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The influence of the healthcare system on optimal economic growth
Economic Modelling 35 (2013) 734–742
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
The influence of the healthcare system on optimal economic growth Maurício Assuero Lima de Freitas a,⁎, Alexandre Stamford da Silva b a b
Department of Accounting and Actuarial Sciences, Federal University of Pernambuco, Brazil Department of Economy, Federal University of Pernambuco, Brazil
a r t i c l e
i n f o
Article history: Accepted 16 August 2013 Keywords: Healthcare system Optimal economic growth Healthcare sector effort function Macroeconomic models for the healthcare sector
a b s t r a c t This paper analyzes the impact of health system in the economic growth, based upon three macroeconomic models. The first one considers the economy with only one sector, but with morbidity; in the others the economy is divided in two sectors, the productive sector and the health sector, considering it intensive in labor and after intensive in capital. The results show that the presence of the health system increases the life expectancy and the aggregate product, but does not modify the per capita product. © 2013 Elsevier B.V. All rights reserved.
1. Introduction This paper presents an introductory study of the macroeconomic impacts of the inclusion of the healthcare sector in the neoclassical model of optimal economic growth. Studies in economics regarding the healthcare sector – usually referred to as the economics of health – are well established in the literature (Arrow, 1963; Grossman, 1972). Some studies deal with the issue from a microeconomic standpoint, focusing more intensively on the public sector (Blomqvist and Léger, 2005; Østerdal, 2005; Buchumuller, 2006). Others discuss the importance of health for economic growth (Aguayo-Rico, 2005; Ainsworth and Over, 1993; Bloom and Canning, 2005; Howitt, 2005; Muskhin, 1964; Sorkin, 1997; Taylor and Hall, 1967; Winslow, 1951), but they tend to be qualitative analyses without a well-established theoretical basis, that is, a basic mathematical model. Besides, most studies deal with health in the context of the practice of healthy habits, diet, leisure, education, etc. (Schultz, 1991; Becker, 1962; Bloom et al. 2004), and not with the healthcare sector as an intermediary productive sector that deploys technology, capital, and labor with impacts in the recovery of the workforce. In order to attempt to measure the effects of the healthcare sector on economic growth it is necessary to adequately insert such sector in a theoretical model capable of predicting the impacts caused by its presence. The empirical aspect has already been adequately addressed by Acemoglu and Johnson (2007). Thus, the goal of the present paper is to evaluate what is the adequate policy for the allocation of resources for the optimal economic
⁎ Corresponding author at: Av. Dos Economistas, S/N, Cidade Universitária, Recife, Pernambuco 50740-290, Brazil. Tel.: +55 81 2126 7739; fax: 55 81 2126 8369. 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.08.023
growth given the explicit presence of the healthcare sector in the economy. It also attempts to answer the question of whether there are economic advantages for society in opting for the healthcare sector in its economy. The main contribution is to discuss the optimal economic growth in an economy not only with the existence of morbidity, but also with a healthcare sector, that needs to solve the problem of the tradeoffs between capital and labor for production versus capital and labor for the recovery of the sick or ailed workforce. This article was divided into four sections beyond this introduction. The first one presents the model of an economy with a single sector – the productive sector – in the presence of morbidity which affects labor by reducing the workforce, that is to say, the amount of hours available for production. The second section is dedicated to the explanation and to the empirical measures of the recovery effort function, which is the function that gives the production of the healthcare sector in terms of the recovery of the hours of work available for production that were subtracted due to the existence of morbidity (representing the activity of recovering the sick or ailed workforce and returning it to the productive sector). The third one is where the economy is divided into two sectors – productive and healthcare – and where, for simplicity of analysis, two types of healthcare sector are considered, one being labor-intensive and the other capital-intensive. Finally, in the fourth section the main conclusions and some comments are presented. 2. The economy with a single sector in the presence of morbidity in labor The model developed in this section considers a simple, closed, economy, with constant scale earnings in the production, without
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
government or healthcare sector, in which the standard hypothesis of the neoclassical models is valid. The social planner's objective is: Z max C
∞ 0
−ρt
U ðC Þe
subject to ˙ I ¼ Y−C; K ð0Þ ¼ K 0 N0 K¼ L˙ ¼ nL−M ¼ ðn−mÞL; Lð0Þ ¼ L0 N0 ˙ gA⇒A ¼ A0 egt ; Að0Þ ¼ A0 N0 A¼ α β Y ¼ F ðK; LÞ ¼ AK L
(5), and (6) were highlighted because they contain interpretations that are absent in the standard models. λ βY μ˙ þ m−n þ ρ: ¼− μ L μ
dt; ρN0;
ð1Þ
L˙ ¼ GðLÞ−M
ð2Þ
where the G(L) function represents the immunological recovery rate and/or the population reproduction rate.2 To make the model simpler, it will be admitted that G(L) = nL, resulting in equation L˙ of the model (1). It is considered that the morbidity is homogeneous, in other words, that it acts indiscriminately upon all people, though it is a fact that older people have a greater tendency towards ailments. This hypothesis of distinct rates of rates of affliction according to age or other population groupings shall be considered in future studies. The utility function, as assumed by Barbier (1999), Groth (2002), and Márquez and Ruiz-Tamarit (2005), will be given by: U ðC Þ ¼
C 1−ε −1 : 1−ε
ð3Þ
The results of the model are classic and the deduction can be found in Appendix A. Only the morbidity is highlighted in the growth equations, without any difference from the standard neoclassical model. Eqs. (4),
1 It should be realized that, alternatively, any other population growth model, including: Gompertz (1825), Smith (1963), Goel, Maitra and Montroll (1971), Ayala, Ehrenfel and Gilpin(1973), all cited by Bassanezi and Jr (1998) – depending on the interest – because the factor of notice is the growth of the population and not the form of that growth; in other words: in the literature the emphasis is on the value of the growth rate and not the pattern. 2 Elíasson and Turnosky (2004) address renewable resources by considering that the natural growth rate of the renewable resource depends on a second degree logistic func tion that is given by S˙ ¼ GðSÞ−X, where, GðSÞ ¼ rS 1−S=S ; r N 0, and X is the rate of harvest. They consider that the resource is forest or fish. The source of this model is Verhulst (1838). The same function could be used here, but the choice was made to use the classic population growth function.
ð4Þ
Eq. (4) represents the growth rate of the contribution of labor to the optimal social well-being, or how much one would pay to have an extra hour of work, or yet, the rate of evolution for the wages, which depends on the relative process of capital and work, and on the rates of impatience and morbidity. In the BGP — Balanced Growth Path, the growth rate of the product is given by: g Y ¼ ðn−mÞ þ
where U(.) is the utility function, ρ is the society's impatience rate, t is the time, K is the capital, I is the investment, Y is the product of the economy, C is the consumption, L is the workforce, and all for an instant t, is omitted due to presentation issues. Also, n is the population's rate of growth, g is the technology rate growth, M = mL is the morbidity, with m being the given rate of morbidity, F(.) is the production function, A is the technology, α and β are the parameters of the production function, and K0, L0 and A0 are the capital, labor and technology stocks that are assumed to be given. The equations in problem (1) regarding capital, technology, and production are classic and well-established in the literature so will not be commented upon. The equation for the movement of labor,L˙, establishes that the rate of variation in labor is reduced as the morbidity rate grows, and grows with the growth in labor.1 Note that one can admit the rate of variation in the workforce as, more generically, increasing with the natural growth of the population or with the recovery by the “immune system”, and diminishing with the morbidity rate, that is:
735
g : β
ð5Þ
A sufficient condition so that g∗Y N 0 is n ≥ m. The sick people would be replaced by new births and the population would grow exponentially at the rate of (n − m). If n b m, then the rate of technology of the productive system should withstand the reduction in the quantity of labor or in the hours available for work, and be such that g N β(m − n). Now, considering the rate of per capita consumption c ¼ CL , one has that:
g c ¼ g C −g L ¼ g Y −g L ¼
g g ¼ : β 1−α
ð6Þ
Therefore, g∗c depends on the rate of technology and on the elasticity of labor (or capital), and is independent of the morbidity rate. Stiglitz (1974a,b), dealing with growth with exhaustible natural resources, found that the per capita rate of consumption converges to α 2 nþλ ð1−α 1 Þðα 1 þα 2 Þ, where λ, α1, and α2, represent the technological growth and the elasticities respectively. Note that, under the hypothesis of a constant return in scale and a null population growth, this is the same rate determined in Eq. (6). Thus, there is an analogy between the growth models in an economy where the morbidity is highlighted and the growth models in the presence of exhaustible resources. One observes here that the economy is viable even in the absence of the healthcare sector. Considering the population growth rate as null, it is evident that fewer and fewer people participate in the economy and that these people will have an increasingly smaller life expectancy until exhaustion. The probability that a given generation will know the next will decrease as time goes on. One can speculate that, in this case, the quality of life is low. Such speculation is valid even for positive population growth rates. 3. The healthcare sector effort function The introduction of the healthcare sector in the economy obliges one to deal now with two sectors: the productive sector, of end goods, and the healthcare sector, an intermediary one, which acts as a system that recovers labor that is ill. The recovery of the health state of the workers or of the labor force has two characteristics. One is renewable, given by the immune system, and the other is recoverable, given by the effort in the treatment of health. In the present paper, we will deal only with the recoverable portion of this resource, so that the recovery of labor is done exclusively by means of a production effort involving capital, labor, and technology. The implicit hypothesis is that the allocation of these production factors in the effort function influences in the recovery of the workers, or of the hours of work, and that the healthcare system works so as to repair and not to prevent. The recoverable part of the healthcare sector is, therefore, represented by a production function that depends on capital KS, labor LS, and technology (B), given by YS = F(B,KS,LS,M), where M is the number of ill workers or the morbidity mL, a necessary input. Regardless of the
