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)
14 Growth through increasing variety
Required reading: Romer (1990)
Secondary reading: BSiM 6
14.1 Ideas: Rivalry, excludability and nonconvexities
• Are knowledge/ideas public goods?
• We should distinguish between two aspects: Rivalry and excludability.
• Rivalry is a technological attribute of the good itself.
• An idea is in general non-rival.
• That is of course not to say that e.g. one firms sales of a product
produced using the idea is not competing with (is rival to) another
firm’s sales.
• Note further that there is a fundamental difference between knowledge
in the form of ideas/designs (our A) and knowledge in the form of the
human capital of individuals.
• The former is in general non-rival and only partially excludable, the
latter has a high degree of both rivalry and excludability.
• Note further that nonrival knowledge can be accumulated without
bound on a per capita basis, while human capital can not because
it is constrained by the life time of each separate individual.
1
• If a nonrival input has productive value this inevitably violates the
replication argument underlying constant returns to scale and leads to
nonconvexities.
• If K and L are all rival inputs and A is a nonrival input such as knowledge
and the production function is F(A, K, L), the standard replication
argument leads to CRS in K, L:
F(A, sK, sL) = sF(A, K, L) (1)
which also implies (as a special case of Euler’s formula)
F(A, K, L) = KFK + LFL (2)
But if A is productive we must have
F(sA, sK, sL) > sF(A, K, L) (3)
and
F(A, K, L) < AFA + KFK + LFL (4)
• Note that (4) implies that a firm having this production function could
not survive as price taker remunerating all inputs at their marginal
product.
• Thus, under perfect competition we can not explain why firms remunerate
A and hence not explain why private agents do research.
• We have looked at two approaches: Either treating A as a proper public
good (provided by the government) or letting production of A follow
as a by-product such as through learning by doing.
• We now look at a new approach: Letting knowledge be partially excludable
and introducing market power.
14.2 Monopolistic competition in horizontally differentiated
products
14.2.1 Production of the consumption good
• The following is a somewhat simplified version of the model in Romer
(1990).
2
• The production function for the representative firm producing the final
good Y is
Y = L1−α
1 X
where L1 is labor input and X is an index of the use of the N other
inputs {xi}N
i=1 (different capital goods)
X =
N
i=1
xα
i
which inserting gives:
Y = L1−α
1
N
i=1
xα
i (5)
• Note that the production function exhibits CRS with resepct to all
inputs L1 and {xi}N
i=1. Thus the assumption of a single representative
price-taking firm is legitimate.
• The marginal product of input xi is then
MPxi
= αL1−α
1 xα−1
i
which is infinite at xi = 0. Hence the firm will want to use all some of
all N inputs available.
• Note that if we regard xi as different capital goods (5) assumes that
these have additively separable effects on production. The marginal
product of capital good i (tractor) does not depend on the amount
used of other capital goods (e.g. computers)
• This contrasts with the traditional case where we measure the total
use of capital goods simply as the sum (X = ( N
i=1 xi)α
= Kα
). This
assumes perfect substitutability between tractors and computers.
• Assume for reasons that will become clearer below that all inputs xi
are measured in the same unit and employed in equal amounts xi = x,
then
Y = L1−α
1 Nxα
= L1−α
1 (Nx)α
N1−α
(6)
Hence, if we associate K = Nx with the total amount of resources used
of the inputs {xi}N
i=1 we see that the production function exhibits CRS
in L1 and K.
3
• Technological progress in this model is captured by an increase in the
number/variety of available inputs to production, that is, through increases
in N.
• Note that for given levels of Li and K production increases in N. An
interpretation: As a broader diversity of inputs become available production
can increase by using more specialized inputs.
14.2.2 Production of intermediate goods
• The demand for intermediate good xi follows from the profit-maximization
of the representative producer of Y , that is
max
L1,{xi}N
i=1
N
i=1
(L1−α
1 xα
i − pixi)
• Which readily gives the inverse demand function from the FOC
αL1−α
1 xα−1
i = pi(xi) (7)
• Each input xi is supplied by a single firm that has monopoly power in
the production of the intermediate good xi.
• This monopoly power is assumed to be based on the ownership/renting
of a patented design for the production of xi.
• To simplify we assume that the firm producing intermediate output xi
can do so simply by converting a unit of capital/consumption good Y
into xi. In this sense, all xi have the same production function as Y .
• This implies that the xi is produced with a fixed unit price equal to
the interest rate r.
• Thus, inserting for (7), each monopolist faces the profit maximizing
problem
π = max
xi
p(xi)xi − rxi = max
xi
αL1−α
1 xα−1
i xi − rxi
giving
αL1−α
1 xα−1
i = r/α (8)
or
pi = r/α (9)
4
• Due to the symmetry of inputs xi in (5), we must have xi = x and
pi = p for all i in equilibrium.
