Energy saving technological progress in overlapping generations economies with renewable and non-renewable resources
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(2) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. :00 :25. ar. ih:. 03. /02. /20. 19. 11. se ], T. .51. .72. er. sit. es. i],. IP. :[ 13. 9.1. 79. ge n:. [B. ilk. en. tÜ niv. il. B 28. ire. 1 There are endogenous growth models with infinitely lived agents (ILA) dealing with sustainability of long run growth under exhaustible resources (Stollery, 1998; Schou, 2000, 2002; Grimaud and Rouge, 2005, 2008; Groth and Schou, 2007). They conclude that under technological progress no matter it is taken as exogenous or endogenous growth is sustainable in the long-run despite the finite resource stock. Although there are numerous papers addressing these issues with non-renewables (Guruswamy Babu and Kavi Kumar, 1997; John and Peccheccino, 1994) and renewables (Gerlagh and Zwan, 2001; Koskela et al., 2008) separately, there are limited studies within OLG framework considering these resources as alternative sources of energy and analyzing their effect on dynamics of growth.. İnd. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. 1. INTRODUCTION As the worldwide energy demand has continuously been increasing, the question of whether the scarcity of natural energy resources limits economic growth receives special attention. The recent fluctuations in the oil prices along with the threat of climate change have further stimulated the interest in the issue of sustainability as well. Among others, substitution of renewable energy resources with non-renewables and developing energy saving technologies are the most prominent suggestions to overcome the problem. While substitution to renewable resources is accepted to contribute to more sustainable economic development paths (Daly, 1990; Andre and Cerda, 2005); energy efficiency programs are also offered as a policy response by several policy makers and environmental groups (Cabinet Office, 2001; DEFRA, 2005; Allan et. al, 2007). This paper aims to answer whether substitution of non-renewable energy resources with renewables and progress in energy saving technologies will bring growth in the long run. We develop a two-period overlapping generations model in which the energy is an essential input to production and exogenous resource augmenting technical change drives long-run growth. To analyze how scarcity limits can be alleviated by technological progress or substitution of non-renewable resources with renewables, we will address the following questions: (i) Under which circumstances will the economy prevail long run growth? (ii) What determines the rate of depletion? (iii) How will the intensity of renewables in energy production affect growth? (iv) What will be the effect of the energy saving technological progress on the long run growth? (v) How will the patience of the generations and the population growth rate affect these results? (vi) Will the long run growth be optimal and sustainable? Although there is a vast literature analyzing the sustainability of growth in the presence of non-renewable or renewable resources, most of these papers focus on just one type of resources1. Few exceptions in the literature are Tahvonen and Salo (2001), Andre and Cerda (2005), Di Vita (2006), Nguyen and Nguyen-Van (2008), Maltsoglou (2009) and Hung and Quyen (2008). Tahvonen and Salo (2001) considers the problem of substitutability between exhaustible and renewable resources in terms of their costs but not in terms of relative scarcity. Although Andre and Cerda (2005) takes natural growth and technological substitution possibilities into account, they focus on the optimal.
(3) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. :00 :25. ar. ih:. 03. /02. /20. 19. 11. se ], T. .51. .72. sit. es. i],. IP. :[ 13. 9.1. 79. ge er. n:. [B. ilk. en. tÜ niv. il ire. İnd. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. combination of these resources if there is no technological progress and no other inputs (such as capital, labor). Di Vita (2006), Nguyen-Van (2008) and Maltsoglou (2009) consider labor, physical capital and both types of energy resources as inputs to production. Yet, these studies analyze the behavior of the economies with infinitely lived agents. They also do not consider the effects of energy saving technological progress2. To our knowledge, Hung and Quyen (2008) is the only study within the OLG framework while considering both inputs. However, they focus on the effects of endogenous fertility decisions when there is no technological progress and also renewable resource is chosen to be solar energy which is produced from backstop capital. It is well known that although improvements in technology lower the energy consumption, through economic growth, they will in turn create further energy demand. In fact, there are several papers studying the tradeoff between energy saving technological progress, energy consumption and growth (see Van Zon and Yetkiner, 2003; Boucekkine and Pommeret, 2004; Azomahou et al., 2004; Perez-Barahona and Zou, 2006; Yuan et al., 2009). Yet none of them gives particular attention to the intergenerational aspects (such as sustainability) or focuses on the presence of the natural resources. Perez-Barahona (2011) investigates the effect of energy saving technological progress on growth under the presence of an exhaustible resource but does not consider an OLG framework and alternative sources of energy. Valente (2005) accounts for a renewable resource in an OLG economy under resource augmenting technology leaving alternative resource aside. This paper tries to fulfill the above mentioned gaps in the literature through studying the presence of both renewable and non-renewable energy resources and resource augmenting technological progress in an analytically solvable exogenous growth overlapping generations model. To analyze the presence of both resources, we differentiate them according to their relative scarcity. Non-renewable resources are scarce whereas renewables are not due to their regeneration property. The reason behind assuming resourcesaving technological progress stems from the evidence that the energy-saving technological progress has proved to be significant in the last two decades3. We prefer OLG framework to infinitely lived agents (ILA) since the latter ignore `generation overlap and treat society in each period as a single generation caring about (and also discounting) the welfare of its immediate descendants, which has complete control over the rate of resource use and the saving rate’ (Mourmouras, 1991, p. 585). In addition, as Agnani et al. (2005) indicates, 2 Nguyen and Nguyen-Van (2008) mentions that if they assume a Cobb-Douglas production function then a parameter could capture the resource saving technological progress yet in the rest of the paper they do not focus on the effects of this parameter. 3 Newell et al. (1999) reveals that increasing energy prices result in energy saving innovations in the USA. Through investigating the sectors of Dutch economy, Kuper and Soest (2006) shows that energy saving technological progress occurs after periods of high and rising energy prices.. 29.
(4) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. .72. .51. ], T. ar. ih:. 03. /02. /20. +0 30 0. :00 :25. 19. 11. se. er. sit. es. i],. IP. :[ 13. 9.1. 79. ge ire. n:. [B. ilk. en. tÜ niv. il. 30. İnd. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. the OLG models can be preferred to ILA in analyzing the sustainability of long-run growth with exhaustible resources since the natural resources may act as stores of values between different generations. We provide an analytical characterization of the balanced growth path (BGP) and analyze the conditions for positive long run growth. Our model is building upon Agnani et al. (2005), which studies the BGP of an OLG economy with exogenous technical progress where non-renewable resource is an essential input to the production. They show that a sufficiently high labor share is necessary for the economy to exhibit a positive steady state growth rate. Our results also reveal that the share of labor in production has to be sufficiently high in order to yield positive growth along the BGP. However, the constraint on labor share is less binding, compared to Agnani et al. (2005). We show that the share of renewables in energy production, the resource saving technological progress and the regeneration factor are among the key variables having an effect on the required labor share and hence the possibility of long run growth. What determines the rate of depletion and how it is determined, is quite important as it paves the way for understanding the limits to growth. To answer this question, we investigate the effects of discount factor, resource augmenting technological progress and intensity of resources in energy production on the depletion rate. As Smulders (2005) emphasizes, the increase in the discount factor and hence the patience of the households, is expected to decrease the depletion rate --which is also confirmed in our results-- whereas the effect of the exogenous resource saving technological progress on the rate of depletion is accepted to be ambiguous due to opposing income and substitution effects. Under the productivity gains, households would attach a greater value to energy resources in future periods since these resources will be more productive. Households, thus, save more on these resources which demonstrates the substitution channel. On the other hand, more output would be obtained given a resource stock when the productivity increases. As a result, the households would know that they will have more income in the future. The income effect works through consumption smoothing and the households will consume more. We find that along the BGP, the substitution effect dominates the income effect as long as the depletion rate is slightly higher than its lower bound. Thus, the higher the resource saving technological progress the economy depletes its energy resources less. As regards the circumstances, increasing/decreasing the resource intensity of energy production promote growth, our main finding is that the effects of an increase in the intensity of the renewable resources in producing energy has positive long run growth effects. We present that the patience of generations has important long run implications in this context. For more.
