The Representative Household Model

Transcription

1 Chapter 3 The Representative Household Model The representative household class of models is a family of dynamic general equilibrium models, based on the assumption that the dynamic path of aggregate consumption is decided in an optimal fashion by identical households. In the most widely used model in this class, the Ramsey model, 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, the Ramsey model has a lot in common with the neoclassical model of economic growth of Solow [1956]. Historically, the representative household growth 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 and access to the capital market. 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 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 demand- 1 The stochastic growth model is presented and analyzed in Chapter

2 72 Ch. 3 The Representative Household Model ing mathematically, than the Ramsey model. The Fisher model belongs to the representative household class of models, but has a restrictive two period time structure. 2 Assumptions about technology and market structure in the representative household growth model are similar to the assumptions of the Solow model. What di ers 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 of a forward looking household. Consequently, savings behavior is determined endogenously and is optimal. 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. 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 ine ciency, 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. One of the main di erences of the Ramsey representative household model from the Solow model is related to the e ects of population growth on 2 The Fisher model has since the 1950s been used as the basic building block of an alternative class of optimizing growth models, the Samuelson [1958] and Diamond [1965] models of overlapping generations, that we shall examine in the next chapter.

3 George Alogoskoufis, Dynamic Macroeconomics 73 per capita income. As we have shown, in the Solow model the rate of population growth has a negative impact on per capita output and income on the balanced growth path. This is because the savings rate is an exogenous constant. In the Ramsey model, because of the optimal savings behavior of the representative household, the rate of population growth does not a ect per capita income on the balanced growth path. This is because the household internalizes the welfare of future generations, and the optimal savings rate responds to the rate of growth of population in a way that does not a ect capital accumulation and therefore per capita capital and income. However, the Ramsey model is also an exogenous growth model, similar in this respect to the Solow model. 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. However, it does not determine the steady state growth rate, which is treated as an exogenous parameter, the exogenous rate of technical progress. 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 long time horizon equal to T, which is the problem posed and solved by Ramsey [1928] Optimal Consumption and Savings 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 o its savings. The household chooses the path of consumption in order to maximize an intertemporal utility function of the form,

4 74 Ch. 3 The Representative Household Model U = U(c 1,c 2 )=u(c 1 ) u(c 2) (3.1) where, is the pure rate of time preference of the household, and, u is a concave, twice di erentiable utility function, for the first two derivatives of which we assume u 0 > 0, u 00 < 0. Savings of period 1, plus interest, can be consumed in period 2. It thus follows that, c 2 apple (1 + r)(w c 1 ) (3.2) where r is the real interest rate. (3.2) describes the budget constraint of the household. It can be rearranged and expressed in the following form: c r c 2 apple 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, )=u(c 1 ) u(c 2) c r c 2 w (3.4) where 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 0 (c 1 ) = 0 (3.5) 1 1+ u0 (c 2 ) 1+r = 0 (3.6) 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,

5 George Alogoskoufis, Dynamic Macroeconomics 75 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 0 (c 2 ) 1+ u 0 (c 1 ) = 1 1+r (3.7) (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 0 (c 2 ) u 0 (c 1 ) = 1+ 1+r (3.8) (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 = r, then,u 0 (c 1 )=u 0 (c 2 ), c 1 = c 2 If >r,then,u 0 (c 1 ) <u 0 (c 2 ), c 1 >c 2 If <r,then,u 0 (c 1 ) >u 0 (c 2 ), c 1 <c 2 Thus, (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. This takes the form, u(c t )= c t where 1/ is the elasticity of intertemporal substitution. (3.9) This utility

