GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.118

Neutrality of an increase in the price of natural
resources to the level of technology

Asuka Ohura and Tadashi Morita

February 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanNeutrality of an increase in the price of natural
resources to the level of technology ∗Asuka Ohura and Tadashi Morita †Graduate School of Economics, Osaka UniversityFebruary 2010

Abstract
This paper constructs an endogenous variety expansion model of a small open
economy based on Grossman and Helpman (1991) to investigate how an increase in
the price of natural resources affects the level of technology. This paper concludes
that an increase in the price of natural resources does not affect the level of technology.
Keywords: Natural Resources, R&D, Economic Growth
JEL classification: F43, O31∗

We are very grateful to Koichi Futagami for his considerable help. We also thank Shinsuke Ikeda
and Kazuo Mino.
†Graduate School of Economics, Osaka University, 1.7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN e-mail: gge013mt@mail2.econ.osaka-u.ac.jp1

1

Introduction

Starting in the 1970s, the price of oil has fluctuated dramatically. Before the 1970s,
the price of oil had been stable at between $10 and $20 per barrel in 2007 US dollars.
Following the oil shock, the price of oil again became stable at around $30 during theperiod 2000-2004. However, the price of oil rose sharply in July 2008 to $145 per barrel.
Therefore, fluctuations in the price of oil profoundly influence countries importing crude
oil.
This paper investigates how an increase in the price of natural resources affects thelevel of technology. We construct an endogenous variety expansion model in a small open
economy based on Grossman and Helpman (1991). In this model, firms produce final
goods using intermediate goods and a natural resource. Therefore, when the price of thenatural resource increases, firms will substitute the natural resource for the intermediate
goods and the demand for the intermediate goods increases. This increases R&D investment and the level of technology. However, the cost of the natural resource also increases,
the expenditure of the country decreases, and the R&D investment then decreases. Thispaper shows that these two effects cancel out each other. Therefore, when the price ofthe natural resource increases, the firms substitute natural resources for the intermediategoods. However, the level of technology does not change. In the steady state, when theprice of a natural resource increases, the consumption level decreases.12The Model

We develop a dynamic general equilibrium model based on Grossman and Helpman (1991).
In this model, there are final goods, intermediate goods, and a natural resource. Individuals consume only the final goods. Each individual lives forever and is endowed with one
unit of labor services, which is inelastically supplied at each point of time. The population size in this economy is constant over time and normalized to unity. To produce thefinal goods, firms use the intermediate goods and the natural resource. This economy isa small open country that trades final goods and the natural resource at an exogenously
given world price. We suppose that the final goods and the natural resource are tradable.
In contrast, we assume that the intermediate goods are not tradable.

1Many researchers investigate the relationship between the price of natural resources and economicgrowth. For instance, Peretto (2009) finds that an optimal tax rate on energy use exists that maximizes
the welfare. In the short run, a tax on energy use then generates the temporary acceleration of totalfactor productivity growth.

2

2.1

Production

Production of the final goods requires variety-specific intermediate goods and the naturalresource. The production function is given by:1

Y = [γDσ + (1 − γ)Gσ ] σ ,(1)where Y denotes the output of final goods, G is the natural resource input, and D denotes
the composite input of the intermediate goods. σ and γ are parameters. The compositeof the intermediate goods is given by:µZ nt
D≡¶ α1xi di,α(2)0where xi denotes the intermediate goods produced by firm i and nt denotes the level oftechnology at time t. α is the elasticity of substitution between any two varieties in agiven sector. If α is close to one, the goods are nearly perfect substitutes. The final goods
firms sell their output to their own country and abroad. The final goods is chosen to be
the numeraire and the price index of the composite intermediate goods is as follows:³Z nα

pi α−1 djPD ≡´ α−1α,(3)0where pi denotes the price of the intermediate goods produced by firm i. We then obtain
the profit-maximization conditions as follows:hσ i σ−11σ− σ−1σ−1
PD = γ 1 − (1 − γ)
Pg
,
1σµ
D=

