A Two-Sector Growth Model with Labour Supply · Zhang: A Two-Sector Growth Model with Labour Supply 249 (3) The disposable balance is used for saving and consumption. At each point

Perry & Wilson: The Accord and Strikes 245

A Two-Sector Growth Modelwith Labour Supply

Wei-Bin ZhangRitsumeikan Asia Pacific University, Japan

AbstractThis paper introduces endogenous time distribution between work and leisure intoa two-sector growth model. We differ from the traditional growth theory in that weintroduce a novel economic mechanism to determine consumers’ decision on timedistribution, consumption and saving. The utility maximization solves the problemthat there is no profound rational decision mechanism for consumers in the Solowgrowth model and avoids the complication that the Ramsey growth theory bringsabout. First, we define the two-sector growth model with the alternative approachto consumer behaviour with endogenous labour supply. Then, we examine dynamicproperties of the model when the production functions are specified with the Cobb-Douglas forms. We also simulate the model and demonstrate effects of changes inthe propensity to save and technology of the capital good sector over time. Theappendix examines dynamic properties of the model when the production functionstake on general forms.

1. IntroductionThere are two main frameworks in modeling economic growth with capitalaccumulation in continuous time. These are Solow’s one-sector growthmodel and Uzawa’s two-sector growth model (see Burmeister and Dobell,1970). The Solow model is the starting point for almost all analyses ofeconomic growth. As consumer behavior is not described by utilityoptimization in the Solow model, it does not have a rational mechanism todeal with issues related to optimal consumption over time. Ramsey’s 1928paper on optimal savings has influenced modeling of consumers’ behaviorsince the mid 1960s (see, Ramsey, 1928; Barro and Sala-I-Martin, 1995,chapter 2). This approach assumes that utility is addable over time. It hasbecome evident from extensive publications in the economic literature basedon this approach over the last fifty years that even a simple model tends tolead to a complicated dynamic system. For instance, Barro and Sala-I-Martin(1995) propose a one-sector growth model with endogenous time withinthe Ramsey framework. As demonstrated by Barro and Sala-I-Martin, themodel even with simple utility and production becomes too complicatedto get explicit conclusions.

The one-sector model has the unrealistic assumption that a unit of outputcan be instantaneously and costlessly transformed into either a capital or aconsumption good. Two-sector models recognise, however, that

Address for correspondence: Ritsumeikan Asia Pacific University, Jumonjibaru,Beppu-Shi, Oita-ken 874-8577 Japan. Tel: 0977-78-1020, Fax: 0977-78-1123, E-mail:[email protected] author is grateful to important comments of Managing Editor, Paul Flatau andtwo anonymous referees.© The Centre for Labour Market Research, 2005.

Australian Journal of Labour Economics, Vol. 8, No. 3, September 2005, pp 245 - 260

Australian Journal of Labour Economics, September 2005246

consumption goods and capital goods are inherently different and are likelyto be produced in different economic sectors with different productiontechnologies. Uzawa’s model from the early 1960s is the starting point fortwo sector models (see Uzawa, 1961; Burmeister and Dobell, 1970). Therehave been many extensions and generalizations of Uzawa’s two-sectormodel over years (see, Diamond 1965; Stiglitz, 1967; Gram, 1976; Mino,1996; and Drugeon and Venditti, 2001). But all of these studies follow theRamsey approach to consumer behaviour.

This study proposes another approach to consumer behaviour to re-examinethe basic issues addressed by the two-sector growth model. The advantageof our approach over the traditional approaches is that it provides amechanism for analysing rational consumer behaviour and at the same timesimplifies dynamic analysis.

The study makes a contribution to labor economics by introducingendogenous labour supply into growth theory. The choice of the optimaldistribution of time between leisure and work may be constantly changingas living conditions and the work environment changes as the economygrows. It is important, therefore, to examine interdependence betweenlabour supply and economic growth. Without a clear understanding of theinterdependence, it is difficult to understand how the labour market changesover time.

The implications of economic rationality in the allocation of time have beenmade explicit in a formal and rigorous theory since Becker published hisseminal work in 1965. There have been studies on the interdependencebetween the productivity of a person’s labour, his income, the price of leisuretime and time allocation. There is an immense body of empirical andtheoretical literature on economic growth with time distribution betweenhome and non-home economic and leisure activities.

