Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path

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1 Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path Ryoji Ohdoi Dept. of Industrial Engineering and Economics, Tokyo Tech This lecture note is mainly based on Ch. 8 of Acemoglu (29). Ch. 2 of Blanchard and Fischer (1989) and the same chapter of Barro and Sala i Martin (24) also provide excellent explanations of the Ramsey Cass Koopmans model. Contents 1 Setup of the Baseline Model Households Firms Competitive Equilibrium Path Market Clearing Condition Definition of Competitive Equilibrium Path The Social Optimum and The First Theorem of Welfare Economics 7 4 Steady State 8 5 Transitional Dynamics Diagrammatic Analysis of Global Dynamics Local Dynamics Summary 14 1

2 1 Setup of the Baseline Model Recall that in the model of Solow (1956), consumption C and savings S are proportional to current income Y : S = sy, C = (1 s)y, where s (, 1) is the saving rate which is assumed to be exogenous. Ramsey-Cass-Koopmans model (or simply, Ramsey model): differs from the Solow s model in the respect that it endogenizes the saving rate by explicitly introducing the consumer s infinite-horizon optimization to the model. 1. How much should a nation save? Ramsey (1928) discussed such a problem of optimal saving by explicitly modeling the decision to save. 2. Cass (1965) and Koopmans (1965) used Ramsey (1928) to extend the neoclassical growth model of Solow (1956), Swan (1956) and Phelps (1961). Ramsey Cass Koopmans model now gives us a benchmark for many areas of modern macroeconomic analysis. 1.1 Households Demographics The economy consists of a set of identical households. L t denotes the size of population at date t. Population L t grows at a constant rate of n: L t ( dl t /dt) = nl t, (1) It is assumed that L = 1. Then L t = exp(nt). ( ) Hereafter, a dot over a variable is a time derivative of this variable. Preferences The preferences of the representative household are represented by where U = L t [e ρt u(c t )]dt, (2) c t : per capita consumption (c t = C t /L t ); C t is aggregate consumption ρ > : the subjective discount rate. u(c): the instantaneous utility function, assumed to satisfy u (c) >, u (c) <. 2

3 Using (1), the utility is rewritten as 1 U = e (ρ n)t u(c t )dt. (2 ) To ensure that there is discounting of future utility streams, we assume Assumption 1. ρ > n. Budget Constraint Each worker (household) is endowed with one unit of time, and inelastically supplies. Therefore the population L t corresponds to amount of labor. The budget constraint of all households: where A t : amount of total assets held by households r t and w t : the interest and wage rate. A t = r t A t + w t L t C t. (3) Defining per capita assets as a t A t /L t, (3) is rewritten as ȧ t = (r t n)a t + w t c t. (4) Intertemporal Utility Maximization The utility maximization problem of the representative household: max U = exp( (ρ n)t)u(c t )dt s.t. ȧ t = (r t n)a t + w t c t (flow budget constraint), a given (initial condition), ( t ) lim a t exp (r s n)ds (no-ponzi-game condition), 1 Barro and Sala i Martin (24) and Acemoglu (29) assume (2) or (2 ) as the objective function of the representative household at date. Blanchard and Fischer (1989), on the other hand, assume that the sum of the representative member s instantaneous utility as her objective function: U = L t [e ρt u(c t)] L t dt = e ρt u(c t )dt. Although such a difference slightly changes the household s first-order-conditions of utility maximization, key features of competitive equilibrium obtained below remain intact whichever we assume. 3

4 Current-value Hamiltonian: H(a t, c t, µ t ) = u(c t ) + µ t [(r t n)a t + w t c t ]. Necessary and Sufficient conditions of utility maximization are H/ c t = u (c t ) = µ t, (5) H/ a t + µ t (ρ n)µ t = µ t /µ t = ρ r t, (6) lim µ ta t e (ρ n)t =. (7) From (5) and (6), we obtain the following Euler equation: c tu (c t ) ċ t u = r t ρ. (8) (c t ) c t On the other hand, from (6) we have µ t = µ exp( t (ρ r s)ds). Substituting this into (7) yields ( t ) lim a t exp (r s n)ds =. (9) In sum, the household s utility-maximizing behavior is characterized by 1. Initial condition: the value of a 2. Flow budget constraint (4) 3. Euler equation (8), 4. TVC (9) 1.2 Firms Technologies Output, Y t, is produced using physical capital, K t, and labor, L t : Y t = F (K t, L t ). (1) Assumption 2. F is homogenous of degree one: F (vk, vl) = vf (K, L) for all v, K and L. There are many identical firms, each with the same constant-returns-to-scale technology as described by (1). A representative firm s profit maximization problem is max K,L F (K t, L t ) R t K t w t L t, (11) where R t is the rental rate of capital: the relationship between this and the interest rate is R t δ = r t. (12) 4