(5) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. :00 :25. se /02. 03. ih:. ar. ], T. .51. .72. IP. i],. es. sit. er. il. :[ 13. 9.1. 79. ge. /20. 19. 11. 2. THE MODEL We consider a two-period overlapping generations model in discrete time with an infinite horizon. At each period t , a generation of agents appears and lives for two periods, young and old. The population in period t consists of N t young and N t −1 old individuals. We assume that the growth rate of population is constant so that N t +1= (1 + n) N t . In comparison with a standard OLG model4, the novel feature of our analysis is to consider energy as an essential input to production and take into account that it is built upon both renewable and non-renewable resources5. At each period, a single final good is produced in the economy by means of physical capital K , labor N, and energy Ξ . This physical good is either consumed or invested to build future capital. The energy input is obtained from the stock of renewable and non-renewable energy resources denoted by R and E respectively. The renewable resource is assumed to regenerate itself with g ( R) where g ′( R ) > 0 at every period. These resources can act as both stores of value and inputs to the production process.. İnd. ire. n:. [B. ilk. en. tÜ niv. 2.1. The optimal behavior of the agents All agents have rational expectations and each generation consists of a single representative agent. Moreover, all agents in this economy are pricetakers and all the markets are competitive. At a given date, young households work, consume and invest a part of their income in physical capital which is rented and used by the firms in the next period. They invest another part of their income to purchase ownership rights for the renewable and the non-renewable energy resources. When old, they consume their entire income generated from the returns on their savings,. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. patient economies, an increase in resource saving technological progress will result in higher growth. With the presence of renewables, it is shown that the constraint on the labor share which is required to guarantee the long run growth is relaxed. We also show that the sustainability of the economy depends on the energy saving effect of the technological progress and the depletion rate of the resources which in turn depend on the rest of the parameters in the economy. Finally, we find that the BGP can turn out to be optimal. The paper is structured as follows: Section 2 presents the model and the equilibrium conditions for a decentralized economy. Section 3 analyzes the existence and the uniqueness of the BGP and investigates the optimality and the sustainability conditions for BGP. Section 4 performs the comparative statics analysis and Section 5 concludes.. 4 See De la Croix and Michel (2002) for a comprehensive treatment of the OLG models. 5 See, for the discussion of energy being an essential input to production, Ayres et al. (2003; 2005; 2007) and Warr et al. (2006; 2008).. 31.
(6) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. :00 :25. se /02. 03. ih:. ar. ], T. .51. .72. ge. /20. 19. 11. 2.2. Consumers In his first period of life (when young at period t ), the representative individual is endowed with one unit of labor that he supplies inelastically to ct allocates + st + Pt r rthis Pt e et +1 among firms. His income is equal to the real wage wt =He t +1 +income r e Pt rt +1 + Pin current consumption ct , w savings and the purchase of t = ct + st +invested t etfirms +1 the ownership rights for the renewable rt +1 and the non-renewable resources et +1. Budget constraint of period t is. wt = ct + st + Pt r rt +1 + Pt e et +1. (1). ,. xt +1 =. X t +1 Nt +1. IP. i],. es. sit. er. tÜ niv. Zt +1 Nt +1. (2). .. en. where zt +1 =. dt +1 =Qt +1st + Pt +r 1 (rt +1 + g (rt +1 )) + Pt e+1et +1. ilk. il. :[ 13. 9.1. 79. where Pt r and Pt e denote the prices of renewable and non-renewable energy resources, respectively. Note that Rt +1 = N t +1rt +1 , and Et +1 = N t +1et +1. In the second period of his life, the agent is retired and he consumes his entire income generated from the returns on his savings Qt +1st = (1 + qt +1 ) st , and the revenue from selling his stock of energy resources to the firms. Accordingly, his consumption is. B. ire. n:. [B. The preferences of the representative agent is defined over his consumption bundle ( ct , dt +1 ). We assume that they can be represented by ,d ) u ( c ) + β u ( d ) , an additively separable life-cycle utility function U ( c= where β ∈ ( 0,1) is the subjective discount factor. In particular, we adopt a. İnd. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. and from selling their stock of energy resources to the firms. Following Koskela et al. (2008), we assume that at the beginning of each period t , the old agents (generation t − 1 ) own the stock of all energy resources and sell them to the firms. As in Dasgupta and Heal (1974), it is assumed that there are no extraction costs. Firms decide on the amount of renewable and non-renewable energy resources that will be used in the production process, Z t and X t , respectively. Before the end of the each period t , firms sell the remaining stock of renewable resources Rt +1 and the non-renewable resources Et +1 to the young agents (generation t )6. Accordingly, the evolution dynamics of the energy resources can be formalized as follows: Rt +1 =+ Rt g ( Rt ) − Z t , Et += Et − X t . 1. 6 In Olson and Knapp (1997), although old agents own the resource stock they do not sell all of the resource to the firms. Instead, they choose how much of their stock will be sold to the production sector. Then through the asset market the unextracted resource stock is transferred from the old generation to the young generation. The resource accumulation equations does not differ by this specification.. 32.
(7) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. /20. 19. 11. wt = ct + st + Pt r rt +1 + Pt e et +1 , dt +1 =Qt +1st + Pt +r 1 (rt +1 + g (rt +1 )) + Pt e+1et +1 , ct ≥ 0, dt +1 ≥ 0, rt +1 ≥ 0,et +1 ≥ 0.. /02. subject to. :00 :25. se. {ct , dt +1 , st , rt +1 , et +1}. 03. ], T .51 .72. 79. 9.1. (4). IP i],. es. (5). sit. il. dt +1 = β Qt +1 , ct. (3). :[ 13. Pt +r 1 Qt +1 = . r (1 + g ′(rt +1 )) Pt. ar. Pt e+1 = Qt +1 , Pt e. ih:. ge. We have then the following first-order conditions for the consumer’s optimization problem:. İnd. ire. n:. [B. ilk. en. tÜ niv. er. Equation (3) gives the equalization of discounted marginal utilities where the marginal rate of substitution between the current and the future consumption is equal to their relative prices. Equations (4) and (5) present no-arbitrage conditions among different types of savings implying that the marginal return on investing in the exhaustible resource is equal to the marginal return on investing in the renewable resource taking the regeneration factor into account. In other words, an increase in the price of the exhaustible resources from period t to t + 1 is higher than that of the renewable resources reflecting the relative scarcity of the non-renewable resources.. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. logarithmic instantaneous utility function u since we are mainly concerned with the existence of the balanced growth path and its qualitative properties7. Taking the prices of the energy resources and wages as given, the representative agent maximizes his life-time utility by choosing the young and the old periods’ consumption and the ownership of the energy resources. The optimization problem of the representative agent born at time t can be formalized as follows:. max ln ct + β ln dt +1. 7 See, among others, King and Rebelo (1993) and Agnani et al. (2005), for the need to assume comsumer’s preferences with CIES in order to have the existence of a BGP.. 33.
