ECON 4350: Growth and Investment
Lecture note 6
Department of Economics, University of Oslo
Made by: K˚are Bævre (kare.bavre@econ.uio.no)
Exploited by: Miroslav Hlouˇsek (hlousek@econ.muni.cz)
11 Learning by doing
Required reading: BSiM: 4.3
Secondary reading: Irwing Klenow (1994), Thompson (2001)
11.1 Technological progress as an externality
• We remember from the last lecture that if the disaggregated production
function is
Yi = F(Ki, ELi)
with CRS in Ki and Li while
E = A(K) (1)
is taken as given by all agents (e.g. a public good) and satisfies
lim
K→∞
A (K)L = b > 0 (2)
then we have
lim
K→∞
˙Y
Y
= FL(1, b)b > 0
and perpetual growth.
• One rationale for (1) is learning by doing effects.
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• That is, knowledge and productivity gains follow from production and
investments. The higher is production, the more experience and knowhow
have been generated. Since the level of production can be associated
with the level of capital, we can postulate a relationship like
(1).
• Implicit in this formulation is an assumption of complete spill-over of
knowledge in the sense that each firms knowledge is a public good to
all other firms.
• There is thus a positive externality from firm i’s use of capital on the
production of all other firms.
• We thus associate the parameter E with the total stock of capital in
the economy. For convenience we use the functional form
E = A(K) = Kσ
(3)
where σ > 0.
• The aggregate production function is then found by summing over all
firms
Y = F(K, EL) = F(K, Kσ
L)
• If σ < 1 we have A(K)/K = Kσ−1
→ 0 as K → ∞, so condition (2) is
not met.
• However this condition was only a sufficient condition, and derived for
a case with no population growth. Assume that Y and K grow at a
common rate g. Then we have
Y0egt
= F(K0egt
, K0L0e(gσ+n)t
)
Due to CRS this is possible if and only if
g = gσ + n ⇒ g = n/(1 − σ)
Hence for σ < 1 we can have growth in Y forever only if n > 0, that
is if we have population growth. If this is the case also have that Y/L
grows at rate g − n = n(1/(1 − σ) − 1) > 0.
• However, if the learning curve does not exhibit decreasing returns to
K, that is σ ≥ 1 the conclusion is altered.
2
• Assume there is no population growth and σ = 1. Then A(K)/K = 1
so so condition (2) is met, and we will get perpetual growth. This is
the case discussed by Romer (1986).
• Notice that BSiM only consider the case σ = 1.
11.2 Implications for policy
• The individual firm takes E as given. Assuming that the production
function F(·, ·) is CD, the private marginal product of capital for firm
i is
MPKp,i =
∂F(Ki, ELi)
∂K
= F1(Ki, ELi) = αKα−1
i (ELi)1−α
(4)
or in the aggregate
MPKp = αKα−1+(1−α)σ
L1−α
(5)
• Notice that if σ = 1
MPKp = αL1−α
i
so the marginal product will be independent of K and we have an
AK-model.
• In this case, increases in the size of the population will shift the constant
marginal product up leading to stronger growth (i.e. like a shift in A
in the AK-model). This is what we call a scale effect.
• A social planner, on the other hand, will evaluate the marginal product
of capital to be
MPKs =
∂F(K, A(K)L)
∂K
= F1(K, A(K)L) + LA (K)F2(K, A(K)L)
= (α + (1 − α)σ)K
α−1+(1−α)σ
i L1−α
i = (α + (1 − α)σ)/α · MPKp > MPKp
and hence choose a path where more capital is accumulated.
• Hence the decentralized solution will end up with to low investments
and sub-optimal growth.
• A subsidy on investments can go at least some way in making the firms
internalize the externality. However, this might have to be financed by
taxes introducing other distortions.
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11.3 Some evidence on learning by doing
• Learning appears to be particularly strong in early phases of a firms
life. This might suggest that σ < 1.
• There are important case studies suggesting substantial learning effects.
