Lecture 4B: The Discrete-Time Overlapping-Generations Model: Tragedy of Annuitization

Transcription

1 Lecture 4B: The Discrete-Time Overlapping-Generations Model: Tragedy of Annuitization Ben J. Heijdra Department of Economics, Econometrics & Finance University of Groningen 21 January 2012 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 1 / 75

2 Outline Model 1 Model Structure Capital intensity Welfare analysis 2 From WE to PA From TY to PA Discussion 3 Knife-edge case: Endogenous growth Conclusions NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 2 / 75

3 Motivation (1) Based on: Heijdra, B.J., Mierau, J.O., & Reijnders, L.S.M. (2009), The tragedy of annuitization. CESifo Working Paper, Nr. 2898, December Although death is one of the true certainties in life, the date at which it occurs is unknown to all but the most desperate individuals. Faced with life-time uncertainty, individuals must balance the risk of leaving unconsumed wealth in the form of unintended (accidental) bequests against the risk of running out of resources in old age. Life annuities are very attractive insurance instruments in the presence of longevity risk. They allow risk sharing between lucky (long-lived) and unlucky (short-lived) individuals (Yaari, 1965; Kotlikoff et al., 1986; Davidoff et al., 2005). NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 3 / 75

4 Motivation (2) From a microeconomic perspective rational non-altruistic agents should fully annuitize their assets. From a macroeconomic perspective matters are less clear. Microeconomic analysis ignores: Redistribution of accidental bequests Endogeneity of factor prices The objective of this paper is to study the macroeconomic effects of life annuities. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 4 / 75

5 Overview (1) Model Main model features: General equilibrium model of a closed economy. Individuals live for two periods but face mortality risk. Generalised investment externality between firms. In the absence of annuities, accidental bequests flow to the government. Revenue recycling modes: WE Wasteful expenditure TO Lump-sum transfers to the old TY Lump-sum transfers to the young Introduce a market for life-annuities and study the transition from the different revenue recycling modes to a perfect annuities (PA) scenario. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 5 / 75

6 Overview (2) Model Although full annuitization of assets is privately optimal it is not socially beneficial due to adverse general equilibrium repercussions. Weak version Going from the TY-mode. Always holds. Strong version Going from the WE-mode. Holds if the inter-temporal substitution elasticity is smaller than 1. Empirical evidence suggests that this is the case (Attanasio and Weber, 1995, JPE). NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 6 / 75

7 Earlier literature Sheshinski and Weiss (1981, QJE) Full annuitisation is also socially optimal. They analyse whether full annuitisation is optimal if an annuity market is present. Abel (1985, AER) and Kotlikoff, Shoven and Spivak (1986, JLE). The introduction of perfect annuity markets decreases the aggregate capital stock. Pecchenino and Pollard (1997, EJ) Study the optimal " size" of the annuity market. Complete annuity market is not optimal due to accidental bequests and variable factor prices. Fehr and Habermann (2008, EL) Annuities can have adverse welfare effects. Fixed factor prices. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 7 / 75

8 Individuals (1) Model Structure Capital intensity Welfare analysis Maximise expected lifetime utility: Subject to: EΛ y t U(Cy t )+ 1 π 1+ρ U(Co t+1 ) C y t +S t = w t +Z y t Ct+1 o = Zo t+1 +(1+r t+1)s t Utility function is of the CRRA type: C 1 1/σ 1 if σ > 0, σ 1 U(C) = 1 1/σ lnc if σ = 1 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 9 / 75

9 Individuals (2) Model Structure Capital intensity Welfare analysis Optimal plans: [ ] C y t = Φ(r t+1) w t +Z y t + Zo t+1 1+r t+1 Ct+1 o [ ] = [1 Φ(r t+1 )] w t +Z y t 1+r + Zo t+1 t+1 1+r t+1 S t = [1 Φ(r t+1 )][w t +Z y t ] Φ(r t+1) Z o t+1 1+r t+1 Φ(r t+1 ) 1+( 1 π 1+ρ 1 ) σ(1+rt+1 0 < Φ( ) < 1 ) σ 1, NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 10 / 75

10 Individual firms Model Structure Capital intensity Welfare analysis Technology available to each individual firm i is given by: Y it = Ω t K α it L1 α it, 0 < α < 1 Profit maximisation conditions: w t = (1 α)ω t k α it, r t = αω t k α 1 it k it K it L it δ All firms face the same factor prices and choose the same level of capital intensity k it = k t Kt L t. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 11 / 75