• The profit for the monopolists producing intermediate inputs is hence
π =
r
α
x − rx =
1 − α
α
rx > 0 (10)
14.2.3 Research and development
• Designs for new intermediate goods are developed through research and
development. The production of new ideas for designs is characterized
by the deterministic production function
˙A = BL2A (11)
where L2 is the number of workers in R&D, satisfying the constraint
L1 + L2 = L, where the total supply of labor is fixed.
• For notational convenience we measure A such that A ≡ N.
• Note that ideas, A, are partially excludable. Inventors own the patent
for the designs on how to produce xi, but the ideas underlying each
design is non-excludable in the production of new ideas and designs.
Hence, A is a public good in (11).
• We do not need to be explicit as to who conduct the R&D activities (be
it the monopolistic producers of intermediate goods or someone else)
as long as the patent can be rented.
• The value Va of inventing one new design is thus equal to the net present
value of receiving the profit in (10) in all future periods, or
Va =
∞
t=0
1
(1 + r)t
π =
π
r
=
(1 − α)
α
x (12)
• One more unit of labor devoted to research gives rise to BA more new
ideas, each having value Va. Hence, the marginal product of labor in
the R&D sector is BAVa.
• Equating the marginal products of labor in the two sectors we then
have (remember that A = N)
BAVa = (1 − α)L−α
1 xα
A =⇒ Va =
1 − α
B
L−α
1 xα
5
which together with (8) and (12) gives
r = BαL1 (13)
i.e. the interest rate as a function of L1.
14.2.4 Consumers
• To close the model we need to add the consumer side. We adopt the
usual case of intertemporal utility maximization for given market prices
(w and r), using a CRRA functional form.
• Note that we can as usual define an accounting stock K = N
i=1 xi
(since xi are produced by converting one unit of capital).In equilibrium
xi = x for all i, so
K = K =
N
i=1
xi = Nx = Ax
• For simplicity we neglect depreciation, so
˙K(t) = Y (t) − C(t)
• We skip the details of familiar utility maximization problem, turning
directly to the Euler equation
˙C
C
=
r − ρ
θ
(14)
where ρ and θ are the now common parameters of the life-time utility
expression.
14.2.5 The balanced growth path
• We will now show that we get a balanced growth path where r and L1
are constant.
• We do not solve for the transitional dynamics because this is difficult,
and not all that interesting.
• Start by assuming that r is constant. Then L1 is also constant according
to (13), and hence x is also constant according to (8). It then follows
from (6) that the growth rate of Y is the same as the growth rate of A
(or N).
6
• Further, since K = Ax and x is constant K also grows at the same
constant rate as A. Hence also K/Y is constant.
• It then follows that
C
Y
=
Y − ˙K
Y
= 1 −
˙K
K
K
Y
is constant since ˙K/K and K/Y are both constant.
• We then finally have
γ =
˙C
C
=
˙Y
Y
=
˙K
K
=
˙A
A
= BL2 = BL − BL1 = BL −
r
α
(15)
• Thus, the existence of this balanced growth path depends on r being
constant. But the constancy of r was what we assumed at the outset
and showed that it implied this balanced growth path. That there
indeed is consistency between these two is seen by setting (14) and
(15) equal which gives
r =
α
α + θ
(θBL + ρ)
i.e. a constant as required.
• Thus, there exists a balanced growth path
γ =
˙C
C
=
˙Y
Y
=
˙K
K
=
˙A
A
=
αBL − ρ
α + θ
(16)
• Note the following about the steady state growth rate:
1. There is positive and constant growth in equilibrium (The model
is misbehaved if αBL < ρ).
2. This growth is driven by private agents doing research (L1 > 0),
so we have endogenized technological progress.
3. The growth depends on the willingness to save reflected by ρ and
θ. Hence there is room for policy to affect the long run growth
rate.
4. The growth rate increases in population L. Hence there is a scale
effect.
5. It can be shown that the two last properties relies heavily on the
properties of the R&D production function (11), or more specifically
from ˙A being linear in A.
7
14.3 Implications for policy
• We now look at the allocation a social planner would have chosen in
the setting with an increasing variety of intermediate inputs.
• Since the aggregate capital indicator X is concave in each xi, and the
price of each capital good xi is one unit of foregone consumption, the
social planner will always choose xi = x for all i.
• Hence production is given by
Y = L1−α
1 Nxα
= L1−α
1 Axα
= A1−α
L1−α
1 Kα
where the second equality follows from using the measurement convention
A = N. In the third equality we have defined K = Nx = Ax as
the total amount of resources used on capital goods.
