, ¯H(t) = m(t)H(t) = m(t)h(t)N is the eﬀective human
capital stock used in physical production (note that N = 1).
◮ Production sector:
Y (t) = K(t)α ¯H(t)1−α
⇒ y(t) = k(t)α
(m(t)h(t))1−α
S.124

◮ The capital stock evolves according to the savings
˙k = y − c = [kα
(mh)1−α
] − c = [rk + wmh] − c (27)
◮ Note that income from physical and human capital is used for
consumption expenditures or for saving. There are no
expenditures for schooling (schooling fees), but these will be
included in the model later on.
◮ We have two diﬀerential equations for ˙h and ˙k which are
constraints for the household’s optimization problem!
S.125

a) Households have the following optimization problem:
max
c,m
U(0) =
∞
0
u(c)e−ρt
dt
conditional to ˙k = y − c
˙h = A(1 − m)h
m ∈ [0, 1], k(0) > 0, h(0) > 0
The Hamiltonian is now
H = u(c)e−ρt
+ µ1[[kα
(mh)1−α
] − c] + µ2[A(1 − m)h]
S.126

The optimality conditions are
∂H
∂c
= uc(c)e−ρt
− µ1 = 0 (28)
∂H
∂m
= µ1(1 − α)kα
h1−α
m−α
− µ2Ah = 0 (29)
−
∂H
∂k
= ˙µ1 = −µ1αkα−1
(mh)1−α
(30)
−
∂H
∂h
= ˙µ2 = −µ1(1 − α)kα
m1−α
h−α
− µ2(1 − m) (31)
The partial derivatives to µ1 and µ2 yields the known diﬀerential
equation for ˙k and ˙h. The transversality conditions for k(t) and
h(t) are deﬁned in the usual way.
S.127

◮ Again, we derive the growth rate for consumption
(Keynes-Ramsey rule) and obtain the growth rates gc, gk, gh
and gy . A steady state is deﬁned where all growth rates are
constant and gm = 0 (constant human capital allocation
between production and schooling).
◮ An equilibrium growth path is characterized by identical
constant growth rates.
◮ Deﬁning q = c/k and z = k/h (capital structure) then an
equilibrium growth path implies
gq = gz = gm = 0 ⇐⇒ gy = gc = gh = gk
S.128

Using the new terms the marginal return to capital can be
rewritten as
y = kα
(mh)1−α
⇒ r = yk = αkα−1
(mh)1−α
= αkα−1
(mk/z)1−α
= α(m/z)1−α
Diﬀerentiating (28) with respect to time and inserting (30) to
substitute ˙µ1 leads to the Keynes-Ramsey rule
gc = σ(r − ρ) = σ(αm1−α
z−(1−α)
− ρ) (32)
S.129

From the diﬀerential equation ˙k and ˙h (using the new terms) we
have
gk = m1−α
z−(1−α)
− q
gh = A(1 − m)
Obviously, gq = gc − gk and gz = gk − gh holds true.
S.130

We have not yet discussed the evolution of m (human capital
allocation):
◮ Differentiating (29) with respect to time and then inserting
(30), (31) and the differential equations (27) and (26) in
order to substitute ˙µ1, ˙µ2, ˙k and ˙h leads to a differential
equation for ˙m.
◮ The resulting growth rates are:
gq = (σα − 1)m1−α
z−(1−α)
+ q − σρ
gz = m1−α
z−(1−α)
− q − A(1 − m)
gm =
(1 − α)A
α
+ mA − q
S.131

◮ An equilibrium growth path with gq = gz = gm = 0 leads to
the steady state:
q∗
= σ(ρ − A) +
A
α
(33)
z∗
=
α
A
1
1−α
·
σρ
A
+ 1 − σ (34)
m∗
=
σρ
A
+ 1 − σ (35)
◮ An economically reasonable (positive) solution requires
σ < A/(A − ρ).
◮ An equilibrium allocation of human capital between schooling
and production sector requires identical marginal returns:
⇒ r = A
◮ Therefore the equilibrium growth rate is (similar AK)
gc = σ(A − ρ) = gy = gk = gh
S.132