5 Here we introduce a new variable indicating capital-labor ratio, k K/L. From the linear homogeneity of the function F, we can define f as the per capita production function: f(k) F (k, 1). Assumption 3. f satisfies (i)f (k) >, (ii) f (k) <, (iii) f() =, (iv) lim k f (k) =, and (v) lim k f (k) =. Example: F (K, L) = AK α L 1 α, where A > and < α < 1. In this case f(k) Ak α. It is easily verified that the Cobb Douglas production function satisfies Assumptions 2 and 3. Profit Maximization under Perfect Competition Profit maximization problem (11) is now rewritten as max k,l (f(k t ) R t k t w t ) L t. (13) Under perfect competition, the firms takes R t and w t as given. The first-order conditions are k t : R t = f (k t ), (14) L t : w t = f(k t ) k t f (k t ). (15) It is quite easy to find that from (14) and (15), R t k t + w t = f(k t ). (16) 2 Competitive Equilibrium Path 2.1 Market Clearing Condition Labor market equilibrium: We have already imposed this condition. ( ) Both supply of labor (in (3)) and demand of labor (in (11)) are denoted by L t. Asset market equilibrium: In practice, household assets, A t can consists of 1. (claims to) Capital stock, K t, which the households rent to firms, 2. Individual bonds (IOUs) due to lending and borrowing within households these bonds are in zero net supply. 5

6 Therefore, the asset market equilibrium is given by A t = K t a t = k t. (17) ( ) Implication of (12): interest rate (r t ) = rate of return on capital (R t ) capital depreciation rate (δ) Goods market equilibrium: Goods market equilibrium is given by F (K t, L t ) }{{} = C }{{} t + K t + δk }{{} t. Gross Domestic Product Consumption Aggregate Investment In per capita terms, k t = f(k t ) (n + δ)k t c t. (18) Caution: 1. Substituting (17) into the household s per capital budget (4): k t = (r t n)k t + w t c t 2. Substituting (12) into the result: k t = (R t n δ)k t + w t c t 3. Finally substituting (16) into the above equation: k t = f(k t ) (n + δ)k t c t. Thus, the goods market equilibrium is endogenously obtained from the other conditions of market equilibria and the economic agents (i.e., household s and firm s) budget constraint. Walras law. 2.2 Definition of Competitive Equilibrium Path As already known, the household s budget constraint (4) is combined with (12), (16) and (17) to give (18): k t = f(k t ) (n + δ)k t c t. (18) This equation describes the dynamics of physical capital k t, which depends on k t and c t. 6

7 On the other hand, from (12) and (14), (12) : r t = R t δ = f (k t ) δ. (14). Substituting this equation into the household s Euler equation (8), we can obtain c tu (c t ) u (c t ) = f (k t ) δ ρ. (19) This equation describes the dynamics of consumption c t, which depends on k t and c t. Furthermore, Imposing a t = k t in (9) and substituting r s = f (k s ) δ into the reuslt, we can express the TVC as follows: ( lim k t exp t The initial condition is given by k (= a ). ) (f (k s ) n δ)ds =. (2) Now we are in position to define the Competitive Equilibrium Path. Definition 1 (Competitive Equilibrium Path). Given k >, the paths of k t and c t which jointly satisfy (18) (2) constitute a competitive equilibrium path. 3 The Social Optimum and The First Theorem of Welfare Economics Before going to further characterization of competitive equilibrium, let us turn to the social planner s optimal growth problem. Briefly, the social planner is the agent who maximizes the representative household s life-time utility (2), directly faces the dynamics of k t, (18), as the constraint of the problem. Any allocation chosen by a social planner is called the socially optimal allocation or command optimum. By definition, this is the first-best. The social planner s problem is given by max U = e (ρ n)t u(c t )dt, s.t. kt = f(k t ) (n + δ)k t c t, k given. 7