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analytical form of the production function, the end product obtained by the healthcare sector can be expressed as YS = bmL, where m and b are real in the unit interval, m being the morbidity as previously defined and b the percentage of ill individuals recovered. The fraction of the population that is ill in time (t − 1) is mL and bmL is the fraction of the ill population that was recovered by the health sector in time t. Since the ill, mL, a priori do not depend on capital and labor, then, without loss of generality, one can assume that b is a measure of the rate of recovery of the healthcare sector that depends only on the percentage of capital and labor allocated to the sector. Therefore, parameter b will be an index of the labor recovery rate due to the allocation of capital and/or labor to the healthcare sector. Mathematically, one can write b = BG(aK,aL) with B indicating the level of technology of the healthcare sector, aK and aL the fractions of capital and labor allocated to it. It is considered that the effort is limited, that is, 0 ≤ b = BM(aK,aL) ≤ 1 and that the Inada conditions (Romer, 1996, p.19) are satisfied. Intuitively, the effort of the healthcare system, represented by the effort function b = BM(aK,aL), is the cause of the recovery of ill individuals, that is to say, people are recovered and returned to the productive system because of the existence of this social effort, represented by the allocation of labor and capital. Since only a fraction of the population, mL, gets ill, then only a portion of that fraction, bmL, can be recovered, and that fraction is exactly the end product, or product, of the healthcare system. The labor's movement equation can be written, thus, as: L˙ ¼ nL−mL þ bmL ¼ ½n−mð1−bÞ:
ð7Þ
This equation can be used so as to empirically measure the value of b. This parameter is an alternative for the comparison of the absolute efficiency of the health sectors of distinct economies. Different from the DEA method (Data Envelopment Analyses: Akazili et al. 2008; AlShammari, 1999; Salinas-Jimenez and Smith, 1996; Sherman, 1984; Tsaprounis, 1997), the measure here is not relative to the content of sample data, but rather measured on an absolute scale, which can greatly improve such comparisons. A possible procedure to have access to a measure of parameter b is described as follows. The basic idea is that the growth rates of the population and of the morbidity are available in various official databases, but the recovery rate of the healthcare sector (b) is not. However, a proxy of the rate of individuals not recovered by the healthcare system, N, can be obtained by the sum of the data on deaths due to disease, retirements due to disability, retirement due to labor accidents, and other similar data that is generally available, measure along a certain period. By definition, N = (1 − b)m, and through simple algebra one can calculate that b ¼ 1−mN . Returning to the theoretical model, the mathematical treatment of a complete effort function, with technology, fraction of capital, and fraction of labor does not present results that are easy to interpret. For this reason, the choice was made to use an effort function initially dependent only on the fraction of labor allocated and then only on the fraction of capital allocated. In the first case one can assume that the amount of capital allocated to the healthcare sector is not large enough for its impacts to be taken into consideration in the recovery effort, thus, the approximation is done by the exclusion of the fraction of the capital. It is as if this worker recovery system was based on the clinic and things of the sort. Whatever the case, the main assumption is that there is impact in the recovery when these clinicians or healers are allocated to the healthcare sector. In the case of the dependency only on the fraction of the capital allocated, the allocated work is what does not participate in a relevant way in the recovery, so that the effort function is dependent only on the fraction of capital. It is the instrumental medicine whose treatments and
diagnosis are done and made available by machines where human participation is minimal. 4. Models with the healthcare sector As previously said, the insertion of the healthcare sector in the economy will be done in two ways: the first is considering a labor-intensive healthcare sector and the second assuming a capital-intensive one. 4.1. Labor-based effort function In the model without a healthcare system (Section 1), all of the population was available to work in production, the morbidity rate being similar to an extraction rate for labor that affects the workforce, and, in the long run, the labor would exponentially tend to exhaustion if there was no population growth (or when the morbidity rate is greater than the population's rate of growth). Now, the fractions aL and 1 − aL of the population are allocated to the healthcare and productive sectors respectively, as done in Aghion and Howitt (1992), Romer (1996, p. 97), and Stamford da Silva (2008), among others, that treated growth with models from two sectors. What one wants to verify is whether this reduction in the workforce allocated to production in favor of the allocation of some relevant effort for the recovery of the health of the workforce is advantageous for the economy as a whole. This is done comparing the economic system that makes the choice for that allocation with the economic system that does not go for that option, as presented in Section 1. The effort function of the healthcare system, based only on labor, is given by b = BaL. The existence of some technology (B) is considered in the healthcare system, such as techniques from traditional medicine, new techniques for well-established curative procedures, curative methods in general, or even some tool with fixed technology. It is considered that there is perfect mobility of labor between the two sectors, that is to say, each worker is capable of executing tasks in both sectors with equal efficiency. If B = 1, the healthcare system recovers exactly the same percentage of labor as that which was allocated to the healthcare sector. For instance, if b = aL = 0.20 only 20% of the ill return to the productive system. This would represent a highly inefficient healthcare sector. The goal of the social planner is the same as before, except for the equations: max C; aL
Z
∞ 0
−ρt
U ðC Þe
dt; ρN0
subject to ˙ I ¼ Y−C; K ð0Þ ¼ K 0 N0 K¼ L˙ ¼ nL−mL þ bmL ¼ ½mðBaL −1Þ þ nL; Lð0Þ ¼ L0 N0 α β Y ¼ F ðK; LÞ ¼ AK ½ð1−aL ÞL :
ð8Þ
Appendix B brings a solution for this model. The wage rate is given by: μ˙ ¼ ρ−mðB−1Þ−n ¼ ρ−mB þ m−n: μ
ð9Þ
The equation above shows the joint effect of the morbidity, of the healthcare system's level of technology, and of the population growth rate on the wage rate. The wage rate increases if the morbidity increases because it reduces the workforce (fewer hours available for work) while the population growth rate and the effect of the healthcare system's technology on the ill reduces the wage rate because they are determinants of the population increase. One notes the absence of the marginal productivity of labor in the equation.
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
If Eqs. (4) and (9) are considered the same, one obtains: βY μ ¼ F L ¼ mB: L λ
ð10Þ
Eq. (10) shows that, for the systems to be equivalent, the marginal productivity of labor is proportional to the amount of workers recovered. In other words, the marginal contribution of the worker to the product must equal the relative cost of the worker times the impact of the allocation on the growth rate of labor. The growth rate of the product is given by: gY
g βmðB−1Þ β β þ n− g B : ¼ þ αz þ α α α α
Y
B
gY
1 g ðn−mÞ þ þ mB−ρ : ¼ ε β
1 ð1−εÞ½g þ βmðB−1Þ þ n ρ− : mBε β
ð12Þ
ð13Þ
∗ L
Therefore, ∂a∂g ¼ 1−ε mBε N0, that is, an increase in the technology of the productive sector generates an increase in the healthcare sector, as expected. For it to be viable, the fraction of labor allocated must obey to the following inequality: 0b
1 ð1−εÞ½g þ βmðB−1Þ þ n ρ− b1 mBε β
that is, β
1 g þ mðB−1Þ þ n α β
ð14Þ
1 α g ρ 1− þ mðB−1Þ þ n þ α ε β ε
ð15Þ
x ¼
aL ¼ 1−
h ρ i ρ−mBε −mðB−1Þ−n b gb β −mðB−1Þ−n : 1−ε 1−ε
Thus, so that the health sector exists the technology of the productive sector stays limited. The amplitude of this interval, ðβmBε 1−εÞ , does not
1 ð1−εÞ½g þ βmðB−1Þ þ n ρ− : mBε β
ð16Þ
The per capita consumption rate, g∗c = g∗C − g∗L, is given by
gc ¼
1 g þ mðB−1Þ þ n−ρ − m BaL −1 þ n : ε β
ð17Þ
Replacing a∗L results in:
4.1.1. Dynamics Besides analyzing the dynamics of the variables z ¼ KY , product per unit of capital, and x ¼ KC, consumption per unit of capital, it is necessary to analyze the labor allocation policy, by means of the dynamics of aL = aLL / L = aL. Appendix B shows that the optimal fraction of labor allocated to the healthcare sector is:
z ¼
gc ¼
One observes, first of all, the dependency of the long term growth rate on the inverse of the elasticity of the marginal utility of the consumption, ε, the social option. One also notes the negative presence of the intertemporal impatience rate, ρ, and the positive presence of the level of technology of the healthcare sector, B, or of the variation of the growth rate of the workforce in relation to the percentage of workers allocated, mB. None of these dependencies occurs without the presence of the healthcare sector, as can be seen by comparison with Eq. (5).
aL ¼ 1−
depend on the growth of the population, but captures the effect of the level of technology of the healthcare sector. The greater B is, the wider will be the technology interval allowed for the production sector. One observes again the presence of the marginal utility of the consumption highlighting its fundamental role in the growth when the healthcare sector is present. The dynamics, shown in Appendix B, shows the existence of an economically viable point given by (z∗,x∗,a∗L) formed by the optimal amounts of each variable:
ð11Þ
This expression shows that, in this model, the existence of a fixed level of technology in the healthcare sector contributes to the increase ¼ −αβb0 , of the product, but that technology cannot grow, for ∂g ∂g compromising the growth of the product. This is an important result. It says that a growing rate in the level of technology of the healthcare sector can make the economy unviable because the variation of the variation of the workforce (the second derivative) is constant, indicating that the workforce will grow more than exponentially and there will be no product for all to consume in the long run. It will be considered here that that the technology of the healthcare system is fixed at a certain level B so that gB = 0. From this results that, in the BGP, the growth rate of the product is:
737
g g ¼ : β 1−α
ð18Þ