◮ In contrast to the previously discussed AK-type model the
Lucas model has a transitory dynamic: The marginal returns
of human capital in schooling and production may diﬀer in the
starting point! This leads to a re-allocation of human capital
(gm = 0) and therefore to diﬀerent (and non-constant)
growth rates gh and gk (and gq, gz, respectively).
◮ The dynamic systems is 3-dimensional and complicated to
analyze. It is convenient to operate with a transformed
version of the model. Let
x = m1−α
z−(1−α)
i.e. z is substituted by x.
◮ Using the equilibrium values (35) and (34) for m and z we
have the equilibrium value
x∗
=
A
α
S.133

◮ The transformed model is
gq = (σα − 1)(x − x∗
) + (q − q∗
) (36)
gx = −(1 − α)(x − x∗
) (37)
gm = A(m − m∗
) − (q − q∗
) (38)
◮ Instead of system (33) – (35) where gq and gz depend
nonlinearly on q, z, m, we have now a linear system of
diﬀerential equations!
◮ The steady state value of the new variable x is stable since
gx > 0 ⇐⇒ x < x∗ and vice versa.
◮ Since gq does not depend on m and gm does not depend on x
it is possible to portray the isoclines in a 2-dimensional
graphic.
S.134

˙q = 0˙m = 0
˙x = 0
S.135

The transitional dynamics and the behavior of growth rates is
extensively studied in Barro/Sala-i-Martin (chapter 5.2) and will
not discussed here. The equilibrium is a saddle point. A stable path
to the equilibrium requires that e.g. for a given q(0) determines
the appropriate choice of x(0) and m(0).
S.136

One famous implication of the Lucas model:
◮ The growth rate for consumption c (and also for y and for the
capital stock K) depends negatively on the capital structure term z
(see eq. (32).
◮ This implies that a disequilibrium z < z∗
, e.g. by destroying physical
capital (“war”) leads to higher (transitory) growth rates for c and
y. The marginal return of the remaining physical capital increases
and this stimulates capital accumulation.
◮ A disequilibrium z > z∗
, e.g. by destroying human capital
(“epidemy”, migration) leads to lower (transitory) growth rates. The
logic is, that the education sector operates only with human capital.
If the latter decreses by a shock, the marginal returns increase. This
reallocates human capital away from the physical sector.
◮ One policy implication is that for low developped countries it is
more important to support the local human capital stock rather
than physical investments. The Lucas model emphasizes the
importance of education policy.
S.137

A version with positive externalities:
◮ Similar to the Arrow (1962) or Romer (1986) model, positive
external effects are modelled by
y = kα
(mh)1−α¯hη
, η ∈ (0, 1)
where a single firm treats ¯h as exogenously given. Hence the
marginal return from physical and human capital are
calculated, neglecting the external effect.
◮ It can be shown that with decentralized planning the steady
state growth rates are (with σ = 1!):
gy = gc = gk =
1 − α + η
1 − α
(A − ρ)
gh = A − ρ < gy
◮ The growth rate gc is larger than in the model without the
externality.
S.138

◮ The growth rates gh and gy are constant but different. The
external effect of human capital enlarges the returns in the
physical production. Hence, the households work too much
but learn too less!
◮ Therefore, gz = gk − gh = η
1−α (A − ρ) > 0 increases, i.e.
physical assets accumulate faster than intellectual assets.
◮ A social planner treats ¯h = h not as exogenously given and
includes the external effect when maximizing the welfare. She
calculates the social return of human capital.
S.139

