A Discrete Two-Sector Economic Growth Model

Abstract

This paper studies a key model in economic theory—the two-sector growth model—with an alternative utility function. We show that the system has a unique stable equilibrium when the production functions take on the Cobb-Douglas form. We also simulate the model and demonstrate effects of changes in some parameters.

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Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2007, Article ID 89464, 13 pages
doi:10.1155/2007/89464
Research Article
A Discrete Two-Sector Economic Growth Model
Wei-Bin Zhang
Received 12 February 2006; Accepted 19 April 2006
Thispaperstudiesakeymodelineconomictheory—thetwo-sectorgrowthmodel—with
an alternative utility function. We show that the system has a unique stable equilibrium
when the production functions take on the Cobb-Douglas form. We also simulate the
model and demonstrate effects of changes in some parameters.
Copyright © 2007 Wei-Bin Zhang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Solow’s one-sector growth model and Uzawa’s two-sector growth model have played the
role of the key models in the neoclassical growth theory (Uzawa [1], Solow [2], Burmeis-
ter and Dobell [3]). These two models and their various extensions and generalizations
are fundamental for the development of new economic growth theories as well (e.g.,
Barro and Sala-i-Martin [4], Aghion and Howitt [5]). Since Uzawa proposed the model
in [1], many works have been published to extend and generalize the model in from the
1960s till today (e.g., Drandakis [6], Diamond [7], Weizs¨ acker, [8], Corden [9], Stiglitz
[10], Gram [11], Mino [12], Drugeon and Venditti [13]). But all these studies follow
the Solow or Ramsey approach to consumer behavior. This study proposes another ap-
proach to consumer behavior to reexamine the basic issues addressed by the two-sector
growth model in discrete time. The paper is organized as follows. Section 2 defines the
two-sector growth with an alternative approach to consumer behavior with saving and
consumption. Section 3 examines dynamic properties of the model when the production
functions are specified with the Cobb-Douglas form and simulates the model. Section 4
carries out comparative dynamic analysis with regard to technological and preference
changes. Section 5 concludes the study.

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2Discrete Dynamics in Nature and Society
2. Thetwo-sector model
This paper reexamines dynamics of the two-sector economic model proposed by Uzawa
[1]. The Uzawa model extends the Solow model by a break down of the productive sys-
tem into two sectors using capital and labor, one of which produces capital goods, the
other consumption goods (Solow [2]). This paper introduces an alternative approach to
consumer decision to examine structural change for a two-sector economy with capital
accumulation. Like in the traditional two-sector growth model, it is assumed that con-
sumption and capital goods are different commodities, which are produced in two dis-
tinct sectors. We develop the model with endogenous saving in discrete time. The econ-
omy has an infinite future. We represent the passage of time in a sequence of periods,
numbered from zero and indexed by t =0,1,2,.... The end of period t −1 coincides with
the beginning of period t; it can also be called period t. We assume that transactions are
made in each period. The population, N, is constant. Most aspects of our model are sim-
ilar to the Solow one-sector growth model and the Uzawa two-sector model. The discrete
versionoftheSolow-modelisreferredtoDiamond[7]andZhang[14].Itisassumedthat
there is only one (durable) good in the economy under consideration. Households own
assets of the economy and distribute their incomes to consume and save. Exchanges take
place in perfectly competitive markets. Production sectors sell their product to house-
holds or to other sectors and households sell their labor and assets to production sectors.
Factor markets work well; the available factors are fully utilized. Saving is undertaken
only by households, which implies that all earnings of firms are distributed in the form
of payments to factors of production, labor, managerial skill, and capital ownership.
We assume perfect competition in all markets and select commodity to serve as nu-
meraire, with all the other prices being measured relative to its price. Let K(t) denote the
capital existing in period t and N the flow of labor services used in period t for produc-
tion. Capital depreciates at a constant exponential rate δk, which is independent of the
manner of use. The two inputs are smoothly substitutable for each other in each sector
and are freely transferable from one sector to the other. Both exogenously determined
labor supply and irrevocably existing capital stock are inelastically offered for employ-
ment. Both sectors use neoclassical technology with the standard Inada conditions. The
production functions are given by Fj(Kj(t),Nj(t)), j = i,s, where the subscripts i and s
denote the capital good sector and the consumption good sector and Fjare the output
of sector j; Kj(t) and Nj(t) are, respectively, the capital and labor used in sector j. For
simplicity, we assume that Fjtakes on the Cobb-Douglas form as follows:
Fj(t) = AjKαj
j(t)Nβj
j(t),
αj,βj>0, αj+βj=1, j = i,s.
(2.1)
We express the above equations by
fj(t) =Ajkαj
j(t),
fj(t) ≡Fj(t)
Nj(t),
kj(t) ≡Kj(t)
Nj(t),
j = i,s.
(2.2)
Markets are competitive; thus labor and capital earn their marginal products, and firms
earn zero profits. We assume that the capital good serves as a medium of exchange and is

