Public Economics The Macroeconomic Perspective Chapter 2: The Ramsey Model. Burkhard Heer University of Augsburg, Germany

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1 Public Economics The Macroeconomic Perspective Chapter 2: The Ramsey Model Burkhard Heer University of Augsburg, Germany October 3, 2018

2 Contents I 1 Central Planner 2 3 B. Heer c Public Economics: Chapter 2 2/65

3 Ramsey Model: Central Planner I Ramsey Model with Inelastic Labor Supply Ramsey (1928): How much should an economy save over an infinite time horizon in order to maximize lifetime utility of its agents? Trade-off: Consumption today or consumption tomorrow? Central planner: Benevolent, maximizes lifetime utility of households: U = β t u(c t ), 0 < β < 1. (1) Infinite lifetime t=0 Utility in period t is discounted by the discount factor β. B. Heer c Public Economics: Chapter 2 3/65

4 Ramsey Model: Central Planner II Instantaneous utility is a function of per-capita-consumption c t : u(c) = Labor supply is inelastic { c 1 σ 1 1 σ σ 1 ln(c) σ = 1, 1/σ denotes the intertemporal elasticity of substitution of consumption. The number of household N t grows at the constant rate n: (2) N t = (1 + n)n t 1. (3) B. Heer c Public Economics: Chapter 2 4/65

5 Ramsey Model: Central Planner III Production uses labor N t and capital K t as inputs: where Z t denotes the level of technology. Y t = Z t F (K t, N t ), (4) Production technology is characterized by constant returns to scale per capita production, y = Y/N, is a function of the capital intensity, k = K/N: y t Y t N t = Z t f(k t ) Z t F (K t /N t, 1). (5) B. Heer c Public Economics: Chapter 2 5/65

6 Ramsey Model: Central Planner IV Constant elasticity of substitution σ p = 1/(1 ρ) in production: Y t = Z t [αk ρ t + (1 α)n ρ t ] 1 ρ. (6) We set σ p = 1 so that (6) reduces to the Cobb-Douglas production function: Y t = Z t K α t N 1 α t. (7) The central planner owns the capital stock K t and saves S t = Y t C t for next period. B. Heer c Public Economics: Chapter 2 6/65

7 Ramsey Model: Central Planner V In equilibrium, savings are equal to investment, I t = S t, and capital accumulates according to K t+1 = (1 δ)k t + I t = (1 δ)k t + Y t C t. (8) Capital depreciates at the rate δ. The resource constraint in per-capita terms (c t C t /N t and k t K t /N t ): (1 + n)k t+1 = (1 δ)k t + Z t f(k t ) c t. (9) B. Heer c Public Economics: Chapter 2 7/65

8 Ramsey Model: Central Planner VI Tirst-order conditions of the central planner s optimization problem follow from the derivation of the Lagrangean [ L = β t c 1 σ ] t 1 + λt (Ztf(kt) + (1 δ)kt ct (1 + n)kt+1) 1 σ t=0 (10) with respect to c t and k t+1 : λ t = c σ t, (11a) (1 + n)λ t = λ t+1 β [ 1 + Z t+1 f (k t+1 ) δ ]. (11b) B. Heer c Public Economics: Chapter 2 8/65

9 Ramsey Model: Central Planner VII Euler condition (also called the Keynes-Ramsey rule): ( ct+1 c t ) σ = β [ 1 + Zt+1 f (k t+1 ) δ ]. (12) 1 + n per capita consumption c t grows if the marginal product of capital less depreciation, Z t+1 f (k t+1 ) δ, is above the rate ˆr = (1 + n)/β 1. B. Heer c Public Economics: Chapter 2 9/65

10 Ramsey Model: Central Planner VIII Steady State: Long-run equilibrium with constant technology level c t = c t+1 = c, k t+1 = k t = k, and Z t = Z t+1 = Z = 1: with y = f(k) = k α 1 + n β k = ( 1 + δ = f (k). 1+n β α 1 + δ ) 1 1 α. Annual parameterization: α = 0.36, β = 0.96, δ = 8%, 1/σ = 1/2 k = 5.447, y = k α = 1.841, and c = y (δ + n)k = in steady state. B. Heer c Public Economics: Chapter 2 10/65