8 The current value Hamiltonian associated with the above problem is given by Ĥ(k t, c t, η t ) = u(c t ) + η t (f(k t ) (n + δ)k t c t ). Then, the optimal path is characterized by (18) and H c = u (c t ) = η t, (21) H k + η t (ρ n)η t = η t = (ρ + δ f (k t ))η t, (22) lim tk t e (ρ n)t =. (23) From (21) and (22), we can obtain the same equation [ as the dynamics of c t, (19). On ] t the other hand, from (22) we have η t = η exp (ρ + δ f (k s ))ds. Substituting this into (23) gives the same condition as (2). Proposition 1. The competitive equilibrium path achieves the first-best allocation. Both of (18) and (19) are nonlinear differential equations. How can we derive the competitive equilibrium path? 1. Examine the existence and uniqueness of steady state: A steady state is defined as an equilibrium path in which k t and c t are constant. 2. Examine the stability of the steady state: Briefly speaking, the stability of steady state refers to the ability of the steady state to converge to it when the initial value of the state variable k is different from the steady state. 3. Check whether or not the path is uniquely determined or not. 4 Steady State A steady state is defined as an equilibrium path in which k t and c t are constant. We denote the steady state values of these variables, k and c, respectively. From (19) with ċ t =, we have f (k ) = ρ + δ. (24) Note that from property (iv) and (v) of Assumption 3, there uniquely exists k > that solves the above equation. Once k is uniquely determined, the dynamics of k t, (18) with k t determines the steady state consumption level: c = f(k ) (n + δ)k. (25) 8

9 Proposition 2. There exists a unique steady state (k, c ) that satisfies the TVC (2) and k > and c >. Proof. Since k > has been already shown, it is sufficient to show that c > and that k satisfies the TVC. Recall that f(k) is strictly concave and f() = are assumed in Assumption 3. Then, it follows that f(k ) > f() + f (k )(k ) k > f(k ) > f (k )k k > Substituting this result into (25) and using (24), we obtain c > f (k )k (n + δ)k = [f (k ) (n + δ)]k = [ρ + δ (n + δ)]k = (ρ n)k. Since ρ n > holds from Assumption 1, we can show that c >. Next we will show that k satisfies the TVC. Substituting k into the left-hand-side (LHS) of (2) and using (24), LHS of (2) = lim k exp [ (f (k ) n δ)t ] = lim k exp [ (ρ + δ n δ)t] = lim k exp [ (ρ n)t] Since ρ n > holds from Assumption 1 and k is finite, we can show the LHS converges to zero. Figure 1 graphically show how k and c are determined. In Figure 1, k g is defined as k g = arg max f(k) (n + δ)k. k That is, k g is called the Golden Rule of Capital Stock. Lemma 1. k < k g under Assumption 1. Proof. Exercise. 9

10 O Figure 1: Steady State 5 Transitional Dynamics (18) and (19) are the system of differential equations of k t and c t. These equations are called the autonomous dynamic system of the model. 2 Since k is historically given, the equilibrium paths of k t and c t are determined, once c is determined. Then, how can we solve the equations for c? 1. Graphical Solution: we depict the phase diagram. 2. Analytical Solution: we linearize the system of equations (18) and (19) in the neighborhood of the steady state, and then solve these approximated equations. 3. Numerical Solution 5.1 Diagrammatic Analysis of Global Dynamics From (18) and (19), k t c t f(k t ) (n + δ)k t, (26) ċ t u (c t ) u (c t ) (f (k t ) δ ρ). (27) 2 Consider the system of differential equations of variables x t and y t. Then the system is called autonomous when it depends on time only through the variables x t and y t. 1

11 Assumption 4. u (c) u (c) = if c =. Example: If we specify u as the CRRA form: u(c) = c1 θ 1 1 θ for θ >, θ 1, then u (c)/u (c) = c/θ. So this function satisfies this assumption. Therefore, (27) is simplified to ċ t f (k t ) ρ + δ k t k when c t >, (28) and ċ t = for all k t when c t =. B E O A A A D Figure 2: Phase Diagram Figure 2 depicts the dynamics of k t and c t given by (26) and (28). Note that all points in the positive orthant are feasible. So there is a continuum of feasible paths. However, the paths are classified mainly into the following three types. 1. The path that converges to the steady state (k, c ). Given k, a representative shape of this type is depicted by the curve AE. 11