Solution with a social planner:
g∗
y = g∗
c = g∗
k =
1 − α + η
1 − α
A − ρ
g∗
h = A −
1 − α
1 − α + η
ρ
The policy advice is to change the incentives in order to reallocate
a part of human capital from the physical to the education sector.
This could be done by a tax-transfer-system.
S.140

A design for a tax-transfer system:
◮ Since we have two production factors with a speciﬁc return,
we have two income taxes:
◮ interest rate tax τr ≥ 0 for physical capital
◮ wage tax τw ≥ 0 for human capital
◮ Furthermore the incentive to allocate human capital to the
education sector depends on the opportunity cost w(1 − m)h.
The government deﬁnes a fees/grants for education which are
proportional to the opportunity cost
ω = θw(1 − m)h
where θ > 0 menas that the household has to pay fees ω > 0
and θ < 0 menas that the household receive grants ω < 0.
S.141

◮ The intertemporal budget constraint can now written as
˙k = (1 − τr )rk + (1 − τw )wuh − θw(1 − m)h − c
◮ Also the government has a budget constraint:
τr rk + τw wuh + θw(1 − m)h = 0
S.142

Solving the model with these additional assumptions leads to:
g∗∗
y = g∗∗
c = g∗∗
k =
1 − α + η
1 − α
1 − τw
1 − τw + θ
A − ρ
g∗∗
h =
1 − τw
1 − τw + θ
A − ρ
Observe that τr has no inﬂuence on these growth rates!
S.143

Result:
◮ For θ > 0 (schooling fee) it is gy > g∗∗
y for all τw . The
dparture from the pareto-efficient solution increases!
◮ For θ = 0 (free acess to education) also the tax on labor
income has no effect on the growth rates. Hence, we have the
same pareto-inefficient result as in the unregulated case.
◮ For θ < 0 (schooling grants) the pareto-efficiency is improved
due to the incentive to allocate more human capital to the
education sector.
S.144

Optimal tax-transfer system:
There is a continuum of (θ, τw )-combinations which internalize the
externalities of human capital and lead to pareto-eﬃciency:
θ∗
= (τ∗
w − 1) ·
ηρ
(1 − α + η)A + ηρ
S.145

3.4 R&D based growth with increasing product variety
Literature:
◮ Romer, P.M. (1990), Endogenous Technological Change.
Journal of Political Economy 98 (5), S71–S102.
Basic Idea:
◮ In the previous models the aim was to uphold a persistent
incentive for capital accumulation by preventing that the
marginal return of capital declines (see Keynes-Ramsey rule).
This has been achieved by
◮ knowledge spillover eﬀects (externalities)
◮ accumulation of human capital (with constant returns)
◮ Now the innovation activities of ﬁrms are addressed (R&D ).
◮ Here: Innovation = development of new products.
S.146

◮ A firm will engage in R&D only if it could earn profits by
generating innovative products:
◮ There must be a kind of intellectual property righst protection
(like patents) which guarantees monopolistic power.
◮ This contradicts the assumption of perfect competition. Hence
the model will be based on monopoly power.
◮ Due to monopoly we have static efficiency losses. Hence
pareto-improving governmental regulation is possible.
◮ The new products are assumed to be intermediate goods =
inputs for the final homogenous good Y .
◮ Three-sector model: R&D sector, sector for intermediate
goods, production sector (final good)
◮ All sectors have identical technology, no population growth.
S.147

An alternative:
◮ Grossman, G.M., Helpman, E. (1991b), Innovation and
Growth in the Global Economy. MIT Press, Cambridge, MA.
◮ R&D increases the variety of consumption goods.
◮ Hence the utility function could not depend on aggregated
consumption but has to account for product variety (variety
preference). This also affects the Keynes-Ramsey rule for the
growth of aggregated consumption.
◮ We will not discuss this approach since the basic logic could
also be studied in the Romer approach (R&D generates
monopoly profits = increasing firm value ⇒ persistant
stimulus for investing a constant share of (increasing) income
into R&D )
S.148