11 Ramsey Model: Central Planner IX Dynamics following a shock The economy is in steady state in period t = 0 Technology level Z t = 1.0 is constant during period t = 0, 1,..., 9. In period t = 10, the technology level increases from 1.0 to 1.1 for three periods, Z 10 = Z 11 = Z 12 = 1.1, and declines to Z t = 1.0 for t > 12 thereafter: { 1.1 t = 10, 11, 12 Z t = 1 else. (13) Two scenarios: 1 Shock in period t = 10 unanticipated. 2 The central planner learns about the 3-period increase already in period t = 0. B. Heer c Public Economics: Chapter 2 11/65

12 Ramsey Model: Central Planner X Methodology: Solving a non-linear equations problem: ( ) Ztkt α σ + (1 δ)k t (1 + n)k t+1 = β [ 1 + αztk α 1 Z t 1kt 1 α t δ ]. + (1 δ)kt 1 (1 + n)kt 1 + n (14) The solution is illustrated in the following figures Ch2 ramsey1.m/ch2 ramsey1.g compute the solution GAUSS and MATLAB computer code are available as downloads from the author s homepage buch/. B. Heer c Public Economics: Chapter 2 12/65

13 B. Heer c Public Economics: Chapter 2 13/65

14 B. Heer c Public Economics: Chapter 2 14/65

15 B. Heer c Public Economics: Chapter 2 15/65

16 Observations: 1 Output increases in period t = 10 by 10% in case 1 (by more than 10% in case 2). 2 Capital stock attains its maximum in period Central planner smoothes consumption. 4 Lifetime utility is higher in case 2 (why?) 5 News-driven cycles: Expected shock implies lower correlation of y t and c t B. Heer c Public Economics: Chapter 2 16/65

17 B. Heer c Public Economics: Chapter 2 17/65

18 Effects of the intertemporal elasticity of substition (IES), 1/σ {1, 1/2}: 1 For a higher IES, 1/σ = 1, the household increases his consumption smoothing over time. 2 This can already be seen from the Euler condition (12): ( ct+1 c t ) = ( β [ 1 + Zt+1 f (k t+1 ) δ ]) 1 σ. 1 + n B. Heer c Public Economics: Chapter 2 18/65

19 Stability Analysis I Stability Analysis and Saddle Path Formally, (14) is a 2nd-order difference equation in k t : ( ) ( kt+1 g 1 ) (kt, x = g(k x t, x t ) = t ) t+1 g 2 (k t, x t ) ( [ β = Z t kα t +(1 δ)k t 1+n 1+αZ t k t α 1 ]) 1 ( δ σ Z t 1 x α ) t +(1 δ)x t (1+n)k t 1+n k t (15) B. Heer c Public Economics: Chapter 2 19/65

20 Stability Analysis II Local Stability: Absolute values of the eigenvalues of the Jacobian matrix, again evaluated at the steady state: ) J(k t, x t ) = ( g 1 (k t,x t) k t g 1 (k t,x t) x t g 2 (k t,x t) k t g 2 (k t,x t) x t (16) B. Heer c Public Economics: Chapter 2 20/65

21 Stability Analysis III Evaluated at the steady state, the numerical value of the Jacobian is equal to J(k, k) = ( with the eigenvalues ρ 1 = and ρ 2 = saddlepoint stable ) (17) B. Heer c Public Economics: Chapter 2 21/65

22 Stability Analysis IV Graphical Analysis of the Saddlepoint Stability Switch presentation to a first-order difference system in the variables k t and c t : ( ) ( kt+1 h = h(k c t, c t ) = 1 ) (k t, c t ) t+1 h 2 (k t, c t ) 1 1+n [Z tkt α + (1 δ)k t c t ] ( [ = β c t 1 + α 1+n ( Ztk α t +(1 δ)kt ct 1+n ) α 1 δ ]) 1 σ Eigenvalues of this difference equations system are equal to ρ 1 = and ρ 2 = the system is saddlepoint stable. (18) B. Heer c Public Economics: Chapter 2 22/65

