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A Tractable Model of Precautionary Reserves, Net Foreign Assets, or Sovereign Wealth Funds Christopher Carroll and Olivier Jeanne Johns Hopkins University March 3, 2013 Introduction Model Motivation Calibration And Simulation Literature

  1. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Connection? Precautionary Motives Commonly Cited In All Three Cases Our Model of Precautionary Net Foreign Assets: Tractable. Tractable! TRACTABLE!!! The Natural Extension of the Ramsey Model Shows Eqbm Relation Between Precautionary, Other Motives Two applications Economic Growth and Capital Flows Impact of Reducing Global Financial Imbalances Carroll, Jeanne A Tractable Model

  2. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Connection? Precautionary Motives Commonly Cited In All Three Cases Our Model of Precautionary Net Foreign Assets: Tractable. Tractable! TRACTABLE!!! The Natural Extension of the Ramsey Model Shows Eqbm Relation Between Precautionary, Other Motives Two applications Economic Growth and Capital Flows Impact of Reducing Global Financial Imbalances Carroll, Jeanne A Tractable Model

  3. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Connection? Precautionary Motives Commonly Cited In All Three Cases Our Model of Precautionary Net Foreign Assets: Tractable. Tractable! TRACTABLE!!! The Natural Extension of the Ramsey Model Shows Eqbm Relation Between Precautionary, Other Motives Two applications Economic Growth and Capital Flows Impact of Reducing Global Financial Imbalances Carroll, Jeanne A Tractable Model

  4. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Literature Aggregated Micro Model “Real” microfoundations! Builds on Toche (2005) Related: Fogli and Perri (2006), Mendoza, Quadrini, and Rios-Rull (2007), Sandri (2008) Other Approaches: Caballero, Farhi, and Gourinchas (2008) Carroll, Jeanne A Tractable Model

  5. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Literature Aggregated Micro Model “Real” microfoundations! Builds on Toche (2005) Related: Fogli and Perri (2006), Mendoza, Quadrini, and Rios-Rull (2007), Sandri (2008) Other Approaches: Caballero, Farhi, and Gourinchas (2008) Carroll, Jeanne A Tractable Model

  6. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Literature Aggregated Micro Model “Real” microfoundations! Builds on Toche (2005) Related: Fogli and Perri (2006), Mendoza, Quadrini, and Rios-Rull (2007), Sandri (2008) Other Approaches: Caballero, Farhi, and Gourinchas (2008) Carroll, Jeanne A Tractable Model

  7. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Literature Aggregated Micro Model “Real” microfoundations! Builds on Toche (2005) Related: Fogli and Perri (2006), Mendoza, Quadrini, and Rios-Rull (2007), Sandri (2008) Other Approaches: Caballero, Farhi, and Gourinchas (2008) Carroll, Jeanne A Tractable Model

  8. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Literature Aggregated Micro Model “Real” microfoundations! Builds on Toche (2005) Related: Fogli and Perri (2006), Mendoza, Quadrini, and Rios-Rull (2007), Sandri (2008) Other Approaches: Caballero, Farhi, and Gourinchas (2008) Carroll, Jeanne A Tractable Model

  9. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Structure Model Calibration and Simulation Applications Growth and Capital Flows Complete World Knowledge (General Equilibrium) Carroll, Jeanne A Tractable Model

  10. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Structure Model Calibration and Simulation Applications Growth and Capital Flows Complete World Knowledge (General Equilibrium) Carroll, Jeanne A Tractable Model

  11. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Structure Model Calibration and Simulation Applications Growth and Capital Flows Complete World Knowledge (General Equilibrium) Carroll, Jeanne A Tractable Model

  12. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Structure Model Calibration and Simulation Applications Growth and Capital Flows Complete World Knowledge (General Equilibrium) Carroll, Jeanne A Tractable Model

  13. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Structure Model Calibration and Simulation Applications Growth and Capital Flows Complete World Knowledge (General Equilibrium) Carroll, Jeanne A Tractable Model

  14. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Overview Small Open Economy Balanced Growth Path With Population And Productivity Growth Accumulate Buffer Stock to Self-Insure Against Unemployment NFA: Aggregate Stock of Wealth Minus Domestic Capital Stock Closed-Form Solutions For Equilibrium Carroll, Jeanne A Tractable Model

