Optimal education and pensions in an endogenous growth model

Optimal Education and Pensions in an Endogenous Growth ModelI Elena Del Reya, Miguel-Angel Lopez-Garcia∗,b aUniversity of Girona, Spain bApplied Economics Department, Autonomous University of Barcelona, 08193 Bellaterra (Barcelona), Spain Abstract In OLG economies with life-cycle saving and exogenous growth, competitive equilibria in general fail to achieve optimality because individuals accumulate amounts of physical capital that differ from the one that maximizes welfare along a balanced growth path (the Golden Rule). With human capital, a second potential source of departure from optimality arises, related to education decisions. We propose to recover the Golden Rule of physical and also human capital accumulation. We characterize the optimal policy to decentralize the Golden Rule balanced growth path when there are no constraints for individuals to finance their education investments, and show that it involves education taxes. Also, when the government subsidizes the repayment of education loans, optimal pensions are positive. Key words: endogenous growth, human capital, intergenerational transfers, education policy JEL classification: D90, H21, H52, H55 IWe gratefully acknowledge the hospitality of CORE, Universit´ e catholique de Louvain and the University of Exeter Business School, as well as financial support from Instituto de Estudios Fiscales (Spanish Ministry of Finances), the Spanish Ministry of Science and Innovation (mobility grants JC2008-00098 and PR2008-0069 and projects ECO2010-16353 and ECO2012-37572) and the Generalitat de Catalunya (contracts 2009SGR189 and 2009SGR600, the XREPP and the Barcelona GSE Research Network). We are indebted to Dolors Berga, Raouf Boucekkine, Jordi Caball´ e, David de la Croix, Christos Koulovatianos, Pierre Pestieau, Jos´ e Ignacio Silva and Xavier Raurich for insightful comments and criticism. We also thank the Editor and an anonymous referee for their valuable comments on our submitted manuscript. We retain responsibility for any remaining error. ∗Corresponding author. Tel +34 93 581 12 29. Fax +34 93 581 22 92. Email addresses: [email protected] (Elena Del Rey), [email protected] (Miguel-Angel Lopez-Garcia) Preprint submitted to Elsevier December 12, 2012 1. Introduction The original approach to endogenous growth theory (Romer (1986), Lucas (1988), Rebelo (1991)) is based on the idea that returns to some capital goods, such as the stock of human capital resulting from education, do not diminish as economies develop. Knowledge spillovers and education externalities are some of the mechanisms that help sustain the returns to the accumulation of capital. In turn, the presence of a positive externality implies that a Pigouvian subsidy is required to attain optimality. Within growth theory, the overlapping generations literature has specifically focused on the existence of intergenerational externalities (see, among others, Azariadis and Drazen (1990), Caball´ e (1995), Boldrin and Montes (2005), Docquier et al. (2007)). These are a consequence of children inheriting a portion of the human capital of their parents, or of the spillovers generated by the aggregate stock of human capital. Because individuals ignore the effect of their educational decision on future human capital, they underinvest in education. In overlapping generations economies (OLG hereafter), this source of departure from optimality adds to the potential discrepancy between laissez-faire and optimal physical capital accumulation well known since Diamond (1965). As a result, two policy instruments are required to attain optimality, typically education subsidies and intergenerational transfers. The persistence in the result that education should be subsidized in a growth context masks the fact that it hinges on the optimality criterion adopted, to wit, the maximization of a discounted sum of utilities defined over consumption levels. In the case of infinitely lived individuals, they are assumed to discount future utility, so there is no difference between individual and social objectives. In an OLG economy with labour productivity growth that translates into increasing consumption levels, the planner cannot maximize individual utility along a balanced growth path simply because it grows without limit. The social planner is then assumed to maximize the sum of present and future utilities of consumption per unit of natural labour, discounting the latter at an arbitrary discount rate. In this paper we assume, on the