George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 The Representative Household Model The representative household model is a dynamic general equilibrium model, based on the assumption that the dynamic path of aggregate consumption is decided in an optimal fashion by identical households. All households are assumed to have an infinite time horizon and free access to the capital market, at a given, competitively determined, real interest rate. In other respects, this model has a lot in common with the neoclassical model of economic growth of Solow (1956). Historically, the representative household model predates the Solow growth model. The first such model is due to Ramsey (1928), who set out to analyze the optimal savings behavior of a household with a long time horizon. However, as the majority of economists at the time were not familiar with the mathematical techniques employed by Ramsey, the Ramsey model remained in relative obscurity for many years. It re-surfaced in the 1960s, with the restatement and the extensions of Cass (1965) and Koopmans (1965), and has since evolved as one of the key models in intertemporal macroeconomics. It is being used widely both in the theory of economic growth and the theory of aggregate fluctuations, in the form of the stochastic growth model. 1 It is also worth mentioning, that around the same time as Ramsey, the problem of the optimal intertemporal choice of consumption by a representative household was also analyzed by Fisher (1930). Fisher employed a two period dynamic model, which was in many respects similar, but much less demanding mathematically, than the Ramsey model. Assumptions about technology and market structure in the representative household growth model are similar to the assumptions of the Solow model. What differs is the assumption about the determination of savings. Instead of the fixed and exogenous saving rate of the Solow model, in the representative household model savings are determined as a result of optimal intertemporal consumption behavior. Consequently, savings behavior is determined endogenously and is optimal. 2 The representative household model is theoretically more satisfactory than the Solow model, as it is based on full intertemporal optimization. It is a dynamic general equilibrium model in which the path of the economy depends solely on parameters related to the preferences of households, the technology of production, population growth and market structure. Instead of assuming an exogenous savings rate, the model explains savings behavior as the outcome of the optimizing behavior of households. Moreover, as the typical form of the model assumes complete and competitive markets, and that all households are alike, the Ramsey model determines the socially optimal savings behavior in the sense of the maximization of social welfare. 1 See Attanasio (2015) for a recent analysis of the original Ramsey paper, from a current perspective. Throughout this book we shall be referring to this growth model as the representative household model or simply the 2 Ramsey model. It has to remembered however that the contributions of Cass and Koopmans were extremely significant for the development of the model. Thus, many authors refer to it as the Ramsey-Cass-Koopmans model.
The savings rate in the Ramsey growth model is not constant, as in the Solow model, but a function of the state of the economy. Given that the savings rate is one of the key determinants of the accumulation of capital and the dynamic evolution of all other real variables, the fact that the savings rate is determined optimally, is extremely important. For example, in the representative household model there is no possibility of dynamic inefficiency, in the sense of an excessively high savings rate that leads the economy to a level of capital beyond the golden rule. The representative household chooses its individually optimal level of savings, which, because of the assumption of full competitive markets, is also socially optimal. As it turns out, the steady state capital stock in this model is below the golden rule capital stock, because of the assumption of a positive pure rate of time preference. This optimal steady state capital stock defines the so called modified golden rule. However, this model is also an exogenous growth model, similar in this respect to the Solow model. It does not determine the steady state growth rate of per capita income, but instead treats it as an exogenous parameter, the exogenous rate of technical progress. As with the Solow model, what the Ramsey model does determine is the level of the per capita capital stock, per capita output and consumption, per capita real wages and the real interest rate, both on the balanced growth path, as well as on the convergence path towards the balanced growth path. 3.1 The Optimal Intertemporal Path of Consumption In order to introduce the problem of the optimal intertemporal choice of consumption, we shall initially assume a household that lives only for two periods, and maximizes an intertemporal utility function which depends on the level of consumption in each of the two periods. This type of two period dynamic model was first analyzed by Fisher (1930), and also forms the basis of the Diamond (1965) overlapping generations model analyzed in Chapter 4. We shall then generalize the analysis for a household with a time horizon equal to T, which is the problem posed and solved by Ramsey (1928). 3.1.1 The Determination of Optimal Consumption in the Two Period Case Assume, following Fisher (1930), a household which lives only for two periods. During period 1 it works and receives labor income w and during period 2 it does not work and lives off its savings. The household chooses the path of consumption in order to maximize an intertemporal utility function of the form,! U = U(c 1,c 2 ) = u(c 1 ) + 1 (3.1) 1+ ρ u(c ) 2 where, ρ is the pure rate of time preference of the household, and, u is a concave, twice differentiable utility function, for the first two derivatives of which we assume u >0, u <0. Savings of period 1, plus interest, can be consumed in period 2. It thus follows that,! c 2 (1+ r)(w c 1 ) (3.2) where r is the real interest rate.!2
! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 (3.2) describes the budget constraint of the household. It can be rearranged and expressed in the following form: c 1 + 1 1+ r c 2 w (3.3) The interpretation of (3.3) is that the present value of consumption, evaluated at the market real interest rate r, cannot exceed the present value of household income, which is equal to its first period labor income w. In this model, labor income in the first period is equal to the total lifetime wealth of the household. In order to derive the first order conditions for the maximization of (3.1) under the constraint (3.3), we define the Lagrange function of this problem, and derive the first order conditions for its maximization. The Lagrange function takes the form,! L(c 1,c 2,w,λ) = u(c 1 ) + 1 (3.4) 1+ ρ u(c ) λ c + 1 2 1 1+ r c w 2 where λ is the relevant Lagrange multiplier of the constraint (3.3). From the first order conditions for the maximization of (3.4) it follows that,! u (c 1 ) λ = 0 (3.5) 1! (3.6) 1+ ρ u (c ) λ 2 1+ r = 0 It also follows that the budget constraint (3.3) is satisfied with equality. The maximization of the lifetime utility of the representative household implies that the present value of consumption cannot be lower than its wealth, which is the present value of labor income. Of course, it also implies that it cannot be higher than its wealth, as this would violate the intertemporal budget constraint. Dividing the first order condition (3.6) by (3.5) it follows that, 1 u (c! 2 ) (3.7) 1+ ρ u (c 1 ) = 1 1+ r (3.7) indicates that in order to maximize its lifetime utility, the household chooses a consumption path for which the marginal rate of substitution between future and current consumption is equal to the opportunity cost (relative value) of future consumption, which depends negatively on the real interest rate. (3.7) can be transformed as, u (c! 2 ) (3.8) u (c 1 ) = 1+ ρ 1+ r!3
(3.8) indicates that the ratio of consumption in the two periods depends solely on the relationship between the real interest rate and the pure rate of time preference of the household. If ρ=r, then,! u (c 1 ) = u (c 2 ) c 1 = c 2 If ρ>r, then,! u (c 1 ) < u (c 2 ) c 1 > c 2 (3.9) If ρ<r, then,! u (c 1 ) > u (c 2 ) c 1 < c 2 (3.8) indicates that the household will smooth its consumption between the two periods, by saving in the first period and by consuming its savings, plus their return, in the second period. If the real interest rate is equal to the pure rate of time preference of the household, then consumption smoothing is complete. If the real interest rate is lower than the pure rate of time preference of the household, then consumption is higher in the first period than in the second period. Finally, if the real interest rate is higher than the pure rate of time preference of the household, then consumption in the first period is lower than in the second period. To examine how the savings rate depends on the real interest rate, we will use a special, but widely used, utility function, the utility function with a constant elasticity of intertemporal substitution of consumption, or constant relative risk aversion. This takes the form, 1 θ 1! u(c t ) = c t (3.10) 1 θ where 1/θ is the elasticity of intertemporal substitution, and θ is the coefficient of constant relative risk aversion. This utility function is quite general and will be used throughout this book. In the case where the elasticity of intertemporal substitution is equal to unity, then, this function is not defined. By using L Hopital s rule, one can show that for θ=1, it takes the form of logarithmic utility,! u(c t ) = lnc t (3.11) For the utility function (3.10), the first order condition (3.8) can be written as, c! 2 (3.12) = 1+ ρ 1+ r c 1 θ From (3.12) it follows that consumption in the first period satisfies, 1! c 1 = 1+ ρ θ (3.13) 1+ r c2 Combining (3.13) with the intertemporal budget constraint (3.3), the savings rate in the first period is determined by,!4
! s(r) = w c 1 (3.14) w = (1+ r) 1 1 θ < 1 (1+ ρ) θ θ + (1+ r) (3.14) indicates that the savings rate is a positive function of the real interest rate r, only if θ is less than unity, that is, if the elasticity of intertemporal substitution 1/θ is greater than unity. Only then does the substitution effect of a change of the real interest rate dominate the income effect. If θ is greater than unity, the savings rate is a negative function of r, as the elasticity of intertemporal substitution 1/θ is smaller than unity, and the income effect dominates. In the special case where θ is equal to unity (logarithmic utility), the savings rate is independent of the real interest rate and is equal to 1/(2+ρ). Moreover, regardless of the value of the elasticity of intertemporal substitution, if the pure rate of time preference is equal to the real interest rate, then, from (3.14), the savings rate equals 1/(2+ρ). 3.1.2 The Optimal Path of Consumption with a Time Horizon Equal to Τ We next turn to the more general Ramsey (1928) model. This model analyzes the problem of the optimal choice of the consumption path by a representative household with a long time horizon. We shall switch to continuous time and assume a representative household with a time horizon equal to Τ. Assume a household that has a exogenous flow of income equal to w per instant, and which can borrow and lend freely in the capital market, at an interest rate r. The household has an finite horizon Τ and initial interest yielding assets equal to a(0). It is assumed to maximize the following intertemporal utility function, T! U = e ρt u(c(t))dt (3.15) subject to, 1 θ θ! a (t) = ra(t) + w c(t) (3.16)! a(0) 0 (3.17)! a(t ) 0 (3.18) u is the instantaneous utility function of the household, which depends on consumption of goods and services. u is twice differentiable and concave. ρ is the pure rate of time preference, the rate at which the household discounts future utilities. (3.16) is the asset accumulation equation. (3.17) defines the initial assets of the household, and (3.18) is a terminal condition which ensures that the household respects its intertemporal budget constraint (transversality condition).!5