  15. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Overview Small Open Economy Balanced Growth Path With Population And Productivity Growth Accumulate Buffer Stock to Self-Insure Against Unemployment NFA: Aggregate Stock of Wealth Minus Domestic Capital Stock Closed-Form Solutions For Equilibrium Carroll, Jeanne A Tractable Model

  16. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Overview Small Open Economy Balanced Growth Path With Population And Productivity Growth Accumulate Buffer Stock to Self-Insure Against Unemployment NFA: Aggregate Stock of Wealth Minus Domestic Capital Stock Closed-Form Solutions For Equilibrium Carroll, Jeanne A Tractable Model

  17. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Overview Small Open Economy Balanced Growth Path With Population And Productivity Growth Accumulate Buffer Stock to Self-Insure Against Unemployment NFA: Aggregate Stock of Wealth Minus Domestic Capital Stock Closed-Form Solutions For Equilibrium Carroll, Jeanne A Tractable Model

  18. Introduction Model Motivation Calibration And Simulation Literature Applications Structure Conclusions Overview References Overview Small Open Economy Balanced Growth Path With Population And Productivity Growth Accumulate Buffer Stock to Self-Insure Against Unemployment NFA: Aggregate Stock of Wealth Minus Domestic Capital Stock Closed-Form Solutions For Equilibrium Carroll, Jeanne A Tractable Model

  19. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Macroeconomic Assumptions Domestic output is produced with the Cobb-Douglas function: K α L t ) 1 − α , Y Y t = K K t ( z t L L (1) Y Labor productivity increases by G in every period, z t +1 = G z t . (2) Capital perfectly mobile internationally, ≡ 1 − δ + α Y Y Y t ���� � = R , (3) K K K t Capital-to-output ratio is constant and equal to, K α K K Y = R − � . (4) Y Y Carroll, Jeanne A Tractable Model

  20. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Macroeconomic Assumptions Domestic output is produced with the Cobb-Douglas function: K α L t ) 1 − α , Y Y t = K K t ( z t L L (1) Y Labor productivity increases by G in every period, z t +1 = G z t . (2) Capital perfectly mobile internationally, ≡ 1 − δ + α Y Y Y t ���� � = R , (3) K K K t Capital-to-output ratio is constant and equal to, K α K K Y = R − � . (4) Y Y Carroll, Jeanne A Tractable Model

  21. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Macroeconomic Assumptions Domestic output is produced with the Cobb-Douglas function: K α L t ) 1 − α , Y Y t = K K t ( z t L L (1) Y Labor productivity increases by G in every period, z t +1 = G z t . (2) Capital perfectly mobile internationally, ≡ 1 − δ + α Y Y Y t ���� � = R , (3) K K K t Capital-to-output ratio is constant and equal to, K α K K Y = R − � . (4) Y Y Carroll, Jeanne A Tractable Model

  22. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Macroeconomic Assumptions Domestic output is produced with the Cobb-Douglas function: K α L t ) 1 − α , Y Y t = K K t ( z t L L (1) Y Labor productivity increases by G in every period, z t +1 = G z t . (2) Capital perfectly mobile internationally, ≡ 1 − δ + α Y Y Y t ���� � = R , (3) K K K t Capital-to-output ratio is constant and equal to, K α K K Y = R − � . (4) Y Y Carroll, Jeanne A Tractable Model

  23. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  24. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  25. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  26. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  27. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  28. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  29. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  30. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  31. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References People and Populations Each worker is part of a single ‘generation’ born at the same time Size of generation born at t : Ξ t . Life Stages: Employment Unemployment/Retirement Death Transitions to unemployment and death are Poisson processes Flow probabilities � and D. Employed and Unemployed Populations: Ξ t +1 = E t Ξ − � � � Ξ t +1 = � ) . U t (Ξ − � D)(Ξ − � Carroll, Jeanne A Tractable Model

  32. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Balanced Growth Capital and output grow at constant rates Real wage grows by factor G in every period. Main variable of interest= N N N t , the aggregate net foreign assets of the economy at the beginning of period t . N N t = B B B t − K K K t . (5) N Carroll, Jeanne A Tractable Model