The algebraic cancelations done in Eq. (16) that resulted in Eq. (17) have economic meanings that are worthy of greater investigation. For example: when the population increases, the product per unit of capital grows by a factor 1ε . But the consumption per unit of capital also increases due to the increase in the population and acts as a reduction factor of the product in exactly the same proportion, making the growth rate of the consumption per capita independent of the population growth rate. On the other hand, the fact that there is no change in the individual wealth with the insertion of a healthcare system may contradict the commonsense. However, Acemoglu and Johnson (2007), in an empirical work dealing with health interventions worldwide from 1940 till date, showing, by means of regression analysis, that such interventions (that is, the action of the healthcare sector) increased the life expectancy of the people and that: There is no evidence that the increase in life expectancy led to faster growth of income per capita. This evidence sheds considerable doubt on the view that health has a first-order impact on economic growth (Acemoglu and Johnson, 2007). So, the main results of the model discussed in this section are in consonance with the results from Acemoglu and Johnson (2007), when it shows that the healthcare system increases the life expectancy of the people, by means of the inclusion of the system as an element of recovery in the labor equation, but that this does not affect individual wealth. Indeed, considering that Y = yL and given that y does not vary in the models, the healthcare sector generates a greater L, therefore, a greater product. With this, society will have a greater well-being, considering the aggregated product and the fact that generations meet due to the increase in life expectancy as given by the equation of the evolution of labor. 4.2. Effort function based on capital In this section, the healthcare sector will be assumed to be capital intensive. The effort function, discussed in Section 2, maintains its
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previously mentioned assumptions, and will be given by 0 ≤ b = BaK ≤ 1. The goal of the social planner is: max C; aK
Z
∞ 0
−ρt
U ðC Þe
Simple algebraic manipulations show that
ð19Þ
The resolution of the model can be seen in Appendix C. The movement equation for the wage rate is: ðα−βÞð1−BaK Þ−βðB−1Þ μ˙ −n: ¼ρþm α μ
ð20Þ
As one can see, the rate of variation of the worker's shadow price diminishes with the increase of both in the technology of the healthcare sector as of the fraction of capital allocated to this sector, as expected. If it happens that α = β Eqs. (20) and (9) become equal and present the same results. The resolution of the model shows, however, that α N β. The growth rate of the product is given by: g α α þ αz− x þ mðB−1Þ þ n− g B β β β
ð21Þ
where one again observes that the growth rate of the product diminished if there is growth in the technology of the healthcare system. It is important to note that, in this model, the impact of the growth rate of the technology of the healthcare system is elastic. Thus, whether the effort of the healthcare system is based in the allocation of labor or on the allocation of capital, the rate of variation of the technology of the healthcare system cannot grow indefinitely, under the penalty of the economy becoming unviable, with not enough products for the amount of workers recovered. Once again, it will be considered that B is constant, that is, the amount of the technology is given. With that, gB = 0. To define the allocation policy for the capital, one must evaluate the dynamics of aK. As may be seen in Appendix C, from the movement equation of the fraction of capital, one has that
α βρ−ð1−εÞ½g þ βðmB−m þ nÞ : mβB β−α ð1−εÞ
ð22Þ
For the healthcare system to have a viable size, it must have: α g þ βðmB−m þ n βρ−ð1−εÞ b1: mβB β−α ð1−ε Þ
0b
Determining the value of in the inequality above, one obtains: β
h ρ i ρ mBðβ−α ð1−εÞÞ −½mðB−1Þ þ n− bgb −½mðB−1Þ þ n : 1−ε α ð1−εÞ 1−ε
ð1−εÞÞ The amplitude of this interval is given by βBðβ−α which, as obα ð1−ε Þ served, is not influenced by the growth rate of the population. Now, using the movement equations for the variables, z, x, and aK, one forms a system of differential equations in which some of the equilibrium points do not have an economic interpretation. In Appendix C, the classification of these points is made. The growth rate of the per capita consumption is given by
gc ¼
˙ I ¼ Y−C; K ð0Þ ¼ K 0 N0 K¼ L˙ ¼ nL−mL þ bmL ¼ ½mðBaK −1Þ þ nL; Lð0Þ ¼ L0 N0 α β Y ¼ F ðK; LÞ ¼ A½ð1−aK ÞK L :
aK ¼ 1−
g c ¼ g C þ ½m−mBaK −n:
dt; ρ N0
subject to
gY ¼
that is,
g c ¼ g C −g L ¼ g C −½mðBaK −1Þ þ n
g g ¼ : β 1−α
ð23Þ
This rate does not differ from the results obtained in the previous models. Mariani, Pérez-Barahona and Raffin (2010) had found that a correlation positive enters the life expectancy and the ambient quality, considering “an infinite-horizon economy that is populated by overlapping generations of agents living for three periods: childhood, adulthood, and old age”. The results show that “environmental quality depends on life expectancy, since agents who expect to live longer have a stronger concern for the future and therefore invest more in environmental care and longevity is affected by environmental conditions”. On the other hand, Blackburn et al. (2002) had considered, too, a model, with overlapping generations and had arrived at a result that shows the effect of the changes in the life expectancy on the way of the development of the economy. They had been emphatic in studying the impact of the increase of the life expectancy on the economy, while our work suggests that it will have greater welfare in the economy due to the presence of the health system, but the health system, does not cause impact on the per capita income. 5. Analysis, results and final considerations The model in which the economy has only one sector brings results similar to the economic growth models with exhaustible resources. In this model, it was found that the existence of a BGP is assured as long as it is true that n N m or, if that does not happen, it is necessary that g N β(m − n). The insertion of the healthcare sector immediately modifies the labor movement equation that now begins to decay into a smaller rate, that is, the presence of the healthcare sector increases people's life expectancy, but this increase in life expectancy does not change g the growth rate of the per capita consumption, g ∗ ¼ 1−α , that depends, with or without the healthcare sector, on the rate of technology of the productive system and on the elasticity of the product in relation to the capital (or of the product in relation to the labor, given that 1 − α = β). This result does not change whether the effort function of the healthcare sector is labor-intensive or capital-intensive, and this can suggest some level of inoperance for the healthcare sector, however, the theoretical result obtained here is corroborated by the empirical results found by Acemoglu and Johnson (2007) who studied the impact of illness on economic development and, using a regression analysis, showed that there is no evidence that the increase in life expectancy, the main result of the healthcare sector, had any influence over the growth in per capita income. One of the results of the work is to highlight that the healthcare sector increases society's well-being because the aggregated product is greater than the aggregate product of an economy with morbidity and without a healthcare system. Therefore, one obtains a result contrary to that of Thomas Malthus (1798), who advocated the positive control of the population suggesting the construction of houses close to mangroves and the narrowing of streets in order to provoke the return of plagues, saying that: … but above all we must repudiate specific medicine for overwhelming diseases …. Ehrlich and Lui (1997), referring to Malthus, said: “Malthus dramatized the idea by identifying population as potentially detrimental to growth. Since that time, work on growth and development has been inextricably linked with population economics”. The authors trace the evolution of the literature on population growth until the recent
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
endogenous growth theory and development and propose a model of dynasty. The results are important, but the approach is different from ours. Here we show that the health system as a productive sector of the economy, has no impact on the increase in per capita income, but it is of fundamental importance in social welfare because it contributes to the increase in life expectancy of people. The models presented here show that Malthus was wrong, for, even if the healthcare sector approaches the effect of snake oil, it brings positive results for the economic system. The present work does not deal with an effort function based simultaneously on capital and labor, due to the initial difficulty in expressing a functional form for the healthcare sector that can be both of the Cobb– Douglas and of the fixed proportions type, allowing for some level of substitution between factors. It also does not consider the randomness of the illness, a fact that would force an analysis in light of dynamic programming. Finally, the endogenization of technology is not done, in spite of the importance of technology for the existence of BGP.
739
Differentiating Eq. (24) logarithmically and using Eq. (25), we get: gC ¼
αz ρ − : ε ε
ð34Þ
And now, substituting Eqs. (28), (31) and (34) in Eq. (27), following: z˙ ¼ −ð1−α Þz þ ð1−α Þx þ βðn−mÞ þ g z
ð35Þ
α ρ x˙ −1 z þ x− : ¼ ε ε x
ð36Þ
We'll assume an interior break-even point, to form the system: (
−ð1−α Þz þ ð1−α Þx þ βðρn−mÞ þ g α −1 z þ x − ε ε
¼0 ¼0
:
ð37Þ
Appendix A Whose solution is given for The Hamiltonian of the problem (1) is: H¼
C
−1 þ λðY−C Þ þ μ ðn−mÞL: 1−ε
1 εg ρ þ εðn−mÞ þ α β
ð38Þ
1 εg g ρ þ εðn−mÞ þ − ðn−mÞ þ : α β β
ð39Þ
ð24Þ x ¼
The condition for an interior solution is: ∂H −ε −ε ¼ C −λ ¼ 0⇒C ¼ λ: ∂C
z ¼
1−ε
ð25Þ
Linearing Eqs. (35) and (36) in the equilibrium point, we find the Jacobian, J z ;x ¼
The equations of motion of the costate variables are given for: ˙ − λ¼
∂H λαY Y λ˙ þ ρλ⇒ ¼ ρ−α ¼− K K λ ∂K
ð26Þ
μ˙ ¼ −
∂H λβY þ ðn−mÞμ þ ρμ: ¼− L ∂L
ð27Þ
"
Let g H ¼ we would want to know about the dynamics of the variables: Y z¼ K C x¼ : K
#
x
g Y ¼ ðn−mÞ þ
g β
ð40Þ
implying ð28Þ
g c ¼ g C −g L ¼ g Y −g L ¼
g g : ¼ β 1−α
ð41Þ
Appendix B
By the definitions: z˙ ¼ g Y −g K z x˙ ¼ g C −gK : x
βz
∗ ∗ whose determinant is given for jJj ¼ −αβ ε z x b0, characterizing a saddle point and the presence of feasible politics to the optimum growth. From Eq. (28) following
H˙ H the rate relative growth of the variable H. In the long-term
α−βz −1 z ε
The Hamiltonian of the problem (8) is given for: ð29Þ H¼
˙ Y−C , L˙ ¼ ðn−mÞL and Y = F(K,L) = AKαLβ, we have, From K¼ respectively g K ¼ z−x
ð30Þ
g L ¼ ðn−mÞ
ð31Þ
g Y ¼ g þ αg K þ βg L :
ð32Þ
C 1−ε −1 þ λðY−C Þ þ μ ½mðBaL −1Þ þ nL: 1−ε
ð42Þ
The conditions for an interior solution are given for: ∂H −ε −ε ¼ C −λ ¼ 0⇒C ¼ λ ∂C
ð43Þ
∂H λβY λβY ¼ ð1−aL ÞμmB: ¼− þ μLmB ¼ 0⇒ 1−aL L ∂aL
ð44Þ
The equations of motion of the costate variables are given for:
Substituting Eqs. (28) and (29) in Eq. (30), following: g Y ¼ g þ α ðz−xÞ þ βðn−mÞ:
ð33Þ
˙ − λ¼
∂H λαY Y λ˙ þ ρλ⇒ ¼ ρ−α ¼− K K λ ∂K
ð45Þ
740
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
μ˙ ¼ −