Nevertheless, it is not easy to model dynamics of labour supply in economicgrowth theory. In order to provide a behavioral mechanism for households’decisions, which the Solow approach lacks, and to make the analysis ofeconomic growth with endogenous time analytically more tractable, thisstudy applies an alternative approach to the modeling of rational consumers’behavior with endogenous saving and labor supply. We demonstrate thatthe resulting dynamics in our approach are more analytically tractable thanmodels based on the Ramsey approach.

The paper is organized as follows. Section 2 defines the two-sector growthwith an alternative approach to consumer behaviour with endogenouslabour supply, saving and consumption. Section 3 examines dynamicproperties of the model when the production functions are specified asCobb-Douglas technology. Section 4 carries out comparative statics analysiswith specified technological change and preference change. Section 5simulates the model. Section 6 concludes the study.

Zhang: A Two-Sector Growth Model with Labour Supply 247

2. The Two-Sector ModelThis paper re-examines dynamics of the traditional two-sector modelinitially proposed by Uzawa (1961). The Uzawa model extends the Solowmodel by disaggregating the production system into two distinct sectors,one of which produces capital goods, the other consumption goods, withboth sectors utilising capital and labour in production but using differentproduction technologies. Labour is assumed to be fixed in quantity andhomogeneous. The assumption of constant population does not affect ouranalysis in the sense that we can get similar results with a constant growthrate (because the two sectors exhibit constant returns to scale). There isonly one, malleable capital good. Capital depreciates at a constantexponential rate δk, which is independent of the manner of use.

Like in the Uzawa model, the two sectors use a single grade of labour anda single type of capital good. The two inputs are smoothly substitutable foreach other in each sector and are freely transferable from one sector to theother. Both exogenously determined labour supply and an irrevocablyexisting capital stock are inelastically offered for employment. Both sectorsuse neoclassical technology with the standard Inada conditions. Theproduction functions are given by

Fj (Kj (t),Nj (t)), j = i, s

where the subscripts i and s denote the capital good sector and theconsumption good sector, and Fj is the output of sector j, Kj and Nj arerespectively the capital and labour used in sector j. Assume Fj to beneoclassical. We have

The functions fj have the following properties:

(i) fj (0) = 0;(ii) fj are increasing, strictly concave on R+, and C2 on R++; fj

′(Kj ) > 0 and fj′′ (k) < 0;

and

(iii) lim and lim .

Markets are competitive; thus labour and capital earn their marginalproducts, and firms earn zero profits. We assume that the capital good servesas a medium of exchange and is taken as the numeraire. The price ofconsumption good is denoted by p(t). The rate of interest, r(t), and wagerate, w(t), are determined by competitive markets. Hence, for any individualfirm r(t) and w(t) are given at each point of time. The production sectorchooses the two variables, Kj (t) and Nj (t), to maximize its profit. The marginalconditions are given by

r + δk = fi′(ki ) = pfs

′(ks), w(t) = fi (ki ) – ki fi

′(ki ) = p(t)[ fs(ks) – ks fs′(ks)] (1)


Total capital stock, K(t), is allocated to the two sectors. As full employmentof labour and capital is assumed, we have

Ki (t) + Ks(t) = K (t), Ni (t) + Ns(t) = T (t)N0

where N0 (= 1)is the fixed population and T (t ) is work time of therepresentative household. We rewrite the above equations as

ni (t)ki (t) + ns(t)ks(t) = k(t), ni (t) + ns(t) = 1 (2)

where

In Uzawa’s model, a representative agent maximizes the present valueutility of the entire consumption stream, while a representative producermaximizes profit at any point of time. Utility levels at different points oftimes are additive over time. We propose an alternative to the Ramseyapproach to consumer behaviour by assuming that a representativeconsumer maximizes the utility at any point of time. Consumers makechoice of consumption levels of services and commodities, as well as onthe amount of saving in each period. In order to provide a proper descriptionof endogenous savings, we should know how individuals perceive thefuture. Different from the optimal growth theory in which utility definedover future consumption streams is used, we assume that we can find thepreference structure of consumers over leisure, consumption and saving inthe current state. The preference of the current and future is reflected bythe consumer’s preference structure over leisure, current consumption, andsaving.