  33. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Balanced Growth Capital and output grow at constant rates Real wage grows by factor G in every period. Main variable of interest= N N N t , the aggregate net foreign assets of the economy at the beginning of period t . N N t = B B B t − K K K t . (5) N Carroll, Jeanne A Tractable Model

  34. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Balanced Growth Capital and output grow at constant rates Real wage grows by factor G in every period. Main variable of interest= N N N t , the aggregate net foreign assets of the economy at the beginning of period t . N N t = B B B t − K K K t . (5) N Carroll, Jeanne A Tractable Model

  35. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem Budget constraint of individual: labor income b b b t +1 � �� � + c c t = b c b b t + ξ t ℓ t W t , (6) R Worker’s labor supply ℓ grows by a factor X per period over his lifetime, ℓ t = X t ℓ 0 , (7) For consumer who remains employed, labor income grows by Γ GX . ≡ Carroll, Jeanne A Tractable Model

  36. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem Budget constraint of individual: labor income b b b t +1 � �� � + c c t = b c b b t + ξ t ℓ t W t , (6) R Worker’s labor supply ℓ grows by a factor X per period over his lifetime, ℓ t = X t ℓ 0 , (7) For consumer who remains employed, labor income grows by Γ GX . ≡ Carroll, Jeanne A Tractable Model

  37. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem Budget constraint of individual: labor income b b b t +1 � �� � + c c t = b c b b t + ξ t ℓ t W t , (6) R Worker’s labor supply ℓ grows by a factor X per period over his lifetime, ℓ t = X t ℓ 0 , (7) For consumer who remains employed, labor income grows by Γ GX . ≡ Carroll, Jeanne A Tractable Model

  38. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem Unemployment: Complete and permanent destruction of h CRRA felicity u ( • ) = • 1 − ρ / (1 − ρ ); geometric discounting at β Unemployed convert their wealth into annuities. Solution to the unemployed consumer’s optimization problem, c u κ u b c = b b t , c t where κ is the marginal propensity to consume, D( β R) 1 /ρ κ u 1 − � . ≡ R Carroll, Jeanne A Tractable Model

  39. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem Unemployment: Complete and permanent destruction of h CRRA felicity u ( • ) = • 1 − ρ / (1 − ρ ); geometric discounting at β Unemployed convert their wealth into annuities. Solution to the unemployed consumer’s optimization problem, c u κ u b c = b b t , c t where κ is the marginal propensity to consume, D( β R) 1 /ρ κ u 1 − � . ≡ R Carroll, Jeanne A Tractable Model

  40. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem Unemployment: Complete and permanent destruction of h CRRA felicity u ( • ) = • 1 − ρ / (1 − ρ ); geometric discounting at β Unemployed convert their wealth into annuities. Solution to the unemployed consumer’s optimization problem, c u κ u b c = b b t , c t where κ is the marginal propensity to consume, D( β R) 1 /ρ κ u 1 − � . ≡ R Carroll, Jeanne A Tractable Model

  41. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem Unemployment: Complete and permanent destruction of h CRRA felicity u ( • ) = • 1 − ρ / (1 − ρ ); geometric discounting at β Unemployed convert their wealth into annuities. Solution to the unemployed consumer’s optimization problem, c u κ u b c = b b t , c t where κ is the marginal propensity to consume, D( β R) 1 /ρ κ u 1 − � . ≡ R Carroll, Jeanne A Tractable Model

  42. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem ‘Growth impatience condition’: ( β R) 1 /ρ Þ Þ Γ Þ ≡ < 1 Γ necessary for finite target ratio of wealth to income (Carroll (2011)) Defining nonbold variables as, e.g., c e c e t = c c t / (W t ℓ t ), we get b e t +1 = (R / Γ) ( b e t − c e t + 1) . (8) � � Þ � ρ � − 1 /ρ c e Þ Þ Γ � 1 /ρ c e c e t Þ Γ � t +1 = Þ Þ 1 − � . (9) t R / Γ( b e t − c e κ u t + 1) Saddle-point stable dynamics. Carroll, Jeanne A Tractable Model