From the maximum principle, the conditions for the maximization of (3.15) subject to the accumulation constraint (3.16) are the same as the first order conditions for the maximization of the current value Hamilton function, which for this problem is defined by, 3 ( )! H (t) = u(c(t)) + λ(t) ra(t) + w c(t) (3.19) where λ(t) is the current value multiplier of the asset accumulation constraint. The first order conditions for the maximization of the current value Hamilton function are given by, H (t)! (3.20) c(t) = 0 u (c(t)) = λ(t) H (t)! (3.21) a(t) = λ (t) ρλ(t) λ (t) = (r ρ)λ(t) H (t)! (3.22) λ(t) = a (t) a (t) = ra(t) + w c(t) From (3.20), on the optimal path, the multiplier λ(t), which is the value of the marginal increase in assets, is equal to the marginal utility of consumption. Thus, the household is indifferent between one extra unit of consumption and one extra unit of savings. From (3.21), on the optimal path, the real interest rate plus the expected capital gain on assets, is equal to the pure rate of time preference. Finally, (3.22) is the asset accumulation equation. We can use the first order conditions (3.20) and (3.21) to characterize the behavior of consumption along the optimal path. From (3.20), differentiating with respect to time, we get,! λ (t) = u (c(t))c (t) (3.23) Substituting (3.20) and (3.23) in (3.21), we get,! c (t) = u (c(t)) (r ρ) (3.24) u (c(t)) (3.24) is known as the Euler equation for consumption. It is nothing more than the expression in continuous time of the typical condition for optimality, that the marginal rate of intertemporal substitution of consumption is equal to the marginal rate of intertemporal transformation of current to future consumption. The interpretation of (3.24) is analogous to the interpretation of condition (3.7) for the two period problem we have already analyzed. 3 See Mathematical Annex 3 for methods of intertemporal optimization in continuous time, and their application to the representative household problem.!6
This interpretation of equation (3.24) is sometimes referred to as the Keynes-Ramsey rule, because this type of Euler equation was presented by Ramsey in his classic Economic Journal article, accompanied by the interpretation above, which Ramsey partly attributed to Keynes, then editor of the journal. Since the second derivative of the instantaneous utility function is negative, the change in consumption will have the same sign as the difference between the real interest rate and the pure rate of time preference. If the real interest rate is higher than the pure rate of time preference, consumption will be continuously increasing. In the opposite case, consumption will be continuously decreasing. If the real interest rate is equal to the pure rate of time preference, consumption will be constant on the optimal path. If we assume that the instantaneous utility function of the household has the form of (3.10), with a constant elasticity of intertemporal substitution 1/θ, then (3.24) takes the form, c (t)! (3.25) c(t) = 1 (r ρ) θ (3.25) implies that optimal consumption of the household increases, remains constant, or decreases, depending on whether the real interest rate exceeds, equals or falls short of the pure rate of time preference. This optimality rule is essential and logical. The higher the real interest rate relative to the pure rate of time preference, the greater the incentive for the representative household to reduce current consumption and invest in assets with a higher rate of return r, in order to enjoy higher future consumption. So if the real interest rate is higher than the pure rate of time preference, consumption per capita will be growing along the optimal path. If the real interest rate is lower than the pure rate of time preference, consumption per capita will be declining along the optimal path. Finally, if the real interest rate is equal to the pure rate of time preference, consumption will be constant along the optimal path. In this latter case there will be full consumption smoothing. (3.25) also highlights the role of the elasticity of intertemporal substitution 1/θ. The higher the elasticity of intertemporal substitution, the easier it is for the household, in utility terms, to substitute consumption over time. So, the easier it is to substitute current for future consumption. Consequently, for a given difference between the real interest rate and the pure rate of time preference, the growth rate of per capita consumption is higher, the higher the elasticity of intertemporal substitution. We can now proceed by incorporating the optimal consumption behavior of a representative household in a full blown growth model. 3.2 The Representative Household Model of Economic Growth As with the Solow model, in the representative household model we shall focus on the following set of endogenous variables: Y Aggregate Output (or y, Output per efficiency unit of labor)!7
!! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 K C r wh Aggregate (physical) Capital Stock (or k, Capital per efficiency unit of labor) Aggregate Consumption (or c, Consumption per efficiency unit of labor) Real interest rate Real wage per worker (or w, Real Wage per efficiency unit of labor) The exogenous variables and exogenous parameters in the model are defined as follows: t L h n g δ ρ Η time (a continuous exogenous variable) Aggregate Population and Labor Force (an exogenous variable that depends on time) efficiency of labor (an exogenous variable that depends on time) rate of growth of population (exogenous parameter) rate of technical progress (exogenous parameter) rate of depreciation of capital (exogenous parameter) pure rate of time preference of households (exogenous parameter) number of households (exogenous parameter) 3.2.1 The Production Function At each instant, the economy has a stock of capital, a given labor force and given labor "efficiency", which are combined to produce output. The production function has the form, Y (t) = F( K(t),h(t)L(t) ) (3.26) The production function has all the properties of the neoclassical production function assumed in the Solow model. The marginal product of all inputs is positive but decreasing, there are constant returns to scale and the Inada conditions are satisfied (see. Chap. 2). As in the Solow model, we shall assume that population growth and the efficiency of labor evolve according to,! L(t) = L 0 e nt (3.27)! h(t) = h 0 e gt (3.28) where L0 and h0 denote population and the efficiency of labor at time 0. Due to the assumption of constant returns to scale, the production function can be expressed in intensive form, i.e. per efficiency unit of labor, as, y(t) = f ( k(t) ) (3.29) where, y = Y/hL k = K/hL f(k) = F(k, 1) output per efficiency unit of labor capital per efficiency unit of labor production function per efficiency unit of labor!8