23 IV I III II B. Heer c Public Economics: Chapter 2 23/65

24 Dynamics for initial capital stock k 0 (next figure): Pathes starting in points A, C, or D violate k 0 or c 0 in finite time only path starting in B obeys optimality conditions! B. Heer c Public Economics: Chapter 2 24/65

25 0 B. Heer c Public Economics: Chapter 2 25/65

26 Numerical example for saddlepoint path: Dynamics from the Ramsey model above for the subperiod t = 13,..., 100 when Z t 1 for all remaining periods. At the beginning of period t = 13, when Z t falls back to its steady state value Z = 1, the capital stock is given by k 13 = For this value of the capital stock, (18) only converges to the steady state (k, c) = (5.447, 1.405) for c 13 = See saddlepoint path in the next figure B. Heer c Public Economics: Chapter 2 26/65

27 B. Heer c Public Economics: Chapter 2 27/65

28 Linear approximation of the policy functions around steady: Dynamic eqs. system k t+1 = h 1 (k t, c t) = n [kα t + (1 δ)k t c t], (19a) ( [ ( ) ]) 1 c t+1 = h 2 β k α α 1 σ (k t, c t) = c t 1 + α t + (1 δ)k t c t δ. 1 + n 1 + n Linear approximation of these two equations at the point (k t, c t ) = (k, c) (19b) k t+1 = k + h1 (k, c) k t (k t k) + h1 (k, c) c t (c t c), (20a) c t+1 = c + h2 (k, c) k t (k t k) + h2 (k, c) c t (c t c), (20b) B. Heer c Public Economics: Chapter 2 28/65

29 or, more formally, ( kt+1 k c t+1 c ) = J ( kt k c t c ) (21) The Jacobian matrix J is equal to J = ( ) 1 αk α δ 1+n c β ( ) 1+n (1+n)σ α(α 1)kα 2 αk α δ 1+n n c β, α(α 1)kα 2 (1+n)σ (22) where we have already used the observation that, in steady state, [ ( β k α ) + (1 δ)k c α α δ] = n 1 + n B. Heer c Public Economics: Chapter 2 29/65

30 Numerical evaluation of the Jacobian: ( J = ) Schur decomposition J = T S T 1 : J = T S T 1 with T = ( ), S = ( ), B. Heer c Public Economics: Chapter 2 30/65

31 Givens rotation: ( T = ) ( , S = ), Again, J = T ST 1. Definition of new auxiliary variables k t and c t : ( ) ( ) kt = T 1 kt k c t c t c (23) so that our system of difference (21) can be rewritten as: ( kt+1 c t+1 ) ( kt = S c t ) = ( ρ ρ 2, ) ( kt c t ) (24) B. Heer c Public Economics: Chapter 2 31/65

32 Consider the second equation of (24): c t+1 = ρ 2 c t. We can rearrange the equation to get c t = 1 ρ 2 c t+1. Iterating this equation forward and substituting it into itself, we derive 1 c t = lim i ρ i c t+i. 2 Assuming that c t+i remains bounded for i, and with ρ 2 > 1, it follows that c t = 0. B. Heer c Public Economics: Chapter 2 32/65

33 Since T 1 is given by we find that T 1 = ( ), c t = (k t k) (c t c) = 0 or c t c = (k t k). (25) B. Heer c Public Economics: Chapter 2 33/65

34 Next, we need to determine k t+1 k as a function of k t k. Using the first difference equation from (24): k t+1 = k t c t = [0.9932(k t k) (c t c)] (k t+1 k) (c t+1 c) = [0.9932(k t k) (c t c)] (k t+1 k) (k t+1 k) = [0.9932(k t k) (k t k)], or k t+1 k = (k t k). (26) Clearly, this equation is stable and k t converges to k for every value k 0 > 0. In order to derive this stability, we needed an eigenvalue ρ 1 with absolute value smaller than one. See next figure for comparison of direct computation with the solution from linear approximation B. Heer c Public Economics: Chapter 2 34/65

35 B. Heer c Public Economics: Chapter 2 35/65

36 Elastic Labor We introduce elastic labor supply in the Ramsey model: ( c ι (1 L) 1 ι) 1 σ 1 u(c, 1 L) =, (27) 1 σ ι and 1 ι denote the weights of consumption and leisure in utility. Intertemporal utility of the individual household at period t = 0: U = β t u(c t, 1 L t ). (28) t=0 Aggregate labor, N t L t : product of the number of households N t and their labor supply L t. B. Heer c Public Economics: Chapter 2 36/65