  43. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem ‘Growth impatience condition’: ( β R) 1 /ρ Þ Þ Γ Þ ≡ < 1 Γ necessary for finite target ratio of wealth to income (Carroll (2011)) Defining nonbold variables as, e.g., c e c e t = c c t / (W t ℓ t ), we get b e t +1 = (R / Γ) ( b e t − c e t + 1) . (8) � � Þ � ρ � − 1 /ρ c e Þ Þ Γ � 1 /ρ c e c e t Þ Γ � t +1 = Þ Þ 1 − � . (9) t R / Γ( b e t − c e κ u t + 1) Saddle-point stable dynamics. Carroll, Jeanne A Tractable Model

  44. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The microeconomic consumer’s problem ‘Growth impatience condition’: ( β R) 1 /ρ Þ Þ Γ Þ ≡ < 1 Γ necessary for finite target ratio of wealth to income (Carroll (2011)) Defining nonbold variables as, e.g., c e c e t = c c t / (W t ℓ t ), we get b e t +1 = (R / Γ) ( b e t − c e t + 1) . (8) � � Þ � ρ � − 1 /ρ c e Þ Þ Γ � 1 /ρ c e c e t Þ Γ � t +1 = Þ Þ 1 − � . (9) t R / Γ( b e t − c e κ u t + 1) Saddle-point stable dynamics. Carroll, Jeanne A Tractable Model

  45. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Phase Diagram Consumption Ratio � � � � � � 0 � � � 0 � � Wealth ratio � Carroll, Jeanne A Tractable Model

  46. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The Growth Impatience Condition Target wealth-to-income ratio: impatience vs prudence. Closed-form solution for the target wealth-to-income ratio  � 1 /ρ  − 1 � Þ − ρ − 1  Γ 1 + Þ Þ ˇ R − 1 + κ u Γ b = . (10)  � ∂ � > 0 , ∂ ˇ ∂ ˇ ∂β > 0 , ∂ ˇ b b b ∂ Γ < 0 . (11) ∂ ˇ b ∂ρ > 0 . (12) The response of ˇ b to R is ambiguous. Carroll, Jeanne A Tractable Model

  47. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The Growth Impatience Condition Target wealth-to-income ratio: impatience vs prudence. Closed-form solution for the target wealth-to-income ratio  � 1 /ρ  − 1 � Þ − ρ − 1  Γ 1 + Þ Þ ˇ R − 1 + κ u Γ b = . (10)  � ∂ � > 0 , ∂ ˇ ∂ ˇ ∂β > 0 , ∂ ˇ b b b ∂ Γ < 0 . (11) ∂ ˇ b ∂ρ > 0 . (12) The response of ˇ b to R is ambiguous. Carroll, Jeanne A Tractable Model

  48. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The Growth Impatience Condition Target wealth-to-income ratio: impatience vs prudence. Closed-form solution for the target wealth-to-income ratio  � 1 /ρ  − 1 � Þ − ρ − 1  Γ 1 + Þ Þ ˇ R − 1 + κ u Γ b = . (10)  � ∂ � > 0 , ∂ ˇ ∂ ˇ ∂β > 0 , ∂ ˇ b b b ∂ Γ < 0 . (11) ∂ ˇ b ∂ρ > 0 . (12) The response of ˇ b to R is ambiguous. Carroll, Jeanne A Tractable Model

  49. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References The Growth Impatience Condition Target wealth-to-income ratio: impatience vs prudence. Closed-form solution for the target wealth-to-income ratio  � 1 /ρ  − 1 � Þ − ρ − 1  Γ 1 + Þ Þ ˇ R − 1 + κ u Γ b = . (10)  � ∂ � > 0 , ∂ ˇ ∂ ˇ ∂β > 0 , ∂ ˇ b b b ∂ Γ < 0 . (11) ∂ ˇ b ∂ρ > 0 . (12) The response of ˇ b to R is ambiguous. Carroll, Jeanne A Tractable Model

  50. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Foreign Assets Ratio of employed workers’ wealth to output,   + ∞ B e  1 − � t = B B � X �   B e t Λ n b e = (1 − α ) t , t − n , (13)   Y Y Y t Ξ  n =0 ���� ≡ Λ where Λ is the factor by which the share of a generation in total labor supply shrinks every period. The Level of Unemployed Workers’ Wealth is B u t +1 = R(1 − κ u ) B B u B e B B B t + � B B t +1 . (14) Carroll, Jeanne A Tractable Model