(8) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. l. Yt = Ξ t α1 K tα 2 N tα3 , α i > 0 and ∑ α i = 1,. (6). i. se. = Ξ t At X tρ Z t1− ρ , 0 ≤ ρ ≤ 1,. +0 30 0. At +1 = (1 + a ) At , a > 0.. (7). (8). :00 :25. where. İnd. /02. 03. ih:. ar. ], T. .51. .72. IP. i],. es. sit. er. ire. n:. [B. ilk. en. tÜ niv. il. :[ 13. 9.1. 79. ge. /20. 19. 11. The energy input, Ξ t is produced from non-renewable ( X t ) and renewable energy resources ( Z t ) by means of a Cobb-Douglas production technology. The intensity of non-renewable resources in producing energy is captured by ρ . As ρ increases the production of energy becomes more intensive in using non-renewable resources than using renewable resources. The productivity of resources in producing energy is represented by At . If the productivity of the resources in producing energy increases, fewer amounts of resources will be needed to produce the same amount of energy. Therefore, the technical progress (increase in productivity) which is captured by a is considered to be energy saving. Generally, technical progress is considered as Hicks-neutral under Cobb-Douglas specification. The importance of distinguishing the energy-saving effect of the technical progress from the input neutral technological progress is that the prospects for sustainability depend on the energy-saving effect of technical progress and not on its global effect on the output levels. Note that the assumption of perfect substitutability between all inputs does not stem from theoretical considerations only. As a matter of fact, the extent to which capital and energy are substitutes or complements in production is highly debated in the literature. Even in the early literature, Hudson and Jorgenson (1973) and Berndt and Wood (1975) found that capital and energy were complements, while Humphrey and Moroney (1975), Griffin and Gregory (1976) and Halvorsen (1977) concluded that they were substitutes. Apostolakis (1990), suggested that the studies based on time-series data reflect short-term relationships and hence these studies concludes capital and energy to be complements. However, he claims that the cross-sectional analysis reflects the long term relationship thus the studies based on cross-. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. 2.3. Firms Firms are owned by the old households and produce a homogenous consumption/investment good under perfect competition. Production at the final good sector is made through Cobb-Douglas constant returns to scale technology8:. 8 Taking into account the use of energy, Ayres (2008) shows that Cobb-Douglas production function fits to the economic growth for the US and Japan economy in the 20th century. Also, Serrenho et al. (2010) show that the inclusion of energy-related variables, increases the explanatory power of the models for a panel data of EU-15 countries between 1995 to 2007.. 34.
(9) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. se /20. 19. 11. :00 :25. where 0 ≤ δ ≤ 1 denotes the depreciation rate of capital. At an interior solution of the firm’s optimization problem, where all variables are expressed in per capita terms (kt = KNtt , zt = NZtt and et = NEtt ) , the following first order conditions are satisfied :. 1 α 2 Atα xt α ρ zt α (1− ρ ) ktα −= qt + δ ,. α 3 Atα xt α ρ zt α (1− ρ ) ktα = wt ,. (12) α1 (1 − ρ ) Atα xt α ρ zt α (1− ρ ) −1ktα = Pt r ,. α1 ρ Atα xt α ρ −1 zt α (1− ρ ) ktα = Pt e .. 1. 1. 1. 1. ih:. 2. 1. 2. (11). (13). 79. 1. 2. ar. 1. (10). ], T. 1. .51. 1. /02. 2. 03. 1. .72. 1. ge. 1. i],. IP. (14). es. (1 − ρ ) xt Pt r = . ρ zt Pt e. il. :[ 13. 9.1. Re-arranging equations (12) and (13), the optimal mix between the exhaustible and renewable energy resources can be obtained as:. {. }. İnd. ire. n:. [B. ilk. en. tÜ niv. er. sit. By Equation (14), we observe that the optimal mix between the renewable and non-renewable resources depend on their prices and the elasticity of substitution between the two sources of energy resources.. 2.4. The competitive equilibrium A dynamic competitive equilibrium of this overlapping generations ∞ , and the economy is determined by the sequence of prices wt , qt , Pt e , Pt r t =0 feasible allocations. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. sectional data imply the perfect substitutability between energy and capital inputs. As with constant returns to scale, the number of firms does not matter and the production is independent of the number of firms that use the same technology, we are concerned with the problem of a representative firm. Under this perfectly competitive environment, the representative firm producing at period t maximizes its profit by choosing the amount of labor, physical capital and the energy inputs that will be utilized in the production process9:. = max π t Atα1 X tρα1 Z t(1− ρ )α1 K tα 2 N tα3 − (qt + δ ) K t − wt N t − Pt r Z t − Pt e X t , (9) {Kt , Nt , Zt , X t }. 9 Maximization problem of the firm is. = max π t Atα1 X tρα1 Z t(1− ρ )α1 K tα 2 N tα3 − (qt + δ ) K t − wt N t − Pt r ( Rt + g ( Rt ) − Rt +1 ) − Pt e ( Et − Et +1 ), if the cash flow going through the firm is the explicitly written. Taking into account the dynamics of the energy resources, the consumption of the old individual at t + 1 can be recast as Equation (Profit).. 35.
(10) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. At +1}t =0 given positive initial values for the state variables {k0 , E0 , R0 , A0 } and the law of motion of At and N t such that the consumers maximize their life-time utility, firms maximize their profits and all markets clear at every period t : t. t. t. t. t. t. t. t. ∞. t +1,. = st kt +1 (1 + n),. (15). (16). et = (1 + n)et +1 + xt ,. (17). +0 30 0. rt + g (rt ) = (1 + n)rt +1 + zt ,. se. /02. 03. ih:. ar. ge. /20. 19. 11. :00 :25. (18) yt = ct + dt (1 + n) −1 + st . Accordingly, a dynamic competitive equilibrium is a solution of the equation system, (1)-(18). Equation (15) indicates that the capital stock at t + 1 is fully determined by saving decisions made at t , since the output is used either for consumption or investment in capital goods. The following two equations, (16) and (17) reveal the resource constraints for the energy resources. Equation (18) is the market clearing condition in the output market which holds by Walras’ law.. 36. İnd. ire. n:. [B. ilk. en. IP. i],. es. sit. er. tÜ niv. il. :[ 13. 9.1. 79. .72. .51. ], T. 3. THE BALANCED GROWTH PATH We aim to analyze the feasibility of positive long run growth in the economy and hence we will focus on the balanced growth path. To guarantee the analytical solution of the balanced growth path we assume that the renewable resource regenerates linearly, i.e., g ( Rt ) = ΠRt for some constant regeneration factor 0 < Π < 1. In order to characterize the balanced growth path of this competitive economy we will first define growth factors of the variables. We denote the growth factor of any variable at by γ a which is the ratio at +1 / at . Along the balanced growth path γ a − 1 will represent the growth rate of the corresponding variable. Then, we will reduce the system in terms of the depletion rates of resources. How much of the energy resources are used in the production compared to the total resource stock is represented by these rates. The depletion rate of the non-renewable resources is defined as τ t = extt t and the depletion rate of the renewables can be defined as ξt = rt (1z+Π ) . Proposition 1: Along a balanced growth path of this economy all variables grow at a constant rate. The balanced growth path is described by the stationary depletion rates, τ= τ= τ t +1 and ξ= ξ= ξt +1 which solve the following non-linear t t equations. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. t. l. {c ,d , s ,r ,e , x , z , y ,Ξ ,k.
(11) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. α 2γ. γ (1 + n) 1−τ. = γ Q 1,= and ξ τ .. /02 03 ih: ar ], T .51. ,. .72. (1 + Π )(1 − ξ ). 79. r. 9.1. γp =. :[ 13. γ (1 + n) , 1−τ γ (1 + n). ge. e. /20. (1 + a )(1 + Π ) ρ (1 − ξ ) ρ (1 − τ )(1− ρ ) , 1+ n = γ z γ x (1 + Π ),. γ en =. γp =. +0 30 0. 19. 11. 1+ n. :00 :25. se. γA = (1 + a ), γ N = (1 + n), 1−τ γ= γ= , e x. IP. i],. es. sit. er. İnd. ire. n:. [B. ilk. en. tÜ niv. il. Proof: See the Appendix. From the dynamics of the non-renewable resource stock (17), it can be inferred that et is decreasing as long as there is a positive amount of extraction. Along the balanced growth path, a constant decrease in nonrenewable resource stock is only possible with a constant depletion rate: τ= τ= τ t +1. Similarly, to guarantee a constant growth rate for the renewable t resource stock, the depletion rate of renewables should also be constant ξ= ξ= ξt +1 along the balanced growth path. Therefore, the energy resources t used in production will decline over time indicating an asymptotic depletion. However, the rate of decrease in renewable resource stock used in production will be smaller than that of the non-renewable resources due to the regeneration factor. As the non-renewable stock is declining along the balanced growth path, the price non-renewables are growing at a higher rate than income and that of the renewable resources. However, the comparison between the price of renewables and income depends on the relationship between the growth. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. + (1 − δ ), (1α+3ββ ) − α(11ρ+(1n−)ττ ) − α1 (1− ρξ )(1−ξ ) where α (1− ρ ) 1 − τ (1α−1αρ2 ) (1 + Π )(1 − ξ ) (11 −α2 ) α1 (1 + Π )(1 − ξ ) , γ= γ= , γ= (1 + a ) (1−α2 ) r z 1+ n 1+ n 1 + n and the following growth rates γ= γ= γ= γ= γ= γ= γ, y k c d s w. 37.