37 Production function is Cobb-Douglas: Y t = Z t K α t (N t L t ) 1 α, (29) so that per-capita production is presented by with k t K t /N t. Lagrangean of the household: L = β t t=0 ( c ι t (1 L t) 1 ι) 1 σ 1 1 σ y t Y t N t = Z t k α t L 1 α t, (30) + λ t ( Z t k α t L1 α ) t + (1 δ)k t c t (1 + n)k t+1 (31) B. Heer c Public Economics: Chapter 2 37/65

38 First-order conditions: λ t(1 α)z tk α t L α t (32a) and (32b) imply: Euler equation: ( ct c t+1 λ t = ιc ι(1 σ) 1 t (1 L t) (1 ι)(1 σ), (32a) = (1 ι)c ι(1 σ) t (1 L t) (1 ι)(1 σ) 1, (32b) (1 + n)λ t = λ t+1β [ 1 + αz t+1k α 1 t+1 L 1 α t+1 δ ]. (32c) ) ι(1 σ) 1 ( 1 Lt (1 α)z t k α t L α t 1 L t+1 = 1 ι ι c t 1 L t (33) ) (1 ι)(1 σ) = β [ 1 + αzt+1k α 1 t+1 L 1 α t+1 δ ]. 1 + n (34) B. Heer c Public Economics: Chapter 2 38/65

39 Steady State: Z t = Z = 1, c t = c t+1 = c, k t = k t+1 = k, and L t = L t+1 = L Euler equation simplifies to or 1 + n β k = 1 + δ = αk α 1 L 1 α, (35) ( 1+n β α 1 + δ ) 1 1 α L. Parameterization: β = 0.96, σ = 2.0, α = 0.36, δ = 0.08, and n = 0. Calibration of ι = with the help of (33) so that L = 0.3 in steady state B. Heer c Public Economics: Chapter 2 39/65

40 Dynamics Again, we consider a temporary increase of productivity Z t from 1 to 1.1 during the periods t = 10, 11, 12. The dynamic system (14) needs to be adjusted to allow for elastic labor supply: β [ 1 + αz t k α 1 t L 1 α t 1 + n ] δ (1 α)z t k α t L α t = Z tkt α L1 α t + (1 δ)k t (1 + n)k t+1 Z t 1 k t 1 α L1 α t 1 ( 1 Lt 1 = 1 ι ι + (1 δ)k t 1 (1 + n)k t 1 ι(1 σ) ) (1 ι)(1 σ), 1 L t (36a) c t. 1 L t (36b) See the following figure for the dynamics uses Ch2 ramsey2.m/ch2 ramsey2.g for the computation B. Heer c Public Economics: Chapter 2 40/65

41 B. Heer c Public Economics: Chapter 2 41/65

42 Observations Substitution effect dominates income effect in the response of labor supply: Leisure declines in period t = 10. In the case of an expected increase of the technology level Z t in period t = 10, the household already increases both consumption and leisure prior to period t = 10 in order to smooth utility intertemporally. B. Heer c Public Economics: Chapter 2 42/65

43 I Households Household budget: w t L t + r t k t = c t + s t. Wealth accumulation: (1 + n)k t+1 = (1 δ)k t + s t, so that the budget constraint becomes w t L t + r t k t + (1 δ)k t = c t + (1 + n)k t+1. (37) B. Heer c Public Economics: Chapter 2 43/65

44 II First-order conditions of the household s optimization problem follow from the derivation of the Lagrangean L = t=0 [ ( c ι β t t (1 L t) 1 ι) 1 σ σ λ t (w tl t + r tk t + (1 δ)k t c t (1 + n)k t+1) with respect to c t, L t, and k t+1 : ] (38) λ t = ιc ι(1 σ) 1 t (1 L t ) (1 ι)(1 σ), (39a) λ t w t = (1 ι)c ι(1 σ) t (1 L t ) (1 ι)(1 σ) 1, (39b) (1 + n)λ t = λ t+1 β [1 + r t+1 δ]. (39c) B. Heer c Public Economics: Chapter 2 44/65