Final good sector:
Y (t) = N1−α
n(t)
0
X(i, t)α
di, α ∈ (0, 1) (39)
◮ n(t) is the “number” of intermediate goods (inputs)
(not population growth rate!).
More precisely, there is a continuum of intemediate goods
[0, n(t)] with i ∈ [0, n(t)] as the index and X(i, t) as the
quantity of the intemediate good i.
◮ There is no physical capital, and labor supply N(t) is
unelastic.
◮ The production function has constant returns to scale.
◮ The price of the ﬁnal good is normalized to 1.
S.149

Intermediate good sector:
◮ All intermediate goods are produced with identical constant
marginal cost (normalized to 1).
◮ The price of each intermediate good i is P(i).
R&D sector:
◮ The innovation process is deterministic!
◮ Developing a new intermediate good has constant costs θ.
There are no economies of scale and no synergy eﬀects.
◮ Firms have an unlimited patent for the innovative
intermediate good. Hence we have n(t) monopolies in the
intermediate good sector.
◮ The incentive to innovate (= being an entrepreneur) depends
on the present value of monopoly proﬁts compared to the
costs of R&D (market entry costs).
S.150

Firms in the ﬁnal good sector:
Firms are price takers. They maximize
max
N,{X(i)}n
i=0
π = N1−α
n
0
X(i)α
di − wN −
n
0
P(i)X(i)di
From FOC we have
w = (1 − α)
Y
N
(40)
and the marginal return of an intermediate good equals its price:
∂Y
∂X(i)
= αN1−α
X(i)α−1
= P(i)
⇒ X(i) = N
α
P(i)
1
1−α
(41)
This is the demand function for intermediate goods which has a
constant price elasticity η = −1/(1 − α).
S.151

Firms in the intermediate good sector:
We have monopolistic price setting ﬁrms which maximize proﬁts:
max
P(i)
π = (P(i) − 1)X(i)
∂π
∂P(i)
= X(i) + (P(i) − 1)
∂X(i)
∂P(i)
= 0
Multiplying with P(i)/X(i) leads to
P(i) − (P(i) − 1)η = 0
⇒ P(i) =
1
α
> 1
Hence the price exceeds the marginal costs (markup: (1 − α)/α).
S.152

Since firms in the intermediate good sector make profits it is
possible to calculate the present value of the profits as
V (i, t) =
∞
t
(P(i) − 1)X(i, t)e−¯r(s)t
ds (42)
where ¯r(s) is the average interest rate in the time interval [t, s].
Recall that firms in the final good sector have zero profits. Total
assets in the economy in t are therefore
n(t)
0 V (i, t)di, and since
households are the owner of the firms, each household has assets
v(t) =
n(t)
0 V (i, t)di
N(t)
The intertemporal budget constraint is then
˙v(t) = w(t) + r(t)v(t) − c(t)
S.153

Remark:
◮ All intermediate goods are close substitues (see production
function).
◮ This limits the monopoly power of a single firm!
⇒ Monopolistic competition! (Dixit/Stiglitz (1977))
◮ In the long run the increasing number of substitutes makes
the residual demand curve more and more elastic and the
market share of a single firm decreases. Therefore, the
monopoly price converges to average cost (= zero profit).
◮ This effect does not take place in the model since the growing
aggergate demand prevents that the demand for a single
intermediate good decreases (possible extension: see
Barro/Sala-i-Martin, chapter 6.1.6).
S.154

Households face the usual maximization program
max U(t) =
∞
0
u(c)e−ρt
dt
conditional to ˙v(t) = w(t) + r(t)v(t) − c(t)
v(0) > 0
and also the TVC limt→∞ λ(t)v(t) = 0 holds true. Total
consumption C = cN must satisfy the market equilibrium
condition (to be explained later on)
C = Y − nX − θ ˙n
The result is the Keynes-Ramsey rule
gc = σ(r − ρ)
S.155