(12) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. :00 :25. ar. ih:. 03. /02. /20. 19. 11. se ], T .51. 79. 9.1. IP. i],. es. sit. er. tÜ niv. en. r. il. γ γ (1 + n) = , 1−τ γx γ γ (1 + n) γ= = , p γ z (1 + Π )(1 − τ ) γr γz 1−τ = = , γ= γx = e e. :[ 13. and the following growth rates: γ= γ= γ= γ= γ= γ= γ, y k c d s w. γ= p. .72. ge. α 2γ γ (1 + n) = + (1 − δ ), α3β 1 −τ (1+ β ) − α(11 +(1n−)ττ ) where α1 α1 α1 (1− ρ ) (1 − τ ) (1−α2 ) (1−α 2 ) γ = (1 + a ) (1 + Π ) (1−α2 ) , (1 + n ) . [B. 1+ n. ire. n:. (1 + a )(1 + Π ) ρ (1 − τ ) , 1+ n. İnd. ρ (1− ρ ) = γ en γ= Aγ x γ z. (1 + Π ). ilk. (1 + Π ). B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. rate of the population and regeneration factor. As the regeneration rate decreases or the population growth rate increases, the increase in the price of renewables will be higher than that of income. In parallel with Agnani et al. (2005), Proposition Exist BGP shows that income, capital, consumption, savings and wages grow at the same rate and the interest rate is constant along the balanced growth path. It should be noted that, in line with Agnani et al. (2005), although we model technological progress as exogenous, the growth rate of the economy depends on all of the parameters of the model, actually a feature of endogenous growth models. In addition to this striking result, we can also analyze the effects of the regeneration factor and the intensity of nonrenewables in energy production explicitly in our set-up. We will now prove that the balanced growth path of this model described by the above system has a unique solution with a constant depletion rate. To do so the system of equations will be recast in terms of a single depletion rate. Corollary 2: Any balanced growth path of this economy is characterized by a stationary depletion rate, τ which is the solution of the following nonlinear equation. γA = (1 + a ) , γ n = (1 + n), and γ Q = 1. α1 Proposition 3: A unique stationary equilibrium exists if <τ (1 + n ) (1+ββ ) + α1 α 3 Proof: See the Appendix.. 38. < 1..
(13) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. :00 :25. ar. ih:. 03. /02. /20. 19. 11. se ], T. .51. .72. 79. ge IP. i],. es. sit. er. İnd. ire. n:. [B. ilk. en. tÜ niv. il. :[ 13. 9.1. Proof: Proposition (3) establishes a lower bound for the depletion rate in α1 the economy by α3 (1+ n ) (1+ββ ) +α1 < τ , necessary for the existence of a balanced growth path. Moreover, the upper bound for the depletion rate is established when γ > 1. Thus, τ ∈ α (1+ nα) 1 β +α ,1 − (1+ a )ρ (1+(1b+)1n−ρ) (1+Π )1−ρ . The economy 3 (1+β ) 1 will contract if the depletion rate is higher than its upper bound. In fact, the economy will not exhibit a positive growth if the lower bound for the depletion rate is higher than its upper bound. We show that the share of labor in production has to be sufficiently high in order to yield positive growth along the balanced growth path. This condition highlights that, a minimum amount of labor share is necessary for the young to earn high enough wages to finance their investments. However, the option of saving on renewables other than just capital and non-renewable resources, relaxes the constraint on the labor share. As a result, compared with Agnani et al. (2005), the constraint on labor share is less binding. This result stems from the fact that Agnani et al. (2005) does not take into account the presence of renewables. With Proposition (2), we demonstrate that the share of renewables. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. Proposition (3) indicates that a unique balanced growth path exists if the depletion rate is higher than some critical level which is positively related with the share of energy input in production and inversely related with the share of the labor in production, the growth rate of population and the discount factor. The positive growth along the balanced growth path is not guaranteed under Proposition (3). In fact, it is shown that even without the incorporation of scarce natural resources the OLG economies may contract. To illustrate, Galor and Ryder (1989) examines an OLG economy without natural resources and show that unless restrictions on the nature of the interaction between technology and preferences are satisfied the economy may contract. There are also studies indicating the possibility of contraction if natural resources are taken into account. For instance, Agnani et al. (2005) show that high enough labors share is a necessary condition for the economy to exhibit positive growth. In particular, they mention that the young generations need to earn high enough wages to do savings on capital and exhaustible resources whose prices are increasing along the balanced growth path. They conclude a minimum amount of labor share is necessary to guarantee such a wage income. Proposition (4) The economy will contract unless the labor share is high enough, i.e., α1 (1 + a ) −1 (1+ββ ) (1 + Π ) ρ −1 α3 ≥ . 1 − (1 + n)(1 + a ) −1 (1 + Π ) ρ −1 . 39.
(14) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. (19). 11. :00 :25. se. 1−τ (1 + a )( )(1 + Π )1− ρ ≥ 1. 1+ n. 10 Specifically, if. Ut. /02. 03. ih:. ar. ], T. .51. .72. IP. i],. es. sit. er. İnd. ire. n:. [B. ilk. en. tÜ niv. il. :[ 13. 9.1. 79. ge. /20. 19. Proof: See the Appendix. The above condition clearly shows that the sustainability of the economy depends on the energy saving technological progress but not on the total factor productivity. In addition, it depends on the depletion rate which in turn depends on all of the structural parameters of the economy. It can be observed that the higher the patience of the individuals, the higher the share of renewables in production and the regeneration rate, the more sustainable is the economy. However if the population growth rate increases, as there are more future generations it will be more difficult to sustain growth. To derive the conditions for intergenerational optimality, we assume the existence of a social planner whose maximizes a discounted sum of utilities for all current and future generations with respect to the resource constraints of the economy. The optimal balanced growth path is characterized by: (a) Income, capital, consumption growing at the same rate γ so that γ= γ= γ= γ= γ. y k c d (b) Energy resources used in production will decline over time indicating 1 an asymptotic depletion: γ= γ= e x 1+ R , where R denotes the subjective discount factor of the social planner. (c)The rate of decrease in renewable resource stock used in production will be smaller than that of the exhaustible resources: γ r = γ z = (1 + Π )γ x . In accordance with these we have the following Proposition on the optimality of the competitive equilibrium. Proposition 6: The competitive balanced growth path is Pareto optimal as long as τ = 1+RR .. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. in energy production, the technological progress in producing energy and the regeneration factor are among the key variables affecting the required labor share and hence the possibility of long run growth. 3.1. Sustainability and Optimality After demonstrating the existence of the unique competitive balanced growth path, the following propositions analyze whether this unique path is sustainable and/or optimal. In line with recent literature, we define a sustainable path a path along which welfare is non-declining over time10. Proposition 5: A necessary and sufficient condition for sustainability in this economy is to yield positive growth along the BGP γ y ≥ 1, so that. denotes the lifetime utility of an agent born in period. U t +1 (ct +1 , dt + 2 ) ≥ U t (ct , dt +1 ) 40. t,. sustainability requires.