To explain our approach to consumer behaviour, first define group j’s percapita current income y(t) from the interest payment (where )and the wage payment w(t) as follows

The sum of money that consumers are using for consuming or saving arenot necessarily equal to current income because consumers can sell wealthto pay for consumption if contemporary income is not sufficient forpurchasing goods and services. Retired people may live not only on theinterest payment but also have to spend some of their wealth. The totalvalue of wealth that a representative household can sell to purchase goodsand to save is equal to . We do not allow borrowing for currentconsumption. We assume that selling and buying wealth can be conductedinstantaneously without any transaction cost. This is evidently a strictconsumption as it may take time to draw savings or to sell one’s assets. Theper capita disposable balance of the household is defined as the sum of thecurrent income and the wealth available for purchasing consumption goodsand saving


(3)

The disposable balance is used for saving and consumption. At each pointof time, a consumer would distribute the total available budget betweensavings s(t) and consumption of goods c(t). The budget constraint is givenby

(4)

Equation (4) means that consumption and savings exhaust the consumers’disposable balance.

Denote Th(t) the leisure time at time t and the (fixed) available time for workand leisure by T0. The time constraint is expressed by

T(t) + Th(t) = T0

Substituting this function into the budget constraint yields

(5)

In our model, at each point of time, consumers have three variables to decide.We assume that utility level U(t) that the consumers obtain is dependent onthe leisure time, Th , the consumption level c(t) of commodity, and the savingss(t) as follows

(6)

where σ is called propensity to use leisure time, ξ, propensity to consume,and, λ propensity to own wealth. This utility function can be applied todifferent economic problems (see, Zhang, 1993, 1997). A detailedexplanation of the approach and its applications to different dynamicproblems are provided in Zhang (2005).

For consumers, the wage rate, w(t), and rate of interest, r(t), are given inmarkets and wealth, , is predetermined before decision-making. Eachperiod maximising U(t) in (6) subject to the budget constraint (5) yields

(7)

We now find dynamics of capital accumulation. According to the definitionof s(t), the change in the household’s wealth is given by

(8)

The output of the consumer goods sector is consumed by the households.That is

c(t)N0 = Fs(t) (9)


As output of the capital goods sector is equal to the depreciation of capitalstock and the net savings, we have

S(t) – K(t) + δkK(t) = Fi(t) (10)

where S(t) – K(t) + δkK(t) is the sum of the net savings and depreciation. It isstraightforward to show that this equation can be derived from the otherequations in the system. We have thus built the dynamic model. In theappendix, we show that the variable ks(t) is determined by the followingone-dimensional differential equation

(11)

where are functions of ks(t )explicitly defined in the appendix. From the appendix, we establish thefollowing lemma.

Lemma 1For a solution ks(t) of the differential equation (11), all the other variables areuniquely determined at any point of time by the following procedure: ki = Ω(ks) r and w by (1) k by (A6) ni and ns by (A2) T by (A4)

fj = fj (kj), j = i, s p = fi′/fs

′ Nj = njT Kj = kjNj Fj = fjNj

=kT by (A3) c and s by (7).

The above lemma guarantees that it is sufficient to analyse dynamicproperties of the differential equation (11) for describing the whole system.

3. Equilibrium and StabilityIt is straightforward to analyse dynamic properties of the one-dimensionaldifferential equation (11). But since it is difficult to interpret results, wefurther specify production functions for illustration. For explicitinterpretation, we specify production functions of the Cobb-Douglas form

(12)

We now analyse dynamic properties of the system under (12). From (A1),we directly obtain

ks = αki

where

The capital intensity of the consumer good sector is proportional to that ofthe capital good sector. By ks = αki and βi fi = βs pfs, we solve


(13)

The price of consumer goods is positively related to the technological levelof the capital good sector but negatively related to that of the consumergood sector. The price is positively or negatively related to the capitalintensity of the capital good sector, depending on the sign of α i – αs. Ifα i = αs, then the price is constant, p = Ai/As. In the remainder of this section,we require α i ≠ α s. The analysis of case α i = α s is straightforward.Corresponding to (A2), we have

(14)

The labour distribution between the two sectors is uniquely determined byk and ki. By the equation (A6), we solve