◮ Recall, that all intermediate good ﬁrms are identical, hence
P(i) = P, X(i) = X.
◮ Inserting the price P = 1/α into the demand function yields
X = Nα
2
1−α
◮ Inserting P and X into the present value of proﬁts (42):
V (t) = N(1 − α)α
1+α
1−α
∞
0
e−¯r(s)t
ds
S.156

Incentive to innovate:
◮ If V (t) > θ the net present value of proﬁt exceeds the
constant cost of innovation. Hence there is an incentive to
re-allocate all ressources in favor of the R&D sector by
detracting them from other sectors. This could not be an
equilibrium.
◮ If V (t) < θ then there is no incentive to innovate.
◮ If V (t) = θ then innovation activities are on an equilibrium
level. The ressource allocation between the sectors is
constant. The creation of innovative products has a positive
constant growth rate gn = ˙n/n > 0.
S.157

The equilibrium condition V (t) = θ = const implies that the
average interest rate ¯r has to be constant in each time interval
[t, s]. Integrating V (t) = θ leads to
r =
N
θ
(1 − α)α
1+α
1−α
A constant interest rate parallels the result uf the AK model.
S.158

Market equilibrium:
◮ We have already determined w and r.
◮ The complete income Y can either be consumed, or used for
the production of intermediate goods, or used in the R&D
sector.
◮ The aggregated demand for (identical) intermediate goods is
nX. Since the price is normalized to 1 this represents the
expenditures for intermediate goods.
◮ The period expenditures for R&D are θ ˙n.
◮ Hence,
C = Y − nX − θ ˙n
S.159

In equilibrium we have the Keynes-Ramsey rule
gc = σ
N
θ
(1 − α)α
1+α
1−α − ρ
which is constant (no transitory dynamics!).
Observe, that we have scale eﬀects since the absolute term N is an
argument of the function (large countries should then grow faster
than small countries!).
S.160

Since all intermediate good ﬁrms are identical, we can rewrite the
production function as
Y = N1−α
Xα
n
and inserting the intermediate goods demnand function (41)
= α
2α
1−α Nn
Hence, we have gY = gn = gc as the equilibrium growth rate.
Also the return to labor
w = (1 − α)
Y
N
= (1 − α)α
2α
1−α n
will grow with the same rate (compatible with the stylized fact).
S.161

Summary:
◮ The economy grows with the same rate as the variety of
intermediate goods grows. This requires a constnt incentive to
invest into R&D and innovation. The interest rate must be
kept on a level that households are willing to finance
monopolistic entrepreneurs in the market of intermediate
goods. Therefore the net present value of monopolistic profits
must equal the R&D costs. The markets for intermediate
goods grows with the same rate as the aggregated demand.
◮ There is no transitory dynamic.
◮ There are scale effects (dependency on N).
S.162

Optimality:
◮ Since the price of intermediate goods exceed the marginal
cost (monopoly due to patents), the result cannot be
pareto-efficient!
◮ Higher price = lower demand for intermediate goods = lower
production of final good.
◮ By increasing the number of intermediate goods, R&D
enhances the productivity of labor in the final good sector.
This externality is not internalized.
◮ The efficient interest rate can be calculated as
r∗
=
N
θ
(1 − α)α
α
1−α
(since α ∈ (0, 1) it is r∗ > r and hence g∗
c > gc)
S.163

Governmental regulation:
Government is able to change the relative prices (incentives) by
taxing or paying subsidies. Each tax-transfer structure requires that
the governmental budget is balanced, e.g. subsidies have to be
financed by allocation-neutral per capita taxes.
a) Subsidies for the demand for intermediate goods:
A subsidy ξ = 1 − α would decrease the price to the level of
marginal cost. Static efficiency is enhanced since the demand
for X and hence the output increases. This also enhances the
flow of profits and therefore the interest rate to its socially
optimal level. This induces incentives to invest into R&D .
Therefore also the dynamic efficiency is increased.
S.164

b) Subsidies for producing the final good:
This provides an incentive to expand the production Y and
therefore the demand for X. The results are the same as in a).
c) Subsidies for R&D :
This would lower the cost of R&D and thus enhance the
interest rate. The dynamic efficiency increases. But this is no
solution for the static efficiency loss due to monopolistic
pricing.
S.165