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Wei-Bin Zhang3
taken as numeraire. The price of consumption good is denoted by p(t). The rate of inter-
est, r(t), and wage rate, w(t), are determined by markets. Hence, for any individual firm,
r(t) and w(t) are given in each period. The production sector chooses the two variables,
Kj(t) and Nj(t), to maximize its profit. The marginal conditions are given by
r(t)+δk= αiAik−βi
i
(t) = p(t)αsAsk−βs
s
(t),
w(t) = βiAikαi
i(t) =αsAsp(t)kαs
s(t).
(2.3)
The total capital stock, K(t), is allocated to the two sectors. As full employment of labor
and capital is assumed, we have
Ki(t)+Ks(t) = K(t),
Ni(t)+Ns(t) = N, (2.4)
where N(= 1) is the fixed population. We rewrite the above equations as
ni(t)ki(t)+ns(t)ks(t) =k(t),
where
ni(t)+ns(t) =1,(2.5)
k(t) ≡K(t)
N
,
nj(t) ≡Nj(t)
N
,
j =i,s.
(2.6)
A representative household’s current income, y(t), from the interest payment, r(t)k(t),
and the wage payment, w(t), in period t are given as follows:
y(t) =r(t)k(t)+w(t).
(2.7)
The sum of money that consumers are using for consuming, saving, or transferring are
not necessarily equal to the current income because consumers can sell wealth to pay, for
instance, current consumption if the temporary income is not sufficient for purchasing
goodsandservices.Retiredpeoplemaylivenotonlyontheinterestpaymentbutalsohave
to spend some of their wealth. The total value of wealth that a representative household
can sell to purchase goods and to save is equal to k(t). Here, we do not allow borrowing
for current consumption. We assume that selling and buying wealth can be conducted
instantaneously without any transaction cost. This is evidently a strict consumption as
it may take time to draw savings from bank or to sell one’s properties. The per capita
disposable income of the household is defined as the sum of the current income and the
wealth available for purchasing consumption goods and saving:
? y(t) = y(t)+k(t) =?1+r(t)?k(t)+w(t).
The disposable income is used for saving and consumption. In each period, a consumer
would distribute the total available budget between savings, s(t), and consumption of
goods, c(t). The budget constraint is given by
(2.8)
p(t)c(t)+s(t) = ? y(t).
(2.9)
Equation (2.9) means that consumption and savings exhaust the consumers’ disposable
income.