• A famous example is that of the manufacturing of the Liberty-class
ships in the US during WW2.
• The amount of worker-days required to build each ship declined dramatically
as did the time used on each ship. Suggesting productivity
gains of up to of 40 per cent per year (averages).
• A large share of this productivity gain has been attributed to learning
by doing.
• However, more recent research show that previous estimations exaggerated
the effect by not taking sufficiently care of increases in capital
and capital deepening. In addition there was a strong tendency for
increasing the output of ships at the expense if quality.
• Other important case studies has focused on US airframe manufactur-
ing.
11.4 The scope of the externality: Local or global?
• An important thing to note is that the Liberty-ship experience only
provides evidence on the learning curve at the level of an individual
firm, and does not tell us much about the degree of spill-over.
• The extent of spill-over effects is of course crucial for the implications
both for how learning by doing affects growth and for the role of policy.
• It can be worth considering how spill-overs might matter to varying
degrees within industries vs across industries, within countries/regions
or between countries/region.
• This in turn has important consequences for the extent of scale effects,
that is how the population/scale of the relevant units affect growth.
12 Government investments
Required reading: BSiM: 4.4
4
12.1 Public provision of non-rival infrastructure
• An alternative justification of (1) is to regarding the efficiency parameter
E as a reflection of public services which enhance labor productivity.
• Assume that these services are produced with the same production
function as private services, and that the flow of services promoting
labor productivity is equal to government spending, i.e. E = G.
• Let G be financed by a proportional output tax, τ and assume for
simplicity that the government always runs a balanced budget. Then
E = G = τY = τF(K, EL)
• Due to CRS we have
1 = τF(K/E, L)
which implicitly determines
E = A(K) = cK (6)
where c is a positive constant (for a given τ) determined by 1 =
τF(1/c, L)
• Equation (6) of course satisfies (2) so we will get perpetual growth in
this model.
12.2 Crowding in and crowding out
• We will now look more closely at a model underlying the argument in
the previous section. The model is due to Barro (1990).
• We endogenize the saving/consumption decision by introducing optimizing
household-producers. The economy is closed.
• For simplicity we abstract from population growth, and any exogenous
growth in technology.
• Instead of having y = f(K), we now incorporate public services in the
production function
y = Φ(k, g)
where g = G/L is the amount of public services per worker. Note that
this implicitly assumes that G is rival for the users, though in a rather
narrow sense. We could use G instead without altering the analysis
very much.
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• The general idea of including g as a separate argument is that it is not a
close substitute for private inputs (k), i.e. public services/activity represents
something that would not be replaced by corresponding private
activities if it vanished. This is again based on non-excludability which
would make private incentives weak/non-existent.
• The production function Φ() is assumed to exhibit CRS to the two
inputs k and g, so
y = Φ(k, g) = kφ(g/k)
with φ > 0 and φ < 0.
• Whenever it is simpler, we work with the Cobb-Douglas case:
y/k = φ(g/k) =
g
k
1−α
• The balanced government budget is:
g = τy = τkφ(g/k)
• It follows directly that both g/y and g/k only depend upon the tax-rate
τ. For a given τ they will therefore be constant.
• The marginal product of capital as seen by the individual agents who
regard g/k as independent of their decision is
∂y
∂k
= φ
g
k
1 − φ
g
y
= φ
g
k
α
where η ≡ φ g
y
= ∂y
∂g
g
y
= 1 − α, i.e. is the elasticity of y with respect to
g which is equal to the constant 1 − α in the CD-case.
• As usual, we consider a representative household-producer who maxi-
mizes:
U =
∞
t=0
e−ρt c(t)1−θ
1 − θ
dt (7)
• Since the after-tax private returns to capital is (1−τ)(∂y/∂k), it should
by now be familiar that the Euler-equation now reads
γ =
˙c(t)
c(t)
=
1
θ
(1 − τ)φ
g
k
α − δ − ρ (8)
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• We can now see how the size of government, as characterized by τ = g/y
impinges on the growth rate.