∂H λβY þ μ ½mðBaL −1Þ þ n þ ρμ: ¼− L ∂L
ð46Þ
necessary to discover if it has some feasible way that has taken the economy to a BGP. Rewriting Eq. (51) in a BGP, with gB = 0, as
Substituting Eq. (44) in Eq. (46), and dividing both sides of Eq. (46) for μ, we get:
αz −g Y ¼ x −
μ˙ ¼ ρ þ m−mB−n: μ
and substituting this results in Eq. (50),with gB = 0, we have:
ð47Þ
The rate growth of the product is obtained by the logarithmic differentiation, with respect to time, of the equation Y = F(K,L) = AKα[(1 − aL)L]β, that is, g Y ¼ g þ αg K þ βg L −
βaL g L: 1−aL a
ð48Þ
But, to determine Eq. (48) it is necessary to know the rate of allocation of the labor. Differentiating logarithmically, with respect to time, Eq. (44) we get: λ˙ a˙ μ˙ þ gY þ L ¼ þ gL þ gB : 1−aL μ λ
g βmðB−1Þ β β þ αz−x þ þ n− g B : α α α α
8 > > z˙ > > > < x˙ > > > > > : a˙L
g mðB−1Þ n þ þ : αβ α α
α g mðB−1Þ n ρ þ þ þ : x ¼ 1− ε αβ α α ε
0b ð50Þ
¼
ð58Þ
1 ð1−εÞ½g þ βðmðB−1Þ þ nÞ ρ− : mBε β
ð59Þ
1 ð1−εÞ½g þ βðmðB−1Þ þ nÞ ρ− b1 mBε β
that is,
ð51Þ
β
ρ ρ−mBε −½mðB−1Þ þ n bgbβ −½mðB−1Þ þ n : 1−ε 1−ε
The linear approximation of Eq. (58) is, of a generalized form, given for: 2 2
3
6 z 6 4 x 5 ¼ 6 6 6 4 a˙L
g þ βmðB−1Þ þ βn −2βz þ α alpha x − 1− ε 0
ð52Þ
3 0
0
x
0
g þ βmðB−1Þ þ βn x − 1−aL α
7 7 7 7: 7 5
In the break-even point (z∗,x∗,a∗L) the linear approximation is: ð53Þ
ð54Þ
Now, Eqs. (53) and (54), when substituted in Eq. (51), considering constant returns of scale and gB = 0, provide, in a BGP: 1 g −ρ þ mðB−1Þ þ n : gY ¼ ε β
1 ½g−βmð1−BÞ þ βnz αh α ρi 2 x − 1− z þ x ε ε β g ð1−aL Þ x− ½mðB−1Þ þ n− −mBð1−aL Þ : α α
This is a feasible point since:
Now, given that, xx˙ ¼ g C −gK and differentiating logarithmically, with respect to time, Eq. (43), we'll get the same Eq. (34) of Appendix A, g C ¼ ρ ρ x˙ αz α ˙ ε − ε, with this x ¼ −ð1− ε Þz− ε þ x, and in a BGP, x ¼ 0, and of the results Eq. (53), it follows that
ð57Þ
2
¼
ð49Þ
In a BGP, z˙ ¼ 0 then z ¼
ð56Þ
−βz þ
¼
In similar way what already was a fact, the technology of the health system will be fixed in one determined way level that gB = 0. Since the rates of growth of z and x are given, respectively, for zz˙ ¼ gY −g K and xx˙ ¼ g C −g K , then, we can write: g βmðB−1Þ β z˙ þ n: ¼ g Y −g K ¼ −ð1−α Þz þ α α α z
βmð1−BÞ β g − n− α α α
These three equations (the equation above and equations zz˙ and xx˙ ), form the system
aL ¼ 1−
Now, substituting Eq. (50) in Eq. (48) and rearranging the terms, we have that: gY ¼
β g a˙L ¼ x − ½mðB−1Þ þ n− −mBð1−aL Þ: α α 1−aL
Substituting the results of Eqs. (45) and (46), in Eq. (48), we get: aL g ¼ αz−mðB−1Þ−n−gY þ g L þ g B : 1−aL aL
Considering a˙L ¼ 0 , following that a∗L = 1 or mB 1−a∗L ¼ x∗ −αg − βmðB−1Þ Þg mð1−εÞðB−1Þ Þn −αn , where x∗ −αg −βmðαB−1Þ−αn ¼ ρε−ð1−ε −ð1−ε α βε − ε ε . So:
This last equation can be written as aL μ˙ λ˙ g ¼ − −g þ g L þ g B : 1−aL aL μ λ Y
ð55Þ
By BGP definition, it must have g∗Y = g∗C = g∗K, where g∗Y is given for Eq. (54). In equal way what it was made in the Appendix A, it is
2 3 −βz z 6 alpha 4 x 5 ¼ 6 − 1− x 4 ε a˙L 0 2
0
x
0
1−aL
0
−mðB−1Þ 1−aL
3 7 7: 5
There are two negative eightvalues, r1 = − mB(1 − a∗)L and r2 = − βz∗, and one positive eightvalue, r3 = x∗, characterizing a saddle point. Appendix C In this appendix equations are deduced referring to the model which the health system's function effort is intensive in capital. The Hamiltonian of the problem (19) is:
H¼
C 1−ε −1 þ λðY−C Þ þ μ ½mðBaK −1Þ þ nL: 1−ε
ð60Þ
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
z and x is given, respectively, for written as:
The conditions for an interior solution are given by: ∂H −ε −ε ¼ C −λ ¼ 0⇒C ¼ λ ∂C
ð61Þ
∂H λαY ¼− þ μLmB ¼ 0: 1−aK ∂aK
ð62Þ
The equations of motion of the costate variables are given of:
∂H λβY þ μ ½mðBaK −1Þ þ n þ ρμ: μ˙ ¼ − þ ρμ ¼ − L ∂L
ð64Þ
Of Eq. (63), we get Y λ˙ ¼ ρ−α : K λ
ð65Þ
That's already a known result and Eq. (62) can be written as μLmb and therefore λβY βμmð1−aK ÞB ¼ : L α
λαY 1−ak
ð66Þ
ð68Þ
ε½g þ βmðB−1Þ þ βn−ðα−βÞρ : α ½β−α ð1−εÞ
ð75Þ
" g þ βðmðB−1Þ þ n þ β2 ρ α ½β−α ð1−εÞ:
ð76Þ
gY ¼
g þ β½mðB−1Þ þ n−αρ : β−α ð1−εÞ
ð77Þ
Now, the idea is to discover if it has feasible path that they lead to the BGP. Of Eq. (71) it elapses that, in a BGP,
αz −g Y ¼ x −
α g x − −mð1−BÞ−n β β
ð78Þ
and substituting this results in Eq. (70), we find: α g mβðB−1Þ mβBð1−aK Þ a˙k x− − −n− : ¼ β β α α 1−aK
ð79Þ
Now, these equations (the equation above and the equations of zz˙ and xx˙ ), form the system:
α g x þ þ mðB−1Þ þ n z 1− hβ α β ρi 2 ¼ x − 1− z þ x ε ε α g mβðB−1Þ mβBð1−aK Þ −n− : ¼ ð1−aK Þ x− − β β α α
ð69Þ
8 > > > z˙ > > < x˙ > > > > > :a˙K
ð70Þ
Substituting a˙K ¼ 0, following that a∗K = 1 or ð α Þ ¼ αβx−βg −mβðαB−1Þ− ðB−1Þþn n, where αβx−βg −mβðαB−1Þ−n ¼ βρ−ð1−εÞ½gþβm . With this, α
This last equation can be written as
From Eqs. (65) and (67), Eq. (69) can be written as:
Substituting this equation in Eq. (69) and rearranging the terms, we get: g α α þ αz− x þ mðB−1Þ þ n− g B : β β β
ð74Þ
x ¼ ðε−α Þ
ð67Þ
λ˙ a˙ μ˙ þ g Y þ K ¼ þ gL þ g B : 1−aK μ λ
gY ¼
ð73Þ
And now, substituting, Eqs. (75) and (76) in Eq. (71), and rearranging the terms, it is obtained:
Here, to specify Eq. (68) it is necessary to know the rate of allocation of the capital, fact that can be obtained by the logarithmic differentiation, with respect to time, of Eq. (62), namely:
aK mðα−βÞð1−aK Þ −g Y þ gL þ g B : g ¼ αz þ α 1−aK aK
g 1 α mðB−1Þ n 1− x þ þ : þ β β β β2 β
With this, Eq. (74) it will can be written as
The growth rate of product is obtained for logarithmic differentiation of equation Y = F(K,L) = A[(1 − aK)K]αLβ, that is,
aK μ˙ λ˙ g ¼ − −g þ g L þ g B : 1−aK aK μ λ Y
ð72Þ
Substituting Eq. (74) in Eq. (73), we get:
with this,
αaK g þ βg L : 1−aK aK
¼ g C −g K , can be
α ρ x ¼ 1− z þ : ε ε
z ¼
ðα−βÞð1−BaK Þ−βðB−1Þ −n −μ m α
g Y ¼ g þ αg K −
x˙ x
Now, again, it is necessary to determine x to know z (or the opposite) and this can be made from the logarithmic differentiation, with respect to time, of Eq. (61) substituted in Eq. (65) that it provides, the same Eq. (34) of Appendix A, that is, g C ¼ αzε −ρε. As xx˙ ¼ g x ¼ g C −gK , it results that xx˙ ¼ αzε −ρε−z þ x. In a BGP, x˙ ¼ 0, and so:
¼
Now, substituting this result in the expression between keys of Eq. (64), we get
ðα−βÞð1−BaK Þ−βðB−1Þ μ˙ −n : ¼ρþ m α μ
¼ g Y −g K and
In a BGP, z˙ ¼ 0, and so then
ð63Þ
z˙ z
g α z˙ x þ mðB−1Þ þ n: ¼ −ð1−α Þz þ 1− β z β
z ¼
∂H ˙ − λ¼ þ λρ ¼ −λ F K þ λρ ∂K
741
ð
2
−β z þ
ð80Þ
mB 1−a∗K
aK ¼ 1−
ð71Þ
This model, also considers that the health system technology it is given, that is, gB = 0. Considering that the growth rate of the variables
¼
α βρ−ð1−εÞ½g þ βðmðB−1Þ þ nÞ : βmB β−α ð1−ε Þ
This a feasible point if
0b
1 βρ−ð1−εÞ½g þ βðmðB−1Þ þ nÞ b1: βmB β−α ð1−εÞ
ð81Þ
742
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
Determining the value of g in the inequality above, it is obtained: β
h ρ i ρ−mBε −½mðB−1Þ þ n bgbβ −½mðB−1Þ þ n : 1−ε 1−ε
ð1−εÞÞ , where, clearly, The amplitude of this interval is given for mBðβ−α αð1−ε Þ only the technology of the health system influences. It observes that the limits are generalizations of the previous results. One more time, the introduction of technological level in the health sector and the population growth, do not attenuate the limitation of the technology rate of the productive sector. Now, the size of the productive sector is given by
1−aK ¼
1 βρ−ð1−εÞ½g þ βðmðB−1Þ þ nÞ : mβB β−α ð1−ε Þ
The linear approximation of Eq. (80) is, of a generalized form, given for: 2
g 6 −2βz þ β þ mðB−1Þ þ n 6
z 6 α 4 x 5 ¼ 6 − 1− x 6 β 6 a˙K 4 0 2
3
α 1− β x
3 0 0
α g α 1−aK x − −mðB−1Þ−n β β β mβBð1−a∗K Þ α
That it has at least a negative eightvalue given for r 1 ¼ − the break-even point (z∗,x∗,a∗K).
7 7 7 7: 7 7 5
in
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Becker, Gary S., 1962. Investment in human capital: a theoretical analysis. The Journal of Political Economy. Vol. 70. n. 5, Part 2: Investment in Humans Begins. 9–49 (Oct). Blackburn, K., Pietro, G., Cipriani, P., 2002. A model of longevity, fertility and growth. J. Econ. Dyn. Control 26, 187–204. Blomqvist, Ake, Léger, Pierre T., 2005. Information asymmetry, insurance, and the decision to hospitalize. J. Heal. Econ. 24, 777–793. Bloom, David E., Canning, David, 2005. Health and Economic Growth: Reconciling Micro and Macro Evidences. Harvard School of Public Health (Fev). Bloom, David, Canning, David, Sevilla, Jayppe, 2004. The effect of health on economic growth: a production function approach. World Dev. vol. 32 (1), 1–13. Buchumuller, Thomas, 2006. Price and the health plan choices of retirees. J. Heal. Econ. 25, 81–101. Ehrlich, Isaac, Lui, Frances, 1997. The problem of population and growth: a review of the literature from Malthus to contemporary models of endogenous population and endogenous growth. J. Econ. Dyn. Control. 21, 205–242. Elíasson, Lúdvik, Turnosky, Stephen, 2004. Renewable resources in an endogenously growing economy: balanced growth and transitional dynamics. J. Environ. Econ. Manag. 48, 1018–1049. Grossman, Michael, 1972. On the concept of health capital and the demand for health care. J. Polit. Econ. 80, 223–235 (mar.). Groth, Cristihian, 2002. Can non-renewable resources alleviate the knife-edge character of endogenous growth? Oxf. Econ. Pap. 53, 386-311. Howitt, Peter, 2005. Health, Human Capital and Economic Growth: A Schumpeterian Perspective. Brown University (Feb). Mariani, F., Pérez-Barahona, A., Raffin, N., 2010. Life expectancy and the environment. J. Econ. Dyn. Control 34, 798–815. Márquez, J. Aznar, Ruiz-Tamarit, J.R., 2005. Renewable natural resources and endogenous growth. Macroecon. Dyn. 9, 170–197. Muskhin, S.J., 1964. Health as investment. J. Polit. Econ. 70 (5 part II), 129–157. Østerdal, Lars Peter, 2005. Axioms for health care resource allocation. J. Heal. Econ. 24, 679–702. Romer, David, 1996. Advanced Macroeconomics. McGraw-Hill, São Paulo. Salinas-Jimenez, Javier, Smith, Peter, 1996. Data envelopment analysis applied to quality in primary health care. J. Ann. Oper. Res. 1 (December). Schultz, Theodore W., 1991. Investment in human capital. Am. Econ. Rev. vol. 51 (1), 1–17 (mar). Sherman, H.D., 1984. Hospital efficiency measurement and evaluation: empirical test of a new technique. Med. Care 22 (10), 922–938. Sorkin, Alan L., 1997. Health Economics in Developing Countries. Lexington Books, Lexington, MA. Stamford da Silva, Alexandre, 2008. Growth with exhaustible resource and endogenous extraction rate. Econ. Model. 25, 1165–1173. Stiglitz, Joseph, 1974a. Growth with exhaustible natural resources: the competitive economy. Rev. Econ. Stud. 136–152. Stiglitz, Joseph, 1974b. Growth with exhaustible natural resources: efficient and optimal growth paths. Rev. Econ. Stud. 123–137. Taylor, C.E., Hall, M.F., 1967. Health population and economic development. Science 157, 651–657. Tsaprounis, D., 1997. The administrative efficiency of hospitals and the effect of electronic data interchange: a critical evaluation of the stochastic frontier and the data envelopment analysis models to efficiency measurement. Dissertation for the Degree of Doctor of Philosophy The City University of New York. Verhulst, P.F., 1838. Notice sur la loi que la population pursuit dans son accroissement. Corresp. Math. Phys. 10, 113–121. Winslow, C.E.A., 1951. The Cost of Sickness and the Price of Health. World Health Organization, Geneva (Monograph Series, n. 7).