3.2.2 The Utility Function of the Representative Household All households in this economy are assumed identical. We therefore focus on the behavior of only one of them, the representative household. Households are indexed by h, where h is uniformly distributed between zero and 1. Thus, h [0,1]. The utility function of household h depends on the level of its per capita consumption. The representative household is assumed to have an infinite time horizon and to maximize the intertemporal utility function, 4! U h = e ρt u(c h (t))l h (t)dt (3.30) where, ch(t) u ρ per capita consumption of household h instantaneous utility function of household h pure rate of time preference of household h (an exogenous parameter) The number of members of the household is given by Lh(t) and is the same for all households. Thus, it follows that the relation between the population of household h and total population is given by,! L h (t) = L(t) 1 = L 0 e nt (3.27 ) dh 0 We assume that the instantaneous utility function of the household takes the form,! u(c h (t)) = c h (t)1 θ, θ > 0, ρ-n-(1-θ)g > 0 (3.31) 1 θ This functional form, is the constant relative risk aversion utility function, or constant elasticity of intertemporal substitution utility function, we have already introduced, where, θ coefficient of relative risk aversion, or 1/θ elasticity of intertemporal substitution of consumption The assumption that ρ-n-(1-θ)g>0 is sufficient in order to ensure that the intertemporal utility function (3.30) is well defined (finite). This assumption is also sufficient to ensure that the economy converges to a balanced growth path. As we have already mentioned, as θ tends to unity, one can prove, using l Hopital s rule, that the utility function tends to the logarithmic utility function. That is, if θ =1, we have that,! u(c h (t)) = lnc h (t) (3.31 ) 4 As in Chapter 2, the analysis is in continuous time. For an analysis of the model in discrete time, see the Annex to Chapter!9
!!! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 Since all households are similar, we shall henceforth drop the subscript h and define consumption per efficiency unit of labor as, c(t) = c h (t) h(t) Thus, consumption per household is given by, c h (t) = c(t)h(t) (3.32) Substituting equation (3.32) into equation (3.31), and the resulting equation in equation (3.30), taking into account that h(t) and L(t) increase at exogenous rates g and n respectively, we arrive at an intertemporal utility function of the representative household, expressed in consumption per efficiency unit of labor. This takes the form, U = B e βt c(t) 1 θ 1 θ dt (3.33) where,! B = h 1 θ 0 L 0 > 0, and β = ρ-n-(1-θ)g > 0. The representative household thus maximizes (3.33), subject to the constraint that its savings result in the accumulation of real assets, which take the form of physical capital. 3.2.3 The Accumulation of Capital by the Representative Household As in the Solow model, the accumulation of capital per efficiency unit of labor in the economy is determined by,! k (t) = f (k(t)) c(t) (n + g + δ )k(t) (3.34) Equation (3.34) shows that the change of aggregate physical capital per efficiency unit of labor is determined by the difference of two terms: Current investment (savings) per efficiency unit of labor, minus equilibrium investment, i.e the investment that is required in order to maintain capital per efficiency unit of labor at its current level. A social planner who would seek to maximize the intertemporal utility of the representative household, subject to the constraint (3.34), would thus determine the savings behavior that maximizes social welfare. The question is whether (3.34) is also the budget constraint facing the representative household itself. If so, the optimal savings behavior of the representative household, would also maximize social welfare. In order to answer this question one needs to consider the budget constraint facing the representative household. We assume a competitive economy, in which each member of the representative household provides a unit of labor and in which savings take the form of investment in physical capital. The real wage per worker equals the real wage per efficiency unit of labor w(t), times the efficiency of labor h(t). Consequently, the per capita labor income of the household at time t equals,!10
!!! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 w(t)h(t) where w(t) is the real wage per efficiency unit of labor. The per capita income from capital of the representative household is given by, ( r(t) + δ )K(t) L(t) where r(t) is the real interest rate. Consequently, the accumulation of capital per efficiency unit of labor by the representative household is given by,! k (t) = ( r(t) + δ )k(t) + w(t) c(t) (n + g + δ )k(t) = r(t)k(t) + w(t) c(t) (n + g)k(t) (3.34 ) The representative household takes the evolution of w(t) and r(t) as given. We shall assume that these are determined in competitive labor and capital markets. Assuming competitive markets, capital and labor are paid their respective marginal product. Thus, r(t) = f (k(t)) δ (3.35)! w(t) = f (k(t)) k(t) f (k(t)) (3.36) Substituting (3.35) και (3.36) in (3.34 ) we end up with (3.34). Thus, the budget constraint faced by the representative household, in the form of the asset accumulation equation (3.34 ), is the same as the budget constraint (3.34), faced by the economy as a whole. 3.2.4 The Efficiency of the Competitive Equilibrium In this model, due to the assumption of competitive markets, maximizing the intertemporal utility function of a representative household is essentially under the same constraint as the one that would be used by a social planner, i.e the economy wide budget constraint (3.34). Thus, the problem of the representative household is the same as the problem of an omnipotent social planner. Consequently, the competitive equilibrium in the model of the representative household would be fully efficient. A decentralized competitive equilibrium in which each household maximizes its own utility function over time, under its private budget constraint, would lead to the same outcome as that of the choice of an omnipotent social planner who had as her objective the maximization of the intertemporal utility function of the representative household, under the appropriate aggregate budget constraint. In the case of the representative household model with full and competitive markets, we have an application of the first theorem of welfare economics, which suggests that when markets are competitive and complete, and there are no externalities, the decentralized equilibrium is efficient as it maximizes social welfare.!11
)! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 3.2.5 Conditions for Utility Maximization by the Representative Household In order to find the first-order conditions for the maximization of (3.33) under the accumulation equation (3.34 ), we define the current value Hamilton function, ) H (t) = c(t)1 θ (3.37) 1 θ + λ(t) ( r(t)k(t) + w(t) c(t) (n + g)k(t) ) where λ(t) is the multiplier of the Hamilton function. λ(t) can be interpreted as the shadow value of the marginal instantaneous change of the capital stock at t (marginal investment). The Hamiltonian is thus defined as the sum of the instantaneous utility of consumption, plus the value of the change in the capital stock, priced at the shadow value λ(t). An optimal plan must maximize the Hamiltonian at each instant t, provided that the shadow value is chosen correctly. The first order conditions for the maximization of the Hamiltonian would coincide with the first order conditions for the maximization of (3.33), subject to the sequence of the accumulation equations (3.34 ). The first order conditions for the maximization of the Hamiltonian are, H (t)! (3.38) c(t) = 0 H (t)! (3.39) λ(t) = k (t) H (t) k(t) = λ (t) βλ(t) (3.40) (3.38) implies, λ(t) = c(t) θ (3.41) (3.40) implies, ( ) = λ(t) ( r(t) ρ θg)! λ (t) = λ(t) r(t) β n g (3.42) Finally, (3.39) implies the capital accumulation equation (3.34 ). (3.41) suggests that consumption must be chosen at each instant so that its marginal utility is equal to the shadow value of the marginal unit of savings, invested in capital. At each instant, goods must be equally valuable at the margin, either as consumption, or as investment. (3.42) implies that capital gains, i.e. the rate of change of the shadow value of capital λ(t), must be equal to the difference between the discount factor of the household β, and the rate of return of!12
! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 capital per efficiency unit of labor, r(t)-n-g. Τhis condition essentially ensures that the total rate of return of a marginal unit of capital, including capital gains, is equal to the discount factor of the household β. This interpretation can be confirmed by rearranging (3.42) as, r(t) + λ (t) λ(t) n g = β (3.42 ) It is straightforward to show that if a social planner maximized the intertemporal utility function of the representative household, subject to the economy wide capital accumulation condition (3.34), the relevant first order conditions would be (3.41) and, ( )! λ (t) = λ(t) f (k(t)) δ ρ θg (3.42 ) By the definition of the real interest rate in (3.35), (3.42) is exactly the same as (3.42 ), which confirms that the competitive equilibrium maximizes social welfare in this model. 3.2.6 The Euler Equation for Consumption We can examine the first order conditions, by substituting out λ(t) between (3.41) and (3.42). Taking the first derivative of (3.41) with respect to time, after using (3.42) and the definition of β, we end up with the following differential equation for c. c (t)! (3.43) c(t) = 1 ( r(t) ρ θg ) θ = 1 ( θ r(t) ρ) g (3.43) is the Euler equation for consumption per efficiency unit of labor. The growth rate of consumption per capita is positive if the real interest rate exceeds the pure rate of time preference of the representative household. In addition, the higher the elasticity of intertemporal substitution of consumption, the higher the growth rate of consumption for a given difference in the real interest rate from the pure rate of time preference of the representative household. g has a negative impact in equation (3.43), because c is consumption per efficiency unit of labor, and its denominator increases at a rate g, the exogenous rate of technical progress. This is why g must be subtracted. If (3.14) were to be re-written in terms of per capita consumption, we would have, c _ (t)! = 1 ( (3.43 ) c _ (t) θ r(t) ρ) which is the same as equation (3.25), and has the same interpretation. In what follows we concentrate on the Euler equation for consumption per efficiency unit of labor (3.43). If the economy is on a balanced growth path, consumption per head will be growing at the!13
exogenous rate of technical change g, and consumption per efficiency unit of labor will be constant. Thus, from (3.43), the real interest rate along the balanced growth path will be equal to ρ+θg. There is a unique constant real interest rate which is consistent with balanced growth in this model. 3.2.7 The Intertemporal Budget Constraint of the Representative Household (3.43) determines the rate of change of consumption on the optimal path. To determine the level of consumption on the optimal path, one must solve the two differential equations (3.43) and (3.34 ) that determine the optimal path of consumption and capital accumulation of the representative household. Let us first solve the capital accumulation equation of the representative household (3.34 ). This is a first order linear differential equation with variable coefficients. As a result, its solution for any T 0 takes the form, 5 T t! e r(v)dv (n+g)t v=0 T r(v)dv (n+g)t k(t ) + e v=0 T c(t) dt = k(0) + e r(v)dv (n+g)s v=0 w(t) dt (3.44) (3.44) describes the intertemporal budget constraint of the representative household with horizon T. This intertemporal budget constraint implies that at time 0, the present value of labor income between time 0 and time T, plus