  51. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Foreign Assets Ratio of employed workers’ wealth to output,   + ∞ B e  1 − � t = B B � X �   B e t Λ n b e = (1 − α ) t , t − n , (13)   Y Y Y t Ξ  n =0 ���� ≡ Λ where Λ is the factor by which the share of a generation in total labor supply shrinks every period. The Level of Unemployed Workers’ Wealth is B u t +1 = R(1 − κ u ) B B u B e B B B t + � B B t +1 . (14) Carroll, Jeanne A Tractable Model

  52. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Foreign Assets (cont) Steady state ratio of net foreign assets to GDP � B � � � B e N N N Y = ΞG � ΞG B α 1 + Y − ΞG . (15) ΞG − � D( β R) 1 /ρ R − � Y Y R Y Y Depends on Employed Workers’ Target Savings Carroll, Jeanne A Tractable Model

  53. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Foreign Assets (cont) Steady state ratio of net foreign assets to GDP � B � � � B e N N N Y = ΞG � ΞG B α 1 + Y − ΞG . (15) ΞG − � D( β R) 1 /ρ R − � Y Y R Y Y Depends on Employed Workers’ Target Savings Carroll, Jeanne A Tractable Model

  54. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References ‘Stakes’ Model with no stakes + ∞ B e B e = B B � Λ n b e ( n ) . = (1 − α )(1 − Λ) (16) Y Y Y n =0 Model with stakes yielding a representative agent B e B B = B ˇ (1 − α )ˇ ˇ ˇ = b . (17) Y Y Y where  � 1 /ρ  − 1 � Þ − ρ  Γ 1 Þ − 1 1 + Þ ˇ ˇ 2 − Λ + κ u Γ b = R − (18)  � Carroll, Jeanne A Tractable Model

  55. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References ‘Stakes’ Model with no stakes + ∞ B e B e = B B � Λ n b e ( n ) . = (1 − α )(1 − Λ) (16) Y Y Y n =0 Model with stakes yielding a representative agent B e B B = B ˇ (1 − α )ˇ ˇ ˇ = b . (17) Y Y Y where  � 1 /ρ  − 1 � Þ − ρ  Γ 1 Þ − 1 1 + Þ ˇ ˇ 2 − Λ + κ u Γ b = R − (18)  � Carroll, Jeanne A Tractable Model

  56. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Advantages Of Model With Stakes Closed-form solution for steady state Simple to characterize transition dynamics Carroll, Jeanne A Tractable Model

  57. Introduction Macroeconomy Model People Calibration And Simulation Balanced Growth Applications The Microeconomic Problem Conclusions Foreign Assets References Advantages Of Model With Stakes Closed-form solution for steady state Simple to characterize transition dynamics Carroll, Jeanne A Tractable Model

  58. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Calibration and Simulation Table 1 β − 1 α δ Ξ G R Φ ρ d � 0.3 0.06 1.01 1.04 1.04 1.04 1.01 0.025 2 0.05 N N N / Y Y Y = 0 . 17 in the model with no stakes N / Y Y = 0 . 79 in the model with stakes N N Y Carroll, Jeanne A Tractable Model

  59. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Calibration and Simulation Table 1 β − 1 α δ Ξ G R Φ ρ d � 0.3 0.06 1.01 1.04 1.04 1.04 1.01 0.025 2 0.05 N N N / Y Y Y = 0 . 17 in the model with no stakes N / Y Y = 0 . 79 in the model with stakes N N Y Carroll, Jeanne A Tractable Model

  60. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Paths 5 Consumption and wealth ratios, � and � Wealth 4 3 2 Consumption 1 0 20 40 60 80 100 Time Carroll, Jeanne A Tractable Model

  61. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Sensitivity analysis 3 4 2 1 2 N � Y N � Y 0 0 � 1 � 2 � 2 0.02 0.03 0.04 0.05 � 3 Risk aversion, Ρ Unemployment probability, � 1.5 2 1.0 1 0.5 0 N � Y N � Y With stakes 0.0 � 1 No stakes � 0.5 � 2 1.0 � 1.5 � 3 1.04 1.06 1.08 1.10 1.00 1.01 1.02 1.03 1.04 1.05 Productivity growth, G Interest factor, R Carroll, Jeanne A Tractable Model