• This definition of the accounting stock K (‘capital’) has the following
law of motion:
˙K = Y − C = A1−α
L1−α
1 Kα
− C
• We now can formulate the problem of finding a socially optimal balanced
growth path (i.e. we again ignore transitional dynamics) by solving
the social planners problem:
max
∞
0
C1−θ
1 − θ
e−ρt
dt (17)
˙K = A1−α
L1−α
1 Kα
− C (18)
˙A = BL2A (19)
L1 + L2 ≤ L (20)
• Notice that (19) is the R&D production function.
• The control variables are C and L2, and there are two state variables K
and A. The extension of optimal control theory to this two dimensional
case is straightforward.
• The current-value Hamiltonian for the problem is
H =
C1−θ
1 − θ
+ p[A1−α
(L − L2)1−α
Kα
− C] + qBL2A (21)
8
• The co-state variables p and q must satisfy
˙p = ρp −
∂H
∂K
(22)
˙q = ρq −
∂H
∂A
(23)
• While we have the usual first order conditions for the control variables
∂H
∂C
= C−θ
− p = 0 (24)
∂H
∂L2
= −(1 − α)pA1−α
(L − L2)−α
Kα
+ qBA = 0 (25)
• Since we are only looking at balanced growth path solutions we must
have γ∗
=
˙C
C
=
˙A
A
. Since (19) gives
˙A
A
= BL2 we have γ∗
= BL2 and
for determining the common growth rate γ∗
we need only to determine
the share of labor used for research L2.
• Note that (dividing by A) (25) implies
(1 − α)pA−α
(L − L2)−α
Kα
= qB
and since
∂H
∂A
= (1 − α)pA−α
(L − L2)1−α
Kα
+ qBL2
we have
∂H
∂A
= qB(L − L2) + qBL2 = qBL
• Joining this with (23) we have
˙q
q
= ρ − BL (26)
• As usual (24) gives
−θ(
˙C
C
) =
˙p
p
• Since on a balanced growth path we have
˙C
C
=
˙A
A
and must also have
˙q
q
= ˙p
p
, this translates to
−θ(
˙A
A
) =
˙q
q
9
Inserting for
˙A
A
= BL2 and (26) we get
−θBL2 = ρ − BL
which we can finally solve for L2
L2 =
L
θ
−
ρ
Bθ
(27)
which in turn implies that the growth rate on the balanced growth path
is
γ∗
=
˙C
C
=
˙A
A
=
BL − ρ
θ
(28)
• Since α ∈ (0, 1) it follows that the growth rate in the decentralized
solution we found last time (γ) falls short of the growth rate in the
socially optimal solution (γ∗
). This, of course, follows from too little
research being done in the decentralized solution.
• There are two reasons why there is too little research in the decentralized
solution
1. There are positive externalities in research. The private agents
do not take into consideration that new inventions also makes it
more easy to come up with yet newer inventions (that is, by doing
research they contribute to A which is a public good in the R&D
production function (19)).
2. There is monopolistic price setting for the intermediate goods.
Thus, mark-up pricing forces a wedge between the marginal social
product of a new invention/design and its market compensation.
Thus, the producer which maximizes the monopoly profit does not
have as strong incentives to come up with a new design as does a
social planner maximizing the social surplus.
• Note that market power is necessary for there to be private incentives
to invest, but that monopoly pricing gives rise to sub-optimal growth.
15 Growth through increasing quality
Required reading: Aghion and Howitt (1998), Chapter 2.1–2.3.
10
15.1 The Schumpeterian perspective
• Rather than assuming monopolistic competition in a variety of intermediate
inputs, we now look at a case where innovations lead to new
and better intermediate inputs which replace the old ones.
• As long as the good is the best one available, its innovator has a
monopoly for sales of inputs (enforced by a patent). But since the
new good only survives until it itself is replaced, a new innovation only
gives a temporary monopoly.
• The profit enjoyed while in the monopoly position secures incentives
for doing R&D.
• This framework reflects Schumpeter’s ideas about creative destruction.
New innovations are creative in increasing productivity. However, they
are destructive in removing the ground for older products. But then
again the fact that one can benefit from a monopoly position after
having destroyed ones competitors is what induces the creative behavior
in the first place.
15.2 A simple model without capital accumulation
• We now formulate a simple model oAghion and Howitt (1998).
• Output of the consumption good per unit of calendar time is given by
Yt = Atxα
t (29)
where xt is the amount used of the newest intermediate input and is
the only input to production.
• Note that the index t does not denote calendar time but keeps track of
the number of innovations.
• There is a given stock of labor L which is either used to do research (n)
or to produce x. One unit of labor produces one unit of the intermediate
input. Thus we have labor market clearing given by
L = n + x (30)
• When n units of labor are used in research this gives an expected
λn new innovations per unit of calendar time. A new innovation
11
gives rise to a new intermediate input xt+1 with a higher productivity
At+1 = γAt, where γ > 1 is a parameter characterizing the magnitude
of innovations.