11 Aggregate production Structure Capital intensity Welfare analysis Inter-firm investment externality: Ω t = Ω 0 k η t, 0 < η 1 α Output and factor prices in aggregate terms: y t Y t = Ω 0 k α+η t, L t w t = (1 α)ω 0 k α+η t r t = αω 0 k α+η 1 t δ NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 12 / 75

12 Aggregation Model Structure Capital intensity Welfare analysis Demography: The population grows at an exogenous rate n > 0 Young cohort at time t is L t = (1+n)L t 1 Total population at time t is P t (1 π)l t 1 +L t Aggregate stock of capital: K t+1 = L t S t (1+n)k t+1 = S t Fundamental difference equation: (1+n)k t+1 = [1 Φ(r t+1 )][w t +Z y t ] Φ(r t+1) Z o t+1 1+r t+1 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 13 / 75

13 Government Model Structure Capital intensity Welfare analysis The government collects the accidental bequests and redistributes them over the surviving agents. Government s budget constraint: π(1+r t )L t 1 S t 1 = (1 π)l t 1 Zt o +L tz y t +G t π(1+r t )k t = 1 π 1+n Zo t +Z y t +g t Revenue recycling modes: TY Zt o = g t = 0, TO Z y t = g t = 0, WE Z y t = Zo t = 0, Z y t = π(1+r t )k t 1 π 1+n Zo t = π(1+r t)k t g t = π(1+r t )k t NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 14 / 75

14 Recapitulation (1) Model Structure Capital intensity Welfare analysis Individual choices: [ ] C y t = Φ(r t+1) w t +Z y t + Zo t+1 1+r t+1 Ct+1 o [ ] = [1 Φ(r t+1 )] w t +Z y t 1+r + Zo t+1 t+1 1+r t+1 S t = [1 Φ(r t+1 )][w t +Z y t ] Φ(r t+1) Z o t+1 1+r t+1 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 15 / 75

15 Recapitulation (2) Model Structure Capital intensity Welfare analysis Factor prices and redistribution scheme: r t = αω 0 k α+η 1 t δ w t = (1 α)ω 0 k α+η t π(1+r t )k t = 1 π 1+n Zo t +Zy t +g t Fundamental difference equation: (1+n)k t+1 = [1 Φ(r t+1 )][w t +Z y t ] Φ(r t+1) Z o t+1 1+r t+1 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 16 / 75

16 Structure Capital intensity Welfare analysis Two assumptions & Exogenous growth A.1 [Dynamic efficiency] The steady-state interest rate ˆr satisfies ˆr > n. A.2 [Admissible values for σ] The inter-temporal substitution elasticity satisfies: 0 < σ σ 2 α η 1 α η. Growth Small investment externality 0 η < 1 α. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 18 / 75

17 The WE equilibrium Model Structure Capital intensity Welfare analysis Fundamental difference equation: [Ψ(k t+1 ) ] Φ(k) is given by: Φ(k) [ 1+ k t+1 1 Φ(k t+1 ) = (1 α)ω 0 1+n kα+η t [ Γ(k t )] ( ) 1 π σ ( 1 σ 1] 1 δ +αω0 k α+η 1) 1+ρ Figure 1 gives phase diagram (for different values of σ) Properties in Proposition 1 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 19 / 75

18 Figure 1: Phase diagram Structure Capital intensity Welfare analysis future capital intensity (k t+1 ) E σ=1 σ=0.5 σ=1.5 k t+1 = k t current capital intensity (k ) t NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 20 / 75

19 Proposition 1 Model Structure Capital intensity Welfare analysis Existence and stability of the WE equilibrium (i) The model has two steady-state solutions; the trivial one features k t+1 = k t = 0, and the economically relevant satisfies k t+1 = k t = ˆk WE, where ˆk WE is the solution to: ˆk WE 1 Φ(ˆk WE ) = (1 α)ω0 1+n (ˆk WE ) α+η. (ii) The trivial steady-state solution is unstable whilst the non-trivial solution is stable: 0 < dkt+1 dk t < 1, for k t+1 = k t = ˆk WE. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 21 / 75