(15) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. Proof: See the Appendix.. +0 30 0. :00 :25. ar. ih:. 03. /02. /20. 19. 11. se ], T. .51. .72. 9.1. 79. ge IP. i],. es. sit. er. İnd. ire. n:. [B. ilk. en. tÜ niv. il. :[ 13. Proof: See the Appendix. As the depreciation rate δ increases, capital becomes scarce compared to the energy resources. This scarcity will result in an increase in the price of capital relative to the prices of energy resources. Thus, the agents will demand more resource assets than capital for their savings. There will be less resource for production which will in turn yield a lower depletion rate. As a result, the economy grows at a higher rate along the balanced growth path. The higher the discount factor β - i.e. the more patient the generations are- when young households will consume less and save more. Since agents save more, the depletion rate along the balanced growth path will decrease and therefore the economy will grow at a higher rate along the balanced growth path. As Smulders (2005) emphasizes, the effect of the exogenous resource saving technological progress on the rate of depletion is accepted to be ambiguous due to the opposing income and substitution effects. Under the productivity gains, households would attach a greater value to resources in future periods. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. 4. COMPARATIVE STATICS We analytically prove the effects of the following parameters on the depletion rates of the resources and on the long run growth: (i) a change in the depreciation rate, (ii) a change in the discount rate, (iii) a change in the resource saving technological progress (iv) a change in the intensity of nonrenewables in energy production, and (v) a change in the regeneration factor. Proposition (i) Higher depreciation of capital brings about lower depletion rates and ∂τ ∂γ < 0, > 0. higher growth along the balanced growth path: ∂δ ∂δ (ii) More patient generations will deplete their natural resources less and ∂τ ∂γ < 0, > 0. benefit from higher growth along the balanced growth path: ∂β ∂β (iii) Economies with higher resource saving technologies grow faster along the balanced growth path ( ∂∂γa > 0) and deplete their resources less ( ∂∂τa < 0). (iv) Economies with higher share of renewables in energy production have lower depletion rates ( ∂∂τρ < 0) and will exhibit higher growth ( ∂∂ργ > 0) . (v) Economies with higher regeneration rates in the renewable resources grow ∂γ faster along the balanced growth path ( ∂Π > 0) and deplete their resources ∂τ less ( ∂Π < 0).. 41.
(16) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. .72. .51. ], T. ar. ih:. 03. /02. /20. +0 30 0. :00 :25. 19. 11. se. er. sit. es. i],. IP. :[ 13. 9.1. 79. ge ire. n:. [B. ilk. en. tÜ niv. il. 42. İnd. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. since these resources will be more productive. Thus, households save more on these resources which demonstrated the substitution channel. On the other hand, when the productivity increases, more output would be obtained given a resource stock. As a result, the households would expect to have more income in the future. The income effect works through consumption smoothing and households will consume more. In the proposition stated above, it is shown that as long as the depletion rate along the balanced growth path is slightly higher than the lower bound of the depletion rate, the higher the resource saving technological progress, the economy will deplete its corresponding energy resource less and have higher growth rate along the balanced growth path. Thus, the higher the resource saving technological progress through the income effect, the economy will deplete its energy resources less. Due to the regeneration property of renewable resources, economies with higher share of renewables in energy production (lower ρ ) have lower depletion rates as long as the depletion rate at the balanced growth path is slightly higher than the existence lower bound. Comparing identical economies with one having a higher share of renewables reveals that the economy which is using renewables more intensively in energy production will deplete its resources less and this will induce higher growth. Economies with higher regeneration rate have more renewable resources and the constraint on growth due to limited energy sources become less binding. Thus, through a lower depletion rate economies grow faster at the balanced growth path. Similar to the cases discussed above, the depletion rate at the balanced growth path must be slightly higher than the lower bound of it for this result to hold. The effect of an increase in the population growth rate (higher n ) on the depletion rate is found to be ambiguous. This creates further uncertainty for the effect of an increase in population growth on the long run growth. As the population is higher, there is a need for higher consumption. This will result in higher depletion rates which indicate a positive relationship. At the same time, through an increase in population growth rate, there will be an increase in the amount of labor utilized in production. As labor substitutes the resources utilized in production, less resources will be exhausted. As a result, the depletion rate will decrease which reflects a negative relationship. Corollary 8 (i) For the more patient economies (higher β ), the increase in resource saving technological progress has a higher growth effect. (ii) If increasing the share of renewable resources in the energy production yields a higher growth rate in the economy, that effect will be even higher for more patient economies. From the comparative statics analysis, it can be observed that if an economy is more patient the effect of increasing the resource saving technological.
(17) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. :00 :25. /20. 19. 11. se /02. 03. ih:. ar. ], T. .51. .72. IP. i],. es. sit. er. İnd. ire. n:. [B. ilk. en. tÜ niv. il. :[ 13. 9.1. 79. ge. 5. Conclusion In this paper, we study the feasibility and the determinants of the long run growth within a two-period overlapping generations model in which the energy is an essential input and technical change is resource augmenting. This paper can be considered as a contribution to the resource economic models literature as the recent literature has mostly focused on just one type of resources or utilizes infinitely lived agents framework. In parallel with the OLG literature (Galor and Ryder, 1989; Agnani et al., 2005) our model necessitates sufficiently high labor share for the economy to exhibit positive growth. However, compared with previous models the condition required is less binding. In fact, we demonstrate that the share of renewables in energy production, the technological progress in producing energy and the regeneration factor are among the key variables affecting the required labor share and hence the possibility of long run growth. We are able to offer a simpler analytical setting (i) to investigate the effects of an increase in productivity of resources in producing energy on the depletion rate and (ii) to observe under which circumstances increasing/ decreasing the resource intensity of energy production will bring about growth. Indeed, we show that the effects of an increase in the intensity of the renewable resources in producing energy promote long run growth. We analyze the sustainability of the economy under the presence of both resources and observe that the sustainability depends on the energy saving effect of the technological progress and the depletion rate of the resources which in turn depend on the rest of the parameters in the economy. Finally, we show the balanced growth path is optimal.. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. progress is higher on growth. To begin with, we see that under technological progress the economies deplete their resources less. If the agents in the economy are saving more when they are young then the economies’ resources will be depleted even lesser on the balanced growth path. Thus, the effect of developments in the technology will benefit growth more. For more patient economies, an increase in the share of renewables in the energy production induces a higher growth rate. This result can be interpreted in the following manner. For economies whose renewable energy resources is highly developed with high regeneration and technological progress rates, increasing the share of renewables will induce higher growth if the economy is more patient. If we consider developed economies as economies with higher saving rates and with improving renewable energy production technologies, we could conclude that it is optimal for that economies to support policies increasing the share of renewables compared to the less patient developing economies.. 43.
(18) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. γ αA γ xα ρ γ zα (1− ρ ) − (1 − δ ) = α 2 ( At +1 xt +1 )α ρ ( At +1 zt +1 )α (1− ρ ) ktα+1−1. 1. 1. 1. 2. +0 30 0. 11. 1. 19. 1. :00 :25. se. Then, using Equation (10) we obtain that. /02. /20. By evaluating this expression at t + 1 and t and taking the ratio we get (21). ih:. ge. 03. 1 = γ αA1 γ xα1ρ γ zα1 (1− ρ )γ kα 2 −1. ar. γ= γ= 1 or Qt +1 = Qt = Q are obtained from taking the ratio of q Q. 44. İnd. ire. n:. [B. ilk. en. IP. i],. es. sit. er. tÜ niv. il. :[ 13. 9.1. 79. .72. .51. ], T. Equation (4) in period t + 1 and t , evaluating on the balanced growth path and then substituting Equation (20). Evaluating Equation (5) along the balanced growth path, we observe that to guarantee a constant growth in renewable prices g ′(rt +1 ) as to be constant. That is why the growth of the renewable resource is g (rt ) is assumed to be a linear function of the previous period’s stock. γ r = γ z is obtained by the ratio of Equation (16) in period t + 1 and t . After evaluating the resulting equation on the balanced growth path we first +Π ) observe ξ= ξ= ξt +1 and then we obtain γ r = (1−ξ1)(1 . t +n We observe that the growth factor of capital is equal to the output per capita i.e. γk = γ y from taking the ratio of the production function in period t + 1 and t and then substituting Equation (21). Similarly, the equality of the growth factor of capital and the wages i.e. γ k = γ w can be shown by taking the ratio of Equation (12) in period t + 1 and t and then substituting Equation (21). The equality of γ k = γ s is obtained through taking the ratio of Equation (15) in period t + 1 and t . The growth factor of the energy resource is obtained through taking the ratio of Equation (7) in period t + 1 :. (22) (1 + a )(1 + Π ) ρ (1 − ξ ) ρ (1 − τ )(1− ρ ) ρ (1− ρ ) = γ Ξ γ= Aγ x γ z 1+ n. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. 6. APPENDIX 6.1. Proof of Proposition (1): The equality of γ A= (1 + a ), γ N= (1 + n) follows from the definition of the technological progress and the population growth rate equations. γ e = γ x is obtained by the ratio of Equation (17) in period t + 1 and t . After evaluating the resulting equation on the balanced growth path we first observe that τ= τ= τ t +1 and then we obtain γ x = 11−+τn . t To find the growth factor of capital per capita first we substitute Equation (13) into Equation (4):. Qt +1 = γ αA1 γ xα1ρ γ zα1 (1− ρ ) . (20).
