ECON6002 Macroeconomics Analysis 1
Week 4: “Romer Model, Growth Empirics”
James Morley
University of Sydney
Semester 1, 2023
Have you read this book?
Class Outline
1. Paul Romer’s microfounded model of endogenous growth
2. Solving the model
3. Using the model
4. Empirics of economic growth
5. Summary
Readings: Romer 3.5-3.7 and Chapter 4; Jones (2005)
Question (pingo.coactum.de → 676307)
What is more important for the profitability of ideas: legal institutions to
enhance excludability or inherent distinctiveness/substitutability?
A. Legal institutions
B. Distinctiveness/substitutability
Romer Model
• Similar to the simple endogenous growth model, but allocation of
resources based on microfounded decisions
• Firms are profit maximizing and households are utility maximizing
• Monopoly power for R&D firms over an output firm’s use of distinct
ideas in producing output good due to patents (excludability)
• No capital, θ = 1, and n = 0 for simplicity (see Jones chapter for
version with physical/human capital accumulation, θ < 1, n > 0)
• No transition dynamics and balanced growth path for output growth
(per capita) depends only on endogenous knowledge accumulation
• Two goods (output, new ideas), one asset (patents), closed
economy, no unemployment
• Ethier (1982) production function for output good
Ethier Production Function
• At time t, there exists a continuum of distinct ideas from 0 to A
• Labour converts a given idea into an input good for production of
output with a one-to-one linear mapping (input X = fX (L) = L)
• The output good production function is
Y =
[∫ A
i=0
L(i)φdi
]1/φ
• 0 < φ < 1 determines the substitutability of input goods (distinct
ideas) in producing output
• Let LY denote number of workers producing input goods
• Suppose number producing each available input is the same, then
L(i) = LY /A and Y =
[
A
(
LY
A
)φ]1/φ
= A(1−φ)/φLY
• I.e., constant returns to scale in labour and output is increasing in
the total stock of knowledge A (same as simple model)
Market Structure
• Assume all markets are perfectly competitive except input market for
using distinct ideas to produce output
• R&D firms hire workers in competitive labour market at wage w(t)
to produce inputs or new ideas
• Due to patents, R&D firms have monopoly power over use of
distinct idea in input market, but no price discrimination or
complicated contracts, just fixed price p(i)
• Output firms buy inputs to produce output
• Output firms take prices of inputs and price of output as given
Output Firms
• Profit maximization taking price of output as given is equivalent to
cost minimization
• Lagrangian for minimizing costs subject to producing a unit of
output:
L =
∫ A
i=0
p(i)L(i)di − λ

[∫ A
i=0
L(i)φdi
]1/φ
− 1

• First-order condition with respect to L(i) given price of input p(i):
p(i) = λL(i)φ−1 ⇒ L(i) =
[
λ
p(i)
] 1
1−φ
• λ > 0 is the marginal cost of output
• Downward-sloping demand curve for distinct idea i as input into
production of output, with elasticity of demand equal to 1/(1− φ)
• The closer φ is to 0, the less substitutability of ideas and the more
inelastic the demand
R&D Firms
• R&D firms choose to employ workers either in producing new ideas
or in producing inputs to sell to output firms:
LA(t) + LY (t) = L¯, where population is fixed at L¯ > 0
• Production function for new ideas is same as simple model with
γ = 1 and θ = 1:
A˙(t) = BLA(t)A(t)
• B > 0, A(0) > 0, and past ideas are freely available in knowledge
sector (i.e., positive R&D externality)
• Free entry into R&D means present value of any profits from selling
inputs is same as cost of producing a new idea:∫∞
τ=t
e−r(τ−t)pi(i , τ)dτ = w(t)BA(t)
• Assumes fixed interest rate r(t) = r in equilibrium (no transition
dynamics)
Price Setting in the Input Market
• From micro, profit maximizing price for monopolist in input market
is η/(η − 1)×marginal cost, where η is the elasticity of demand
• In this case, the elasticity of demand is constant and equal to
1/(1− φ)
• Given one-to-one mapping of labour to input good, marginal cost of
producing input is w(t)
• Thus, each monopolist charges a marked-up price for an input:
p(i) = 1φw(t) > w(t)
• Profits are quantity of firm’s ideas used times the markup:
pi(t) = L(i)
[
w(t)
φ
− w(t)
]
Households
• Same setup as Ramsey model, but instantaneous utility is assumed
to be logarithmic for simplicity (and notation, i.e., CRRA θ = 1)
• Thus, the representative individual’s lifetime utility is∫∞
t=0
e−ρt lnC (t)dt
• Again assuming fixed interest rate r(t) = r in equilibrium, the
intertemporal budget constraint is∫∞
t=0
e−rtC (t)dt ≤ X (0) + ∫∞
t=0
e−rtw(t)dt
• X (0) is initial wealth per person (i.e., individual share of present
value of future profits from ideas already invented and patented)
• Euler equation (given IES=1):
C˙ (t)
C (t)
= r(t)− ρ
Small Group Discussion #1
• Groups of 3-5 in classroom and breakout rooms on Zoom
1. Do you think the substitutability of distinct ideas used in producing
output has increased or decreased over time?