45 III (39a) and (39b) can be combined to yield: w t = 1 ι ι c t 1 L t. (40) The Euler equation is derived from inserting (39a) into (39c): ( ct c t+1 ) ι(1 σ) 1 ( 1 Lt 1 L t+1 ) (1 ι)(1 σ) = β 1 + n [1 + r t+1 δ]. (41) B. Heer c Public Economics: Chapter 2 45/65

46 IV Production: Profit maximization: Π t = Y t w t N t L t r t K t = Z t K α t (N t L t ) 1 α w t N t L t r t K t. (42) implies w t = (1 α)z t K α t (N t L t ) α, (43a) r t = αz t K α 1 t (N t L t ) 1 α, (43b) Production function is linear-homogenous Euler s theorem applies and total production is equal to total costs: Y t = w t N t L t + r t K t = Z t K α t (N t L t ) 1 α Profits are zero, Π t = 0. B. Heer c Public Economics: Chapter 2 46/65

47 V Resource constraint of the economy: y t = c t + (1 + n)k t+1 (1 δ)k t. (44) First Theorem of Welfare Economics holds: Allocation in the market economy with perfect competition is equal to the solution of the central planner. Allocation in the competitive equilibrium (with no externalities) is pareto-efficient (first welfare theorem). B. Heer c Public Economics: Chapter 2 47/65

48 I forms the basic building block of modern business cycle models. Real Business Cycle models King, Plosser, and Rebelo (1988) Kydland and Prescott (1982) New Keynesian Business models emphasizes nominal frictions e.g. Christiano, Eichenbaum, and Evans (2005) see Chapter 4 B. Heer c Public Economics: Chapter 2 48/65

49 II The model In the standard RBC model, the technology level Z t is the exogenous stochastic variable: ln Z t = ρ ln Z t 1 + ɛ t, ɛ t N(0, σ Z ), (45) where the innovation ɛ t is normally distributed with the standard deviation σ Z. Solow residual: Z t = Y t K α t L1 α t Estimate by Cooley and Prescott (1995) for quarterly US data: ρ = 0.95, σ Z = B. Heer c Public Economics: Chapter 2 49/65

50 III Households maximizes expected utility in period 0 U = E 0 t=0 β t u(c t, 1 L t ) (46) subject to to the budget constraint (37). Instantaneous utility u(c, 1 L) is a function of consumption c and leisure, 1 L, as presented by (27). Euler equation: 1 + n β = E 0 { ( ct+1 c t ) ι(1 σ) 1 ( ) (1 ι)(1 σ) 1 Lt+1 [1 + r t+1 δ]}. 1 L t (48) B. Heer c Public Economics: Chapter 2 50/65

51 IV The equilibrium conditions of the RBC model are summarized by the following 8 equations in the 8 variables k t, Z t, c t, L t, y t, w t, r t, and i t : B. Heer c Public Economics: Chapter 2 51/65

52 V 1 + n β = E t { ( ct+1 c t ) ι(1 σ) 1 ( ) (1 ι)(1 σ) 1 Lt+1 [1 + r t+1 δ]}, 1 L t y t = c t + (1 + n)k t+1 (1 δ)k t, (49a) (49b) ln Z t = ρ ln Z t 1 + ɛ t, (49c) w t = 1 ι c t, ι 1 L t (49d) w t = (1 α)z tkt α L α t, (49e) r t = αz tk α 1 t L 1 α t, (49f) y t = Z tkt α L 1 α t, (49g) i t = (1 + n)k t+1 (1 δ)k t. (49h) B. Heer c Public Economics: Chapter 2 52/65

53 VI Deterministic steady state: variables are constant with ɛ t = 0 and Z t = 1. The steady-state values are identical to those in the decentralized model of the previous section. Parameterization for quarterly data: β = 0.99, δ = 0.02 Linear approximation: ( ) ( ĉt = ˆL t ) ˆk t + ( ) Ẑ t. (50) consumption increases by 0.49% if the technology level is equal to Z t = 1.01 ceteris paribus (and, hence, 1.0% above its steady state level, Ẑt = 0.01) consumption increases if capital is above its steady-state value due to the wealth effect. B. Heer c Public Economics: Chapter 2 53/65