6 76 Ch. 3 The Representative Household Model function is quite general and will be used throughout this book. 3 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 = 1, it takes the form of logarithmic utility, u(c t )=lnc t (3.10) For the utility function (3.9), the first order condition (3.8) can be written as, c2 c 1 = 1+ 1+r (3.11) For the utility function (3.10), with an elasticity of intertemporal substitution equal to unity, the first order condition (3.8) can be written as, c2 c 1 = 1+ 1+r From (3.11) it follows that consumption in the first period satisfies, c 1 = (3.12) 1 1+ c2 (3.13) 1+r Combining (3.13) with the intertemporal budget constraint (3.3), the savings rate in the first period is determined by, s(r) = w c 1 w = (1 + r) 1 (1 + ) 1 1 +(1+r) < 1 (3.14) (3.14) indicates that the savings rate is a positive function of the real interest rate r, only if is less than unity, that is, if the elasticity of intertemporal substitution 1/ is greater than unity. Only then does the substitution e ect of a change of the real interest rate dominate the income e ect. If is greater than unity, the savings rate is a negative function of r, as the elasticity of intertemporal substitution 1/ is smaller than unity, and the income e ect dominates. In the special case where is equal to unity (logarithmic utility), the savings rate is independent of the real interest rate and is equal to 1/(2 + ). 3 In the economics of uncertainty this utility function is also known as the constant relative risk aversion utility function, and is referred to as the coe cient of constant relative risk aversion

7 George Alogoskoufis, Dynamic Macroeconomics 77 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 + ) The Optimal Path of Consumption with a Finite Time Horizon We next turn to the more general Ramsey [1928] model. This model addresses 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 finite time horizon equal to T. 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 time horizon T and initial interest yielding assets equal to a(0). It is assumed to maximize the following intertemporal utility function, subject to, U = Z T t=0 e t u(c(t))dt (3.15) ȧ(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 di erentiable and concave. 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). The household cannot end up with negative assets. From the maximum principle, the conditions for the maximization of (3.15) subject to the accumulation constraint (3.16) are the same as the

8 78 Ch. 3 The Representative Household Model first order conditions for the maximization of the current value Hamilton function, which for this problem is defined by, 4 H(t) =u(c(t)) + (t)(ra(t)+w c(t)) (3.19) where (t) is the current value multiplier of the asset accumulation constraint. (t) can be interpreted as the shadow price of the marginal instantaneous change of household assets at t. The Hamiltonian is thus defined as the sum of the instantaneous utility of consumption, plus the value of the change in the assets of the household, priced at the shadow price (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 (3.35). The first order conditions for the maximization of the current value Hamilton function are =0) u0 (c(t)) = = (t) (t) ) (t) = (r ) (t) =ȧ(t) ) ȧ(t) =ra(t)+w c(t) (3.22) From (3.20), on the optimal path, the multiplier (t), which is the value of the marginal increase in assets, is equal to the marginal utility of consumption. Thus, the household is indi erent 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 maginal increase in the value of assets (capital gain), is equal to the pure rate of time preference of the household. 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), di erentiating with respect to time, we get, (t) =u 00 (c(t))ċ(t) (3.23) 4 See Appendix D for the mathematics of intertemporal optimization in continuous time, and their application to the representative household problem.

9 George Alogoskoufis, Dynamic Macroeconomics 79 Substituting (3.20) and (3.23) in (3.21), we get, ċ(t) = u0 (c(t)) u 00 (r ) (3.24) (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. 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 1928 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 assumed negative, i.e marginal utility is declining, the change in consumption will have the same sign as the di erence 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/, then (3.24) takes the form, ċ(t) c(t) = 1 (r ) (3.25) (3.25) implies that the 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 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

10 80 Ch. 3 The Representative Household Model 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/. 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 di erence 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 e ciency unit of labor). K, aggregate stock of (physical) capital (or k, capital per e ciency unit of labor), C, aggregate consumption (or c, consumption per e ciency unit of labor), r, real interest rate, ŵ = wh real wage per worker (or w, real wage per e ciency unit of labor). The exogenous variables and exogenous parameters in the model are defined as follows: t, time (a continuous exogenous variable), L, aggregate population and employment (an exogenous variable that depends on time), h, e ciency of labor (an exogenous variable that depends on time), n, rate of growth of population (exogenous parameter), g, rate of technical progress (exogenous parameter),, rate of depreciation of capital (exogenous parameter),, pure rate of time preference of households (exogenous parameter) The Production Function At each instant, the economy has a stock of capital, a given labor force and given labor e ciency, which are combined to produce output. The production function has the form, Y (t) =F (K(t),h(t)L(t)) (3.26)