• There are two offsetting effects. The direct effect of an increase in τ
is to reduce the incentive to invest. However, increasing g/y will also
increase g/k and hence increase the marginal product of capital. This
promotes incentives to invest.
• In the Cobb-Douglas case we get
dγ
d(τ)
=
1
θ
φ
g
k
(φ − 1)
thus we get the following hump where the growth rate increases in the
size of government when g/k is so low that φ > 1 and declines when
g/k is so high that φ < 1
• Note that the hump is a type of Laffer-curve. We can refer to the
raising part of the curve as a case where public expenses are ‘crowdingin’
by increasing private productivity, while in the declining part they
are ’crowding-out’ by taxation affecting the marginal product of capital
too strongly.
• Note that φ = 1 is a natural condition for productive efficiency. (What
are the terms of trade between y and g?).
• When η is constant as in the CD case it can be shown that increasing
the growth rates monotonically rises life-time utility. So the benevolent
government should indeed choose τ = g/y such that φ = 1.
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• However, this is only a second best solution.
• Consider a social planner who fixes τ = g/y and then is able to dictate
each household-producers consumption over time. For a given value of
g/y, the social marginal returns to capital is (1 − g/y)φ(g/k), where
the adjustment −g/y is required for holding g/y constant.
• Hence the growth rate of consumption chosen by the social planner is
γs =
˙c(t)
c(t)
=
1
θ
1 −
g
y
φ
g
k
− δ − ρ (9)
• Since τ = g/y, the only difference between (8) and (9) is the absence
of the term α in (9). Hence, for all levels of τ the growth rate is lower
in the decentralized case.
• The private agents invest sub-optimally in capital because they do not
internalize the externality of also being able to increase g and hence
increase production. The public provision of these services are financed
by a distortional tax on income (τ reduces the private returns to capi-
tal).
13 Human capital accumulation reconsid-
ered
Required reading: BSiM 5.1, 5.3
Secondary reading:BSiM 5.2.1-5.2.2, Lucas(1988)
13.1 The one-sector model
– In our previous encounters with human capital we have considered
production functions of the following type
Y = F(K, H, L) (10)
with CRS in K, H and L.
– We know that as long as Y , K and H are all produced by the
same technology (10), the inclusion of human capital does not
alter the qualitative conclusions of the neo-classical model (but
significantly affects the quantitative predictions).
8
– We have seen this both in models where equality of returns to the
two types of capital requires that H/K is constant, and in the
model in MRW.
– Consider now instead a production function of the form
Y = F(K, H) (11)
with CRS in broad capital K and H.
– Note that we now should perhaps think of human capital as H =
hL, i.e. as the number of workers, L, multiplied by the human
capital of the typical worker, h. Implicit in the formulation of
(11) as opposed to (10) is then the assumption that it is only the
total stock of human capital that matters for production, not how
this stock is made up of L and h. Stated loosely, labor quantity
(L) and quality (h) are perfect substitutes.
– An important consequence of this is that we now have CRS in
accumulated factors. This gives us a source of perpetual growth
of the AK-type.
– To see this rewrite (11) as
Y = K · f(H/K) (12)
– We then have
r + δK = RK =
∂Y
∂K
= f(H/K) − (H/K) · f (H/K)
r + δH = RH =
∂Y
∂H
= f (H/K)
or
f(H/K) − f (H/K) · (1 + H/K) = δK − δH
which determines a fixed value for the ratio H/K = 1/ω∗
.
– Defining A ≡ f(1/ω∗
) the production function (12) can be written
Y = AK
so we have a model of the AK-type. See BSiM 5.1.1 for additional
details.
– In conclusion, we see that the consequences of adding human capital
to the (one sector) model can depend crucially on which of
the technologies (10) or (11) is the relevant one.
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13.2 Imbalance effects
– In several of the models we have encountered it transpires that the
ratio ω = K/H should be constant to secure equality of returns.