Contents lists available at ScienceDirect
Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
The influence of the healthcare system on optimal economic growth Maurício Assuero Lima de Freitas a,⁎, Alexandre Stamford da Silva b a b
Department of Accounting and Actuarial Sciences, Federal University of Pernambuco, Brazil Department of Economy, Federal University of Pernambuco, Brazil
a r t i c l e
i n f o
Article history: Accepted 16 August 2013 Keywords: Healthcare system Optimal economic growth Healthcare sector effort function Macroeconomic models for the healthcare sector
a b s t r a c t This paper analyzes the impact of health system in the economic growth, based upon three macroeconomic models. The first one considers the economy with only one sector, but with morbidity; in the others the economy is divided in two sectors, the productive sector and the health sector, considering it intensive in labor and after intensive in capital. The results show that the presence of the health system increases the life expectancy and the aggregate product, but does not modify the per capita product. © 2013 Elsevier B.V. All rights reserved.
1. Introduction This paper presents an introductory study of the macroeconomic impacts of the inclusion of the healthcare sector in the neoclassical model of optimal economic growth. Studies in economics regarding the healthcare sector – usually referred to as the economics of health – are well established in the literature (Arrow, 1963; Grossman, 1972). Some studies deal with the issue from a microeconomic standpoint, focusing more intensively on the public sector (Blomqvist and Léger, 2005; Østerdal, 2005; Buchumuller, 2006). Others discuss the importance of health for economic growth (Aguayo-Rico, 2005; Ainsworth and Over, 1993; Bloom and Canning, 2005; Howitt, 2005; Muskhin, 1964; Sorkin, 1997; Taylor and Hall, 1967; Winslow, 1951), but they tend to be qualitative analyses without a well-established theoretical basis, that is, a basic mathematical model. Besides, most studies deal with health in the context of the practice of healthy habits, diet, leisure, education, etc. (Schultz, 1991; Becker, 1962; Bloom et al. 2004), and not with the healthcare sector as an intermediary productive sector that deploys technology, capital, and labor with impacts in the recovery of the workforce. In order to attempt to measure the effects of the healthcare sector on economic growth it is necessary to adequately insert such sector in a theoretical model capable of predicting the impacts caused by its presence. The empirical aspect has already been adequately addressed by Acemoglu and Johnson (2007). Thus, the goal of the present paper is to evaluate what is the adequate policy for the allocation of resources for the optimal economic
⁎ Corresponding author at: Av. Dos Economistas, S/N, Cidade Universitária, Recife, Pernambuco 50740-290, Brazil. Tel.: +55 81 2126 7739; fax: 55 81 2126 8369. 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.08.023
growth given the explicit presence of the healthcare sector in the economy. It also attempts to answer the question of whether there are economic advantages for society in opting for the healthcare sector in its economy. The main contribution is to discuss the optimal economic growth in an economy not only with the existence of morbidity, but also with a healthcare sector, that needs to solve the problem of the tradeoffs between capital and labor for production versus capital and labor for the recovery of the sick or ailed workforce. This article was divided into four sections beyond this introduction. The first one presents the model of an economy with a single sector – the productive sector – in the presence of morbidity which affects labor by reducing the workforce, that is to say, the amount of hours available for production. The second section is dedicated to the explanation and to the empirical measures of the recovery effort function, which is the function that gives the production of the healthcare sector in terms of the recovery of the hours of work available for production that were subtracted due to the existence of morbidity (representing the activity of recovering the sick or ailed workforce and returning it to the productive sector). The third one is where the economy is divided into two sectors – productive and healthcare – and where, for simplicity of analysis, two types of healthcare sector are considered, one being labor-intensive and the other capital-intensive. Finally, in the fourth section the main conclusions and some comments are presented. 2. The economy with a single sector in the presence of morbidity in labor The model developed in this section considers a simple, closed, economy, with constant scale earnings in the production, without
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
government or healthcare sector, in which the standard hypothesis of the neoclassical models is valid. The social planner's objective is: Z max C
∞ 0
−ρt
U ðC Þe
subject to ˙ I ¼ Y−C; K ð0Þ ¼ K 0 N0 K¼ L˙ ¼ nL−M ¼ ðn−mÞL; Lð0Þ ¼ L0 N0 ˙ gA⇒A ¼ A0 egt ; Að0Þ ¼ A0 N0 A¼ α β Y ¼ F ðK; LÞ ¼ AK L
(5), and (6) were highlighted because they contain interpretations that are absent in the standard models. λ βY μ˙ þ m−n þ ρ: ¼− μ L μ
dt; ρN0;
ð1Þ
L˙ ¼ GðLÞ−M
ð2Þ
where the G(L) function represents the immunological recovery rate and/or the population reproduction rate.2 To make the model simpler, it will be admitted that G(L) = nL, resulting in equation L˙ of the model (1). It is considered that the morbidity is homogeneous, in other words, that it acts indiscriminately upon all people, though it is a fact that older people have a greater tendency towards ailments. This hypothesis of distinct rates of rates of affliction according to age or other population groupings shall be considered in future studies. The utility function, as assumed by Barbier (1999), Groth (2002), and Márquez and Ruiz-Tamarit (2005), will be given by: U ðC Þ ¼
C 1−ε −1 : 1−ε
ð3Þ
The results of the model are classic and the deduction can be found in Appendix A. Only the morbidity is highlighted in the growth equations, without any difference from the standard neoclassical model. Eqs. (4),
1 It should be realized that, alternatively, any other population growth model, including: Gompertz (1825), Smith (1963), Goel, Maitra and Montroll (1971), Ayala, Ehrenfel and Gilpin(1973), all cited by Bassanezi and Jr (1998) – depending on the interest – because the factor of notice is the growth of the population and not the form of that growth; in other words: in the literature the emphasis is on the value of the growth rate and not the pattern. 2 Elíasson and Turnosky (2004) address renewable resources by considering that the natural growth rate of the renewable resource depends on a second degree logistic func tion that is given by S˙ ¼ GðSÞ−X, where, GðSÞ ¼ rS 1−S=S ; r N 0, and X is the rate of harvest. They consider that the resource is forest or fish. The source of this model is Verhulst (1838). The same function could be used here, but the choice was made to use the classic population growth function.