(15)

where

By the equations (14) and (15) and according to the definitions of A and A0,we find

(16)

Hence, for ni(t) to satisfy 1 > ni(t) > 0, it is sufficient to have

(17)

Take derivatives of the equation (15) with respect to t

(18)

From (A4), w = βi fi and ni = (αki – k)/ki(α – 1) we get

(19)

where we use (18) and


By (A2), we obtain

(20)

Insert = Tk into equation (8)

Substituting equations (18), (19), and (20) into the above equation yields

(21)

in which

We neglect the trivial solution of ki = 0. It is straightforward to show thatthe dynamic system has a unique (positive) equilibrium point given by

(22)

This guarantees a unique positive solution. The system is stable because

(23)

where we use the definition of A0 in judging the sign of the derivative at theequilibrium point.

In summary, we have the following proposition.

Proposition 1The dynamic system with the Cobb-Douglas production functions has aunique stable equilibrium.

4. Comparative Statics AnalysisWe can examine the impact of changes in parameter values on theequilibrium structure by standard comparative statics analysis as we haveguaranteed the existence of a unique equilibrium point. We now examineimpact of change in some parameters for the case where the productionfunctions are of Cobb-Douglas form. First, we examine technological changein the capital good sector. Taking derivatives of (22) with respect to Ai yields


As technology is improved, the capital intensity, ki, is increased. From ks = αki

and (15), we obtain

As the technology is improved, the capital intensity of the consumer goodsector increases as well. From the equations (14) and (15), we obtain

Hence dni/dAi = 0. The change in the technology has no impact on the labourdistribution. From (1)

We see that the rate of interest is not affected and the wage rate increases.From

we obtain dT/dAi = 0. The time distribution will not be affected by thetechnological change in the long term. By = kT, we have

The effects on the output levels in terms of per unit of work time are given by

We now examine impact of change in preference. As α + ξ + λ = 1, changein the propensity to use leisure has to be associated with change in otherpropensities. For simplicity, we specify the preference change as follows:dσ = – dλ and dξ = 0. Taking derivatives of (22) with respect to σ yields

As the propensity to use leisure increases and the propensity to savedeclines, the capital intensity of the capital sector increases in the long term.It is important to note that if dσ = – dξ and dλ = 0, then we have dk/dσ < 0.That is, if the propensity to use leisure time increases and the propensity toconsume declines, then capital intensity falls. As preference may change indifferent ways, its impact on ki is dependent on the specified pattern ofpreference change. In what follows, we limit our discussion to the case ofdσ = – dλ and dξ = 0. From ks = αki and (15), we obtain


As the sign of 1 – α is the same as that of α i – αs, we see that if α i ≥ αs, kdefinitely increases. Otherwise, the impact on k is ambiguous. We candirectly analyse the impact on the other variables.

5. Dynamic Processes by SimulationWe now simulate the model for illustrating dynamic processes. First, wespecify the parameter values as follows

α i = 0.35, αs = 0.3, δk = 0.05, λ = 0.50, σ = 0.25, Ai = 1.1, As = 0.9 (24)

We note that the productivity of the capital good sector is higher than thatof the consumption good sector. We simulate the motion of the economicsystem over 20 years with ki(0) = 2.8. The equilibrium value of ki

* is 3.589.

Figure 1a shows that the capital intensities in the two sectors and k increasesover time. Figure 1b shows the growth of the per worker output levels ofthe two sectors. The two sectors exhibit positive growth rates. Figure 1cdescribes the motion of the labour distribution and the output ratio betweenthe two sectors. It can be seen that the ratio asymptotically falls over thesimulation period. The labour participation rate in the capital good sectorof the total labour force also declines asymptotically over the simulationperiod. The consumption good sector absorbs more and more labour force.Figure 1d shows how the price, the real wage rate, and the rate of interestchange over time. The price of consumption good (in comparison to capitalgood) changes slowly over the period. The real wage rate increases; therate of interest falls during the simulation period. Figure 1e demonstratesthe current income per capita, consumption per capita, work time and percapita wealth. The work time declines and consumption increases. Figure1f depicts the dynamics of the shares of the two sectors in the GNP. Theshare of the capital good sector, denoted by yi ≡ Fi/(Fi + pFs) falls, and that ofthe consumption good sector, denoted by ys ≡ pFs/(Fi + pFs), rises. Thesimulation demonstrates that by the end of the study period, the systemachieves equilibrium.