contrary, that the social planner purposively wants to treat all generations alike, while being respectful with individual preferences. One way to do this is to maximize a utility function whose arguments are consumption levels per unit of efficient labour. In a three overlapping generation economy where human capital accumulation is the engine of growth, we search for the balanced growth path that maximizes this notion of lifetime welfare subject to the constraint that the welfare of the representative individual of every other generation is fixed at the same level. This optimality criterion would appear as the counterpart in the current framework of the (two-part) Golden Rule (Diamond, 1965, Samuelson, 1968). As will be made clear, embracing this criterion implies that it is no longer wrong for individuals to ignore the (positive) effect of their decisions on the human capital of others. What is now wrong from the point of view of the social planner is for individuals to ignore the effect of their decisions on the growth rate of the economy. This effect can generally be positive or negative, but it is definitely negative when the individual faces the Golden Rule wage and interest rates. Thus, an education tax is required to maintain the economy on its optimal balanced growth path. Furthermore, when the government subsidises the repayment of education loans, the optimal policy also involves positive pensions. In our model, like in Boldrin and Montes (2005) and Docquier et al. (2007), agents are born 1 with an endowment of knowledge equal to the human capital of their parents, and borrow when young in order to invest in education. Credit markets are perfect. Inherited and acquired human capital interact to produce human capital of the middle-aged. As middle-aged, agents work, pay back the loan, save and pay taxes or receive subsidies. Finally, as old-aged, they consume and pay taxes or receive subsidies. We use this model to identify the optimum, compare it to the laissez faire and identify the optimal policy. The objective of the social planner is to maximize a welfare index defined in terms of consumption per unit of efficient labour, and her decision variables are also defined in terms of output per unit of efficient labour. The Golden Rule thus obtained consists of five conditions that need to be satisfied simultaneously. One of the conditions of the optimal allocation is that the marginal product of physical capital (per unit of efficient labour) equals the (endogenous) growth rate of the economy. This is like in models with exogenous productivity growth (Buiter, 1979). As far as human capital accumulation is concerned, a condition equating marginal benefits and marginal costs of investing in education is obtained. Unlike the case of exogenous growth models, where laissez-faire and socially optimal physical capital accumulation can coincide by chance, we show that, in our framework, the laissez faire with perfect credit markets cannot possibly attain the Golden Rule. Individuals will always over or under invest either in physical capital, human capital, or both. To be more precise, as illustrated in Table 1 below, when the laissez-faire accumulation of physical capital is smaller than the optimal one, the laissez-faire expenditure in education can either exceed or fall short of the optimal one. However, a laissez-faire accumulation of physical capital greater than or equal to the optimal one can only coexist with a too large expenditure in education. As a consequence, if the laissez-faire amount of physical capital coincides with the socially optimal level, there will be over-accumulation of human capital. Therefore, as stated above, in order to attain the Golden Rule, education should be taxed. The reason is that individuals choose their human capital investments accounting only for the effects on their earnings and loan repayment costs, not on the growth rate of the economy. If faced with the optimal (i.e., Golden Rule) factor prices in the laissez faire, they would ignore the costs associated with maintaining these factor prices at their Golden Rule level as human capital increases due to their investments. Under these circumstances, they would over-invest in education. For this reason, a tax is required to maintain the economy on the optimal balanced growth path. The rest of the paper is organized as follows. Sections 2 and 3 respectively present the model and the social optimum. Section 4 derives the decentralized market equilibrium in the presence of government. The tax instruments available are lump-sum taxes on both the working and the retired population and education subsidies. The laissez-faire can be easily characterized by setting the tax parameters equal to zero. Section 5 compares the laissez-faire balanced growth path with the Golden Rule. Finally, Section 6 characterizes the optimal policy and Section 7 concludes. 