– So far we have assumed that it is always possible to achieve this
instantly. However, this requires that we at any time can convert
existing K to H and vice versa. This is obviously not very realistic.
– Assume this was not possible, and that changes to the ratio ω
could only be made by distributing positive gross investments IH
and IK on ˙H = IH − δHH and ˙K = IK − δKK. In such situations
we would get so called imbalance effects.
– More precisely we are now constrained by IH ≥ 0 and IK ≥ 0.
– If we initially had too much human capital relative to physical
capital, so that ω < ω∗
the fastest way we can increas ω is by
setting IH = 0 (i.e. the constraint on IH is binding), so that
˙H/H = −δH .
– This process will lead to a transitional phase where ω is increasing
towards it steady state value. It reaches this value in finite time,
after which Y grows at a constant rate γ∗
Y according to the AKmodel
behavior discussed above.
– During this transition H essentially takes the role of L in the
neo-classical growth model, and since the marginal product of K
is decreasing in K/H we get the usual property that the growth
rate of output falls during the transition. Hence, the growth rate
falls monotonically towards γ∗
Y .
– If ω > ω∗
we get the same story only that the roles of H and K
are reversed.
– In sum we get that the growth rate γY falls monotonically towards
γ∗
Y as ω is approaching ω∗
from either side.
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– This implies that if we get an imbalance such as a reduction of
ω from a war that destroys physical capital rather than human
capital (WW2), we get a transitional phase with strong growth.
– According to the model we should get the same effect from e.g. a
plague which destroys H but leaves K intact. This is, however,
not in line with what we see (cf. also WW1).
– An important source of possible asymmetries in the imbalance
effect is that adjustment costs are typically higher for human capital.
The effects also tend to be asymmetric in more elaborate
models such as the two-sector model below.
13.3 A two-sector model. Uzawa-Lucas.
– As stated several times earlier in the course we should also be
sceptical to the assumption that human capital is produced by
the same technology as consumption and physical capital.
– In particular we are lead to expect that production of new human
capital is considerably more intense in the use of existing human
capital than is the production of physical capital.
– To highlight this effect consider the following two-sector model
Y = C + ˙K = F(K, uH) (13)
˙H = B(1 − u)H (14)
where u is the share of human capital used in production of consumption
goods/physical capital. We neglect depreciation for sim-
plicity.
– Here the schooling sector produces new human capital using only
human capital (teachers, student time/vigilance) as an input. This
is obviously an extreme version of what we pointed at above.
– Note that (14) can also be interpreted as primarily capturing that
individuals accumulate human capital more easily when the economy’s
level of human capital is larger, i.e. an externality.
– The important new feature of this model is that one of the capital
inputs (H) is produced using only reproducible inputs.
– Also now we can get perpetual growth, even if there are no changes
in technology over time. To see this assume that we are on a
balanced growth path with
γ = ˙C/C = ˙K/K = ˙H/H = B(1 − u)
11
which is possible if u and the savings rate s are chosen such that
γK = ˙K = Y − C = sY = sF(K, uH)
at all times.
– The full analysis of the model requires that we endogenize allocation
of resources, and let the savings rate s and the fraction u
follow from optimization of a representative household-producer.
– This analysis will primarily teach us something new about the
short run by giving a richer description of transitionary adjustments.
Note however, that since u∗
determines the long run
growth rate, determining u∗
will also tell us something about what
determines growth.
– The results are primarily able to tell us something interesting
about how the economy reacts to imbalances in the ratio of the
two stocks of capital, ω.
– The full analysis is quite involved and is left for cursory self-study.
(See BSiM 5.2).
– The most important new result is that there is now an asymmetry
in the imbalance effect. If ω > ω∗
(i.e. we have too little human
capital) the marginal product of human capital in the goods sector
is high, implying a high wage rate. This makes operation costs in
the sector that is intensive in the use of this input, i.e. education,
high. This motivates an allocation of resources to production of
goods rather than education. This retards the production of the
scarce factor (H), thus retarding the economy’s growth rate.