Figure 1 Simulating the Two-Sector Model

a) capital intensity and per capita b) the per capita output levelswealth


c) the labour distribution and the d) the price, rate of interest, andoutput ratio wage rate

e) the income and consumption f) the share of the outputs in GNPper capita

We now examine impact of changes on dynamic processes of the system.First, we examine the case that all the parameters, except the propensity touse leisure, σ, are the same as in (24). We increase the propensity to useleisure from 0.25 to 0.30. The simulation results are demonstrated in figure2. The solid lines in figure 2 are the same as in figure 1, representing thevalues of the corresponding variables when σ = 0.25; the dashed lines infigure 2 represent the new values of the variables when σ = 0.30 with thepropensity to save, λ, fixed. As the propensity to use leisure rises and thepropensity to consume the consumer good falls, the capital intensities ofthe two sectors and wealth per capita per unit of time increase as shown infigure 1a. Figure 2b shows that the per capita output level of the consumersector falls and that of the capital good sector does not change in the longterm. The labour participation ratio in the capital good sector and the outputratio of the capital good sector and the consumption good sector increase,as illustrated in figure 2c. Figure 2d shows that the price of consumptiongood slightly increases, the wage rate increases, and the rate of interestfalls. From figure 2e, we observe that the work time and the consumptionlevel of consumer goods decline, per capita wealth and per capita currentincome increase. Figure 2f demonstrates that the share of output of theconsumer good sector in the GNP falls, and that of the capital good sectorrises.


Figure 2 As σσσσσ Increases from 0.25 (with the solid lines) to 0.30 (thedashed lines)

a) per worker inputs and per capita b) the per capita output levelswealth



We now examine the case that all the parameters, except Ai, are the same asin (24). We increase the productivity of the capital good sector from 1.1 to1.4. The simulation results are demonstrated in figure 3. The solid lines infigure 3 are the same as in figure 1, representing the values of thecorresponding variables when Ai = 1.1; the dashed lines in figure 3 representthe new values of the variables when Ai = 1.4. As the productivity rises, thecapital intensities of the two sectors and k increase as shown in figure 3a.Figure 3b shows that the per worker output levels of the two sectors rise.The labour participation ratio in the capital good sector and the outputratio of the capital good sector and the consumption good sector becomehigher in the initial stage; but remains the same as before in the same, asillustrated in figure 3c. Figure 3d shows that the price of consumption good


and the wage rate rise, and the rate of interest is not affected in the longterm. From figure 3e, we observe that the work time is not affected in thelong term, the consumption level, per capita current income, and per capitawealth rise as the productivity rises. Figure 3f demonstrates that the shareof output of the capital good sector in the GNP rises, and that of theconsumption good sector remains invariant in the long term.

Figure 3 As Ai Rises from 1.1 (with the solid lines) to 1.4 (the dashedlines)

a) per worker inputs and per capita b) the per capita output levelswealth



6. ConclusionsThis paper re-examined the Uzawa two-sector growth model with analternative utility function to the traditional Ramsey approach. We used autility function, which determines labour supply, saving and consumptionwith utility optimisation but without leading to a higher dimensional


dynamic system like the traditional approach. The dynamics are controlledby a one-dimensional differential equation. We showed that the system hasa unique, stable equilibrium when the production functions take on theCobb-Douglas form. We also simulated the model and demonstrated theeffects of changes in the propensity to save and in the technology of thecapital good sector. As mentioned in the introduction, Uzawa’s two-sectormodel has been generalized and extended in many directions. It is notdifficult to generalize our model along these lines. It would also be feasibleto analyse the behaviour of the model with other forms of production orutility functions. Other extensions could include multiple production sectorsand households that are not homogenous. Our model may exhibit nonlinearbehaviour if possible externalities, such as human capital accumulationthrough education, are considered.