20 From WE to TO Model Structure Capital intensity Welfare analysis Fundamental difference equation: [Ψ(k t+1,z 1 ) ] 1+z 1 π 1 π Φ(k t+1) 1 Φ(k t+1 ) k t+1 = Γ(k t ) Impact and long run effects: dk t+1 dz 1 = Ψ z 1 < 0, kt=ˆk Ψ WE k dk t+ dz 1 = Ψ z 1 kt=ˆk Ψ WE k Γ < 0 Individual choices in Figure 2 Transitional dynamics in Figure 3(a) (ignore σ 1 for now) Properties in Proposition 2 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 22 / 75

21 Structure Capital intensity Welfare analysis Figure 2: Transfers to the old (TO) o C t+1 E 4!! E 1!! E N E 0 o Z t+1 Ẑ o! A 4 wˆ TO!! ˆ w WE A 0 C t y NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 23 / 75

22 Structure Capital intensity Welfare analysis Figure 3(a): capital intensity (k t+τ, σ = 1) WE=PA TY TO post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 24 / 75

23 Proposition 2 Model Structure Capital intensity Welfare analysis Existence and stability of the TO model (i) The model has two steady-state solutions; the trivial one features k t+1 = k t = 0, and the economically relevant one satisfies k t+1 = k t = ˆk TO, where ˆk TO is the solution to: 1+ π 1 π Φ(ˆk TO ) 1 Φ(ˆk TO ) ˆk TO = (1 α)ω0 (ˆk TO ) α+η. 1+n (ii) The trivial steady-state solution is unstable whilst the non-trivial solution is stable: 0 < dkt+1 dk t < 1, for k t+1 = k t = ˆk TO. (iii) The steady-state capital intensity satisfies the following inequality: 0 < ˆk TO < ˆk WE. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 25 / 75

24 From WE to TY Model Structure Capital intensity Welfare analysis Fundamental difference equation: Ψ(k t+1 ) = [1 α(1 z 2π)]Ω 0 k α+η t 1+n Impact and long run effects: dk t+1 dz 2 = Γ z 2 kt=ˆk Ψ > 0, WE Individual choices in Figure 4 dk t+ dz 2 Transitional dynamics in Figures 3(a) Properties in Proposition 3 +z 2 π(1 δ)k t [ Γ(k t,z 2 )] = Γ z 2 kt=ˆk Ψ > 0 Γ WE k NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 26 / 75

25 Structure Capital intensity Welfare analysis Figure 4: Transfers to the old (TY) o C t+1 E 0! E N! E 1! E 4 A!!! wˆ WE w WE y ˆ +Z wˆ TY +Zˆ y t C t y NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 27 / 75

26 Structure Capital intensity Welfare analysis Figure 3(a): capital intensity (k t+τ, σ = 1) WE=PA TY TO post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 28 / 75

27 Proposition 3 Model Structure Capital intensity Welfare analysis Existence and stability of the TY model (i) The model has two steady-state solutions; the trivial one features k t+1 = k t = 0, and the economically relevant satisfies k t+1 = k t = ˆk TY, where ˆk TY is the solution to: ˆk TY 1 Φ(ˆk TY ) = [1 α(1 π)]ω0(ˆk TY ) α+η +π(1 δ)ˆk TY 1+n (ii) The trivial steady-state solution is unstable whilst the non-trivial solution is stable: 0 < dkt+1 dk t < 1, for k t+1 = k t = ˆk TY. (iii) The steady-state capital intensity satisfies the following inequality: 0 < ˆk WE < ˆk TY.. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 29 / 75

28 Social planner solution (1) Structure Capital intensity Welfare analysis The social planner maximises: Subject to: EΛ y U(C y )+ 1 π 1+ρ U(Co ) f (k) (δ +n)k = C y + 1 π 1+n Co +g, f (k) = Ω 0 k α+η. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 31 / 75

29 Social planner solution (2) Structure Capital intensity Welfare analysis The first-best social optimum (FBSO) has the following first-order conditions: U ( C y ) U ( C o ) = 1+n 1+ρ, f ( k) = n+δ, g = 0. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 32 / 75

30 Structure Capital intensity Welfare analysis Characteristics of the WE equilibrium First-order conditions: U (Ĉy ) (1 π)(1+ ˆr) = U (Ĉo ) 1+ρ α α+η f (ˆk) = ˆr +δ ĝ = π(1+ ˆr)ˆk f (k) (δ +n)k = C y + 1 π 1+n Co +g Compared to the FBSO the WE features four distortions: π affects the MRS between C y t and C o t+1: missing insurance market ˆr > n: not in Golden Rule point ĝ > g = 0: wasteful government spending if η > 0: capital externality not internalized NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 33 / 75