PDγ

1
¶ σ−1Y,1¶ σ−1Pg
Y,
G=1−γ1µ ¶ 1−αPD
xi =D,pi

(4)

(5)

µ(6)
(7)

where Pg denotes the world price of the natural resource.
The intermediate goods firms buy a patent from the R&D sector and sell the intermediate goods exclusively to the final goods firm. The production of one unit of eachintermediate good requires one unit of labor. Therefore, we can write the profit of the3

intermediate goods firm i as follows:
πi = pi xi − wxi ,(8)where πi denotes the profit of the intermediate goods firm i and w denotes the wage rateof labor. The monopoly price and the profit level are then:pi =w,α−σ−1(9)α−σσπi = (1 − α)α σ−1 γ σ−1 w σ−1 n α(σ−1) Y.

(10)Consequently, we obtain the total output of intermediate goods and the price index ofintermediate goods PD as follows:µχ ≡ nxi =w
αγ1¶ σ−1σ(α−1)

n α(σ−1) Y,

w α−1n α ,αwhere χ denotes the total output of intermediate goods.
PD =2.2(11)(12)

R&D sector

The R&D activities of the present model follow Grossman and Helpman (1991). Theintermediate goods producers enter into the R&D race and finance the cost of R&D by
issuing equity in the stock market. The equity is bought by individuals. The stock valueof the intermediate goods producers at time t is equal to the present discounted sum of its
profit stream subsequent to t. Then, the stock value of the intermediate goods producers
at time t is given by:
Z∞v=Rse− t rv dv πi ds,

(13)

t

where rs denotes the interest rate on a riskless loan at time s. Differentiating (13) withrespect to time t yields the following no-arbitrage condition:v̇ = −π + rt .

(14)

The intermediate goods producers hire labor to develop blueprints. In this model, anincrease in the number of intermediate goods implies an increase in the efficiency of thenatural resource. We assume that La units of labor for R&D activity over a time interval

4dt produce a new variety of intermediate goods according to:
dn =Ladt,a(15)

where a−1 denotes the productivity of R&D. The cost of R&D activities is wLa dt. The
blueprints create value for the intermediate goods producers of vdn as each blueprint hasa market value of v. We assume there is free entry into the R&D race. Therefore, thefollowing free-entry condition must hold:v ≤ aw

2.3with equality whenever ṅ ≡dn> 0.
dt

(16)

Consumers

A representative agent has the following preference:
Z ∞
U0 =

e−ρt log ct dt,

(17)

0

where ct stands for the consumption of the final goods at time t and ρ > 0 is the constantsubjective discount rate. The budget constraint is represented byZ ∞R− 0t rs ds

Et e

Z ∞
dt = a0 +0Rtwt e− 0 rs ds dt,

(18)

0

where Et denotes expenditure at time t, and a0 ≡ n0 v0 denotes the economy’s aggregate
equity value at time 0. Then, from the first-order conditions of the maximization problem,
we obtain the following Euler equation:Ėt= rt − ρ.
Et

2.4

(19)

Labor market equilibrium and trade balance conditions

Each individual supplies one unit of labor services over any time t. These labor services
are supplied to the R&D sector and to the production of intermediate goods. Thus, thelabor market equilibrium condition is given by:
1 = χ + aṅ.

(20)

In this small open economy, final goods are exported to foreign countries and thenatural resource is imported from abroad. Therefore, the trade balance condition becomes
as follows:
Y − E = Pg G.
5(21)The left-hand side of this equation represents the volume of exports and the right-handside represents the volume of imports.