(19) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. +0 30 0. :00 :25. /02. 03. ih:. ar. ge. /20. 19. 11. se. From dividing Equation (12) to Equation (13) and substituting the growth factors of the prices of the energy resources we find that = γ z γ x (1 + Π ) and ξ =τ . We observe that γ d = γ c from taking the ratio of Equation (3) in period t + 1 and t and evaluating it on the balanced growth path. Moreover we show γ= γ= γ k . First we substitute Equations (12), (13) and (15) into Equation d c (2) and obtain. 1 1 dt +1 − Qt +1kt +1 (1 + n) α1 Atα+11ρ xt +1 α1ρ Atα+11(1− ρ ) zt +1 α1 (1− ρ ) ktα+21 (1 − ρ ) + ρ . (23) τ ξ Then we take the ratio of Equation (23) in period t + 1 and t and evaluate it on the balanced growth path:. 9.1. 79. .72. .51. ], T. dt +1 − Qt +1kt +1 (1 + n) = γ αA1 γ xα1ρ γ zα1 (1− ρ ) −1γ kα 2 dt − Qt kt (1 + n) Using Equation (21) and the definition of growth factors we get dt γ d − Qt +1kt (1 + n) = dt − Qt kt (1 + n).. :[ 13. γk. IP. i],. es. sit. er. tÜ niv. αβ γe γr − α1 ρ Atα1 xt α1ρ zt α1 (1− ρ ) ktα 2−1 3 − α1 (1 − ρ ) (1 + Π ) − γ r (1 + n) (1 − γ e (1 + n) (1 + β ). en. γ k (1 + n). il. Since Qt +1 = Qt = Q we have γ d = γ . As a final step we characterize the growth factor of capital as follows. Substituting Equations (15), (4), (5) and (prod) into Equation (2) and dividing both sides by kt we obtain;. From Equation (20) and Equation (4) we have. [B. ilk. B. ire. n:. 1 αβ 1− ξ 1−τ = − α1 ρ γ k (1 + n) γ αA1 γ xα1ρ γ zα1 (1− ρ )γ kα 2 − (1 − δ ) 3 − α1 (1 − ρ ) α 2 (1 + β ) ξ (1 + n) τ (1 + n) . Then we obtain. İnd. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. For the growth factor of price of the exhaustible resources we take the ratio of Equation (14) in period t + 1 and t , evaluate it on the balanced growth path γk γ (1+ n ) and then substitute Equation (21). Thus we obtain γ P= e γ= 1−τ . In parallel x t with the growth factor of the price of the exhaustible resources, we take the ratio of Equation (13) in period t + 1 and t , evaluate it on the balanced γ (1+ n ) growth path and then substitute Equation (21) to obtain γ pr = (1+Π . )(1−ξ ). . α 2γ + (1 − δ ) (1α+ ββ ) − α1 (1 − ρ ) ξ 1(1−+ξn ) − α1 ρ τ (11−+τn ) . γ k (1 + n) . (. 3. ). 45.
(20) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. 6.2. Proof of Proposition (3): Substituting γ , the system can be solved from solving the following equation involving only τ : α1. α1 (1−α 2 ). α1 (1− ρ ). α1. α1. α1 (1− ρ ). se. l. (11−+τn ) (1−α2 ) (1 + Π ) (1−α2 ) (1 + n) α 2 (1 + a ) (1−α2 ) (11−+τn ) (1−α2 ) (1 + Π ) (1−α2 ) ( ) ( ) = + (1 − δ ). (24) 1 −τ (1α+3ββ ) − α(11 +(1n−)ττ ) . (1+ β ). 1. ). α 3 yt. ct =. 19. /20. /02 03 ih: ar. .72. .51. . (26) (1 + β ) Plugging Equation (26) and Equation (10) into Equation (25) we have. :[ 13. (1 + β ) kt +1. 2. es. i],. α3 α 2β. (1 + β ) 2. ) + (1 + β ) log yt + β log yt +1 − β log kt +1 ,. sit. log( Then, it is clear that U= t (ct , d t +1 ). IP. α 2α 3 yt yt +1. il. dt = β. 9.1. 79. (25). ], T. ge. 6.3. Proof of Proposition (5): Using Equations (2), (4) and (5) we have. dt +1= Qt +1 st + Pt r rt +1 + Pt e et +1 . Through substituting Equations (1) and (4) we get. :00 :25. 3. 11. (. +0 30 0. It could be easily checked that left hand side of the above equation is increasing with respect to τ in [0,1) and right hand side of the equation is decreasing with respect to τ in [ α (1+ nα) 1 β +α ,1]. Thus, there exists a unique 3 1 (1+ β ) τ * ∈ α (1+ nα) 1 β +α ,1 .. [B. ilk. en. tÜ niv. er. and the sustainability condition U t +1 (ct +1 , dt + 2 ) ≥ U t (ct , dt +1 ) reduces to (1 + β ) log γ y > 0 along the balanced growth path. A necessary and sufficient condition for sustainability in this economy is to yield positive growth along the BGP so that 1−τ γ y ≥ 1, (1 + a )( )(1 + Π )1− ρ ≥ 1. 1+ n. B A. ire. planner solves the following ∞ 1 t max ∞ β ln d 0 + ∑ ( ) ln ct + (1 + R ) −1 β ln dt t = 0 (1 + R ) {ct , dt , xt , zt , kt +1}t =0 46. social. n:. 6.4. Proof of Proposition (6):. İnd. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. (1 + a ). problem:.
(21) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. et = (1 + n)et +1 + xt ∞. e0 ≥ ∑ xt t =0. ∞. l. rt + g (rt ) = (1 + n)rt +1 + zt. (29). ∞. r0 ≥ ∑ zt − ∑ (1 + Π )rt −1. =t 0=t 0. 19. /20. 03. (31). ih:. ge. /02. dt +1 = β (1 + n) Rt +1 ct. 11. The first order conditions with respect to ct and dt yield. (30). +0 30 0. (28). :00 :25. yt = At ( xt ) ρ ( zt )(1− ρ ) ktα 2. (27). se. .72. .51. ], T. ar. then using Equation (27) and (28) we get (1 + R) − t −1 (32) Atα1 (et − et +1 )α1ρ (rt (1 + Π ) − rt +1 )α1 (1− ρ ) ktα 2 − (1d+tn ) − kt +1 (1 + n) − (1 − δ )kt. IP. i],. es. sit. il. :[ 13. 9.1. 79. = (1 + R) − t −1−1 β dt−1.. From the first order conditions with respect to kt +1 we get. (1 + R) − t −1 = Atα1 ktα 2 entα1 − (1d+tn ) − kt +1 (1 + n) − (1 − δ )kt. ilk. en. tÜ niv. er. Atα+11ktα+21−1 entα+11 + (1 − δ ) (1 + R) − t −1−1 . Atα+11 (et +1 − et + 2 )α1ρ (rt +1 (1 + Π ) − rt + 2 )α1 (1− ρ ) ktα+21 − (1d+t +n1 ) − kt + 2 (1 + n) − (1 − δ )kt +1. B. [B. From the first order conditions with respect to et +1 and rt +1 we obtain. ire. n:. (1 + R) − t −1 Atα1 (et − et +1 )α1ρ −1 (rt (1 + Π ) − rt +1 )α1 (1− ρ ) ktα 2 =. Atα1 ktα 2 entα1 − (1d+tn ) − kt +1 (1 + n) − (1 − δ )kt. İnd. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. subject to the aggregate resource constraints of the economy. yt = ct + dt (1 + n) −1 + (1 + n)kt +1 − (1 − δ )kt. α1 ρ Atα+11 (et +1 − et + 2 )α1ρ −1 (rt +1 (1 + Π ) − rt + 2 )α1 (1− ρ ) ktα+21 (1 + R) − t −1−1 , Atα+11ktα+21entα+11 − (1d+t +n1 ) − kt + 2 (1 + n) − (1 − δ )kt +1. (33) 47.