3.5 R&D based growth with increasing product quality
Literature:
◮ Grossman, G., Helpman, E. (1991), Quallity Ladders in the
Theory of Growth. Review of Economic Studies 58, 43–61.
◮ Aghion, P., Howitt, P. (1992), A Model of Growth through
Creative Destruction. Econometrica 60 (2), 323–351.
◮ Schumpeter, J.A. (1912), Theorie der wirtschaftlichen
Entwicklung. Leipzig: Duncker & Humblot.
S.166

Basic Idea:
◮ Romer: increasing variety = “horizontal innovation”,
now: increasing quality = “vertical innovation”
◮ If R&D leads to a better product then the “quality leader” is
the monopolist, the previous incumbant has to leave the
market (Schumpeter’s “creative destruction”). The proﬁt ﬂow
from innovation terminates if a quality-leading entrepreneur
enters the market.
◮ In contrast to the Romer model, innovation is a stochastic
process.
S.167

◮ There is a ﬁxed number of intermediate goods i = 1..n.
◮ The quality of each good is measured by a discrete quality
index ki = 0, 1, 2, ....
◮ Successful R&D leads to an incremental increase of the
prevalent quality index ki + 1.
◮ This implies that a potential entrepreneur (follower) “stands
on the shoulders” of the preceeding innovator.
⇒ This is an important intertemporal spillover eﬀect
(externality).
S.168

(Source: Barro/Sala-i-Martin (1995), p.241)
S.169

(Source: Barro/Sala-i-Martin (1995), p.243)
S.170

From the quality index to the quality adjusted input of good i:
◮ Index ki = 0, 1, 2, ...
◮ Current quality is qki , that means quality evolves with
1, q, q2, ..., qki
◮ A quality adjusted input of an intermediate good i is qki Xi .
S.171

Final good sector:
Y = N1−α
n
i=1
[qki
Xi,ki
]α
(43)
Firms in the competitive ﬁnal good sector maximize proﬁts
(price normalized to 1):
max
N,{Xi }n
i=1
π = N1−α
n
i=1
[qki
Xi,ki
]α
− wN −
n
i=1
Pi,ki
Xi,ki
(44)
where w is the wage and Pi,ki
is the price for input i with quality
ki .
S.172

From FOC we have (similar to the Romer model)
w = (1 − α)
Y
N
(45)
∂Y
∂Xi,ki
= αN1−α
qki
Xi,ki
= Pi,ki
(46)
⇒ Xi,ki
= N
αqki
Pi,ki
1
1−α
(47)
which is the demand function for intermediate goods. Observe,
that without quality improvement (ki = 0) this is the same result
as in the Romer model (equation (41)).
S.173

Intermediate good sector:
The current quality leader is the monopolist. As in the Romer
model we assume constant marginal cost which are normalized to
1. Again, maximization of the proﬁts leads to the optimal price
Pi,ki
=
1
α
Employing this price into the demand function yields the optimal
inputs of intermediate goods:
Xi,ki
= Nα
2
1−α q
ki α
1−α
(with ki = 0 this is the same result as in Romer)
S.174

Substituting Xi,ki
in the production function by its optimal input
levels leads to
Y = α
2α
1−α N
n
i=1
q
ki α
1−α
Let Q be an aggregated quality measure deﬁned as
Q =
n
i=1
q
αki
1−α
Then we can write:
Y = α
2α
1−α NQ
X =
n
i=1
Xi,ki
= α
2
1−α NQ
Since labor force N is constant, it follows
gY = gX = gQ
S.175