  62. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Social Insurance Many countries have social transfers to unemployed/retired New assumption: labor income tax on the employed in order to finance transfers to the unemployed. Unemployed receive transfer whose value is a multiple ς of the labor income that they would have received if they had remained employed. New formula for target wealth-to-income ratio. Going through the same steps as before, we get   � 1 /ρ   � Þ − ρ  1 + Þ Þ − 1   � ` ˇ ˇ Γ ˇ Ξ + κ u b ( ς ) =  1 − ς b , (19)  �  Carroll, Jeanne A Tractable Model

  63. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Social Insurance Many countries have social transfers to unemployed/retired New assumption: labor income tax on the employed in order to finance transfers to the unemployed. Unemployed receive transfer whose value is a multiple ς of the labor income that they would have received if they had remained employed. New formula for target wealth-to-income ratio. Going through the same steps as before, we get   � 1 /ρ   � Þ − ρ  1 + Þ Þ − 1   � ` ˇ ˇ Γ ˇ Ξ + κ u b ( ς ) =  1 − ς b , (19)  �  Carroll, Jeanne A Tractable Model

  64. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Social Insurance Many countries have social transfers to unemployed/retired New assumption: labor income tax on the employed in order to finance transfers to the unemployed. Unemployed receive transfer whose value is a multiple ς of the labor income that they would have received if they had remained employed. New formula for target wealth-to-income ratio. Going through the same steps as before, we get   � 1 /ρ   � Þ − ρ  1 + Þ Þ − 1   � ` ˇ ˇ Γ ˇ Ξ + κ u b ( ς ) =  1 − ς b , (19)  �  Carroll, Jeanne A Tractable Model

  65. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Social Insurance Many countries have social transfers to unemployed/retired New assumption: labor income tax on the employed in order to finance transfers to the unemployed. Unemployed receive transfer whose value is a multiple ς of the labor income that they would have received if they had remained employed. New formula for target wealth-to-income ratio. Going through the same steps as before, we get   � 1 /ρ   � Þ − ρ  1 + Þ Þ − 1   � ` ˇ ˇ Γ ˇ Ξ + κ u b ( ς ) =  1 − ς b , (19)  �  Carroll, Jeanne A Tractable Model

  66. Introduction Model Parameter Values Calibration And Simulation Paths Applications Sensitivity Analysis Conclusions Social Insurance References Social insurance 1 Foreign assets � GDP, N � Y 0 � 1 � 2 � 3 � 4 0 1 2 3 4 5 Social insurance benefit in years of wage, ̣ Carroll, Jeanne A Tractable Model

  67. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References Growth And Saving Theory: Good Growth Prospects → Should Borrow to Invest Data: Fast-Growing Countries Export Capital Carroll and Weil (1994); Loayza, Schmidt-Hebbel, and Serv´ en (2000); Attanasio, Picci, and Scorcu (2000); Gourinchas and Jeanne, 2007, Prasad, Rajan and Subramanian (2007); Sandri (2008) Can this model shed light on this puzzle? Yes, if growth take-off entails idiosyncratic risk (both G and � go up). Carroll, Jeanne A Tractable Model

  68. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References Growth And Saving Theory: Good Growth Prospects → Should Borrow to Invest Data: Fast-Growing Countries Export Capital Carroll and Weil (1994); Loayza, Schmidt-Hebbel, and Serv´ en (2000); Attanasio, Picci, and Scorcu (2000); Gourinchas and Jeanne, 2007, Prasad, Rajan and Subramanian (2007); Sandri (2008) Can this model shed light on this puzzle? Yes, if growth take-off entails idiosyncratic risk (both G and � go up). Carroll, Jeanne A Tractable Model

  69. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References Growth And Saving Theory: Good Growth Prospects → Should Borrow to Invest Data: Fast-Growing Countries Export Capital Carroll and Weil (1994); Loayza, Schmidt-Hebbel, and Serv´ en (2000); Attanasio, Picci, and Scorcu (2000); Gourinchas and Jeanne, 2007, Prasad, Rajan and Subramanian (2007); Sandri (2008) Can this model shed light on this puzzle? Yes, if growth take-off entails idiosyncratic risk (both G and � go up). Carroll, Jeanne A Tractable Model