ð4Þ
Eq. (4) represents the growth rate of the contribution of labor to the optimal social well-being, or how much one would pay to have an extra hour of work, or yet, the rate of evolution for the wages, which depends on the relative process of capital and work, and on the rates of impatience and morbidity. In the BGP — Balanced Growth Path, the growth rate of the product is given by: g Y ¼ ðn−mÞ þ
where U(.) is the utility function, ρ is the society's impatience rate, t is the time, K is the capital, I is the investment, Y is the product of the economy, C is the consumption, L is the workforce, and all for an instant t, is omitted due to presentation issues. Also, n is the population's rate of growth, g is the technology rate growth, M = mL is the morbidity, with m being the given rate of morbidity, F(.) is the production function, A is the technology, α and β are the parameters of the production function, and K0, L0 and A0 are the capital, labor and technology stocks that are assumed to be given. The equations in problem (1) regarding capital, technology, and production are classic and well-established in the literature so will not be commented upon. The equation for the movement of labor,L˙, establishes that the rate of variation in labor is reduced as the morbidity rate grows, and grows with the growth in labor.1 Note that one can admit the rate of variation in the workforce as, more generically, increasing with the natural growth of the population or with the recovery by the “immune system”, and diminishing with the morbidity rate, that is:
735
g : β
ð5Þ
A sufficient condition so that g∗Y N 0 is n ≥ m. The sick people would be replaced by new births and the population would grow exponentially at the rate of (n − m). If n b m, then the rate of technology of the productive system should withstand the reduction in the quantity of labor or in the hours available for work, and be such that g N β(m − n). Now, considering the rate of per capita consumption c ¼ CL , one has that:
g c ¼ g C −g L ¼ g Y −g L ¼
g g ¼ : β 1−α
ð6Þ
Therefore, g∗c depends on the rate of technology and on the elasticity of labor (or capital), and is independent of the morbidity rate. Stiglitz (1974a,b), dealing with growth with exhaustible natural resources, found that the per capita rate of consumption converges to α 2 nþλ ð1−α 1 Þðα 1 þα 2 Þ, where λ, α1, and α2, represent the technological growth and the elasticities respectively. Note that, under the hypothesis of a constant return in scale and a null population growth, this is the same rate determined in Eq. (6). Thus, there is an analogy between the growth models in an economy where the morbidity is highlighted and the growth models in the presence of exhaustible resources. One observes here that the economy is viable even in the absence of the healthcare sector. Considering the population growth rate as null, it is evident that fewer and fewer people participate in the economy and that these people will have an increasingly smaller life expectancy until exhaustion. The probability that a given generation will know the next will decrease as time goes on. One can speculate that, in this case, the quality of life is low. Such speculation is valid even for positive population growth rates. 3. The healthcare sector effort function The introduction of the healthcare sector in the economy obliges one to deal now with two sectors: the productive sector, of end goods, and the healthcare sector, an intermediary one, which acts as a system that recovers labor that is ill. The recovery of the health state of the workers or of the labor force has two characteristics. One is renewable, given by the immune system, and the other is recoverable, given by the effort in the treatment of health. In the present paper, we will deal only with the recoverable portion of this resource, so that the recovery of labor is done exclusively by means of a production effort involving capital, labor, and technology. The implicit hypothesis is that the allocation of these production factors in the effort function influences in the recovery of the workers, or of the hours of work, and that the healthcare system works so as to repair and not to prevent. The recoverable part of the healthcare sector is, therefore, represented by a production function that depends on capital KS, labor LS, and technology (B), given by YS = F(B,KS,LS,M), where M is the number of ill workers or the morbidity mL, a necessary input. Regardless of the
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analytical form of the production function, the end product obtained by the healthcare sector can be expressed as YS = bmL, where m and b are real in the unit interval, m being the morbidity as previously defined and b the percentage of ill individuals recovered. The fraction of the population that is ill in time (t − 1) is mL and bmL is the fraction of the ill population that was recovered by the health sector in time t. Since the ill, mL, a priori do not depend on capital and labor, then, without loss of generality, one can assume that b is a measure of the rate of recovery of the healthcare sector that depends only on the percentage of capital and labor allocated to the sector. Therefore, parameter b will be an index of the labor recovery rate due to the allocation of capital and/or labor to the healthcare sector. Mathematically, one can write b = BG(aK,aL) with B indicating the level of technology of the healthcare sector, aK and aL the fractions of capital and labor allocated to it. It is considered that the effort is limited, that is, 0 ≤ b = BM(aK,aL) ≤ 1 and that the Inada conditions (Romer, 1996, p.19) are satisfied. Intuitively, the effort of the healthcare system, represented by the effort function b = BM(aK,aL), is the cause of the recovery of ill individuals, that is to say, people are recovered and returned to the productive system because of the existence of this social effort, represented by the allocation of labor and capital. Since only a fraction of the population, mL, gets ill, then only a portion of that fraction, bmL, can be recovered, and that fraction is exactly the end product, or product, of the healthcare system. The labor's movement equation can be written, thus, as: L˙ ¼ nL−mL þ bmL ¼ ½n−mð1−bÞ:
ð7Þ
This equation can be used so as to empirically measure the value of b. This parameter is an alternative for the comparison of the absolute efficiency of the health sectors of distinct economies. Different from the DEA method (Data Envelopment Analyses: Akazili et al. 2008; AlShammari, 1999; Salinas-Jimenez and Smith, 1996; Sherman, 1984; Tsaprounis, 1997), the measure here is not relative to the content of sample data, but rather measured on an absolute scale, which can greatly improve such comparisons. A possible procedure to have access to a measure of parameter b is described as follows. The basic idea is that the growth rates of the population and of the morbidity are available in various official databases, but the recovery rate of the healthcare sector (b) is not. However, a proxy of the rate of individuals not recovered by the healthcare system, N, can be obtained by the sum of the data on deaths due to disease, retirements due to disability, retirement due to labor accidents, and other similar data that is generally available, measure along a certain period. By definition, N = (1 − b)m, and through simple algebra one can calculate that b ¼ 1−mN . Returning to the theoretical model, the mathematical treatment of a complete effort function, with technology, fraction of capital, and fraction of labor does not present results that are easy to interpret. For this reason, the choice was made to use an effort function initially dependent only on the fraction of labor allocated and then only on the fraction of capital allocated. In the first case one can assume that the amount of capital allocated to the healthcare sector is not large enough for its impacts to be taken into consideration in the recovery effort, thus, the approximation is done by the exclusion of the fraction of the capital. It is as if this worker recovery system was based on the clinic and things of the sort. Whatever the case, the main assumption is that there is impact in the recovery when these clinicians or healers are allocated to the healthcare sector. In the case of the dependency only on the fraction of the capital allocated, the allocated work is what does not participate in a relevant way in the recovery, so that the effort function is dependent only on the fraction of capital. It is the instrumental medicine whose treatments and
diagnosis are done and made available by machines where human participation is minimal. 4. Models with the healthcare sector As previously said, the insertion of the healthcare sector in the economy will be done in two ways: the first is considering a labor-intensive healthcare sector and the second assuming a capital-intensive one. 4.1. Labor-based effort function In the model without a healthcare system (Section 1), all of the population was available to work in production, the morbidity rate being similar to an extraction rate for labor that affects the workforce, and, in the long run, the labor would exponentially tend to exhaustion if there was no population growth (or when the morbidity rate is greater than the population's rate of growth). Now, the fractions aL and 1 − aL of the population are allocated to the healthcare and productive sectors respectively, as done in Aghion and Howitt (1992), Romer (1996, p. 97), and Stamford da Silva (2008), among others, that treated growth with models from two sectors. What one wants to verify is whether this reduction in the workforce allocated to production in favor of the allocation of some relevant effort for the recovery of the health of the workforce is advantageous for the economy as a whole. This is done comparing the economic system that makes the choice for that allocation with the economic system that does not go for that option, as presented in Section 1. The effort function of the healthcare system, based only on labor, is given by b = BaL. The existence of some technology (B) is considered in the healthcare system, such as techniques from traditional medicine, new techniques for well-established curative procedures, curative methods in general, or even some tool with fixed technology. It is considered that there is perfect mobility of labor between the two sectors, that is to say, each worker is capable of executing tasks in both sectors with equal efficiency. If B = 1, the healthcare system recovers exactly the same percentage of labor as that which was allocated to the healthcare sector. For instance, if b = aL = 0.20 only 20% of the ill return to the productive system. This would represent a highly inefficient healthcare sector. The goal of the social planner is the same as before, except for the equations: max C; aL
Z
∞ 0
−ρt
U ðC Þe
dt; ρN0
subject to ˙ I ¼ Y−C; K ð0Þ ¼ K 0 N0 K¼ L˙ ¼ nL−mL þ bmL ¼ ½mðBaL −1Þ þ nL; Lð0Þ ¼ L0 N0 α β Y ¼ F ðK; LÞ ¼ AK ½ð1−aL ÞL :
ð8Þ
Appendix B brings a solution for this model. The wage rate is given by: μ˙ ¼ ρ−mðB−1Þ−n ¼ ρ−mB þ m−n: μ
ð9Þ
The equation above shows the joint effect of the morbidity, of the healthcare system's level of technology, and of the population growth rate on the wage rate. The wage rate increases if the morbidity increases because it reduces the workforce (fewer hours available for work) while the population growth rate and the effect of the healthcare system's technology on the ill reduces the wage rate because they are determinants of the population increase. One notes the absence of the marginal productivity of labor in the equation.
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
If Eqs. (4) and (9) are considered the same, one obtains: βY μ ¼ F L ¼ mB: L λ
ð10Þ
Eq. (10) shows that, for the systems to be equivalent, the marginal productivity of labor is proportional to the amount of workers recovered. In other words, the marginal contribution of the worker to the product must equal the relative cost of the worker times the impact of the allocation on the growth rate of labor. The growth rate of the product is given by: gY
g βmðB−1Þ β β þ n− g B : ¼ þ αz þ α α α α
Y
B
gY
1 g ðn−mÞ þ þ mB−ρ : ¼ ε β
1 ð1−εÞ½g þ βmðB−1Þ þ n ρ− : mBε β
ð12Þ
ð13Þ
∗ L
Therefore, ∂a∂g ¼ 1−ε mBε N0, that is, an increase in the technology of the productive sector generates an increase in the healthcare sector, as expected. For it to be viable, the fraction of labor allocated must obey to the following inequality: 0b
1 ð1−εÞ½g þ βmðB−1Þ þ n ρ− b1 mBε β
that is, β
1 g þ mðB−1Þ þ n α β
ð14Þ
1 α g ρ 1− þ mðB−1Þ þ n þ α ε β ε
ð15Þ
x ¼
aL ¼ 1−
h ρ i ρ−mBε −mðB−1Þ−n b gb β −mðB−1Þ−n : 1−ε 1−ε
Thus, so that the health sector exists the technology of the productive sector stays limited. The amplitude of this interval, ðβmBε 1−εÞ , does not
1 ð1−εÞ½g þ βmðB−1Þ þ n ρ− : mBε β
ð16Þ
The per capita consumption rate, g∗c = g∗C − g∗L, is given by
gc ¼
1 g þ mðB−1Þ þ n−ρ − m BaL −1 þ n : ε β
ð17Þ
Replacing a∗L results in:
4.1.1. Dynamics Besides analyzing the dynamics of the variables z ¼ KY , product per unit of capital, and x ¼ KC, consumption per unit of capital, it is necessary to analyze the labor allocation policy, by means of the dynamics of aL = aLL / L = aL. Appendix B shows that the optimal fraction of labor allocated to the healthcare sector is:
z ¼
gc ¼
One observes, first of all, the dependency of the long term growth rate on the inverse of the elasticity of the marginal utility of the consumption, ε, the social option. One also notes the negative presence of the intertemporal impatience rate, ρ, and the positive presence of the level of technology of the healthcare sector, B, or of the variation of the growth rate of the workforce in relation to the percentage of workers allocated, mB. None of these dependencies occurs without the presence of the healthcare sector, as can be seen by comparison with Eq. (5).
