Theory of economic growth
Slides made by: K˚are Bævre, Department of Economics, University of Oslo
Slight adjustment: Miroslav Hlouˇsek, ESF MU
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.
• That is, knowledge and efficiency follow from production and investments.
The higher is production, the more experience and know-how
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.
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• 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.
• 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)
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• 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 with higher growth (of ˙c/c and ˙k/k).
• Hence the decentralized solution will end up with too low investments
and sub-optimal growth.
• A subsidy on investments (or production) 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.
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.
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• 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
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)
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• 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 Public goods model
• G is nonrival and nonexcludable (pure public good)
• production function
Yi = AL1−α
i Kα
i G1−α
= Li[Akα
i G1−α
]
• proportional tax on output
G = τY
• Firms have to maximize after tax income (per capita income)
max
ki
(1 − τ)Akα
i G1−α
− (r + δ)ki − w
F.O.C.
(1 − τ)αAkα−1
i G1−α
= r + δ = R (7)
(1 − τ)(1 − α)Akα−1
i G1−α
= w (8)
• Household’s optization is unchanged (taxation only affects firms). Equilibrium
conditions. Every firm choosed the same capital-labor ratio.
We can rewrite (aggregated) production function as
G = τLAkα
G1−α
= (τLA)1/α
k
• After inserting into (7) for G we get
R = (1 − τ)α(τL)
1−α
α A1/α
which is independent of K and therefore constant over time. There is
again scale effect. After inserting into Euler equation we get
γc = γk = γy =
1
θ
(1 − τ)α(τL)
1−α
α A1/α
− δ − ρ
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• If the government wants to maximiye the growth rate, what is the
optimal tax rate? Maximum growth rate is attained when the first
derivative wrt τ is equal to 0.
• The result is
τ = 1 − α
Therefore, equilibrium in decentralized economy with benevolent government
is
γc = γk = γy =
1
θ
α2
(1 − α)
1−α
α L
1−α
α A1/α
− δ − ρ
• Larger population causes larger growth. The economy benefits from
greater scale because government services are assument to be nonrival.
That is at odd with the data.
12.3 Crowding in and crowding out
• We will now look more closely at slightly different model where govenment
services are partly rival. The model is due to Barro (1990).
• For simplicity we abstract from population growth, and any exogenous
growth in technology. The economy is closed.
• We incorporate public services in the production function in the following
way
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.
• 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.
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• 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 − η) = 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 (9)
• 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 (10)
• 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)
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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.
• 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.
• 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 (11)
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• Since τ = g/y, the only difference between (10) and (11) is the absence
of the term α in (11). 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 reconsidered
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) (12)
with CRS in K, H and L.
• We know that as long as Y , K and H are all produced by the same
technology (12), the inclusion of human capital does not alter the qualitative
conclusions of the neo-classical model (but significantly affects
the quantitative predictions).
• 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) (13)
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 (13) as opposed
to (12) is then the assumption that it is only the total stock of human
9
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 (13) as
Y = K · f(H/K) (14)
• 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 (14) 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
(12) or (13) is the relevant one.
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.
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• 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 AK-model
behavior discussed above.
• During this transition H essentially takes the role of L in the neoclassical
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.
• 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.
11
• 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) (15)
˙H = B(1 − u)H (16)
where u is the share of human capital used in production of consumption
goods/physical capital. We neglect depreciation for simplicity.
• 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 (16) 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)
12
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
(17)
˙H + δH = B · [(1 − v)K]η1
[(1 − u)H]η2
(18)
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• 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.
• Divide (18) by H and transform both sides to growth rates. This gives
η1γ∗
K + (η2 − 1)γ∗
H = 0 (19)
• Divide (17) 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 (20)
• 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 (20) simplifies to
(α1 − 1)γ∗
K + α2γ∗
H = 0 (21)
• Equations (19) and (21) 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) (22)
• It is worth considering some special cases where the condition is satis-
fied
– 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 (22) 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 (22) can be satisfied provided η1 + η2 > 1, and vice
versa.
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– The AK-model with α1 = 1 and α2 = 0. Notice that H is just a
waste here.
– Notice that as long as α1 = 1 we have from (21) 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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