2. The Model The basic framework of analysis is the overlapping generations model with both human and physical capital developed in Boldrin and Montes (2005) and Docquier et al. (2007). At period t, Lt+1individuals are born. They coexist with Ltmiddle-aged and Lt−1old-aged. Population grows at the exogenous rate nso that Lt=(1 +n)Lt−1with n>−1. Agents are born with an endowment 2 of basic ”knowledge”. This knowledge is assumed to be equal to the level of human capital of their parents ht−1and is measured in units of efficient labour per unit of natural labour. Human capital in period tis produced out of the amount of output invested in education dt−1and basic knowledge ht−1according to the production function ht=E(dt−1,ht−1). Assuming constant returns to scale, the production of human capital can be written in intensive terms as ht/ht−1=e(˜ dt−1), where e(.) satisfies the Inada conditions and ˜ dt−1=dt−1/ht−1is the amount of output devoted to education per unit of inherited human capital. A single good Ytis produced by means of physical capital Ktand human capital Ht, according to a constant returns to scale production function Yt=F(Kt,Ht). As explained below, only the middle-aged work and they inelastically supply one unit of natural labour, so that Ht=htLt. Physical capital is assumed to fully depreciate each period. If we define kt=Kt/Ltas the physical capital per unit of natural labour ratio and ˜ kt=Kt/Ht=kt/htas the physical capital per unit of efficient labour ratio, the technology can be described as Yt/Ht=f(˜ kt), where f(.) also satisfies the Inada conditions. The lifetime utility function of an individual born at period t−1 is Ut=U(cm t,co t+1),(1) where cm tand co t+1denote her consumption levels as middle-aged and old-aged, respectively. This utility function is assumed to be strictly quasi-concave and homogeneous of degree j>0. The reason why we only impose strict quasi-concavity instead of strict concavity of the individual utility function is that, as it will be made clearer shortly, we are only interested in ordinal preferences. Total output produced in period t,F(Kt,Ht),can be devoted to consumption, cm tLt+co tLt−1, investment in education (or human capital), dtLt+1, and investment in physical capital, Kt+1. Thus, the aggregate feasibility constraint writes F(Kt,Ht)=cm tLt+co tLt−1+dtLt+1+Kt+1(2) or, expressed in units of natural labour: htf(kt/ht)=cm t+co t 1+n+(1 +n)dt+(1 +n)kt+1(3) Alternatively, we can divide (3) by ht, which is given at time t, and obtain the aggregate feasibility constraint in period tmeasured in terms of output per unit of efficient labour: f(˜ kt)=˜cm t+˜co t e(˜ dt−1)(1 +n)+(1 +n)˜ dt+e(˜ dt)(1 +n)˜ kt+1(4) where ˜cm t=cm t/htand ˜co t=co t/ht−1denote respectively consumption when middle-aged and consumption when old-aged per unit of efficient labour.1Note that ht+1/ht=e(˜ dt)=1+gt+1, where gt+1is the growth rate of productivity from period tto period t+1. Along a balanced growth path, all variables expressed in terms of output per unit of natural labour are growing at rate g. In consequence, all variables expressed in terms of output per unit of efficient labour remain constant: ˜cm t=˜cm t+1=˜cm, ˜co t=˜co t+1=˜co,˜ kt=˜ kt+1=˜ kand ˜ dt=˜ dt+1=˜ d. 1Note that cm tLtand co tLt−1are expressed in units of output. Since middle-aged individuals supply one unit of natural labour, cm tand co tare expressed in units of output per unit of natural labour. The interpretation of ˜cm tand ˜co tin terms of units of output per unit of efficient labour follows naturally. 