(22) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. and. (1 + R) − t −1 Atα1 (et − et +1 )α1ρ (rt (1 + Π ) − rt +1 )α1 (1− ρ ) −1 ktα 2 . =. Atα1 ktα 2 entα1 − (1d+tn ) − kt +1 (1 + n) − (1 − δ )kt. se. :00 :25 α1. At +1 At . 19. (35). /20. α1 (1− ρ ) −1. zt +1 zt . ar. α1ρ. ], T. α2. α1 (1− ρ ). zt +1 zt . (34). α1. At +1 (36) At . .51. ge. α1ρ −1. /02. ct +1 (1 + Π ) kt +1 xt +1 = (1 + R) kt xt ct. α2. 03. ct +1 1 kt +1 xt +1 = (1 + R) kt xt ct. ih:. 1 (1 + δ ) + α 2 Atα+11ktα+21−1entα+= 1. 11. ct +1 (1 + R), ct. +0 30 0. After some algebra from first order conditions we get. IP. i],. es. sit. er. ilk. 1 = γ αA1 γ xα1ρ γ zα1 (1− ρ )γ kα 2 −1 ,. [B. en. tÜ niv. il. :[ 13. 9.1. 79. .72. The equality of γ A= (1 + a ), γ N= (1 + n) follows from the definition of the technological progress and the population growth rate equations. In addition along the balanced growth path γ R = 1 or Rt +1 = Rt = R . Equality of γ e = γ x is obtained by the ratio of Equation (29) in period t + 1 and t . After evaluating the resulting equation on the balanced growth path we first observe τ= τ= τ t +1 and then we obtain γ x = 11−+τn . The equality of t γ r = γ z is obtained by the ratio of Equation (30) in period t + 1 and t . After evaluating the resulting equation on the balanced growth path we first observe +Π ) ξ= ξ= ξt +1 and then we obtain γ r = (1−ξ1)(1 . By means of Equation (34), t +n we also obtain that γ c (1 + R) − (1 − δ ) = α 2 ( At +1 xt +1 )α1ρ ( At +1 zt +1 )α1 (1− ρ ) ktα+21−1. By evaluating this expression at t + 1 and t and taking the ratio we get. B. α1 (1−α 2 ). α1ρ (1−α 2 ). α1 (1− ρ ) (1−α 2 ). (37). 48. ire. n:. γ ≡ γk = γA γx γz .. Observe that the growth factor of capital is equal to the output per capita, i.e., γ k = γ y from taking the ratio of Equation (28) in period t + 1 and t and then substituting in Equation (37). İnd. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. α1 ρ Atα+11 (et +1 − et + 2 )α1ρ (rt +1 (1 + Π ) − rt + 2 )α1 (1− ρ ) −1 ktα+21 (1 + R) − t −1−1 . Atα+11ktα+21entα+11 − (1d+t +n1 ) − kt + 2 (1 + n) − (1 − δ )kt +1.
(23) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. The growth factor of the energy resource is obtained by taking the ratio of Equation (1 + a )(1 + Π ) ρ (1 − ξ ) ρ (1 − τ )(1− ρ ) ρ (1− ρ ) (7) in period = t +1 : γ = . γ γ γ A x z Ξ 1+ n. +0 30 0. :00 :25. se. α1 (1− ρ ). α1. α1. ar α1 (1− ρ ). .72. .51. (1−α ) (1 + a ) ((11−+τn)) 2 (1 + Π ) (1−α2 ) (1 + n) Aτ = 1−τ α3 (1+ β )2 α1α 2 − − 2 2 (1−τ ) (1−α 2 ) ( (1+ β )( −1+τ )α1 + (1+ n )α3βτ ) (1+ β )τα1α 2 − ( −1+τ )( −1+α 2 )( (1+ β )( −1+τ )α1 + (1+ n )α3βτ ) . IP. τ >. i],. if. es. Aτ < 0. sit. il. :[ 13. 9.1. 79. ], T. α1. α1 (1−α 2 ). α1 (1− ρ ). ih:. Thus, we have A[δ , β , a,α1 ,α 2 ,α 3, ρ ,Π, n,τ ] = 0. and. /20. α1. (1−τ ) (1−α2 ) (1 + Π ) (1−α2 ) (1 + n) α 2 (1 + a ) (1−α2 ) (11−+τn ) (1−α2 ) (1 + Π ) (1−α2 ) (1+ n ) ( ) − − (1 − δ ). 1−τ (1α+3ββ ) − α(11 +(1n−)ττ ) . /02. α1 (1−α 2 ). 03. A(τ ). (1 + a ). ge. as. 19. 11. 6.5. Proof of Proposition (7): Equation (24) can be reduced into a implicit equation involving only τ. α1. α 3 (1 + n) (1+ββ ) + α1. .. en. tÜ niv. er. Taking the total derivative and looking for the comparative statistics with respect to any parameter z we have: ∂∂τz = − AAτz . Accordingly, we only need to check the sign of Az .. İnd. ire. n:. [B. ilk. ∂γ 1 ∂τ = −γα1 > 0.. ∂δ ( −1 + τ ) (−1 + α 2 ) ∂δ. (i) Since Aδ = −1 we have ∂∂δτ < 0. Moreover, α1 α1 α1 (1− ρ ) (1−τ ) (1−α 2 ) (1−α 2 ). (1 + a ) (1+ n) (1 + Π ) (1−α2 ) (1 + n)2 α 3α 2τ 2 Aβ = − , 2 ( (1 + β ) ( −1 + τ ) α1 + (1 + n)α 3 βτ ). B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. By dividing Equation (35) to Equation (36), we find that = γ z γ x (1 + Π ) and ξ =τ. Note that γ d = γ c by taking the ratio of Equation (31) in period t + 1 and t and evaluating it on the balanced growth path. Moreover by Equation (35) it is clear that γ = γ= γk. d c From Equation (35) and using the equality of γ c and γ k we have γ x = (1+1R ) and τ = 1+RR .. 49.