aL ¼ 1−
depend on the growth of the population, but captures the effect of the level of technology of the healthcare sector. The greater B is, the wider will be the technology interval allowed for the production sector. One observes again the presence of the marginal utility of the consumption highlighting its fundamental role in the growth when the healthcare sector is present. The dynamics, shown in Appendix B, shows the existence of an economically viable point given by (z∗,x∗,a∗L) formed by the optimal amounts of each variable:
ð11Þ
This expression shows that, in this model, the existence of a fixed level of technology in the healthcare sector contributes to the increase ¼ −αβb0 , of the product, but that technology cannot grow, for ∂g ∂g compromising the growth of the product. This is an important result. It says that a growing rate in the level of technology of the healthcare sector can make the economy unviable because the variation of the variation of the workforce (the second derivative) is constant, indicating that the workforce will grow more than exponentially and there will be no product for all to consume in the long run. It will be considered here that that the technology of the healthcare system is fixed at a certain level B so that gB = 0. From this results that, in the BGP, the growth rate of the product is:
737
g g ¼ : β 1−α
ð18Þ
The algebraic cancelations done in Eq. (16) that resulted in Eq. (17) have economic meanings that are worthy of greater investigation. For example: when the population increases, the product per unit of capital grows by a factor 1ε . But the consumption per unit of capital also increases due to the increase in the population and acts as a reduction factor of the product in exactly the same proportion, making the growth rate of the consumption per capita independent of the population growth rate. On the other hand, the fact that there is no change in the individual wealth with the insertion of a healthcare system may contradict the commonsense. However, Acemoglu and Johnson (2007), in an empirical work dealing with health interventions worldwide from 1940 till date, showing, by means of regression analysis, that such interventions (that is, the action of the healthcare sector) increased the life expectancy of the people and that: There is no evidence that the increase in life expectancy led to faster growth of income per capita. This evidence sheds considerable doubt on the view that health has a first-order impact on economic growth (Acemoglu and Johnson, 2007). So, the main results of the model discussed in this section are in consonance with the results from Acemoglu and Johnson (2007), when it shows that the healthcare system increases the life expectancy of the people, by means of the inclusion of the system as an element of recovery in the labor equation, but that this does not affect individual wealth. Indeed, considering that Y = yL and given that y does not vary in the models, the healthcare sector generates a greater L, therefore, a greater product. With this, society will have a greater well-being, considering the aggregated product and the fact that generations meet due to the increase in life expectancy as given by the equation of the evolution of labor. 4.2. Effort function based on capital In this section, the healthcare sector will be assumed to be capital intensive. The effort function, discussed in Section 2, maintains its
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M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
previously mentioned assumptions, and will be given by 0 ≤ b = BaK ≤ 1. The goal of the social planner is: max C; aK
Z
∞ 0
−ρt
U ðC Þe
Simple algebraic manipulations show that
ð19Þ
The resolution of the model can be seen in Appendix C. The movement equation for the wage rate is: ðα−βÞð1−BaK Þ−βðB−1Þ μ˙ −n: ¼ρþm α μ
ð20Þ
As one can see, the rate of variation of the worker's shadow price diminishes with the increase of both in the technology of the healthcare sector as of the fraction of capital allocated to this sector, as expected. If it happens that α = β Eqs. (20) and (9) become equal and present the same results. The resolution of the model shows, however, that α N β. The growth rate of the product is given by: g α α þ αz− x þ mðB−1Þ þ n− g B β β β
ð21Þ
where one again observes that the growth rate of the product diminished if there is growth in the technology of the healthcare system. It is important to note that, in this model, the impact of the growth rate of the technology of the healthcare system is elastic. Thus, whether the effort of the healthcare system is based in the allocation of labor or on the allocation of capital, the rate of variation of the technology of the healthcare system cannot grow indefinitely, under the penalty of the economy becoming unviable, with not enough products for the amount of workers recovered. Once again, it will be considered that B is constant, that is, the amount of the technology is given. With that, gB = 0. To define the allocation policy for the capital, one must evaluate the dynamics of aK. As may be seen in Appendix C, from the movement equation of the fraction of capital, one has that
α βρ−ð1−εÞ½g þ βðmB−m þ nÞ : mβB β−α ð1−εÞ
ð22Þ
For the healthcare system to have a viable size, it must have: α g þ βðmB−m þ n βρ−ð1−εÞ b1: mβB β−α ð1−ε Þ
0b
Determining the value of in the inequality above, one obtains: β
h ρ i ρ mBðβ−α ð1−εÞÞ −½mðB−1Þ þ n− bgb −½mðB−1Þ þ n : 1−ε α ð1−εÞ 1−ε
ð1−εÞÞ The amplitude of this interval is given by βBðβ−α which, as obα ð1−ε Þ served, is not influenced by the growth rate of the population. Now, using the movement equations for the variables, z, x, and aK, one forms a system of differential equations in which some of the equilibrium points do not have an economic interpretation. In Appendix C, the classification of these points is made. The growth rate of the per capita consumption is given by
gc ¼
˙ I ¼ Y−C; K ð0Þ ¼ K 0 N0 K¼ L˙ ¼ nL−mL þ bmL ¼ ½mðBaK −1Þ þ nL; Lð0Þ ¼ L0 N0 α β Y ¼ F ðK; LÞ ¼ A½ð1−aK ÞK L :
aK ¼ 1−
g c ¼ g C þ ½m−mBaK −n:
dt; ρ N0
subject to
gY ¼
that is,
g c ¼ g C −g L ¼ g C −½mðBaK −1Þ þ n
g g ¼ : β 1−α
ð23Þ
This rate does not differ from the results obtained in the previous models. Mariani, Pérez-Barahona and Raffin (2010) had found that a correlation positive enters the life expectancy and the ambient quality, considering “an infinite-horizon economy that is populated by overlapping generations of agents living for three periods: childhood, adulthood, and old age”. The results show that “environmental quality depends on life expectancy, since agents who expect to live longer have a stronger concern for the future and therefore invest more in environmental care and longevity is affected by environmental conditions”. On the other hand, Blackburn et al. (2002) had considered, too, a model, with overlapping generations and had arrived at a result that shows the effect of the changes in the life expectancy on the way of the development of the economy. They had been emphatic in studying the impact of the increase of the life expectancy on the economy, while our work suggests that it will have greater welfare in the economy due to the presence of the health system, but the health system, does not cause impact on the per capita income. 5. Analysis, results and final considerations The model in which the economy has only one sector brings results similar to the economic growth models with exhaustible resources. In this model, it was found that the existence of a BGP is assured as long as it is true that n N m or, if that does not happen, it is necessary that g N β(m − n). The insertion of the healthcare sector immediately modifies the labor movement equation that now begins to decay into a smaller rate, that is, the presence of the healthcare sector increases people's life expectancy, but this increase in life expectancy does not change g the growth rate of the per capita consumption, g ∗ ¼ 1−α , that depends, with or without the healthcare sector, on the rate of technology of the productive system and on the elasticity of the product in relation to the capital (or of the product in relation to the labor, given that 1 − α = β). This result does not change whether the effort function of the healthcare sector is labor-intensive or capital-intensive, and this can suggest some level of inoperance for the healthcare sector, however, the theoretical result obtained here is corroborated by the empirical results found by Acemoglu and Johnson (2007) who studied the impact of illness on economic development and, using a regression analysis, showed that there is no evidence that the increase in life expectancy, the main result of the healthcare sector, had any influence over the growth in per capita income. One of the results of the work is to highlight that the healthcare sector increases society's well-being because the aggregated product is greater than the aggregate product of an economy with morbidity and without a healthcare system. Therefore, one obtains a result contrary to that of Thomas Malthus (1798), who advocated the positive control of the population suggesting the construction of houses close to mangroves and the narrowing of streets in order to provoke the return of plagues, saying that: … but above all we must repudiate specific medicine for overwhelming diseases …. Ehrlich and Lui (1997), referring to Malthus, said: “Malthus dramatized the idea by identifying population as potentially detrimental to growth. Since that time, work on growth and development has been inextricably linked with population economics”. The authors trace the evolution of the literature on population growth until the recent
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
endogenous growth theory and development and propose a model of dynasty. The results are important, but the approach is different from ours. Here we show that the health system as a productive sector of the economy, has no impact on the increase in per capita income, but it is of fundamental importance in social welfare because it contributes to the increase in life expectancy of people. The models presented here show that Malthus was wrong, for, even if the healthcare sector approaches the effect of snake oil, it brings positive results for the economic system. The present work does not deal with an effort function based simultaneously on capital and labor, due to the initial difficulty in expressing a functional form for the healthcare sector that can be both of the Cobb– Douglas and of the fixed proportions type, allowing for some level of substitution between factors. It also does not consider the randomness of the illness, a fact that would force an analysis in light of dynamic programming. Finally, the endogenization of technology is not done, in spite of the importance of technology for the existence of BGP.