3 3. The Social Optimum: the Golden Rule in the presence of Endogenous Growth In the presence of productivity growth that translates into consumption growth (as is, of course, the case along a balanced growth path), consumption levels will grow without limit. It is for this reason that, in order to characterize the balanced growth path, consumptions (and all the other variables) have to be expressed in terms of output per unit of efficient labour. As argued in the introduction, an obvious consequence of cm tand co t+1growing to infinity is that a social planner will be unable to choose the consumption levels that maximize Ut=U(cm t,co t+1) along a balanced growth path, because Uttends to infinity. The standard way to sidestep this problem is to assume that the planner maximizes a discounted sum of utilities and to search for the sequence of consumptions leading to the optimal balanced growth path. Assume, on the contrary, that the social planner wants to treat all generations equally while being respectful with individual preferences. One possibility is for her to consider a valuation function defined over ˜cm tand ˜co t+1. Since the utility function (1) is homogeneous, we can write: ˜ Ut=U(˜cm t,˜co t+1)=Ucm t/ht,co t+1/ht=(1/hj t)U(cm t,co t+1)=(1/hj t)Ut(5) Thus, a ”new” utility function is obtained by means of a monotonic transformation of the first one, this ensuring that ordinal preferences are respected. The arguments of this utility function are measured in terms of consumption per unit of efficient rather than natural labour. Notice that (5) and (1) have the same functional form, and are both homogeneous of degree j. Also, the slope, curvature and higher derivatives of indifference curves in (˜cm t,˜co t+1) space are the same as those of the corresponding indifference curves in (cm t,co t+1) space. We can now posit that the social planner’s objective is to choose the balanced growth path that maximizes the welfare of a representative individual, as measured by (5), subject to the constraint that everyone attains the same utility level, ˜ U=U(˜cm,˜co). We adopt this approach, which is reminiscent of Diamond (1965)’s original treatment of the Golden Rule in an OLG framework with productive capital.2Note that, as shown by (5), although both Utand hj ttend to infinity, the ratio ˜ U=(1/hj t)Utis well defined along a balanced growth path. Then, the social planner will choose (˜cm,˜co,˜ k,˜ d) that maximize U(˜cm,˜co) subject to the balanced growth path version of (4) and the technological relationship 1 +g=e(˜ d). We obtain, from the first order conditions: ∂U(˜cm ∗,˜co ∗)/∂˜cm ∂U(˜cm ∗,˜co ∗)/∂˜co=(1+g∗)(1 +n) (6) f0(˜ k∗)=(1+g∗)(1 +n) (7) e0(˜ d∗)      ˜co ∗ (1+g∗)(1 +n)2−˜ k∗     =1 (8) 2Phelps’ Golden Rule identifies the amount of physical capital that maximizes consumption per capita (Phelps, 1961). A wider view of the Golden Rule concept, suggested by Diamond (1965) and Samuelson (1968, 1975a and 1975b) in an OLG model without productivity growth, focuses on the resource allocation that maximizes the lifetime welfare of a representative individual subject to the constraint the everyone else’s welfare is fixed at the same level. The resulting two-part Golden Rule encompasses both Phelps’ Golden Rule and Samuelson(1958)’s biological interest rate. 4 ˜cm ∗+˜co ∗ (1+g∗)(1 +n)=f(˜ k∗)−(1+g∗)(1 +n)˜ k∗−(1 +n)˜ d∗(9) 1+g∗=e(˜ d∗) (10) Definition 1. The Golden Rule balanced growth path (˜cm ∗,˜co ∗,˜ k∗,˜ d∗)provides the maximum level of welfare ˜ U=U(˜cm,˜co)that can be achieved by a representative individual subject to the feasibility constraint and the additional constraint that everyone else attains the same level. It is characterized by expressions (6)-(10). The interpretation of these equations is simpler if we start from the exogenous productivity growth setting, that we obtain for a given gand (without loss of generality) ˜ d=0. Then, the Golden Rule is characterized by (6), (7) and (9) with ggiven and ˜ d=0 (see Buiter, 1979). Equation (6) is the equality of the marginal rate of substitution between second and third period consumptions and the counterpart in the current model of the so-called ”biological” interest rate, i.e., the economy’s growth rate. Equation (7) is the equality of the marginal product of physical capital (per unit of efficient labour) and the growth rate of labour measured in efficiency units (i.e., the sum of the rates at which the efficiency of labour and the natural units of labour respectively grow). Of course, if g=0 we are back to Diamond’s framework and we obtain the original two-part Golden Rule. Turning now to the endogenous growth framework, (6) and (7) continue to hold with the same interpretation, with g∗being obtained from (10). Equation (8), however, requires a careful explanation. It points out that, along the optimal balanced growth path, the marginal benefit of an increase in the amount of output devoted to education (again per unit of efficient labour) must equal its marginal cost. This can be seen using the aggregate feasibility constraint (9). A rise in ˜ dhas a direct cost in terms of the third term in the right hand side of (9), as it reduces consumption possibilities by (1 +n). It also has an indirect cost, given by e0(.)