(24) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. α1. α1 (1− ρ ). α1 (1− ρ ). (1−α ) (1 + a ) (1−α2 ) ((11−+τn)) 2 (1 + Π ) (1−α2 ) (1 + n) ρα1 (1 − α 2 ). 1 (1 + β )τα 2 + −1 + τ (1 + β ) ( −1 + τ ) α1 + (1 + n)α 3 βτ. (1 + β )τα 2 1 + < 0. −1 + τ (1 + β ) ( −1 + τ ) α1 + (1 + n)α 3 βτ. 11. we have. . +0 30 0. −1. se. Aa = −. (1 + a ) (1−α2 ). :00 :25. α1ρ. α1 + (1 − τ )α 2 .. ih:. τ. ar. ge (1 + β ). This condition can be recast as. 03. (1 − τ ). >. ], T. β. α 3 (1 + n). /02. /20. 19. Aa < 0 if. .72. 79. 9.1. i],. es sit >. (1 − τ ). (1 + β ). τ. α1 + (1 − τ )α 2 .. ilk. en. (iii) Since. β. er. α 3 (1 + n). tÜ niv. Moreover,. IP. :[ 13. α1 ρ ∂γ 1 ∂τ −1 = −γα1 + (1 + a ) γ > 0 if ∂a ( −1 + τ ) (−1 + α 2 ) ∂a (1 − α 2 ). il. Thus, we have. .51. ∂τ β (1 − τ ) < 0 if α 3 (1 + n) > α1 + (1 − τ )α 2 . ∂a (1 + β ) τ. B. ire. n:. [B. 1 γ (1 + n)α1 log ( (1+Π ) ) ( (1 + β ) ( −1 + τ ) α1 + τ ((1 + β ) ( −1 + τ ) α 2 + (1 + n)α 3 β )) Aρ = − (1 − α 2 ). / −1 + τ (−1 + α ) (1 + β ) −1 + τ α + (1 + n)α βτ , ( ) ( ) 1 ) 2 ( 3. 1 β (1 − τ ) α 3 (1 + n) > α1 + (1 − τ )α 2 , and log <0 (1 + β ) τ (1 + Π ) . 50. İnd. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. (ii) Since. l. (ii) Since we have Aβ < 0 and hence ∂∂βτ < 0. Moreover, ∂γ 1 ∂τ. = −γα1 > 0. ∂β ( −1 + τ ) (−1 + α 2 ) ∂β.
(25) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012. we have Aρ < 0 if (1 + β ). >. (1 − τ ). τ. . 1 < 0. (1 + Π ) . α1 + (1 − τ )α 2 , and log . < 0 if. ∂τ ∂ρ. 1 (1 + a ) γ log α1 2 (−1 + α 2 ) (1 + a )(1 + Π ) . +0 30 0. :00 :25. γρ =. 11. Moreover,. se. 1 α1 ∂τ 1 −γα1 + γρ = γ log . ( −1 + τ ) (−1 + α 2 ) ∂ρ (1 − α 2 ) (1 + Π ) . (1 + β ). (1 − τ ). τ. /02. 03. ih: ar ], T. >. . 1 < 0. (1 + Π ) . α1 + (1 − τ )α 2 , and log α1. α1. .51. β. .72. α 3 (1 + n). 79. so that γ ρ > 0 if. ge. /20. 19. γ (1 + n) ( (1 + β ) ( −1 + τ ) α1 + τ ((1 + β ) ( −1 + τ ) α 2 + (1 + n)α 3 β )) 1 − α 2 + 2 ( −1 + τ ) ( (1 + β ) ( −1 + τ ) α1 + (1 + n)α 3 βτ ). α1 (1− ρ ). IP. il. :[ 13. 9.1. (1−α ) −1 (1 + a ) (1−α2 ) ((11−+τn)) 2 (1 + Π ) (1−α2 ) (1 + n)(1 − ρ )α1 AΠ = − (1 − α 2 ). n:. ire. ∂τ ∂Π. τ β. es. sit. er α1 + (1 − τ )α 2 ,. (1 + β ). >. (1 − τ ). τ. α1 + (1 − τ )α 2 .. < 0 . Moreover, under the above mentioned assumption,. İnd. Thus, we have. (1 − τ ). [B. we have AΠ < 0 if α 3 (1 + n). , . tÜ niv. (1 + β ). >. en. β. ilk. (iv) Since α 3 (1 + n). i],. 1 (1 + β )τα 2 + −1 + τ (1 + β ) ( −1 + τ ) α1 + (1 + n)α 3 βτ. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. Thus, we have. β. l. α 3 (1 + n). ∂γ 1 ∂τ α1 (1 − ρ ) −1 = −γα1 + (1 + Π ) γ > 0. ∂Π ( −1 + τ ) (−1 + α 2 ) ∂a (1 − α 2 ) 51.
(26) İktisat İşletme ve Finans 27 (318) Eylül / Semptember 2012 References. .72. .51. ], T. ar. ih:. 03. /02. /20. +0 30 0. :00 :25. 19. 11. se. er. sit. es. i],. IP. :[ 13. 9.1. 79. ge ire. n:. [B. ilk. en. tÜ niv. il. 52. İnd. B. İndiren: [Bilkent Üniversitesi], IP: [139.179.72.51], Tarih: 03/02/2019 11:00:25 +0300. l. Agnani, B., Gutiérrez, M., Iza, A. (2005). Growth in Overlapping Generations Economies with Non-Renewable Resources. Journal of Environmental Economics and Management, 50(2), 387-407. http://dx.doi.org/10.1016/j.jeem.2004.12.003 Allan, G., Hanley, N., Mcgregor, P., Swales, J., Turner, K. (2006, May 15). The macroeconomic rebound effect and the UK economy. Final report to the Department of Environment, Food and Rural Affairs (DEFRA), 4-100. Andre, F., & Cerda, E. (2005). On natural resource substitution. Resources Policy, 30(4), 233-246. http://dx.doi.org/10.1016/j.resourpol.2005.10.001 Apostolakis, B. E. (1990). Energy-Capital Substitutability/Complementarity. Energy Economics, 12(1), 48-58. http://dx.doi.org/10.1016/0140-9883(90)90007-3 Ayres, R., & Bergh, J. (2005). A theory of economic growth with material/energy resourses and dematerialization: Interaction of three growth mechanisms. Ecological Economics, 55, 96-118. http://dx.doi.org/10.1016/j.ecolecon.2004.07.023 Ayres, R., & Li, J. (2008). Economic growth and development: towards a catchup model. Environmental and Resource Economics, 40, 1-36. http://dx.doi.org/10.1007/s10640-007-9138-z Ayres, R., Ayres, L., Warr, B. (2003). Exergy, power and work in the US economy, 19001998. Energy, An International Journal, 28, 219-273. Ayres, R., Turton, H., Casten, T. (2007). Energy efficiency, sustainability and economic growth. Energy, 32, 634-648. http://dx.doi.org/10.1016/j.energy.2006.06.005 Azomahou, T., Boucekkine, R., Nguyen, V. P. (2006). Energy Consumption and Vintage Effect: A Sectoral Analysis.. Mimeo, CORE, Université catholique de Louvain (Belgium). Berndt, E. R., & Wood, D. (1975). Technology, Prices and the Derived Demand for Energy. The Review of Economics and Statistics, 57, 376-384. http://dx.doi.org/10.2307/1923910 Boucekkine, R., & Pommeret, A. (2004). Energy saving technological progress and optimal capital stock: the role of embodiment. Economic Modelling, 21(3), 429-444. http://dx.doi.org/10.1016/S0264-9993(03)00039-7 Cabinet Office , P. I. U. (2001, November 01). Resource Productivity: Making More With Less. Retrieved 2012, March 14, from http://webarchive.nationalarchives.gov.uk/+/http:// www.cabinetoffice.gov.uk/media/cabinetoffice/strategy/assets/resource%20productivity.pdf Daly, H. (1990). Commentary: Toward some operational principles of sustainable development. Ecological Economics, 2, 1-6. http://dx.doi.org/10.1016/0921-8009(90)90010-R Dasgupta, P., & Heal, G. (1979). Economic theory and exhaustible resources. Cambridge, Birleşik Krallık: Cambridge University Press. De La Croix, D., & Michele, P. (2002). A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations. Cambridge, Birleşik Krallık: Cambridge University Press. Department Of Environment Food And Rural Affairs , D. (2005, March 07). One Future-Different Paths: The UK’s Shared Framework for Sustainable Development. Retrieved 2012, March 14, from http://www.sd-commission.org.uk/data/files/publications/050307One%20 Future%20-%20Different%20Paths.pdf Di Vita, G. (2001). Natural resources dynamics: Exhaustible and renewable resources, and the rate of technical substitution. Resources Policy, 31, 172-182. http://dx.doi. org/10.1016/j.resourpol.2007.01.003 Field, B. C., & Grebenstein, F. (1980). Capital-Energy Substitution in U.S. Manufacturing. The Review of Economics and Statistics, 62(2), 207-212..
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