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4Discrete Dynamics in Nature and Society
In our model, in each period, consumers have two variables, the level of consumption
and saving, to decide. We assume that utility level, U(t), that the consumers obtain is
dependent on the consumption level of commodity, c(t), and the savings, s(t), as follows:
U(t) =cξ(t)sλ(t),
ξ,λ>0, ξ +λ = 1,(2.10)
where ξ is called the propensity to consume, and λ the propensity to own wealth. This
model was proposed by Zhang in the early 1990s. A comprehensive explanation of the
model is referred to Zhang [15]. In particular, the economic growth theory based on
Zhang’s approach is systematically compared with traditional economic growth theo-
ries. This type of utility functions has also been applied to different economic problems
formed with difference equations [14].
For consumers, wage rate, w(t), and rate of interest, r(t), are given in markets and
wealth, k(t), is predetermined before decision. Maximizing U(t) in (2.10) subject to (2.9)
yields
p(t)c(t) =ξ? y(t),
s(t) =λ? y(t).
(2.11)
According to the definitions, the household’s wealth in period t+1 is equal to the savings
made in period t:
k(t+1) =s(t) = λ? y(t).
(2.12)
This equation describes the accumulation of the households’ wealth. The output of the
consumption good sector is consumed by the households. That is,
c(t)N = Fs(t).
(2.13)
As output of the capital good sector is equal to the depreciation of capital stock and the
net savings, we have
S(t)−K(t)+δkK(t) = Fi(t), (2.14)
where S(t)−K(t)+δkK(t) is the sum of the net saving and depreciation. It is straightfor-
ward to show that this equation can be derived from the other equations in the system.
We have thus built the dynamic model.
3. Thedynamics, equilibrium,andstability
The dynamic system consists of many equations. Before analyzing dynamic properties of
the system, we first show that the motion of the entire system is actually controlled by a
single difference equation. First, from (2.3), we obtain
ks(t) =αki(t),(3.1)

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Wei-Bin Zhang5
where α ≡ βiαs/βsαi. Thecapital intensityofthe consumptiongood sectoris proportional
to that of the capital good sector. By ks=αkiand βifi= βspfs, we solve
βiAi
βsααsAski(t)αi−αs.
p(t) =
(3.2)
The price of consumption goods is positively related to the technological level of the cap-
italgoodsectorbutnegativelyrelatedtothatoftheconsumptiongoodsector.Thepriceis
positively or negatively related to the capital intensity of the capital good sector, depend-
ingonthesignofαi−αs.Ifαi= αs,thenthepriceisconstant, p = Ai/As.Intheremainder
of this section, we require αi?= αs. The analysis of case αi= αsis straightforward. From
(2.5) and ks= αki(t), we solve the labor distribution as follows:
ni(t) =αki(t)−k(t)
(α−1)ki(t),
ns(t) =k(t)−ki(t)
(α−1)ki(t).
(3.3)
The labor distribution between the two sectors is uniquely determined by k(t) and ki(t).
According to the definitions of S(t), K(t), s(t), and k(t), we have
S(t)−δK(t) =?s(t)−δk(t)?N,
where δ ≡1−δk. From the above equation, (2.14), and s = λ? y, we obtain
? y(t) =ni(t)fi(t)+δk(t)
From pc = ξ? y, c = nsfs, and p = f?
ns(t)f?
(3.4)
λ
.
(3.5)
i/f?
s, we get ? y = nsfsf?
i(t)f∗(t) = ni(t)fi(t)+δk(t),
i/ξ f?
s. From this equation and
(3.5), we have
(3.6)
where f∗(ks(t)) ≡ λks(t)/ξαs. Substituting ni(t) = 1−ns(t) and ns(t) in (3.3) into (3.6)
yields
k(t) =Φ?ki(t)?ki(t),
where
(3.7)
Φ?ki(t)?≡
1
A0
?1+Akβi
i(t)?,
A0≡1+λ0
α+λ0>0,
A ≡(1−α)δ/Ai
1+λ0
,
λ0≡βiλ
βsξ.
(3.8)
By (3.6) and (3.7) and according to the definitions of A and A0, we solve
ni(t) =Φ?ki(t)??α1−α2kβi
where α1≡ λ0/(α+λ0) and α2≡ αδ/Ai(α+λ0). Hence, for ni(t) to satisfy 1 > ni(t) > 0, it
is sufficient to have
i(t)?,(3.9)
0<ki(t) <
?α1
α2
?1/βi
.
(3.10)