11 George Alogoskoufis, Dynamic Macroeconomics 81 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. 5 As in the Solow model, we shall assume that population growth and the e ciency of labor evolve according to, L(t) =L 0 e nt (3.27) h(t) =h 0 e gt (3.28) where L 0 and h 0 denote population and the e ciency 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 e ciency unit of labor, as, y(t) =f (k(t)) (3.29) where, y = Y/hL, output per e ciency unit of labor, k = K/hL, capital per e ciency unit of labor, and f(k) =F (k, 1) is the production function per e ciency unit of labor The Utility Function of the Representative Household All households in this economy are assumed identical. Thus, we shall eventually focus our attention on the behavior of only one of them, the representative household. Households are indexed by j, where j is uniformly distributed between zero and 1. Thus, j 2 [0, 1]. The utility function of household j 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, 6 U j = Z 1 t=0 e t u(c j (t))l j (t)dt (3.30) 5 See the discussion on the neoclassical production function in Chapter 2. 6 The assumption of the infinite horizon is justified on the grounds that households are dynasties in economies that go on forever. Without this assumption, the model does not possess a steady state.

12 82 Ch. 3 The Representative Household Model where, c h (t), denotes the per capita consumption of household j at instant t, u, the instantaneous utility function of household j and, thepure rate of time preference of household j, an exogenous preference parameter. The number of members of the household is given by L j (t) and is the same for all households. Thus, it follows that the relation between the members of individual households j and total population is given by, L j (t) = L(t) R 1 0 dj = L 0e nt We assume that the instantaneous utility function of the household takes the form, u(c j (t)) = c j(t) 1 1 (3.31) where, >0and n (1 )g >0. This functional form, is the constant elasticity of intertemporal substitution utility function, we have already introduced in the previous section, with 1/ the elasticity of intertemporal substitution of consumption. The assumption that n (1 )g >0 is su cient in order to ensure that the intertemporal utility function (3.30) is well defined and converges to a finite value. This assumption is also su cient to ensure that the economy eventually converges to a steady state, or balanced growth path. As we have already mentioned, as tends to unity, one can prove, using l Hopital s rule, that the utility function tends to the logarithmic utility function. That is, in the case = 1, we have that, u(c j (t)) = ln c j (t) Since all households are the same, we shall henceforth drop the subscript j and concentrate not on average household consumption, but on consumption per e ciency unit of labor. If c j (t) is the average consumption per member of household j, and all households are the same, then it follows that, c j (t) =ĉ(t), 8j where ĉ(t) is per capita consumption at instant t. Then, consumption per e ciency unit of labor c(t) isdefinedby, c(t) = ĉ(t) h(t) (3.32)

13 George Alogoskoufis, Dynamic Macroeconomics 83 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 terms of consumption per e ciency unit of labor. This takes the form, Z 1 U = B e t=0 t c(t)1 dt (3.33) 1 where, B = h 1 0 L 0 > 0 and = n (1 )g >0. The representative household thus maximizes (3.33), which is equivalent to (3.30), subject to the constraint that its savings result in the accumulation of real assets, which take the form of physical capital The Accumulation of Capital and the Optimality of the Decentralized Competitive Equilibrium As in the Solow model, the accumulation of capital per e labor in the economy is determined by, ciency unit of k(t) =f(k(t)) c(t) (n + g + )k(t) (3.34) Equation (3.34) shows that the change of aggregate physical capital per e ciency unit of labor is determined by the di erence of two terms: Current investment (savings) per e ciency unit of labor, minus equilibrium investment, i.e the investment that is required in order to maintain capital per e ciency unit of labor at its current level. A social planner who would seek to maximize the intertemporal utility of the representative household (3.33), 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 in a decentralized competitive equilibrium. If so, the optimal savings behavior of the representative household, would also maximize social welfare, and the decentralized equilibrium achieve the same outcome as a social planner. In order to answer this question one needs to consider the budget constraint facing the representative household in a decentralized competitive economy. 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