31 From WE to TO (1) Model Structure Capital intensity Welfare analysis Shock-time old: Shock-time young: deλ y t (z 1) dz 1 deλ y t 1 (z 1) dz 1 = 1+n 1+ρ U (Ĉo )π(1+ ˆr)ˆk > 0 = U (Ĉy )(1+n)ˆk [ π 1 π ˆr ] dr t+1 > 0 dz 1 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 34 / 75

32 From WE to TO (2) Model Structure Capital intensity Welfare analysis Future steady-state generations: deλ y t+ (z 1) dz 1 = U (Ĉy ) π(1+n)ˆk 1 π where Θ is: [ ] η ˆr n 1+ ˆr Θ + α(1 α η) 1+ ˆr 1+n Simulations show that Θ > 1 σ 3 2. [1 Θ] 0, ˆr+δ 1+ˆr Φ(ˆk) 1 (1 σ) ˆr+δ 1+ˆr Φ(ˆk) 0 The switch from WE to TO is welfare decreasing because it induces a decrease in the capital intensity (move away from the FBSO). Transitional dynamics in Figures 4(a) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 35 / 75

33 Structure Capital intensity Welfare analysis Figure 4(a): expected lifetime utility (EΛ y t+τ, σ = 1) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 36 / 75

34 From WE to TY (1) Model Structure Capital intensity Welfare analysis Shock-time old: deλ y t 1 (z 2) = 0 dz 2 Shock-time young: deλ y t (z [ 2) = U 1+ ˆr dz (Ĉy )(1+n)ˆk π 2 1+n ˆr ] dr t+1 > 0 dz 2 Future steady-state generations: ( is a positive constant) deλ y t+ (z [ 2) = U dz (Ĉy ) π(1+ ˆr)ˆk+ dk ] t+ > deλy t (z 2) > 0 2 dz 2 dz 2 The switch from WE to TY is welfare increasing because it induces a increase in the capital intensity (move toward the FBSO). Transitional dynamics in Figures 4(a) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 37 / 75

35 Structure Capital intensity Welfare analysis Figure 4(a): expected lifetime utility (EΛ y t+τ, σ = 1) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 38 / 75

36 PA equilibrium Model From WE to PA From TY to PA Discussion Zero-profit condition of annuity firms: 1+r t+1 = (1 π)(1+rt+1 A ) which implies: 1+r A t+1 = 1+r t+1 1 π Full annuitization is optimal as r A t+1 > r t+1. No accidental bequests by definition. Fundamental difference equation: (1+n)k t+1 = [ 1 Φ ( r A t+1)] wt NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 40 / 75

37 From WE to PA; capital intensity From WE to PA From TY to PA Discussion Fundamental difference equation: where Φ(k,z 3 ) is: Φ(k,z 3 ) [Ψ(k t+1,z 3 ) ] k t+1 1 Φ(k t+1,z 3 ) = Γ(k t) [ ( ) 1+(1 z 3 π) 1 σ 1 π σ ( 1 σ 1] 1 δ +αω0 k α+η 1) 1+ρ Individual choices in Figure 5 (for σ = 1) Transitional dynamics in Figures 3(a), (b), and (c) Properties in Proposition 4 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 41 / 75

38 Figure 5: From WE to PA (σ = 1) From WE to PA From TY to PA Discussion o C t+1 E 4! E 0!! ˆ w PA C t y NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 42 / 75

39 From WE to PA From TY to PA Discussion Figure 3(a): capital intensity (k t+τ, σ = 1) WE=PA TY TO post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 43 / 75

40 From WE to PA From TY to PA Discussion Figure 3(b): capital intensity (k t+τ, σ = 1 2 ) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 44 / 75

41 From WE to PA From TY to PA Discussion Figure 3(c): capital intensity (k t+τ, σ = 3 2 ) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 45 / 75