54 VII Labor supply, however, declines with higher capital stock and increases with technology. A one percent rise of Z t results in an increase in labor L by 0.65%. Dynamics of the capital stock k t ˆk t+1 = ˆk t Ẑt. (51) B. Heer c Public Economics: Chapter 2 54/65

55 1.5 5 % Dev. from Trend Output TFP Shock % Dev. from Trend Investment Period Period Real Wage Consumption % Dev. from Trend 1 0 Real Interest Rate Marginal Utility % Dev. from Trend Hours Capital Stock Period Period B. Heer c Public Economics: Chapter 2 55/65

56 Impulse Responses illustrate the percentage deviation of policy functions from the steady state if the economy is in steady state in period t = 1 and is hit by a shock ɛ 2 = 0.01 in period t = 2 In the next period, t = 3, the shock falls back to zero so that ɛ t = 0 for t = 3,.... Since technology Z t is modeled as an autoregressive process with autoregressive parameter ρ = 0.95, technology increases by 0.01 in period t = 2 and slowly decreases in the periods thereafter, e.g. Z 3 = and Z 4 = wages w t, the interest rate r t, and income y t increase consumption c t and investment i t rise by 0.48% and 4.3% output increases stronger than technology, by 1.4% capital k t : hump-shaped response B. Heer c Public Economics: Chapter 2 56/65

57 Time Series Behavior Simulation of time series: In period t = 0, economy is in steady state. 500 random variable simulation: over T periods implying {ɛ t } T t=1 {Z t } T t=0 Elimination of long-run cycles with the help of the HP filter of Hodrick and Prescott (1997) min {g t} T t=1 T T 1 (y t g t ) 2 + λ [(g t+1 g t ) (g t g t 1 )] 2. (52) t=1 t=2 B. Heer c Public Economics: Chapter 2 57/65

58 If λ = 0, the solution y t = g t is simply the original series. For λ, the series g t is chosen so that the growth rate is constant. We choose λ = B. Heer c Public Economics: Chapter 2 58/65

59 B. Heer c Public Economics: Chapter 2 59/65

60 B. Heer c Public Economics: Chapter 2 60/65

61 B. Heer c Public Economics: Chapter 2 61/65

62 Table 2.1: US statistics Variable s x (in %) s x/s Y r xy r xl Output Y Hours L Consumption C Investment I Wage w Risk-free rate r f Equity return r e B. Heer c Public Economics: Chapter 2 62/65

63 Table 2.1, cont.: Time series statistics for the RBC model Variable s x s x/s Y r xy r xl Output Y Hours L Consumption C Investment I Real wage w Real interest rate r B. Heer c Public Economics: Chapter 2 63/65

64 RBC Model I Observations 1 The RBC model replicates the fact that consumption is less volatile than output, while labor is not volatile enough. 2 The correlation of output, consumption, and hours is too high compared with the data. 3 The basic RBC model is not able to replicate the dynamics of wages and interest rates: Both factor prices are much too pro-cyclical. 4 Possible solutions to improve the shortcomings of the basic RBC model: 1 sticky real wages 2 labor market frictions such as search unemployment 3 introduction of other shocks like demand shocks B. Heer c Public Economics: Chapter 2 64/65

65 Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005): Nominal rigidities and the dynamic effects of a shock to monetary policy, Journal of political Economy, 113(1), Cooley, T. F., and E. C. Prescott (1995): Economic growth and business cycles, in Frontiers of Business Cycle Research, ed. by T. F. Cooley, pp Princeton University Press. Hodrick, R. J., and E. C. Prescott (1997): Postwar US business cycles: An empirical investigation, Journal of Money, Credit, and Banking, 29, King, R. G., C. I. Plosser, and S. Rebelo (1988): Production, growth and business cycles I, the basic neoclassical model, Journal of Monetary Economics, 21, Kydland, F. E., and E. C. Prescott (1982): Time to build and agregate fluctuations, Econometrica, 50, Ramsey, F. (1928): A mathematical theory of saving, Economic Journal, 38, B. Heer c Public Economics: Chapter 2 65/65