the initial capital stock at time 0, must be equal to the present value of consumption between time 0 and time T, plus the present value of the capital stock at time T. The term that includes the integral of interest rates is a term that converts one unit of income, consumption or capital in time t, to its present value at time 0. If the real interest rate was fixed at r, the term would simplify to -rt. We can define the average real interest rate between time 0 and time t as, t! r _ (t) = 1 r(v)dv (3.45) t v=0 With this definition of the average real interest rate, (3.44) can be written as, t _ (T ) n g! e r T T k(t ) + e r_ (t ) n g t T c(t) dt = k(0) + e r t w(t) dt (3.46) _ (t ) n g If the horizon of the household was T, then the optimal capital stock at instant T would be equal to zero. If the capital stock at T was positive, the household could increase its utility by consuming the rest of its capital just before T. Thus, a positive capital stock at T would not be optimal. If the capital stock at T was negative, then the household would be accumulating unsustainable debts (negative capital) along the optimal path, which is not consistent with its intertemporal budget constraint. We should therefore assume that on the optimal path, k(t)=0. This type of condition is called a transversality condition, and ensures that the present value of consumption of the household cannot exceed, or fall short of, its total wealth. Total wealth consists 5 See Mathematical Annex 1 for solution methods to ordinary linear differential equations including equations of the form of (3.34 ) and (3.43).!14
of the initial capital stock of the household, plus the present value of its labor income. Thus, since we must have that k(t)=0, for a finite time horizon T, the intertemporal budget constraint of the representative household would take the form, _ (t ) n g T! e r t T c(t) dt = k(0) + e r t w(t) dt (3.46 ) _ (t ) n g Taking into account the transversality condition, the intertemporal budget constraint implies that at time 0, the present value of labor income between time 0 and time T, plus the initial capital stock at time 0, must be equal to the present value of consumption between time 0 and time T. The question that arises is what is the transversality condition when the horizon of the household is infinite, as we have assumed. 3.2.8 The Transversality Condition with an Infinite Time Horizon If the time horizon of the household is infinite, then we should take the limit of (3.46) as T tends to infinity. In this case the term on the left of (3.46) should tend to zero. That is, we should have that, r _ (T ) n g T e! lim T k(t ) = 0 (3.47) If this condition is not satisfied, for example if the above limit is positive, then the household could along the optimal path increase its intertemporal utility by consuming a larger part of its capital. If the above limit is negative, then the household would be accumulating unsustainable debts (negative capital) along the optimal path, which is not consistent with its intertemporal budget constraint. Therefore, the only optimal path consistent with the intertemporal budget constraint of the representative household is the one that satisfies (3.47), which requires that the present value of its capital stock tends to zero as time tends to infinity. Condition (3.47) is the infinite horizon transversality condition. It is satisfied as long as the capital stock per efficiency unit of labor does not increase (or decrease) at a rate faster than r-n-g, which is the same as saying that the aggregate capital stock does not increase (or decrease) at a rate faster than r. As we have already mentioned, and will prove explicitly in the next section, the real interest rate on the balanced growth path is equal to ρ+θg. As a result, if the economy is on the balanced growth path, the transversality condition takes the form, ( ) = lim! lim (3.47 ) T e βt k(t ) u c(t ) e βt k(t ) = 0 T given that u (c(t))>0. (3.47 ) is the transversality condition on the balanced growth path. Given that the transversality condition (3.47) must be satisfied, the intertemporal budget constraint of a representative household with an infinite time horizon takes the form, _ (t ) n g! e r t c(t) dt = k(0) + e r t w(t) dt (3.48) _ (t ) n g!15
(3.48) implies that the present value of consumption of a representative household with an infinite time horizon equals its total wealth, which is defined by its initial capital stock (physical capital), plus the present value of current and future labor income (human capital). 3.2.9 The Consumption Function of the Representative Household If we solve (integrate) the differential equation describing the Euler equation for consumption (3.43), then we find that consumption at time t is defined by, 1 θ r_ (t ) ρ θg t! c(t) = c(0)e (3.49) Substituting (3.49) in the intertemporal budget constraint (3.48), and solving for c(0),! c(0) = γ (0) k(0) + w ~ (0) (3.50) where, e r _ (t ) n g! w ~ (0) = t w(t) dt (3.51) is the present value of labor income, and, 1 r _ (t )(1 θ ) ρ+θn θ t! γ (0) = e dt (3.52) is the share of total wealth in period 0 that is consumed. (3.50), with the definitions (3.51) and (3.52), determines the level of consumption for the representative household. Consumption at time 0 is a proportion γ(0) of total wealth. (3.50) allows us to deduce the properties of the consumption function of the representative household. The representative household consumes a share of its total wealth γ(0), that depends on the evolution of the average future real interest rates, the pure rate of time preference rate ρ, the elasticity of intertemporal substitution of consumption 1/θ, and the population growth rate n. The impact of the average real interest rate on the proportion of total wealth that is consumed depends on the elasticity of intertemporal substitution of consumption 1/θ. An increase in average real interest rates has two kinds of effects on the average consumption to total wealth ratio: an intertemporal substitution effect, and an income effect. First, it induces the household to substitute current for future consumption, and increases the cost of current consumption relative to future consumption. This is the intertemporal substitution effect in consumption, which tends to decrease current consumption. Second, an increase in interest rates increases income from capital, and tends!16
! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 to increase both current and future consumption. This is the income effect, which tends to increase current consumption. If the elasticity of intertemporal substitution of consumption is greater than one (θ < 1), then consumption as a proportion of total wealth decreases when real interest rates rise and savings increase, because the negative substitution effect is stronger than the positive income effect, and thus prevails. If the elasticity of intertemporal substitution is less than unity (θ > 1), then consumption as a proportion of total wealth increases when interest rates rise, because the positive income effect is stronger than the negative substitution effect. Finally, if θ = 1, which is the case with logarithmic preferences, the two results cancel each other out, and consumption as a proportion of total wealth is independent of the path of real interest rates. It is worth deriving the consumption function when the real interest rate is fixed at r, as happens in the steady state. In this case, (3.52) takes the form, γ (0) = e 1 ( r(1 θ ) ρ+θn θ )t 1 r(1 θ) ρ +θn = θ 1 r(1 θ ) ρ+θn θ lim e ( )t 1 θ lim e t t 0 ( r(1 θ ) ρ+θn )t = 1 ( ρ θn r(1 θ) ) θ For this to be positive, we must have that r<(ρ-θn)/(1-θ). On the balanced growth path, the real interest rate is constant and equal to r=ρ+θg. As a result, on the balanced growth path, the share of total wealth that is consumed is equal to, ( )! γ (0) == ρ n (1 θ)g (3.52 ) From the assumptions we have made in order to have a well defined intertemporal optimization problem for the representative household (see eq. (3.4)), this share is positive. In the case where θ=1, i.e logarithmic preferences, from (3.52), the share of consumption to total wealth is given by, ( ) 1 ρ n! γ (0) = e (ρ n)t = = ρ n (3.53) lim e (ρ n)t + lim e (ρ n)t t t 0 Given that we have assumed that ρ>n, with a unitary elasticity of intertemporal substitution of consumption, the share of consumption in total wealth is equal to the difference between the pure rate of time preference and the population growth rate. Finally, it is important to note that the overall impact of real interest rates on consumption is not limited to the impact on the propensity to consume out of total wealth. An increase in real interest rates leads to a decrease in the present value of future labor income, reducing the overall wealth of the representative household, and leading to a reduction in consumption, even if the elasticity of intertemporal substitution is equal to one. Essentially, the effects of real interest rates on the present value of income from employment, i.e the wealth effects of real interest rates, reinforce the negative substitution effect on current consumption.!17
! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 3.3 Dynamic Adjustment and the Balanced Growth Path We now move on to the question of how the balanced growth path is determined, and the dynamic adjustment of consumption, capital and other real variables in the Ramsey model of the representative household. The dynamic adjustment of the economy in the model is described by equations (3.34) for the accumulation of capital and (3.43) for the rate of growth of consumption. We have two first-order differential equations in two variables, k and c. We can use the two differential equations (3.34) and (3.43) to fully analyze the dynamic adjustment of the economy.! k (t) = f (k(t)) c(t) (n + g + δ )k(t) (3.34) c (t)! (3.43) c(t) = 1 ( r(t) ρ θg ) θ = 1 ( f (k(t) δ ρ θg) θ On the right hand side of (3.43) we have have used (3.35) to substitute for the real interest rate it terms of the marginal product of capital minus the depreciation rate of capital δ. Once we determine the path of the capital stock and consumption, the paths of all other real variables, namely output, the real interest rate and the real wage, follow from the production function (3.29) and the marginal productivity conditions (3.35) and (3.36), which only depend on capital per effective unit of labor. The solution of the second order system of non-linear differential equations (3.34) και (3.43) can be described diagrammatically with the help of a phase diagram. 3.3.1 Dynamic Adjustment towards the Balanced Growth Path The capital stock (per efficiency unit of labor) that ensures! c = 0, i.e constant consumption per efficiency unit of labor, is determined by (3.43), from the equalization of the marginal product of capital with the pure rate of time preference, plus the depreciation rate, plus the rate of technical progress g multiplied by θ. This defines the steady state real interest rate and the steady state capital stock in this model. f (k) = ρ + δ +θg (3.54) (3.54) is depicted as the vertical line! c = 0 in Figure 3.1. It defines the steady state capital stock k*. If the capital stock is lower than k*, then the real interest rate is higher than ρ+θg, and from (3.43), consumption per efficiency unit of labor is rising. Hence the vertical arrows depicting rising consumption to the left of k*. If the capital stock is higher than k*, then the real interest rate is lower than ρ+θg, and from (3.43), consumption per efficiency unit of labor is falling. Hence the vertical arrows depicting falling consumption to the right of k*.!18
! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chapter 3 From (3.34), for! k = 0, one can derive the relation between consumption and the capital stock (per efficiency unit of labor) that ensures a constant capital stock (per efficiency unit of labor). This is given by, c = f (k) (n + g + δ )k (3.55) (3.55) is depicted as the! k = 0 curve in Figure 3.1. Because of the properties of the production function f(k), it is upward sloping up to the point f (k) =n+g+δ, and downward sloping after that point. Thus, it achieves its maximum at the golden rule capital stock for which the real interest rate is equal to the steady state growth rate n+g. If consumption is higher than the level implied by (3.55), then savings are lower than the savings required to maintain a constant capital stock per efficiency unit of labor, and the capital stock per efficiency unit of labor is falling. Hence the left pointing horizontal arrows above the constant capital curve in Figure 3.1. If consumption is lower than the level implied by (3.55), then savings are higher than the savings required to maintained a constant capital stock per efficiency unit of labor, and the capital stock per efficiency unit of labor is rising. Hence the right pointing horizontal arrows above the constant capital curve in Figure 3.1. When (3.54) and (3.55) are satisfied simultaneously, the economy is on a balanced growth path, as both the capital stock and consumption per efficiency unit of labor are constant, which is equivalent to saying that per capita consumption and the per capita capital stock grow at the exogenous rate of technical progress g. The remaining real variables, such as output, the real wage and the real interest rate are also on a balanced growth path, as, through the production function and the marginal productivity conditions, they depend solely on the capital stock per efficiency unit of labor. The balanced growth path, or steady state equilibrium, at (k*, c*) is unique, and is depicted in Figure 3.1, which represents both (3.54) and (3.55) geometrically. Figure 3.1 also represents the adjustment path leading to the steady state. In each section of Figure 3.1, adjustment paths are determined by the direction of the resultant of the changes in consumption and capital. Thus, if both consumption and capital are rising in a section of the graph, then the economy moves towards the north east. If both consumption and capital are falling in a section of the graph, then the economy moves towards the south west. If consumption is rising and capital falling, the economy moves towards the north west, and in the opposite case towards the south east. Both of these directions move the economy away from the steady state. The steady state is a saddle point. There is a unique adjustment path, a saddle path, leading to this equilibrium, as the capital stock k is a predetermined (state) variable, and consumption c is a non predetermined (control) variable. The saddle path goes through the north east and the south west part of the diagram. For any initial value of k, consumption adjusts immediately to ensure that the economy is put on the unique saddle path leading to the steady state (the balanced growth path). All the other adjustment paths, some of which are also depicted in Figure 3.1, eventually diverge and lead the economy away from the balanced growth path, not satisfying the transversality condition (3.47). On the adjustment path, if the capital stock (per efficiency unit of labor) is lower that k*, consumption is also lower than c*, and the economy accumulates capital at a rate higher than n+g.!19
During this process, capital and consumption per efficiency unit of labor are rising. The opposite happens if the initial capital stock (per efficiency unit of labor) is higher than k*. The capital stock and consumption per efficiency unit of labor are falling. Consequently, the behavior of the economy on the adjustment path resembles in many ways the behavior of the economy in the Solow model. There is convergence towards a unique long-run equilibrium (steady state or balanced growth path) regardless of initial conditions. The difference is that in the Ramsey model savings behavior is not arbitrary but optimal. At any point in time, the representative household chooses its consumption to maximize its intertemporal utility. Due to competitive markets, the optimal individual behavior of the representative household also maximizes social welfare. Both the short run equilibrium, on the adjustment path, and the long run equilibrium, on the balanced growth path, are not only Pareto efficient, but also consistent with the maximization of social welfare. It is worth looking at the properties of the steady state, or balanced growth path, into more detail. While the capital stock, output and consumption per efficiency unit of labor are constant, the per capita capital stock, output and consumption increase continuously at a rate g, the exogenous rate of technical progress. The real interest rate is fixed on the balanced growth path, as is the real wage per efficiency unit of labor. However, the real wage per employee w(t)h(t) grows at a rate g, the exogenous rate of technical progress, which causes a continuous increase in labor efficiency. In the process of adjustment towards the balanced growth path from the left, i.e when the initial capital stock per efficiency unit of labor is less than k*, per capita output and per capita consumption are rising faster than g, the real interest rate is on a downward path, because of the falling marginal product of capital, and the real wage per employee is rising faster than g, because of the rising marginal product of labor. The opposite happens in the process of adjustment to the balanced growth path from the right, i.e when the initial capital stock per efficiency unit of labor is higher than k*. 3.3.2 The Balanced Growth Path and the Modified Golden Rule The balanced growth path in the representative household model is similar to the balanced growth path in the Solow model. The capital stock, output and consumption per efficiency unit of labor are constant. Consequently, the savings ratio (y-c)/y, is also constant on the balanced growth path. The total capital stock, total output and total consumption are growing at a rate n+g. The per capita capital stock, per capita output and per capita consumption are growing at a rate g. Consequently, the central predictions of the Solow model regarding the determinants of long-term growth are not dependent on the assumption of a constant exogenous saving rate. Even when the savings rate is endogenous, as in the Ramsey model, the exogenous rate of technical progress remains the sole determinant of long-term growth of output, consumption and wages per capita. The main difference of the Ramsey model from the Solow model, regarding the balanced growth path, is that in the Ramsey model it is not possible for the capital stock on the balanced growth path to exceed the capital stock that corresponds to the golden rule. In the Solow model, with a!20