Profits and present value of the intermediate good firm:
Inserting equilibrium prices and quantities into the profit function
leads to the momentum profits
πi,ki
= N
1 − α
α
α
2
1−α q
ki α
1−α (48)
Recall, that the monopolist earns profits only until a new quality
leader with ki + 1 enters the market. The time duration of the
monopoly is therefore
Ti,ki
= tiki +1 − ti,ki
In equilibrium there will be a constant (= average) interest rate.
The present value of the profit flow is then
Vi,ki
=
Ti,ki
0
πi,ki
e−rt
dt = πi,ki
·
1 − exp(−rTi,ki
)
r
Duration Ti,ki
is unknown and depends on a stochastic innovation
process! S.176

Modelling the R&D process:
◮ In this version of the model, the incumbant does not engage
in R&D ! He will be replaced by an entrepreneur which is the
new quality leader.
◮ R&D requires a ressource input Zi,ki
(measured in units of Y )
of all researchers in sector i.
◮ The probability of achieving a higher quality level ki + 1 (=
successful innovation) depends on the input level Zi,ki
:
pi,ki
= Zi,ki
φ(ki ) (49)
where dφ/dki < 0 (since ki is an index number, this is a slight
abuse of notation!) denotes that with growing quality =
complexity of the product the probability of further
improvements decrease.
S.177

◮ With these assumptions about the stochastic innovation
process it is possible to determine the expected value of Vi,ki
(for details see Barro/Sala-i-Martin, chapter 7.2.2):
E[Vi,ki
] =
πi,ki
r + pi,ki
◮ The higher the R&D effort of all firms in sector i, the higher
is the probability pi,ki
of a successful innovation and the lower
is the expected duration of the monopoly (and therefore the
present value of profits).
◮ We have not yet determined the optimal R&D effort!
S.178

Incentives for R&D effort:
◮ We assume risk neutrality, i.e. firms respond to the expected
value of profits, not to the risk.
◮ There is free market entry. This implies that firms enter the
market as long as there is a positive expected profit. Hence, in
equilibrium the zero profit condition must hold true.
pi,ki
E[Vi,ki +1] − Zi,ki
= 0
pi,ki
πi,ki +1
r + pi,ki +1
− Zi,ki
= 0 (50)
Rearranging (50) und using (49) leads to
r + pi,ki +1 = N
1 − α
α
α
2
1−α · φ(ki ) · q
α(ki +1)
1−α (51)
S.179

◮ To make things more convenient we will now adopt a speciﬁc
form of φ(·):
φ(ki ) =
1
ξ
· q
−α(ki +1)
1−α (52)
(Observe the negative dependency on ki ).
◮ Using this speciﬁc form of φ(ki ) in the free-entry condition
(51) the very last term is cancelled out and we have:
r + p =
N
ξ
·
1 − α
α
α
2
1−α (53)
(Observe that p doesn’t depend on ki anymore.)
S.180

Now we are able to calculate the R&D eﬀort in equilibrium
(free-entry condition):
◮ Recall, that the probability of success was deﬁned as
p = Zi,ki
φ(ki ).
◮ Solving for Zi,ki
and inserting p from (53) and φ(ki ) from
(52) we have
Zi,ki
= q
α(ki +1
1−α N
1 − α
α
α
2
1−α − rξ (54)
and aggregating all R&D expenditures:
Z =
n
i=1
Zi,ki
= Q · q
α
1−α N
1 − α
α
α
2
1−α − rξ (55)
◮ Hence,
gZ = qQ = gY = gX
S.181

Using (53) for p the expected ﬁrm value is
E[Vi,ki
] = ξ · q
αki
1−α
and aggregation of all ﬁrms leads to
E[V ] = ξ · Q
Therefore, also the expected value of total assets grows with the
same rate:
gV = gQ = gY = gX = gZ
S.182

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