(1 +n)˜ kas a consequence of the effect of a rise in ˜ don the rate of growth g: indeed, the greater the productivity growth rate g, the greater the amount of output that must be devoted to investment in physical capital in order to keep ˜ kconstant. However, a rise in ˜ d(and thus in g) also has a benefit, since a greater gimplies a greater marginal rate of transformation between second and third period consumption in the LHS of (8). This amounts to an expansion of consumption possibilities that is captured by e0(.)(1+n)˜co ∗/(1+g∗)(1 +n)2. At the optimum, the marginal benefit and the marginal costs must be equal: (1 +n)e0(˜ d∗)˜co ∗ (1+g∗)(1 +n)2=(1 +n)e0(˜ d∗)˜ k∗+(1 +n) (11) The terms involving (1 +n) in (11) cancel out and (8) emerges. For later use, (8) can be rewritten, using (7) and (9), as e0(˜ d∗)f(˜ k∗)−˜ k∗f0(˜ k∗)−Λ∗(˜ k∗,˜ d∗)=f0(˜ k∗) (12) with Λ∗(˜ k∗,˜ d∗)=(1+g∗)(1 +n)˜ k∗+(1 +n)˜ d∗+˜cm ∗>0. Notice that, in the exogenous growth framework, ˜ k∗is univocally determined by (7) and the optimization problem can be solved sequentially. In contrast, with endogenous growth, (7) and (8) 5 are not enough to determine ˜ k∗and ˜ d∗because (8) incorporates also ˜co ∗, which can only be obtained from (6) and (9). A sequential solution of the optimization problem is now impossible: all optimal variables are determined simultaneously. To conclude this section, note that the definition of the social optimum that is posited in this paper differs in several ways from the standard one. First, the optimal allocation resulting from maximizing P∞ t=0γtU(cm t,co t+1) depends upon the choice of a particular social discount factor γ. Second, the inter-temporal trajectory leading to the optimal balanced growth path depends on the initial conditions. Third, the optimum corresponding to the standard approach is contingent upon the particular degree of homogeneity of the utility function, and thus the precise cardinalization of preferences, (a point stressed in Del Rey and Lopez-Garcia, 2012). In contrast, when one adheres to the social planner’s objective underlying Definition 1, the social optimum is independent of the choice of an arbitrary social discount rate and there is no need to choose a specific cardinalization of utility. Finally, the focus is on the choice of the best optimal balanced growth path, this implying that the role of the initial conditions is not an issue. 4. Decentralized Market Equilibrium with Government In this section we analyse the behaviour of the economy in the presence of education subsidies and intergenerational transfers and characterize the equilibrium balanced growth path for given values of the policy parameters. The properties of the laissez-faire balanced growth path are discussed in section 5. Since individuals not only decide about the allocation of their resources along their life cycle but also how much to invest in education, there are now two potential sources of divergence between socially optimal and individual choices. This is the reason why we posit that the government has two policy instruments at its disposal: education subsidies and intergenerational transfers from the middle-aged to the elderly. Among the different ways of tackling education subsidies, we choose to model them as subsidies to the repayment, in the second period of life, of the loans taken in the first one to pay for education. Let zm t>0 [resp. <0] be the lump-sum tax [transfer] the middle aged pay [receive], zo t>0 [<0] the lump-sum tax the old pay [the pension they receive] and let θtbe the subsidy rate, all of them in period t. The laissez-faire equilibrium can then be retrieved by setting zm t=zo t=θt=0.3 Factor prices are determined under perfect competition by their marginal products, so that, if 1+rtand wtare respectively the interest factor and the wage rate per unit of efficient labour, 1+rt=f0(˜ kt) (13) wt=f(˜ kt)−˜ ktf0(˜ kt) (14) Individuals choose, in their first period, the amount of education that maximizes their lifetime resources. They do so by borrowing any amount they wish in perfect credit markets. Concerning 3It is worth emphasizing that, in this paper, the government subsidises the repayment of education loans. Alternatively the government can directly subsidise expenditures in education in the first period. Main results concerning the optimal education subsidy are however robust to alternative designs of this policy. See below. 