– Thus the model predicts that we should see faster recovery after
an event that destroys physical capital, than if it had destroyed
human capital.
13.4 Conditions for perpetual/endogenous growth
– We now generalize the setup of the Uzawa-Lucas model to
Y = C + ˙K + δK = A(vK)α1
(uH)α2
(15)
˙H + δH = B · [(1 − v)K]η1
[(1 − u)H]η2
(16)
– We now look for a steady state where u and v are constant, and C,
K, H, and Y grow at constant (but not necessarily equal) rates.
12
– Divide (16) by H and transform both sides to growth rates. This
gives
η1γ∗
K + (η2 − 1)γ∗
H = 0 (17)
– Divide (15) by K and transform both sides to growth rates. This
gives
C/K
C/K + γ∗
K + δ
· (γ∗
C − γ∗
K) = (α1 − 1)γ∗
K + α2γ∗
H (18)
– It can be shown that we must have γ∗
K = γ∗
C. This in turn implies
that γ∗
K = γ∗
Y , so Y , K and C grow at the same rate.
– Therefore (18) simplifies to
(α1 − 1)γ∗
K + α2γ∗
H = 0 (19)
– Equations (17) and (19) constitutes a homogenous system of two
linear equations in the two unknown γ∗
K and γ∗
H. It should be
a familiar result that the only way the solution to this problem
could be different from γ∗
K = γ∗
H = 0 is if the characteristic matrix
is singular.
– This implies that the only way we can have endogenous growth
(γ∗
K > 0,γ∗
H > 0 as postulated above) is if the parameters satisfy
α2η1 = (1 − η2)(1 − α1) (20)
– It is worth considering some special cases where the condition is
satisfied
∗ CRS in both sectors (α1 + α2 = η1 + η2 = 1). Then we also
have γ∗
H = γ∗
K so K/H is constant.
∗ Uzawa-Lucas: η1 = 0, η2=1. Then (20) holds irrespective of
the size α1+α2, so we can have both decreasing and increasing
returns to scale in the final goods sector.
∗ Decreasing returns in one sector is offset by increasing returns
in the other. Assume all elasticities are positive. If α1 +α < 1
condition (20) can be satisfied provided η1 + η2 > 1, and vice
versa.
∗ The AK-model with α1 = 1 and α2 = 0. Notice that H is
just a waste here.
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∗ Notice that as long as α1 = 1 we have from (19) that
γ∗
K =
α2
1 − α1
γ∗
H
so γ∗
K > γ∗
H if α1 + α2 > 1 and vice versa. In particular,
only with constant returns to scale (α1 + α2 = 1) do we have
γ∗
K = γ∗
H and a constant H/K on the balanced growth path.
∗ It follows that the only situation where we get growth and
where H/K is constant is if we have CRS in both sectors.
– So what have we learned? There are many ways we can get perpetual/endogeneous
growth in this two-sector setup. However, the
predictions depend crucially on the values of the parameters describing
the technologies. Since we know rather little about these
values, the predictive power of the model is limited. This is a
serious short-coming, and a challenge for future research.
References
[1] Barro, R.J. Government Spending in a Simple Model of Endogenous
Growth, Journal of Political Economy, October 1990, 98 (5), pp. 103-
126. Part 2.
[2] Irwin D.A. and P.J. Klenow Learning -by-Doing Spillovers in the
Semiconductor Industry, Journal of Political Economy, December 1994,
102 (6), pp. 1200-1227.
[3] Lucas, R.E. Jr On the Mechanics of Economic Development, Journal
of Monetary Economics, July 1988, 22 (1). pp. 3-42.
[4] Thompson, P. How much did the Liberty shipbuilders learn? New
evidence for an old case study, Journal of Political Economy, 2001, 109
(1) . Pp. 103-137.
[5] Romer, Paul M. Increasing Returns and Long-Run Growth, The Journal
of Political Economy, Vol. 94, No. 5, October 1986, pp. 1002-1037.
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