• The most essential relation of the model is the arbitrage equation
wt = λVt+1 (31)
This says that labor must have the same expected value in its two
uses. It can be used in producing the intermediate good giving the
wage wt per hour, or to do research which gives an expected λ new
innovations per extra hour. Each new innovation leads to the creation
of a monopoly market for the new intermediate input xt+1 which gives
rise to an income stream with net present value Vt+1.
• Since we do neglect capital accumulation in the model the interest rate,
r, is exogenous.
• The discounted value Vt+1 must then obey
rVt+1 = πt+1 − λnt+1Vt+1 (32)
A patent on good number t + 1 is an asset with expected return rVt+1
per unit calendar time. On the right hand side we have the profit
from production and monopoly sales of xt+1 given by πt+1 per unit
calendar time. At some future point the asset will become worthless,
since the good is obsolete after a yet newer innovation. This occurs
with probability λnt+1 per unit of calendar time, so λnt+1Vt+1 is the
expected capital loss per unit of calendar time. Equality between the
two sides is thus a standard asset equation.
• Equation (32) can be reorganized to
Vt+1 =
πt+1
r + λnt+1
(33)
that is, the income stream is discounted by a factor which is larger than
the interest rate r because at some future time (characterized by the
probability λnt+1) the income stream will stop.
• We now have to determine the profit flow πt and the allocation of
labor xt. This is found from the profit maximization of the monopolist
producer of the intermediate good
πt = max
xt
[pt(xt)xt − wtxt]
12
• Since the final good sector is competitive it follows that the price of xt
will equal its marginal product in production of Y , giving an inverse
demand function for the input xt
pt(xt) = Atαxα−1
t
Note that α is the elasticity of this demand curve, thus also characterizing
the market power of the intermediate monopolist.
• The monopolist then chooses
xt =
α2
wt/At
1/(1−α)
(34)
and hence
πt =
1
α
− 1 wtxt = At˜π(ωt) (35)
where ωt = wt/At is the productivity adjusted wage rate, and ˜π(ωt) ≡
(1/α − 1)Atα2/(1−α)
ω
−α/(1−α)
t , so ˜π (ωt) < 0. Note from (34) that also
xt(ωt) < 0.
• Inserting in (33) and rearranging (remember that At+1 = γAt) we get
ωt = λ
γ˜π(ωt+1)
r + λnt+1
(36)
which together with equilibrium in the labor market
L = nt + ˜x(ωt) (37)
determine the model by determining the paths of ωt and nt.
• We focus on the steady state where ωt = ω and nt = n for all t. Since
(36) is downward sloping and (37) upward sloping in the (n, ω)-plane
we can determine a unique ˆn.
13
• From straightforward algebra it follows that the steady state value ˆn
satisfies
1 = λ
γ 1−α
α
(L − ˆn)
r + λˆn
(38)
• Notice that the amount of research (n) determines the growth. Aghion
and Howitt (1998) (Section 2.2.2).
15.3 Incentives for R&D and implications for policy
• We have the following results
1. An decrease in the interest rate raises the benefit to research (future
income more valuable) and raises ˆn.
2. An increase in available labor reduces the wage and hence the
marginal cost of research as well as raising the expected monopoly
profit (increased demand).
3. The more likely are new innovations the less costly is research.
But the less worth is a new innovation because it increases creative
destruction. As specified here, the former effect is dominant so an
increased λ leads to a rise in ˆn.
4. Increasing the size of each innovation (γ) raises the next intervals
monopoly profit relative to today’s productivity and hence raises
the incentives to do research.
5. The amount of research decreases in the elasticity of the demand
curve facing the monopolist (α). So product market competition
is bad for growth.
• When comparing the decentralized solution with that chosen by a social
planner there are three effects.
1. The intertemporal spillover effect. When evaluating the stream
of future profit from a new innovation a private agent does not
take into consideration that all future innovations are derived by
standing on his shoulders. Hence, by having pushed the technology
one step forward future monopolists are enabled to earn a
yet higher profit with the yet better technology. That is, the private
agent discounts πt too heavily. This effects lead to too little
private research.
14
2. The appropriability effect. As in the Romer model, the private monopolist
can not appropriate the entire increase in social surplus
from the new intermediate good and hence has too weak incentives
for research.
3. The business-stealing effect. The private firm does not internalize
that the monopolist it replaces looses its entire profit. This leads
to too strong incentives for research.
• Which effect dominates is an empirical question. However, this model
has an important difference form the Romer model in that it opens up
the possibility that laissez-faire growth can be too high.
References
[1] Aghion, P., Howitt, P. Endogenous growth theory, Cambridge and
London: MIT Press, 1998, Chapter 2.
[2] Romer, Paul M. Endogenous Technological Change, Journal of Political
Economy, October 1990, 98 (5) . Pp. 71-102. Part 2.
15