14 84 Ch. 3 The Representative Household Model real wage per e ciency unit of labor w(t), times the e ciency of labor h(t). Consequently, the per capita labor income of the household at time t equals, w(t)h(t) where w(t) is the real wage per e ciency unit of labor. The per capita income from capital of the representative household is given by, (r(t)+ ) K(t) L(t) where r(t) + is the rental price of capital in a competitive capital market. Consequently, in a decentralized competitive economy, the accumulation of capital per e ciency unit of labor by the representative household is given by, k(t) =(r(t) + ) k(t) + w(t) c(t) (n + g + )k(t) = r(t)k(t)+ w(t) c(t) (n + g)k(t) (3.35) 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)+ w(t) =f(k(t)) = f 0 (k(t)) k(t)f 0 (k(t)) (3.36) Substituting (3.36) in (3.35) we end up with (3.34). Thus, the budget constraint faced by the representative household, in the form of the asset accumulation equation (3.35), is the same as the budget constraint (3.34), faced by the economy as a whole. 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.

15 George Alogoskoufis, Dynamic Macroeconomics 85 Consequently, the competitive equilibrium in the model of the representative household would be fully optimal. 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 optimal as it maximizes social welfare 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 (3.35), we define the current value Hamilton function,! H(t) = c(t)1 + (t)(r(t)k(t)+ w(t) c(t) (n + g)k(t)) (3.37) 1 where (t) is the multiplier of the Hamilton function. (t) can be interpreted as the shadow price 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 price (t). An optimal plan must maximize the Hamiltonian at each instant t, provided that the shadow value is chosen correctly. From the maximum principle, 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 (3.35). The first order conditions for the maximization of the = (t) = k(t) (3.39)

16 86 Ch. 3 The Representative Household Model = (t) (t) (3.40) (3.40) implies, (t) =c(t) (3.41) (t) = (t)(r(t) n g) = (t)(r(t) g) (3.42) Finally, (3.39) implies the capital accumulation equation (3.35). (3.41) suggests that consumption must be chosen at each instant so that its marginal utility is equal to the?shadow price? 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 price? of capital (t), must be equal to the di erence between the discount factor of the household, and the rate of return of capital per e ciency unit of labor, r(t) n g.?his 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. This interpretation can be confirmed by rearranging (3.42) as, r(t)+ (t) (t) 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, n g! = (t) = (t) f 0 (k(t)) g By the definition of the real interest rate in (3.35), (3.42) is exactly the same as the equation above, which confirms that the competitive equilibrium maximizes social welfare in this model.

17 George Alogoskoufis, Dynamic Macroeconomics The Euler Equation for Consumption We can examine the first order conditions, by substituting out (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, we end up with the following di erential equation for c. ċ(t) c(t) = 1 (r(t) g) =1 (r(t) ) g (3.43) (3.43) is the Euler equation for consumption per e ciency 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 di erence 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 e ciency 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, ĉ(t) ĉ(t) = 1 (r(t) ) where ĉ is per capita consumption. This is the same as equation (3.25), and has the same interpretation. In what follows we concentrate on the Euler equation for consumption per e ciency unit of labor (3.43). If the economy is on a balanced growth path, consumption per head will be growing at the exogenous rate of technical change g, and consumption per e ciency unit of labor will be constant. Thus, from (3.43), the real interest rate along the balanced growth path will be equal to + g. As we shall see below, there is a unique constant real interest rate which is consistent with a balanced growth path in this model 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

18 88 Ch. 3 The Representative Household Model two di erential equations (3.43) and (3.35) 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 (3.35). This is a first order linear di erential equation with variable coe cients. As a result, its solution for any T 0 takes the form, e (R T v=0 r(v)dv (n+g)t) Z T k(t )+ Z T k(0) + e (R t t=0 t=0 e (R t v=0 r(v)dv (n+g)t) c(t)dt = v=0 r(v)dv (n+g)s) 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, as, r(t) = 1 t Z t v=0 r(v)dv (3.45) With this definition of the average real interest rate, (3.44) can be written Z T e ( r(t ) n g)t k(t )+ k(0) + Z T t=0 t=0 e ( r(t) n g)t c(t)dt = e ( r(t) n g)t w(t)dt (3.46) 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, and hence the path would not be optimal. Thus, a positive capital

19 George Alogoskoufis, Dynamic Macroeconomics 89 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 would be violating 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 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, Z T t=0 Z T e ( r(t) n g)t c(t)dt = k(0) + e ( r(t) n g)t w(t)dt t=0 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 The Transversality Condition with an Infinite Time Horizon If the time horizon of the household is infinite, as we have been assuming, 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, lim e ( r(t ) n g)t k(t ) = 0 (3.47) T!1 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.