12 2. The paths that eventually hit the vertical axis, and move to the origin. Given k, a representative shape of this type is depicted by the curve A B. Once the path hits the point B, then it jumps to the point O. 3. The paths that converge to the steady state (k, ), where k is the steady state capital stock with zero consumption: f(k) = (n + δ)k. Given k, a representative shape of this type is depicted by the curve A D. Proposition 3. Only the path converging to the steady state (k, c ) is optimal. Proof. We show that on all other paths, either the Euler equaiton (19) eventually falls or the TVC (2) is not satisfied. At first, consider the paths which eventually hit the vertical axis. Note that on this type of paths, ċ t > and the sign of k t becomes negative sooner or later. Differentiating (18) yields d k t = [f (k t ) (n + δ)] dt k t + ċ t >. This implies that these paths hit the vertical axis in finite time. When a path reaches the vertical axis, k t becomes zero. Then the economy has to jump to the origin. However, such a jump violates the Keynes-Ramsey rule, which imposes c t to continuously Therefore, all the paths hitting the the vertical axis cannot be optimal. move over time. Second, consider the paths converging to the steady state (k, ). Note that in this steady state it follows that The above inequality shows f (k) < n + δ = f (k g ). lim k exp[(n + δ f (k))t] =. Thus, the steady state (k, ) does not satisfy the TVC. 5.2 Local Dynamics Linearization of the Dynamic System (18) and (19) Linearization of the dynamic system yields further insights into the dynamic behavior. A first-order Taylor expansion of (18) and (19) around the steady state (k, c ) gives k t (ρ n)(k t k ) (c t c ), and ċ t f (k )u (c ) u (c (k t k ). ) 12

13 Then, the local dynamics is given by ( ) ( kt ρ n 1 = f u /u ċ t ) ( k t k c t c ). Let J denote the Jacobian matrix of the dynamics: ( ρ n 1 J f u /u ). So the eigenvalues of J are given by the values of ω that solve the following quadratic equation: det(j ωi) = ω 2 (ρ n)ω f u /u = The above equation is called the characteristic equation. Generally, in the two variable two equation dynamic system, steady state is called 1. Saddle point, if the system has one positive- and one negative eigenvalues, 2. Source, if the system has two positive eigenvalues, 3. Sink, if the system has two negative eigenvalues. Since f u /u < represents the product of the two roots and the sign of this is negative, there are two real eigenvalues, one negative and one positive. Lemma 2. The steady state (k, c ) is a saddle point. Without any loss of generality, let ω 1 > and ω 2 < respectively denote the positive and the negative eigenvalues. Therefore, the general solution is given by ( ) ( ) ( k t k = Z 1 exp(ω 1 t) + Z 2 c t c where v 11 v 21 v 12 v 22 ) exp(ω 2 t), (29) v j (v 1j, v 2j ) T : the eigenvector corresponding to the eigenvalue ω j, Z j : a constant value still to be determined. Determination of Initial Consumption 13

14 Now we will show that when the steady state is a saddle point, the initial consumption is uniquely determined. Accordingly, the competitive equilibrium path is uniquely determined. Since ω 1 >, the economy never converges to the steady state (k, c ) if Z 1 would not be zero. Since (29) holds at t =, we have k k = Z 1 v 11 + Z 2 v 12, c c = Z 1 v 21 + Z 2 v 22. (3) Thus, the initial consumption, c, is determined such that 1. Z 1 = : otherwise the economy diverges from the steady state, and such a path violates either the Keynes-Ramsey rule or the TVC; 2. Z 2 = (k k )/v 12 : otherwise (3) does not hold given k. In sum, Lemma 3. The initial consumption is determined as c = c + v 22 v 11 (k k ). Therefore, from (29), we can analytically obtain the optimal growth path as follows: k t k = (k k ) exp(ω 2 t), c t c = (v 22 /v 12 )(k k ) exp(ω 2 t). (31) (31) is the competitive equilibrium path near around the steady state. Proposition 4. There exists a unique competitive equilibrium path. 6 Summary The Ramsey Cass Koopmans model is based on the economic agents intertemporal optimization. The competitive equilibrium path in this model corresponds to the social planner s optimal path which achieves the first-best allocation. There exists a unique steady state where both of physical capital and consumption are positive. Saddle point stability of the steady state means the uniqueness of competitive equilibrium path in this model. 14

15 References [1] Acemoglu, D. (29) Introduction to Modern Economic Growth, Princeton University Press. [2] Barro, R. J. and X. Sala-i-Martin (24) Economic Growth, Second Edition, Cambridge, MIT Press. [3] Blanchard, O. and S. Fischer (1989) Lectures on Macroeconomics, Cambridge, MIT Press. [4] Cass, D. (1965) Optimum Growth in an Aggregate Model of Capital Accumulation, Review of Economic Studies 32, pp [5] Koopmans, T. J. (1965) On the Concept of Optimal Economic Growth, in The Econometric Approach to Development Planning. Amsterdam: North Holland, pp [6] Phelps, E. S. (1966) Golden Rules of Economic Growth, New York, W. W. Norton. [7] Ramsey, F. (1928) A Mathematical Theory of Saving, Economic Journal 38, pp [8] Solow, R. M. (1956) A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics 7, pp [9] Swan, T. W. (1956) Economic Growth and Capital Accumulation, Economic Record 32, pp