2. Explain your answer and how it would affect markups charged by
those with patents?
• Noting which group/room you are in, type at least one answer to
each question into the first Ed thread for Week 4
Solving the Model
• We posit an equilibrium with constant LA = L¯− LY and (can)
confirm it is unique
• Given constant LA and production functions, there is constant
growth of ideas and output:
A˙/A = BLA ⇒ Y˙ /Y = [(1− φ)/φ]BLA
• In general equilibrium, consumption equals output, so C (t)L¯ = Y (t)
since all households are identical
⇒ C˙/C = [(1− φ)/φ]BLA
• Because markup rate is constant, payments to workers in input
production is constant fraction of revenues, thus constant LA implies
wages must grow at same rate as output
⇒ w˙/w = [(1− φ)/φ]BLA
Equilibrium Profits
• Note: L¯−LAA(t) = LYA = L(i) given all firms are the same
• Given constant LA and same price for all inputs, profits will be
pi(t) =
L¯− LA
A(t)
[
w(t)
φ
− w(t)
]
=
1− φ
φ
L¯− LA
A(t)
w(t)
• Thus, given growth rates for w and A, profits will grow at constant
rate [(1− 2φ)/φ]BLA
Equilibrium Real Interest Rate and Present Value of
Future Profits = Cost of New Ideas (free entry)
• From the Euler equation and given growth rate for C , the constant
real interest rate, r , will be
r = ρ+
1− φ
φ
BLA
• Therefore, present value of profits from a new idea at time t and the
cost of creating it (payment to labour in knowledge sector) are∫ ∞
τ=t
e−r(τ−t)pi(τ)dτ =
1− φ
φ
L¯− LA
ρ+ BLA
w(t)
A(t)
=
w(t)
BA(t)
=
w(t)LA
A˙(t)
• See math on next slides...
The Math
• Note: given constant growth of profits:
pi(τ) = pi(t)e [(1−2φ)/φ]BLA(τ−t)
• Present-value of profits:∫ ∞
τ=t
e−r(τ−t)pi(τ)dτ = pi(t)
∫ ∞
τ=t
e−r(τ−t)e [(1−2φ)/φ]BLA(τ−t)dτ
=
pi(t)
ρ+ 1−φφ BLA − 1−2φφ BLA
=
pi(t)
ρ+ BLA
=
1− φ
φ
L¯− LA
ρ+ BLA
w(t)
A(t)
The Math Cont.