20 90 Ch. 3 The Representative Household Model Condition (3.47) is the infinite horizon transversality condition. Itis satisfied as long as the capital stock per e ciency 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 + g. As a result, if the economy is on the balanced growth path, the transversality condition takes the form, lim e T!1 T k(t )u 0 (c(t )) = lim T!1 e T k(t ) = 0 (3.48) given that u 0 (c(t )) > 0. (3.48) 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, Z 1 t=0 Z 1 e ( r(t) n g)t c(t)dt = k(0) + e ( r(t) n g)t w(t)dt (3.49) t=0 (3.49) 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) The Consumption Function of the Representative Household If we solve (integrate) the di erential equation describing the Euler equation for consumption (3.43), then we find that consumption at time t is defined by, c(t) =c(0)e 1 ( r(t) g)t (3.50) Substituting (3.50) in the intertemporal budget constraint (3.49), and solving for c(0), where, c(0) = (0) (k(0) + w(0)) (3.51)

21 George Alogoskoufis, Dynamic Macroeconomics 91 w(0) = Z 1 t=0 is the present value of labour income, and, Z 1 (0) = e t=0 e ( r(t) n g)t w(t)dt (3.52) r(t)(1 ) + n t dt 1 (3.53) is the share of total wealth that is consumed in period 0. (3.51), with the definitions (3.52) and (3.53), determines the level of consumption for the representative household. Consumption at time 0 is a proportion (0) of total wealth. (3.51) allows us to deduce the properties of the consumption function of the representative household. The representative household consumes a share of its total wealth equal to (0). This share depends on the evolution of the average future real interest rates, the pure rate of time preference rate, the elasticity of intertemporal substitution of consumption 1/, 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/. An increase in average real interest rates has two kinds of e ects on the average consumption to total wealth ratio: an intertemporal substitution e ect, and an income e ect. First, it induces the household to substitute current for future consumption, as it increases the cost of current consumption relative to future consumption. This is the intertemporal substitution e ect in consumption, which tends to decrease current consumption. Second, an increase in interest rates increases income from capital, and tends to increase both current and future consumption. This is the income e ect, which tends to increase current consumption. If the elasticity of intertemporal substitution of consumption is greater than one ( <1), then consumption as a proportion of total wealth decreases when real interest rates rise and savings increase, because the negative intertemporal substitution e ect is stronger than the positive income e ect, and thus the intertemporal substitution e ect prevails. If the elasticity of intertemporal substitution is less than unity ( >1), then consumption as a proportion of total wealth increases when interest rates rise, because the positive income e ect is stronger than the negative intertemporal substitution e ect. Finally, if ( = 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. These e ects are exactly the same as in the two period model we analyzed in Section 3.1.

22 92 Ch. 3 The Representative Household Model 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, (3.53) takes the form, = (0) = = 1 R 1 t=0 e 1 (r(1 ) + n)t 1 r(1 ) + n lim e 1 (r(1 ) + n)t lim e 1 (r(1 ) + n)t t!1 t!0 ( n r(1 )) For this to be positive, we must have that r<( n)/(1 ). On the balanced growth path, the real interest rate is constant and equal to r = + g. As a result, on the balanced growth path, the share of total wealth that is consumed is equal to, (0) == ( n (1 )g) = From the assumptions we have made in order to have a well defined intertemporal optimization problem for the representative household (see eq. (3.33)), this share is positive. In the case where = 1, i.e logarithmic preferences, from (3.53), the share of consumption to total wealth is given by, Z 1 1 (0) = e ( n)t = t=0 n lim e ( n)t +lime t!1 t!0 ( n)t = n Given that we have assumed that >n, with a unitary elasticity of intertemporal substitution of consumption, the share of consumption in total wealth is equal to the di erence 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 e ects of real interest rates on the present value of income from employment, i.e the wealth e ects of real interest rates, reinforce the negative substitution e ect on current consumption.