3The Equilibrium Path

The equilibrium conditions are the no-arbitrage condition of (14), the free-entry conditionof (16), the Euler equation of (19), the labor market equilibrium condition of (20), andthe trade balance condition of (21). We can derive the two differential equations for thenumber of intermediate goods, n, and expenditure, E, as follows:
ṅt
ĖtEt

·¸1E α−1
α=1−n,aPDµ¶
1 − α E −1=n α − ρ.
aαPD

(22)
(23)

These two equations constitute the dynamic system of this economy.

.

E.

A

E*.

E

E=0

E=0.

A

E*n=0

E**

n=0
BE(0)

E(0)

Gnnn(0)n*n(0) n(T)n*= n**

Figure 2: An increase in Pg at time T

Figure 1: The phase diagram

Figure 1 depicts the phase diagram for the system in the (n, E) plane. The intersection
point of the two curves ṅ = 0 line and Ė = 0 corresponds to the steady state of this system.
The ṅ = 0 line is given by:
1−α

E = PD n α .

(24)When α < 12 , the ṅ = 0 line is convex. On the other hand, when α > 12 , the ṅ = 0 line is6

concave. Hereafter, we focus on the case of α > 12 .2 The Ė = 0 locus is given by:
E=1aαρ
PD n α .
1−α(25)The Ė = 0 locus is convex. Thus, there exists a unique steady state. The steady statevalue of n∗ is given by:
1−αn∗ =.
(26)aαρ
We can show the steady state A becomes saddle path stable (see Appendix for the proof).
Figure 2 depicts the phase diagram when the price of the natural resource increases
suddenly at time T . We assume that the agent does not expect the sudden increase in
the price of the natural resource and has rational expectations. Suppose that the initial
number of intermediate goods is sufficiently small at G in Figure 2. The agent expects
that the economy follows the stable path to A in Figure 2 in the future. Then, at time T ,the price of the natural resource increases suddenly. The agent decreases his expenditure
immediately in response and induces the economy to follow the stable path to B in Figure
2.
The effects of a sudden increase in the price of the natural resource on the steady statevalue of the number of intermediate goods and expenditure are summarized as follows:∂n∗= 0,∂Pg1 hσ i −1−111∂E ∗1 − α 1−α σ−1σ− σ−1σ−1σ−1σα
1 − (1 − γ)= −γ (1 − γ) () Pg
Pg
< 0.
∂Pg
aαρ
1(27)(28)σBecause PD > 0, 1 − (1 − γ)− σ−1 Pgσ−1 > 0.
Accordingly, in the steady state, an increase in the price of the natural resource decreases expenditure and does not change the level of technology. Thus, we can summarize
these results as the following proposition.
PROPOSITION 1. In the long run, an increase in the price of the natural resource
decreases expenditure without affecting the level of technology.

4Conclusion

This paper constructs an endogenous variety expansion model in a small open economy
based on Grossman and Helpman (1991). We focus on the effect of a sudden increase in
the price of natural resources. We conclude that an increase in the price of the naturalresource decreases expenditure but does not affect the level of technology.
2When α < 12 , the following discussion can be applied.

7References[1] Grossman, Gene M., and Elhanan Helpman. (1991). ’Trade, Knowledge Spillovers,and Growth,’ European Economic Review, 35, 517-526.
[2] Peretto, P. F. (2009). ’Energy taxes and endogenous technological change,’ Journal ofEnvironmental Economics and Management, 57, 269-283.
[3] Smith, James L. (2009), ’World Oil: Market or Mayhem?’, Journal of Economic
Perspectives, 23:3, 145-164.

AAppendix

A.1

Proof of the stable saddle pathWe derive that the steady state of this economy is saddle point. The linearized system of(22) and (23) around the steady state is given by:
Ãṅt
Ėt!

Ã=

ρ
1−3α− 1−αP n∗ αaα D− aE1 ∗!Ãnt − n∗Et − E ∗ρ!
.

2αρThe determinant of the characteristic matrix is − 1−α< 0. Then, the determinant takesa negative value. Therefore, the two eigenvalues of the system have opposite signs andthe steady state is saddle point.

8