42 Proposition 4 Model From WE to PA From TY to PA Discussion Existence and stability of the PA model (i) The model has two steady-state solutions; the trivial one features k t+1 = k t = 0, and the economically relevant satisfies k t+1 = k t = ˆk PA, where ˆk PA is the solution to: ˆk PA 1 Φ(ˆk PA,1) = (1 α)ω0(ˆk PA ) α+η 1+n (ii) The trivial steady-state solution is unstable whilst the non-trivial solution is stable: 0 < dkt+1 dk t < 1, for k t+1 = k t = ˆk PA. (iii) The steady-state capital intensity satisfies the following inequality: ˆk PA ˆk WE σ 1. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 46 / 75

43 From WE to PA; welfare From WE to PA From TY to PA Discussion Shock-time old: Shock-time young: deλ y t (z 3) dz 3 deλ y t 1 (z 3) dz 3 = 0 = U (Ĉy )(1+n)ˆk [ π ˆr ] dr t+1 > 0 dz 3 Future steady-state generations: deλ y t+ (z [ 3) = U dz (Ĉy ) π(1+n)ˆk+ dk ] t+ 3 dz 3 0 For σ < 1 capital is crowded out and deλy t+ (z 3) dz 3 < 0; move away from the FBSO. Transitional dynamics in Figures 4(a), (b), (c) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 47 / 75

44 From WE to PA From TY to PA Discussion Figure 4(a): expected lifetime utility (EΛ y t+τ, σ = 1) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 48 / 75

45 From WE to PA From TY to PA Discussion Figure 4(b): expected lifetime utility (EΛ y t+τ, σ = 1 2 ) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 49 / 75

46 From WE to PA From TY to PA Discussion Figure 4(c): expected lifetime utility (EΛ y t+τ, σ = 3 2 ) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 50 / 75

47 Tragedy; part 1 Model From WE to PA From TY to PA Discussion Although full annuitization of assets is privately optimal it is not socially beneficial due to adverse general equilibrium repercussions. Strong version Going from the WE-mode. Holds if the inter-temporal substitution elasticity is smaller than 1. Empirical evidence suggests that this is the case (Attanasio and Weber, 1995, JPE). Theory of the second best. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 51 / 75

48 From WE to PA From TY to PA Discussion Theory of the second best (Lipsey and Lancaster, 1956, RES) Compared to the WE equilibrium the PA eliminates the distortion: Of unproductive government spending. Of the missing insurance market. But, we end up in a worse equilibrium because we are driven away from the golden rule interest rate. Eliminating less than all imperfections in an imperfect market setting does not necessarily lead to a better outcome! NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 52 / 75

49 From TY to PA (1) Model From WE to PA From TY to PA Discussion Fundamental difference equations: Ψ(k t+1,z 3 ) = Γ(k t,1), Ψ(k t+τ+1,z 3 ) = Γ(k t+τ ), τ = 2,3,..., Composite shock Shock-time young still receive transfers but also the higher interest rate on assets. Individual choices in Figure 6 (for σ = 1) Transitional dynamics σ = 1 case in Figures 7(a) and (b) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 54 / 75

50 Figure 6: From TY to PA (σ = 1) From WE to PA From TY to PA Discussion o C t+1 E 4!! E 1! E 0! ˆ w PA! wˆ TY +Zˆ y C t y NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 55 / 75

51 From WE to PA From TY to PA Discussion Figure 7(a): capital intensity (k t+τ, from TY to PA) TY PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 56 / 75

52 From WE to PA From TY to PA Discussion Figure 7(b): expected lifetime utility (EΛ y t+τ, from TY to PA) TY PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 57 / 75

53 From TY to PA; welfare From WE to PA From TY to PA Discussion Shock-time old: Shock-time young: deλ y t (z 3) dz 3 deλ y t 1 (z 3) dz 3 = 0 = U (Ĉy )(1+n)ˆk [ π ˆr ] dr t+1 > 0 dz 3 Future steady-state generations: deλ y t+ (z [ 3) = U dz (Ĉy ) π(ˆr n)ˆk + dk ] t+ 3 dz 3 0 For σ < 1 capital is crowded out and deλy t+ (z 3) dz 3 < 0; move away from the FBSO. Simulations show that even for σ = 3 2 long-run welfare falls. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 58 / 75

54 Tragedy; part 2 Model From WE to PA From TY to PA Discussion Although full annuitization of assets is privately optimal it is not socially beneficial due to adverse general equilibrium repercussions. Weak version Going from the TY-mode. Always holds; for σ < 1 the Tragedy can be shown analytically, for σ > 1 by simulations. Again, theory of the second best. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 59 / 75