  70. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References Growth And Saving Theory: Good Growth Prospects → Should Borrow to Invest Data: Fast-Growing Countries Export Capital Carroll and Weil (1994); Loayza, Schmidt-Hebbel, and Serv´ en (2000); Attanasio, Picci, and Scorcu (2000); Gourinchas and Jeanne, 2007, Prasad, Rajan and Subramanian (2007); Sandri (2008) Can this model shed light on this puzzle? Yes, if growth take-off entails idiosyncratic risk (both G and � go up). Carroll, Jeanne A Tractable Model

  71. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References Growth And Saving Theory: Good Growth Prospects → Should Borrow to Invest Data: Fast-Growing Countries Export Capital Carroll and Weil (1994); Loayza, Schmidt-Hebbel, and Serv´ en (2000); Attanasio, Picci, and Scorcu (2000); Gourinchas and Jeanne, 2007, Prasad, Rajan and Subramanian (2007); Sandri (2008) Can this model shed light on this puzzle? Yes, if growth take-off entails idiosyncratic risk (both G and � go up). Carroll, Jeanne A Tractable Model

  72. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References Growth and capital flows Capital outflows to GDP, � N t � N t � 1 �� Y t C 0.05 A Increasing risk B 0.00 � 0.05 � 0.10 � 0.15 Constant risk � 0.20 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 Productivity growth factor, G Carroll, Jeanne A Tractable Model

  73. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References World General Equilibrium Small economy assumption not appropriate to study global savings glut or adjustment of global financial imbalances. Study steady state equilibria in two-country extension of the model. Global interest rate R endogenous N N h + N N N f = 0 , (20) N Carroll, Jeanne A Tractable Model

  74. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References World General Equilibrium Small economy assumption not appropriate to study global savings glut or adjustment of global financial imbalances. Study steady state equilibria in two-country extension of the model. Global interest rate R endogenous N N h + N N N f = 0 , (20) N Carroll, Jeanne A Tractable Model

  75. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References World General Equilibrium Small economy assumption not appropriate to study global savings glut or adjustment of global financial imbalances. Study steady state equilibria in two-country extension of the model. Global interest rate R endogenous N N h + N N N f = 0 , (20) N Carroll, Jeanne A Tractable Model

  76. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References General Equilibrium Two countries identical except for size (h=20%, f=80%) and level of social insurance ( ς h = 1 . 5, ς f = 0 . 75). This implies N N N h = − 0 . 5 (21) Y Y h Y N N N f = 0 . 125 (22) Y Y Y f What is impact of increasing foreign social insurance to the home level? Carroll, Jeanne A Tractable Model

  77. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References General Equilibrium Two countries identical except for size (h=20%, f=80%) and level of social insurance ( ς h = 1 . 5, ς f = 0 . 75). This implies N N N h = − 0 . 5 (21) Y Y h Y N N N f = 0 . 125 (22) Y Y Y f What is impact of increasing foreign social insurance to the home level? Carroll, Jeanne A Tractable Model

  78. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References General Equilibrium Two countries identical except for size (h=20%, f=80%) and level of social insurance ( ς h = 1 . 5, ς f = 0 . 75). This implies N N N h = − 0 . 5 (21) Y Y h Y N N N f = 0 . 125 (22) Y Y Y f What is impact of increasing foreign social insurance to the home level? Carroll, Jeanne A Tractable Model

  79. Introduction Model Calibration And Simulation Growth And Saving Applications Resorbing Global Imbalances Conclusions References General equilibrium 5.0 4.5 Wealth and capital ratios � Capital ratio 4.0 3.5 Wealth ratio, ̣ � 0 3.0 Wealth ratio, ̣ � 1 2.5 2.0 Wealth ratio, ̣ � 2 1.5 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 Interest factor, R Carroll, Jeanne A Tractable Model

  80. Introduction Model Calibration And Simulation Applications Conclusions References Conclusions Tractable model of net foreign assets of small open economy Two applications Relationship between growth and capital flows Long-run implications of reducing global imbalances. Extensions for future research: portfolio choice, real exchange rates, asset prices, etc. Carroll, Jeanne A Tractable Model

  81. Introduction Model Calibration And Simulation Applications Conclusions References Conclusions Tractable model of net foreign assets of small open economy Two applications Relationship between growth and capital flows Long-run implications of reducing global imbalances. Extensions for future research: portfolio choice, real exchange rates, asset prices, etc. Carroll, Jeanne A Tractable Model

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