• Set present value of profits in input market from idea to costs of
producing idea in knowledge sector:
1− φ
φ
L¯− LA
ρ+ BLA
w(t)
A(t)
=
w(t)
BA(t)
⇒ L¯− LA = φ
1− φ
ρ+ BLA
B
⇒ LA = (1− φ)L¯− φρ
B
Equilibrium Labour Allocation and Output Growth
• Because firms cannot set LA < 0, they will only engage in R&D if
present value of profits is such that LA > 0. Thus, in equilibrium,
LA = max
{
(1− φ)L¯− φρ
B
, 0
}
• ⇒ Endogenous growth if present value of profits high enough:
Y˙
Y
= max
{
(1− φ)2
φ
BL¯− (1− φ)ρ, 0
}
• Note: intertemporal budget constraint holds with equality because
present value of infinite-horizon wealth is zero given interest rate
exceeds growth rate of economy (given r = ρ+ 1−φφ BLA)
Using the Model
Y˙
Y
= max
{
(1− φ)2
φ
BL¯− (1− φ)ρ, 0
}
• Microfoundations tell us that, through endogenous determination of
LA, long-run growth depends positively on:
1. patience (↓ ρ⇒ Y˙ /Y ↑)
2. monopoly power (↓ φ⇒ Y˙ /Y ↑)
3. R&D productivity (↑ B ⇒ Y˙ /Y ↑)
4. population (↑ L¯⇒ Y˙ /Y ↑)
• Recall aL was exogenous parameter in simple endogenous growth
model
• Unlike Ramsey model, decentralized equilibrium is suboptimal due to
externalities
• It can be shown that LA = (1− φ)L∗A, where L∗A is the optimal
number of workers in the knowledge sector to maximize utility
• Patents allow for growth, but don’t lead to social optimum – maybe
governments should subsidize basic research to increase LA?
Small Group Discussion #2
• Groups of 3-5 in classroom and breakout rooms on Zoom
1. What are the potential externalities in the Romer model that
determine the gap from the social optimum? (Hint: there is a
consumer-surplus effect, a business-stealing or business-creating
effect, and an R&D effect.)
2. What if population growth were endogenous to economic factors?
How and what might it depend on?
• Noting which group/room you are in, type at least one answer to
each question into the second Ed thread for Week 4
Question (pingo.coactum.de → 676307)
What is most important in explaining differences in incomes across
countries?
A. levels of physical capital (e.g., machinery, factories)
B. levels of human capital (e.g., average years of education)
C. adaptation/diffusion of technologies
D. legal and political institutions
E. geography (e.g., endowments of natural resources, climate, distance)
Growth Empirics
• Physical and human capital only explain a fraction of cross-country
variation in output per capita (Hall and Jones, 1999, QJE)
• Geography and institutions seem important (diffusion and
implementation of ideas)
• Trade and knowledge diffusion (Frankel and D. Romer, 1999, AER)
• British Empire versus other colonies in the tropics (Acemoglu,
Johnson, and Robinson, 2001, AER)
• North/South Korea, East/West Germany (Olsen, 1996, JEP)
• But England and France (North, 1981, and Nye, 1991)
• North and South of the United States
• What is the role of population growth in explaining long-run
growth? (Kremer, 1993, QJE)
Endogenous Population Growth
• Kremer (1993) develops a Malthusian endogenous growth model in
which new ideas lead to population growth
• New ideas production function: A˙(t) = BL(t)A(t)θ
• But land is fixed factor of production and population endogenously
adjusts such that output per capita equals subsistence level
Y (t)/L(t) = y¯
• Equilibrium implies population growth is positively related to
population even if θ < 1
Population Growth Empirics
• Kremer shows there has been a positive, approximately linear
relationship between population growth and population historically
since 1 million B.C. (see graph)
• Recent deviations from trend could be due to a de-linking from
subsistence or slow adjustment of population growth when y > y¯
• Kremer also considers natural experiment of disappearance of
intercontinental land bridges after last ice age that separated
populations in Eurasia-Africa, the Americas, Australia, Tasmania,
and Flinders Island until about 500 years ago
• Technological progress and population density were higher in regions
with higher initial populations (humans died out on Flinders Island
in 3000 B.C.)
Is it reasonable to assume n > 0?
Some Key Points
• Nonrival ideas are a possible engine of growth, but require deviation
from perfect competition to motivate their creation
• Patents can deliver endogenous growth in a microfounded model, but
decentralized equilibrium has suboptimal investment in knowledge
• Empirical sources of cross-country variation in output per capita and
growth remain elusive, although institutions are clearly important
Next time
• The Real-Business-Cycle Model (Romer, Chapter 5)