Appendix 1

Proving Lemma 1We show that the motion of the system is determined by a differentialequation. First, from r + δk = fi

′(ki ) = pfs′(ks) in (1), we get

Substituting this equation into fi (ki ) – ki fi

′(ki ) = p( fs(ks) – ks fs′(ks)) in (1) yields

(A1)

From the properties of fi (ki ), we can show that the function Ψi

(ki ) has thefollowing properties:

The function Ψs (ks) has the same properties in ks. We see that for any given ks ≥ 0,

the equation (A1) determines ki ≥ 0 as a unique function of ks, denoted byki = Ω(ks). We have Ω(0) = 0 and Ω′0 > 0. From (2), we solve the labourdistribution as functions of, k, ks and kias follows

(A2)

The labour distribution is uniquely determined by k and ks. According tothe definitions of S, K, s and k, we have

S(t) – δK(t) = (s(t) – δ (t))N0

where δ ≡ 1 – δk. From this equation, the equation (10), and s = λ , we obtain

(A3)


From (A3), wT0 – wT = σ , and = kT, we solve

(A4)

where w(ki ) = fi (ki ) – ki fi′(ki ). We see that for any given positive ki and k, the

equation uniquely determines the work time. As 0 < T < T0, we alwayshave: 0 < Th (= T0 – T) < T0.

From pc = ξy, c = nsTfs, and p = fi′/fs, we get = Tns fs fi

′/ξ fs′. From this equation

and (A3), we have

ns fi′fi

∗ = ni fi + δk (A5)

where we use = kT and f ∗ ks ≡ λfs/ξ fs

′. Substituting ni = 1 – ns and ns in (A2)into equation (A5) yields

(A6)

where ki = Ω(ks). This equation guarantees that for any given ks(t), k(t)isuniquely determined. From = kT, we have

(A7)

From equations (A6), we obtain

(A8)

where

in which f ∗ ′(ks) = ( fs

′/fs – fs′′/fs

′) f ∗ > 0 for any ks > 0. From equation (A4), we

obtain

(A9)

where we use

and equation (A8). From the definition of , we have

(A10)


where f ′, w, k and T are functions of ks as shown before. Combining equations(A10), (A7), (A8), (A9), we obtain

(A11)

The differential equation (11) contains a single variable ks(t). For given initialconditions, the equation has solutions. We thus have proved Lemma 1.

ReferencesBarro, R.J. and Sala-i-Martin, X. (1995), Economic Growth, McGraw-Hill, New

York.Becker, G.S. (1965), ‘A Theory of Allocation of Time’, Economic Journal, 75,

493-517.Burmeister, E. and Dobell, A.R. (1970), Mathematical Theories of Economic

Growth, Collier Macmillan Publishers, London.Diamond, P.A. (1965), ‘Disembodied Technical Change in a Two-Sector

Model’, Review of Economic Studies, 32, 161-68.Drugeon, J.P. and Venditti, A. (2001), ‘Intersectoral External Effects,

Multiplicities & Indeterminacies’, Journal of Economic Dynamics andControl, 25, 765-787.

Gram, H.G. (1976), ‘Two-Sector Models in the Theory of Capital andGrowth’, The American Economic Review, 66, 891-903.

Mino, K. (1996), ‘Analysis of a Two-Sector Model of Endogenous Growthwith Capital Income Taxation’, International Economic Review, 37, 227-251.

Ramsey, F. (1928), ‘A Mathematical Theory of Saving’, Economic Journal, 38,543-559.

Solow, R.M. (1962), ‘Note on Uzawa’s Two-Sector Model of EconomicGrowth’, Review of Economic Studies, 29.

Stiglitz, J.E. (1967), ‘A Two Sector Two Class Model of Economic Growth’,Review of Economic Studies’, 34, 227-238.

Uzawa, H. (1961), ‘On A Two-Sector Model of Economic Growth’, Review ofEconomic Studies, 29, 47-70.

Zhang, W.B. (1993), ‘Woman’s Labor Participation and Economic Growth -Creativity, Knowledge Utilization and Family Preference’, EconomicsLetters, 42, 105-110.

Zhang, W.B. (1997), ‘A Two-Region Model with Endogenous Capital andKnowledge - Locational Amenities and Preferences’, InternationalReview of Economics and Finance, 6, 1-16.

Zhang, W.B. (2005), Economic Growth Theory, Ashgate, London.

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A Two-Sector Growth Model with Labour Supply · Zhang: A Two-Sector Growth Model with Labour Supply 249 (3) The disposable balance is used for saving and consumption. At each point