6 savings, they behave as pure life-cyclers, i.e., they save to transfer purchasing power from the second to the third period. Then, for an individual born at t−1, consumption when middle-aged and consumption when old can be written respectively cm t=wtht−(1 +rt)dt−1(1 −θt)−zm t−st(15) co t+1=(1 +rt+1)st−zo t+1(16) where stare the savings of a middle-aged. Thus, the lifetime budget constraint of an individual born at period t−1 is: cm t+co t+1 1+rt+1 =wtht−(1 +rt)dt−1(1 −θt)−zm t− zo t+1 1+rt+1 (17) The first order conditions associated with the individual decision variables, dt−1,cm tand co t+1, are: wte0(dt−1/ht−1)=(1 +rt)(1 −θt) (18) ∂U(cm t,co t+1)/∂cm t ∂U(cm t,co t+1)/∂co t+1 =(1 +rt+1) (19) where use has been made of the homogeneity of degree one of the Efunction, i.e., ht=e(dt−1/ht−1)ht−1. Equation (18) shows that the individual will invest in education up to the point where the marginal benefit in terms of second period earnings equals the marginal cost of investing in human capital allowing for subsidies. Rewriting (18) as e0(˜ dt−1)=(1 −θt) (1 +r(˜ kt))/w(˜ kt),this expression implicitly characterizes the optimal ratio ˜ dt−1as a function of ˜ ktand θt, i.e., ˜ dt−1=φ(˜ kt, θt).Since e00 <0 it can readily be shown that the greater ˜ ktand θtare, the greater ˜ dt−1is. The government finances education subsidies with revenues obtained from taxing the middleaged and/or the old-aged: zm tLt+zo tLt−1=θt(1 +rt)dt−1Lt(20) which plugged into (17) yields cm t+co t+1 1+rt+1 =ωt(21) where ωtis the present value of the net lifetime income of an individual born at t−1: ωt=wtht−(1 +rt)dt−1(1 −θt)−zm t−1+n 1+rt+1 [θt+1(1 +rt+1)dt−zm t+1] (22) The homogeneity assumption on preferences implies that the co t+1/cm tratio is a function of rt+1only. Substituted into the budget constraint (22) this allows us to write consumption in the second period as a fraction of lifetime income, cm t=π(rt+1)ωt.Equilibrium in the market for physical capital is achieved when the (physical) capital stock available in t+1,Kt+1, equals gross savings made by the middle-aged in t,stLt, minus the amount of output devoted to human capital investment by the young in t,(1 +n)dtLt,i.e., when Kt+1=stLt−(1 +n)dtLt(23) 7 Using (23), the budget constraints (15) and (16), the government budget constraint (20) and the equilibrium factor prices (13) and (14), one can obtain the feasibility constraint (3). Since st=wtht−cm t−(1 +rt)dt−1(1 −θt)−zm t, equilibrium condition (23) can be written in terms of output per unit of efficient labour: ˜ kt+1=(1−π(rt+1))˜ωt e(φ(˜ kt+1, θt+1))(1 +n) − ˜zm t+1 1+rt+1 − (1−θt+1)φ(˜ kt+1, θt+1) e(φ(˜ kt+1, θt+1)) (24) where ˜ωt=ωt/htand use has been made of the fact that dt=φ(˜ kt+1, θt+1)ht. This expression implicitly provides ˜ kt+1as a function Ψ(˜ kt; ˜zm t,˜zm t+1, θt, θt+1). Along a balanced growth path, we can delete the time subscripts and write ˜ k= Ψ(˜ k; ˜zm, θ). An equilibrium ratio of physical capital to labour in efficiency units along a balanced growth path in the presence of government intervention, ˜ kG, will then be a fixed point of the Ψfunction, i.e., ˜ kG= Ψ(˜ kG; ˜zm, θ). Such an equilibrium will be locally stable provided that 0 < ∂Ψ(˜ kG; ˜zm, θ)/∂˜ k<1.In what follows, we will focus on situations where the equilibrium is unique and stable so that the relationship between ˜ kand the tax parameters can be written, with an obvious notation, as ˜ k=˜ k(˜zm, θ) (25) We can now turn to the determination of ˜ dor, what is the same, the growth rate g. The amount of output devoted to education per unit of inherited human capital along a balanced growth path will be governed by the relationship arising from the education decision (18), i.e., ˜ d=φ(˜ k, θ). Using (25) we can write ˜ d=φ˜ k(˜zm, θ), θor, for short, ˜ d=˜ d(˜zm, θ) (26) Letting gGbe the growth rate of any variable expressed in terms of output per unit of natural labour, since 1 +g=e(˜ d), we have 1 +gG=eφ(˜ kG, θ). Finally, the growth rate of all variables expressed in absolute terms (physical capital, human capital and output) is (1 +gG)(1 +n). This follows from writing H[K] as hL [kL] and observing that h[k] grows at rate gGwhile Lgrows at rate n. Summarizing, the balanced growth path in the presence of government intervention is characterized by (25) and (26) satisfying ∂U(˜cm,˜co)/∂˜cm ∂U(˜cm,˜co)/∂˜co=(1 +r) (27) we0(˜ d)=(1 +r)(1 −θ) (28) ˜cm+˜co 1+r=˜ω(29) where ˜ω≡ω/his given by ˜ω=w−(1 +r)˜ d (1 +g)+"(1 +r)−(1 +g)(1 +n) 1+g#θ˜ d−"(1 +r)−(1 +g)(1 +n) 1+r#˜zm(30) and represents the present value of lifetime resources expressed in terms of output per unit of efficient labour. 8