23 George Alogoskoufis, Dynamic Macroeconomics Dynamic Adjustment and the Balanced Growth Path We may 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 di erential equations in two variables, k and c. We can use the two di erential equations (3.34) and (3.43) to fully analyze the dynamic adjustment of the economy. (3.34) is given by, k(t) =f(k(t)) c(t) (n + g + )k(t), and is the equation describing the accumulation of capital per e ciency unit of labor. (3.43) is given by, ċ(t) c(t) = 1 (f 0 (k(t) g), and is the Euler equation for consumption per e ciency unit of labor. 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. 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 (3.36), which only depend on capital per e ective unit of labor. The solution of the second order system of non-linear di erential equations (3.34) and (3.43) can be described diagrammatically with the help of a phase diagram Dynamic Adjustment towards the Balanced Growth Path The capital stock (per e ciency unit of labor) that ensures ċ(t) = 0, i.e constant consumption per e ciency 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. This defines the steady state real interest rate and the steady state capital stock in this model. f 0 (k) = + + g (3.54) Note that (3.54) implies that the steady state capital stock, depends only

24 94 Ch. 3 The Representative Household Model on the parameters of the production function and four other parameters. The pure rate of time preference, the depreciation rate, theinverse of the elasticity of intertemporal substitution in consumption and the rate of technical progress g. Only technological and preference parameters a ect the steady state per capita capital stock, and, through the production function, the steady state per capita income. Note that the rate of growth of population n, unlike the Solow model, does not a ect steady state per capita income. (3.54) is depicted as the vertical line ċ(t) = 0 in Figure 3.1. We shall call it the steady state consumption line, because along it, consumption per e ciency unit of labor is constant. It defines the steady state capital stock k. If the capital stock is lower than k, then the real interest rate is higher than + g, and, from (3.43), consumption per e ciency 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 + g, and from (3.43), consumption per e ciency unit of labor is falling. Hence the vertical arrows depicting falling consumption to the right of k. c c=0 c * k=0 k * k Figure 3.1: The Balanced Growth Path and Dynamic Adjustment in the Ramsey Model

25 George Alogoskoufis, Dynamic Macroeconomics 95 From (3.34), for k(t) = 0, one can derive the relation between consumption and the capital stock (per e ciency unit of labor) that ensures a constant capital stock (per e ciency unit of labor). This is given by, c = f(k) (n + g + )k (3.55) (3.55) is depicted as the k(t) = 0 curve in Figure 3.1. We shall call it the steady state capital curve, because along it, the capital stock per e - ciency unit of labor is constant. Because of the properties of the production function f(k), it is upward sloping up to the point f 0 (k) =n + g +, 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 e ciency unit of labor, and the capital stock per e ciency 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 e ciency unit of labor, and the capital stock per e ciency 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 e ciency 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 e ciency unit of labor. The balanced growth path, or steady state equilibrium, at (k,c )is unique, and is depicted in Figure 3.1, which depicts both (3.54) and (3.55) geometrically. Figure 3.1 also depicts 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

26 96 Ch. 3 The Representative Household Model 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, the saddle path, leading to this saddle point, 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, violating the transversality condition (3.47). On the adjustment path, if the capital stock (per e ciency 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. During this process, capital and consumption per e ciency unit of labor are rising. The opposite happens if the initial capital stock (per e ciency unit of labor) is higher than k.the capital stock and consumption per e ciency 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 di erence is that in the Ramsey model savings behavior is not arbitrary but optimal. At any point in time, the representative household chooses its consumption in order 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 e cient, 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 e ciency 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 e ciency 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 e ciency. In the process of adjustment towards the balanced growth path from the left, i.e when the initial capital stock per e ciency unit of labor is less than k, per capita output and per capita consumption are rising faster than g,