55 From WE to PA From TY to PA Discussion What about the switch from TO to PA? Fundamental difference equation: [Ψ(k t+1,z 1,z 3 ) ] 1+ z 1π 1 π Φ(k t+1) 1 Φ(k t+1,z 3 ) k t+1 = Γ(k t ) Composite shock Shock-time old lose transfers Shock-time young lose anticipated transfers but can annuitize their savings Individual choices in Figure 8 (σ = 1) Transitional dynamics σ = 1 case in Figure 8(a) and (b) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 61 / 75

56 Figure 8: From TO to PA (σ = 1) From WE to PA From TY to PA Discussion o C t+1! E 1! E 0! E 4 Ẑ o!!! ˆ ˆ w TO w PA C t y NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 62 / 75

57 From WE to PA From TY to PA Discussion Figure 9(a): capital intensity (k t+τ, from TO to PA) TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 63 / 75

58 From WE to PA From TY to PA Discussion Figure 9(b): expected lifetime utility (EΛ y t+τ, from TO to PA) TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 64 / 75

59 Endogenous growth (1) Knife-edge case: Endogenous growth Conclusions Endogenous growth model: η = 1 α Output and factor prices: y t Y t L t = Ω 0 k t w t = (1 α)ω 0 k t r t = r αω 0 δ NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 66 / 75

60 Endogenous growth (2) Knife-edge case: Endogenous growth Conclusions Growth rates: 1+γ WE = 1 Φ( r) 1+n (1 α)ω 0 1+γ TY = 1 Φ( r) 1+n [(1 α)ω 0 +π(1+ r)] 1+γ TO 1+γ WE = π 1+Φ( r) 1 π 1+γ PA = 1 Φ( ra ) (1 α)ω 0 1+n Comparison WE and revenue recycling modes: γ TY > γ WE > γ TO NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 67 / 75

61 Endogenous growth (3) Knife-edge case: Endogenous growth Conclusions Growth performance of PA relative to other scenarios depends on the value of σ Different values of σ γ TY > γ WE > γ PA > γ TO 0 < σ < 1 γ TY > γ PA = γ WE > γ TO σ = 1 γ PA > γ WE > γ TO,γ PA γ TY σ > 1 NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 68 / 75

62 Endogenous growth (4) Knife-edge case: Endogenous growth Conclusions Welfare effect is time- and scenario-dependent: [ Φ( r) 1/σ θ i( 1+γ i) ] τ 1 1/σ 2+ρ π wt 1+ρ 1 1/σ ÊΛ y,i t+τ for σ > 0 σ 1 Ξ ρ π [θ i( 1+γ i) ] τ wt + 1 π ln(1+ r) for σ = 1 1+ρ 1+ρ Figures 10(a), (b), (c) show different cases. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 69 / 75

63 Knife-edge case: Endogenous growth Conclusions Figure 10(a): welfare in the endogenous growth model (σ = 1 2 ) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 70 / 75

64 Knife-edge case: Endogenous growth Conclusions Figure 10(b): welfare in the endogenous growth model (σ = 1) WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 71 / 75

65 Knife-edge case: Endogenous growth Conclusions Figure 10(c): welfare in the endogenous growth model (σ = 3 2 ) 5 4 WE TY TO PA post shock time (τ) NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 72 / 75

66 Main findings Model Knife-edge case: Endogenous growth Conclusions With an uncertain life expectancy and in the absence of annuities, non-altruistic agents engage in precautionary saving to avoid running out of assets in old age. While they refrain from leaving intentional bequests to their offspring, they will generally make unintended bequests which we assume to flow to the government. Form of recycling scheme is very important. Starting from WE a move to TO may be welfare decreasing in the long run (capital crowding out) In the presence of annuities, full annuitization is attractive to agents. NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 74 / 75

67 Main findings and future extensions Knife-edge case: Endogenous growth Conclusions Two forms of the Tragedy of Annuitization Strong version: WE welfare-dominates PA in the long run Weak version: TY welfare-dominates PA in the long run Eliminating less than all imperfection in an imperfect market does not necessarily lead to a better outcome! Future extensions: Endogenize labour supply / retirement decision. PAYG systems. Incorporate adverse selection effects NAKE Dynamic Macroeconomic Theory Lecture 4B: (January 20, 2012) 75 / 75