739
Differentiating Eq. (24) logarithmically and using Eq. (25), we get: gC ¼
αz ρ − : ε ε
ð34Þ
And now, substituting Eqs. (28), (31) and (34) in Eq. (27), following: z˙ ¼ −ð1−α Þz þ ð1−α Þx þ βðn−mÞ þ g z
ð35Þ
α ρ x˙ −1 z þ x− : ¼ ε ε x
ð36Þ
We'll assume an interior break-even point, to form the system: (
−ð1−α Þz þ ð1−α Þx þ βðρn−mÞ þ g α −1 z þ x − ε ε
¼0 ¼0
:
ð37Þ
Appendix A Whose solution is given for The Hamiltonian of the problem (1) is: H¼
C
−1 þ λðY−C Þ þ μ ðn−mÞL: 1−ε
1 εg ρ þ εðn−mÞ þ α β
ð38Þ
1 εg g ρ þ εðn−mÞ þ − ðn−mÞ þ : α β β
ð39Þ
ð24Þ x ¼
The condition for an interior solution is: ∂H −ε −ε ¼ C −λ ¼ 0⇒C ¼ λ: ∂C
z ¼
1−ε
ð25Þ
Linearing Eqs. (35) and (36) in the equilibrium point, we find the Jacobian, J z ;x ¼
The equations of motion of the costate variables are given for: ˙ − λ¼
∂H λαY Y λ˙ þ ρλ⇒ ¼ ρ−α ¼− K K λ ∂K
ð26Þ
μ˙ ¼ −
∂H λβY þ ðn−mÞμ þ ρμ: ¼− L ∂L
ð27Þ
"
Let g H ¼ we would want to know about the dynamics of the variables: Y z¼ K C x¼ : K
#
x
g Y ¼ ðn−mÞ þ
g β
ð40Þ
implying ð28Þ
g c ¼ g C −g L ¼ g Y −g L ¼
g g : ¼ β 1−α
ð41Þ
Appendix B
By the definitions: z˙ ¼ g Y −g K z x˙ ¼ g C −gK : x
βz
∗ ∗ whose determinant is given for jJj ¼ −αβ ε z x b0, characterizing a saddle point and the presence of feasible politics to the optimum growth. From Eq. (28) following
H˙ H the rate relative growth of the variable H. In the long-term
α−βz −1 z ε
The Hamiltonian of the problem (8) is given for: ð29Þ H¼
˙ Y−C , L˙ ¼ ðn−mÞL and Y = F(K,L) = AKαLβ, we have, From K¼ respectively g K ¼ z−x
ð30Þ
g L ¼ ðn−mÞ
ð31Þ
g Y ¼ g þ αg K þ βg L :
ð32Þ
C 1−ε −1 þ λðY−C Þ þ μ ½mðBaL −1Þ þ nL: 1−ε
ð42Þ
The conditions for an interior solution are given for: ∂H −ε −ε ¼ C −λ ¼ 0⇒C ¼ λ ∂C
ð43Þ
∂H λβY λβY ¼ ð1−aL ÞμmB: ¼− þ μLmB ¼ 0⇒ 1−aL L ∂aL
ð44Þ
The equations of motion of the costate variables are given for:
Substituting Eqs. (28) and (29) in Eq. (30), following: g Y ¼ g þ α ðz−xÞ þ βðn−mÞ:
ð33Þ
˙ − λ¼
∂H λαY Y λ˙ þ ρλ⇒ ¼ ρ−α ¼− K K λ ∂K
ð45Þ
740
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
μ˙ ¼ −
∂H λβY þ μ ½mðBaL −1Þ þ n þ ρμ: ¼− L ∂L
ð46Þ
necessary to discover if it has some feasible way that has taken the economy to a BGP. Rewriting Eq. (51) in a BGP, with gB = 0, as
Substituting Eq. (44) in Eq. (46), and dividing both sides of Eq. (46) for μ, we get:
αz −g Y ¼ x −
μ˙ ¼ ρ þ m−mB−n: μ
and substituting this results in Eq. (50),with gB = 0, we have:
ð47Þ
The rate growth of the product is obtained by the logarithmic differentiation, with respect to time, of the equation Y = F(K,L) = AKα[(1 − aL)L]β, that is, g Y ¼ g þ αg K þ βg L −
βaL g L: 1−aL a
ð48Þ
But, to determine Eq. (48) it is necessary to know the rate of allocation of the labor. Differentiating logarithmically, with respect to time, Eq. (44) we get: λ˙ a˙ μ˙ þ gY þ L ¼ þ gL þ gB : 1−aL μ λ
g βmðB−1Þ β β þ αz−x þ þ n− g B : α α α α
8 > > z˙ > > > < x˙ > > > > > : a˙L
g mðB−1Þ n þ þ : αβ α α
α g mðB−1Þ n ρ þ þ þ : x ¼ 1− ε αβ α α ε
0b ð50Þ
¼
ð58Þ
1 ð1−εÞ½g þ βðmðB−1Þ þ nÞ ρ− : mBε β
ð59Þ
1 ð1−εÞ½g þ βðmðB−1Þ þ nÞ ρ− b1 mBε β
that is,
ð51Þ
β
ρ ρ−mBε −½mðB−1Þ þ n bgbβ −½mðB−1Þ þ n : 1−ε 1−ε
The linear approximation of Eq. (58) is, of a generalized form, given for: 2 2
3
6 z 6 4 x 5 ¼ 6 6 6 4 a˙L
g þ βmðB−1Þ þ βn −2βz þ α alpha x − 1− ε 0
ð52Þ
3 0
0
x
0
g þ βmðB−1Þ þ βn x − 1−aL α
7 7 7 7: 7 5
In the break-even point (z∗,x∗,a∗L) the linear approximation is: ð53Þ
ð54Þ
Now, Eqs. (53) and (54), when substituted in Eq. (51), considering constant returns of scale and gB = 0, provide, in a BGP: 1 g −ρ þ mðB−1Þ þ n : gY ¼ ε β
1 ½g−βmð1−BÞ þ βnz αh α ρi 2 x − 1− z þ x ε ε β g ð1−aL Þ x− ½mðB−1Þ þ n− −mBð1−aL Þ : α α
This is a feasible point since:
Now, given that, xx˙ ¼ g C −gK and differentiating logarithmically, with respect to time, Eq. (43), we'll get the same Eq. (34) of Appendix A, g C ¼ ρ ρ x˙ αz α ˙ ε − ε, with this x ¼ −ð1− ε Þz− ε þ x, and in a BGP, x ¼ 0, and of the results Eq. (53), it follows that
ð57Þ
2
¼
ð49Þ
In a BGP, z˙ ¼ 0 then z ¼
ð56Þ
−βz þ
¼
In similar way what already was a fact, the technology of the health system will be fixed in one determined way level that gB = 0. Since the rates of growth of z and x are given, respectively, for zz˙ ¼ gY −g K and xx˙ ¼ g C −g K , then, we can write: g βmðB−1Þ β z˙ þ n: ¼ g Y −g K ¼ −ð1−α Þz þ α α α z
βmð1−BÞ β g − n− α α α
These three equations (the equation above and equations zz˙ and xx˙ ), form the system
aL ¼ 1−
Now, substituting Eq. (50) in Eq. (48) and rearranging the terms, we have that: gY ¼
β g a˙L ¼ x − ½mðB−1Þ þ n− −mBð1−aL Þ: α α 1−aL
Substituting the results of Eqs. (45) and (46), in Eq. (48), we get: aL g ¼ αz−mðB−1Þ−n−gY þ g L þ g B : 1−aL aL
Considering a˙L ¼ 0 , following that a∗L = 1 or mB 1−a∗L ¼ x∗ −αg − βmðB−1Þ Þg mð1−εÞðB−1Þ Þn −αn , where x∗ −αg −βmðαB−1Þ−αn ¼ ρε−ð1−ε −ð1−ε α βε − ε ε . So:
This last equation can be written as aL μ˙ λ˙ g ¼ − −g þ g L þ g B : 1−aL aL μ λ Y
ð55Þ
By BGP definition, it must have g∗Y = g∗C = g∗K, where g∗Y is given for Eq. (54). In equal way what it was made in the Appendix A, it is
2 3 −βz z 6 alpha 4 x 5 ¼ 6 − 1− x 4 ε a˙L 0 2
0
x
0
1−aL
0
−mðB−1Þ 1−aL
3 7 7: 5
There are two negative eightvalues, r1 = − mB(1 − a∗)L and r2 = − βz∗, and one positive eightvalue, r3 = x∗, characterizing a saddle point. Appendix C In this appendix equations are deduced referring to the model which the health system's function effort is intensive in capital. The Hamiltonian of the problem (19) is:
H¼
C 1−ε −1 þ λðY−C Þ þ μ ½mðBaK −1Þ þ nL: 1−ε
ð60Þ
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
z and x is given, respectively, for written as:
The conditions for an interior solution are given by: ∂H −ε −ε ¼ C −λ ¼ 0⇒C ¼ λ ∂C
ð61Þ
∂H λαY ¼− þ μLmB ¼ 0: 1−aK ∂aK
ð62Þ
The equations of motion of the costate variables are given of:
∂H λβY þ μ ½mðBaK −1Þ þ n þ ρμ: μ˙ ¼ − þ ρμ ¼ − L ∂L
ð64Þ
Of Eq. (63), we get Y λ˙ ¼ ρ−α : K λ
ð65Þ
That's already a known result and Eq. (62) can be written as μLmb and therefore λβY βμmð1−aK ÞB ¼ : L α
λαY 1−ak
ð66Þ
ð68Þ
ε½g þ βmðB−1Þ þ βn−ðα−βÞρ : α ½β−α ð1−εÞ
ð75Þ
" g þ βðmðB−1Þ þ n þ β2 ρ α ½β−α ð1−εÞ:
ð76Þ
gY ¼
g þ β½mðB−1Þ þ n−αρ : β−α ð1−εÞ
ð77Þ
Now, the idea is to discover if it has feasible path that they lead to the BGP. Of Eq. (71) it elapses that, in a BGP,
αz −g Y ¼ x −
α g x − −mð1−BÞ−n β β
ð78Þ
and substituting this results in Eq. (70), we find: α g mβðB−1Þ mβBð1−aK Þ a˙k x− − −n− : ¼ β β α α 1−aK
ð79Þ
Now, these equations (the equation above and the equations of zz˙ and xx˙ ), form the system:
α g x þ þ mðB−1Þ þ n z 1− hβ α β ρi 2 ¼ x − 1− z þ x ε ε α g mβðB−1Þ mβBð1−aK Þ −n− : ¼ ð1−aK Þ x− − β β α α
ð69Þ
8 > > > z˙ > > < x˙ > > > > > :a˙K
ð70Þ
Substituting a˙K ¼ 0, following that a∗K = 1 or ð α Þ ¼ αβx−βg −mβðαB−1Þ− ðB−1Þþn n, where αβx−βg −mβðαB−1Þ−n ¼ βρ−ð1−εÞ½gþβm . With this, α
This last equation can be written as
From Eqs. (65) and (67), Eq. (69) can be written as:
Substituting this equation in Eq. (69) and rearranging the terms, we get: g α α þ αz− x þ mðB−1Þ þ n− g B : β β β
ð74Þ
x ¼ ðε−α Þ
ð67Þ
λ˙ a˙ μ˙ þ g Y þ K ¼ þ gL þ g B : 1−aK μ λ
gY ¼
ð73Þ
And now, substituting, Eqs. (75) and (76) in Eq. (71), and rearranging the terms, it is obtained:
Here, to specify Eq. (68) it is necessary to know the rate of allocation of the capital, fact that can be obtained by the logarithmic differentiation, with respect to time, of Eq. (62), namely:
aK mðα−βÞð1−aK Þ −g Y þ gL þ g B : g ¼ αz þ α 1−aK aK
g 1 α mðB−1Þ n 1− x þ þ : þ β β β β2 β
With this, Eq. (74) it will can be written as
The growth rate of product is obtained for logarithmic differentiation of equation Y = F(K,L) = A[(1 − aK)K]αLβ, that is,
aK μ˙ λ˙ g ¼ − −g þ g L þ g B : 1−aK aK μ λ Y
ð72Þ
Substituting Eq. (74) in Eq. (73), we get:
with this,
αaK g þ βg L : 1−aK aK
¼ g C −g K , can be
α ρ x ¼ 1− z þ : ε ε
z ¼
ðα−βÞð1−BaK Þ−βðB−1Þ −n −μ m α
g Y ¼ g þ αg K −
x˙ x
Now, again, it is necessary to determine x to know z (or the opposite) and this can be made from the logarithmic differentiation, with respect to time, of Eq. (61) substituted in Eq. (65) that it provides, the same Eq. (34) of Appendix A, that is, g C ¼ αzε −ρε. As xx˙ ¼ g x ¼ g C −gK , it results that xx˙ ¼ αzε −ρε−z þ x. In a BGP, x˙ ¼ 0, and so:
¼
Now, substituting this result in the expression between keys of Eq. (64), we get
ðα−βÞð1−BaK Þ−βðB−1Þ μ˙ −n : ¼ρþ m α μ
¼ g Y −g K and
In a BGP, z˙ ¼ 0, and so then
ð63Þ
z˙ z
g α z˙ x þ mðB−1Þ þ n: ¼ −ð1−α Þz þ 1− β z β
z ¼
∂H ˙ − λ¼ þ λρ ¼ −λ F K þ λρ ∂K
741
ð
2
−β z þ
ð80Þ
mB 1−a∗K
aK ¼ 1−
ð71Þ
This model, also considers that the health system technology it is given, that is, gB = 0. Considering that the growth rate of the variables
¼
α βρ−ð1−εÞ½g þ βðmðB−1Þ þ nÞ : βmB β−α ð1−ε Þ
This a feasible point if
0b
1 βρ−ð1−εÞ½g þ βðmðB−1Þ þ nÞ b1: βmB β−α ð1−εÞ
ð81Þ
742
M.A.L. de Freitas, A. Stamford da Silva / Economic Modelling 35 (2013) 734–742
Determining the value of g in the inequality above, it is obtained: β
h ρ i ρ−mBε −½mðB−1Þ þ n bgbβ −½mðB−1Þ þ n : 1−ε 1−ε
ð1−εÞÞ , where, clearly, The amplitude of this interval is given for mBðβ−α αð1−ε Þ only the technology of the health system influences. It observes that the limits are generalizations of the previous results. One more time, the introduction of technological level in the health sector and the population growth, do not attenuate the limitation of the technology rate of the productive sector. Now, the size of the productive sector is given by
1−aK ¼
1 βρ−ð1−εÞ½g þ βðmðB−1Þ þ nÞ : mβB β−α ð1−ε Þ
The linear approximation of Eq. (80) is, of a generalized form, given for: 2
g 6 −2βz þ β þ mðB−1Þ þ n 6
z 6 α 4 x 5 ¼ 6 − 1− x 6 β 6 a˙K 4 0 2
3
α 1− β x
3 0 0
α g α 1−aK x − −mðB−1Þ−n β β β mβBð1−a∗K Þ α
That it has at least a negative eightvalue given for r 1 ¼ − the break-even point (z∗,x∗,a∗K).
7 7 7 7: 7 7 5
in
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