1. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 1 ITFD Growth and Development UPF 2008-2009 LECTURE SLIDES SET 3 Professor Antonio Ciccone

2. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 2 II. ECONOMIC GROWTH WITH ENDOGENOUS SAVINGS

3. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 3 1. Household savings behavior

4. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 4 1. “ Keynesian theory ” of savings and consumption So far we assumed a “ Keynesian ” savings function where s is the marginal propensity to save. 1. The Keynesian consumption (savings) function

5. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 5 Because of the BUDGET CONSTRAINT this implies the “ Keynesian ” consumption function where c is the marginal propensity to consume.

6. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 6 2. Limitations CONCEPTUAL The consumption behavior is assumed to be “ mechanic ” and “ short- sighted ” : –Are households really only looking at CURRENT income when deciding consumption? Not really. Many households borrow from banks in order to be able to consume more today because they know they will be able to pay the money back in the future. –If people save, presumably they are doing this for future consumption. Hence, savings is a FORWARD-LOOKING decision and must take into account what happens in the future.

7. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 7 Assuming savings as a function of current income therefore appears to contradict the use that households make of their savings. EMPIRICAL “ Consumption smoothing: ” – Empirically, we observe that households smooth consumption. To put it differently, the income of households is often more volatile than their consumption. This suggests that households look forward and try to stabilize consumption (their standard of living) as much as they can.

8. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 8 time HOUSEHOLD INCOME OF FARMER FIGURE 1: CONSUMPTION SMOOTHING: A VOLATILE INCOME PATH

9. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 9 time HOUSEHOLD INCOME OF FARMER HOUSEHOLD CONSUMPTION OF FARMER (“KEYNESIAN” theory) FIGURE 2: INCOME AND "KEYNESIAN CONSUMPTION"

10. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 10 time HOUSEHOLD INCOME OF FARMER HOUSEHOLD CONSUMPTION OF FARMER (EMPIRICAL OBSERVATION) FIGURE 3: CONSUMPTION SMOOTHING

11. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 11 time HOUSEHOLD INCOME CONSUMPTION SMOOTHING SAVE FOR “RAINY DAYS” DIS-SAVE TO MAINTAIN CONSUMPTION LEVELS FIGURE 4: SAVINGS AND DIS-SAVINGS IN CONSUMPTION SMOOTHING MODELS

12. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 12 INTERESTINGLY: The Keynesian theory of consumption seems to do better at the aggregate level than at the level of individual households. For example: –Keynesian theory does well in describing relationship between consumption and income of a country at different in different years –Theory does also well in describing relationship between consumption and income across different countries

13. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 13 INCOME CONSUMPTION A PUZZLE? AGGREGATE LEVEL INDIVIDUAL HOUSE- HOLD LEVEL Germany 1950 Or Country 1 Germany 1960 Or Country 2 Germany 1980 Or Country 3 Mr A Ms B Mr C Ms D

14. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 14 2. The permanent income theory of consumption and savings 1. Basic idea and two-period model Households make consumption decisions: LOOKING FORWARD to future USING SAVINGS AND LOANS from BANKS to maintain their living standards STABLE in time to the extent possible

15. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 15 SIMPLEST POSSIBLE formal model (2 PERIODS) INGREDIENTS: – Household lives 2 periods and tries to maximize INTERTEMPORAL utility –Understands that will earn LABOR income Lw[0] in period 0 and Lw[1] in period 1 –Starts with 0 WEALTH –Can save and borrow from bank at interest rate r

16. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 16 MATHEMATICAL MAXIMIZATION PROBLEM: by choosing C 0 and C 1 subject to S=Lw 0 -C 0 C 1 =Lw 1 +(1+r)S DISCOUNT APPLIED TO FUTURE UTILITY NOTE that S can be NEGATIVE (which means the household is BORROWING or DISSAVING)

17. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 17 MATHEMATICAL FORMULATION Maximize INTERTEMPORAL UTILITY by choosing C subject to INTERTEMPORAL BUDGET CONSTRAINT C 1 =Lw 1 +(1+r)S= Lw 1 +(1+r)(Lw 0 -C 0 )

18. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 18 INTERTEMPORAL BUDGET CONSTRAINT can also be written: IMPORTANT TERMINOLOGY: PERMANENT INCOME (PI) PRICE OF FUTURE CONSUMPTION RELATIVE TO CURRENT CONSUMPTION

19. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 19 C[0] C[1] Lw[0] Lw[1] GRAPHICALLY: INCOME LEVELS AND CONSUMTION

20. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 20 C[0] C[1] Lw[0] Lw[1] 1+r THE INTERTEMPORAL BUDGET CONSTRAINT

21. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 21 C[0] C[1] Lw[0] Lw[1] 1+r INTERTEMPORAL UTILITY MAXIMIZATION

22. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 22 C[0] C[1] Lw[0] Lw[1] 1+r C[0] C[1]

23. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 23 C[0] C[1] Lw[0] Lw[1] 1+r C[0] C[1] BORROWING FOR CURRENT CONSUMPTION BORROW REPAY

24. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 24 2. Closed form solution in a simple case SUPPOSE THAT INTEREST RATE is ZERO: r = 0 FUTURE UTILITY DISCOUNT is ZERO: MAXIMIZATION PROBLEM BECOMES: with respect to C subject to

25. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 25 FIRST ORDER MAXIMIZATION CONDITIONS: First-order conditions can be obtained from with respect to C 0 where we have substituted the budget constraint. TAKE DERIVATIVE WITH RESPECT TO C[1] AND SET EQUAL ZERO: OR C1C1

26. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 26 EQUALIZE MARGINAL UTILITY AT DIFFERENT POINTS IN TIME THIS IMPLIES  “ PERFECT CONSUMPTION SMOOTHING ” Using the INTERTEMPORAL BUDGET CONSTRAINT yields consumption as a function of PERMANENT INCOME

27. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 27 Lw[0] C[0] 0.5*Lw[1] 0.5*Lw[0]+0.5*Lw[1] "CONSUMPTION FUNCTION"

28. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 28 Lw[0] C[0] 0.5*Lw[1] 0.5*Lw[0]+0.5*Lw[1] “TEMPORARY” INCREASE IN INCOME INCREASE In first-period income THE EFFECT OF AN INCREASE IN INITIAL-PERIOD INCOME ON C[0]

29. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 29 Lw[0] C[0] 0.5*Lw[0]+0.5*Lw[1] “PERMANENT” INCREASE IN INCOME INCREASE Lw[0] INCREASE Lw[1] THE EFFECT OF AN INCREASE IN INITIAL AND FUTURE INCOME

30. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 30 DISCOUNTING OF FUTURE UTILITY, AND INTEREST MAXIMIZATION WITH DISCOUNTING&INTEREST with respect to C subject to INTERTEMPORAL BUDGET CONSTRAINT

31. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 31 FIRST-ORDER CONDITIONS “ EFFECTIVE TIME DISCOUNTING ”  CONSTANT CONSUMPTION DISCOUNTING OF FUTURE UTILITY AND POSTITIVE INTEREST RATE JUST OFFSET

32. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 32 UPWARD SLOPING CONSUMPTION PATHS IN TIME:  INCREASING CONSUMPTION OVER TIME POSITIVE INTEREST MORE THAN OFFSETS UTILITY DISCOUNTING DOWNWARD SLOPING CONSUMPTION PATHS IN TIME:  DECREASING CONSUMPTION OVER TIME UTILITY DISCOUNTING MORE THAN OFFSETS POSITIVE INTEREST

33. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 33 AN EXAMPLE Take the following utility function: with FIRST-ORDER CONDITION BECOMES or

34. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 34 3. The case of 3 and more periods -- Timing -- Intertemporal budget constraint -- Optimality conditions -- Time consistency

35. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 35 PRESENT-VALUE INCOME AND CONSUMPTION - PERMANENT INCOME - PRESENT VALUE CONSUMPTION t=0t=1 Q[0]w[0]Lw[1]Lw[2]L C[0]C[1]C[2] t=2 YOU ARE HERE interest discounting interest discounting interest discounting

36. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 36 INTERTEMPORAL BUDGET CONSTRAINT

37. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 37 BUDGET CONTRAINT AND TIME EVOLUTION OF WEALTH t=0t=1t=2 Q[0]w[0]Lw[1]Lw[2]L C[0]C[1] C[2]

38. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 38 INTERTEMPORAL BUDGET CONSTRAINT

39. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 39 MAXIMIZE BETWEEN ADJACENT PERIODS OPTIMAL SOLUTION OF CONSUMPTION PROBLEM plus BUDGET CONSTRAINT WITH EQUALITY

40. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 40 Shortest way from A to B? A B

41. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 41 Shortest way from A to B A B

42. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 42 Must be the shortest way between ANY two points A B C D

43. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 43 A B C D Must be the shortest way between ANY two points

44. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 44 INFINITE HORIZON =TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT (end of) PERIOD t

45. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 45 INTERTEMPORAL BUDGET CONSTRAINT NO-PONZI-GAME condition

46. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 46 TIME T  0 WHAT IF: NO PONZI GAME CONDITION VIOLATED?

47. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 47 INTERTEMPORAL BUDGET CONSTRAINT WITH EQUALITY NO-PONZI-GAME condition

48. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 48 TIME T  0 WHAT IF:

49. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 49 CAN INCREASE TIME-0 CONSUMPTION  CONSUMPTION PLAN NOT OPTIMAL! NECESSARY FOR OPTIMALITY:

50. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 50 TIME CONSISTENCY of HOUSOLD CONSUMPTION PLANS

51. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 51 TIME 0 CONSUMPTION PLANS t=0t=1 Q[0]w[0]Lw[1]Lw[2]L C[0]C[1]C[2] t=2 YOU ARE HERE interest discounting interest discounting interest discounting t=0t=1 Q[0] Q(1) w[1]Lw[2]L C[1]C[2] t=2 interest discounting interest discounting YOU ARE HERE TIME 1 CONSUMPTION PLANS (NO NEW INFO)

52. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 52 ***** TIME CONSISTENCY ***** t=0t=1 Q[0]w[0]Lw[1]Lw[2]L C[0]C[1]C[2] t=2 YOU ARE HERE interest discounting interest discounting interest discounting t=0t=1 Q(1) w[1]Lw[2]L C[1]C[2] t=2 interest discounting interest discounting YOU ARE HERE TIME 1 CONSUMPTION PLANS (NO NEW INFO)

53. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 53 3. Optimal consumption and savings in continuous time 1. Infinite horizon subject to = TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT TIME t

54. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 54 2. Intertemporal budget constraint Wealth in discrete time Wealth in continuous time

55. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 55 Intertemporal budget constraint in continuous time satisfied with equality if

56. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 56 3. Interpretation of  and r r is the interest rate that is received between two very close periods in time is the discount rate applied PER UNIT OF TIME between two very close periods in time TO SEE THAT is the discount rate applied PER UNIT OF TIME between two very close periods in time 1)Note that the utility discount between period 0 and t is:

57. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 57 2) Hence the utility discount per unit of time is: 3) What is the limit as t  0? Hopital ’ s rule yields

58. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 58 4. First-order condition where: is INTERTEMPORAL RATE OF TIME PREFERENCE and measures how IMPATIENT people are is the INTERTEMPORAL ELASTICITY OF SUBSTITUTION and measures how much future consumption increases when the interest rate goes up (how much people “ respond to interest rates ” )

59. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 59 TIME OPTIMAL CONSUMPTION PATH r =  C(t) C(0) CONSTANT CONSUMPTION IN TIME

60. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 60 TIME OPTIMAL CONSUMPTION PATH r >  C(t) C(0) INCREASING CONSUMPTION IN TIME

61. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 61 TIME OPTIMAL CONSUMPTION PATH r <  C(t) C(0) DEACREASING CONSUMPTION IN TIME

62. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 62 5. Closed form solution in special case ASSUME (consumers have an INFINITE HORIZON) SOLUTION CHARACERIZED BY  PEOPLE WANT CONSTANT CONSUMPTION OVER TIME ( “ PERFECT CONSUMPTION SMOOTHING ” CASE)

63. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 63 THE INTERTEMPORAL BUDGET CONSTRAINT without initial wealth HENCE

64. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 64 6. Deriving the continuous time first-order condition MAXIMIZATION BETWEEN ANY TWO PERIODS SEPARATED BY TIME x subject to = TOTAL SPENDING IN TWO PERIODS

65. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 65 Take the following utility function: with

66. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 66 FIRST ORDER CONDITIONS FOR THE TWO PERIODS IN TIME making use of the utility function

67. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 67 REWRITING THIS CONDITIONS YIELDS subtracting 1 from both sides

68. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 68 DIVIDE BY x (the TIME BETWEEN THE TWO PERIODS) to get CONSUMPTION GROWTH PER UNIT OF TIME What happens when the two periods get closer and closer (x  0)?

69. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 69 Apply Hopital ’ s rule

70. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 70 HENCE as two periods become VERY CLOSE WHICH IS WHAT WE WANTED TO SHOW

71. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 71 SUMMARIZING QUESTION: What characterizes the optimal consumption PATH that solves subject to

72. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 72 ANSWER: and or

73. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 73 2. The Ramsey-Cass-Koopmans model

74. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 74 We will now integrate a household that chooses consumption optimally over an infinite horizon in the Solow model. The results is often refereed to as the Cass- Koopmans model. The Cass-Koopmans model is exactly like the SOLOW MODEL only that the household does NOT behave mechanically but instead chooses consumption and savings to maximize: subject to where 1. Equilibrium growth with infinite-horizon households

75. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 75 In order to NOT complicate things too much we will simplify the model by assuming: 1.no technological changes (i.e. a=0 in Solow model) 2.no population growth (i.e. n=0 in Solow model)

76. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 76 WHAT WE CAN KEEP FROM THE SOLOW MODEL CONSTANT RETURNS PRODUCTION FUNCTION E(1) E(2) CAPITAL ACCUMULATION EQUATION E(3) PRODUCTION FUNCTION 1. Technology and the capital market

77. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 77 CAPITAL MARKET EQUILIBRIUM E(4) E(5)

78. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 78 WHAT WE CANNOT KEEP IS INSTEAD: E(6) E(7) INTERTEMPORAL BUDGET CONSTRAINT WITH EQUALITY where c[t] is CONSUMPTION per PERSON 2. Household behaviour

79. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 79 WE WILL TRY TO CHARACTERIZE THE EQUILIBRIUM OF THIS ECONOMY IN TERMS OF THE EVOLUTION OF c and k. The goal is to reduce the equations above to a TWO- DIMENSIONAL DIFFERENTIAL EQUATION SYSTEM WHERE CHANGE in CONSUMPTION c=FUNCTION OF k and c CHANGE IN CAPITAL k=FUNCTION OF k and c (E6) and (E5) imply E(8) 3. Dynamic equilibrium system

80. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 80 (E3) and (E4) imply recall that there is NO population growth and therefore E(9)

81. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 81 SO WE HAVE OUR TWO EQUATIONS: and

82. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 82 THESE CAN BE BEST ANALYZED IN A PHASE DIAGRAM Start with capital accumulation equation FIRST: Find ISOCLINE, which are the (c, k) combinations such that INTERPRETATION: capital per worker does NOT grow IF the economy consumes all of the output net of capital depreciation. In this case, investment is just enough to cover the depreciation of capital. 2. Equilibrium growth and optimality

83. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 83 k c k-ISOCLINE: CAPITAL DOES NOT GROW k-ISOCLINE

84. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 84 k c k-ISOCLINE: CAPITAL DOES NOT GROW CHANGES IN k in PHASE DIAGRAM

85. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 85 Continue with the optimal consumption equation FIRST: Find ISOCLINE, which are the (c, k) combinations such that or

86. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 86 k c c-ISOCLINE: CONSUMPTION DOES NOT GROW k* is the k such that f’(k)=  c-ISOCLINE 0

87. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 87 k c c-ISOCLINE: CONSUMPTION DOES NOT GROW k* is the k such that f’(k)=  CHANGES IN c in PHASE DIAGRAM 0

88. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 88 k c c-ISOCLINE: CONSUMPTION DOES NOT GROW k* is the k such that f’(k)=  CHANGES IN c in PHASE DIAGRAM 0

89. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 89 k c k-ISOCLINE: CAPITAL DOES NOT GROW c-ISOCLINE: NO CONSUMPTION GROWTH k* PUTTING CHANGES in k and c TOGETHER 0

90. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 90 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* 0

91. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 91 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* 0

92. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 92 All these paths satisfy by construction: -period-by-period consumer maximization -capital market equilibrium They DO NOT necessarily satisfy constraints like: -non-negative capital stock k[t]>=0 -intertemporal budget constraint with EQUALITY

93. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 93 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) PATHS that violate NON-NEGATIVE capital stock (consume too much in beginning) 0

94. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 94 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH EQUALITY (consume too little in beginning) 0 k_bar

95. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 95 Q(t)=K(t) or q(t)=k(t) (1) Wealth=Capital (2) Intertemporal budget constraint with equality

96. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 96 k c f(k)-  k c-ISOCLINE: NO CONSUMPTION GROWTH k*k(0) PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH EQUALITY f’(k)-  =r=0 NEGATIVE INTEREST RATEPOSITIVE INTEREST k_bar

97. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 97 time t NEGATIVE INTEREST RATE

98. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 98 k c c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) PATHS THAT DO NO SATISFY BUDGET CONSTRAINT WITH EQUALITY 0 k_bar YOU ARE NOT SPENDING ALL YOUR PERMANENT INCOME!!!!!!!

99. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 99 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) EQUILIBRIUM ( “ SADDLE ” ) PATH 0

100. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 100 SADDLE PATH SATISFIES INTERTEMPORAL BUDGET CONSTRAINT WITH EQUALITY Capital market equilibrium: Income per worker=Labor income + Capital income: Hence:

101. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 101 Moreover: As: given that interest rates>0 for k<=k*

102. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 102 OPTIMALITY -- What would social planner do? - Social planner: dictator who decides allocation according to HH welfare subject to physical contraints

103. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 103 MRS=MRT The GLOBALLY OPTIMAL PATH MUST SATISFY If not satisfied, the planner could increase utility between adjacent periods by either: -- consuming one unit less today, investing that unit, and consuming the resulting additional output tomorrow -- consuming one unit more today, invest one unit less today, and reducing future consumption accordingly (A)

104. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 104 RESOURCE CONSTRAINT The GLOBALLY OPTIMAL PATH MUST SATISFY To see why, suppose first that -- in this case the planner must be throwing away goods (investment goods) because the increase in the number of machines is LESS THAN the machines built less depreciation : BUT THROWING AWAY GOODS CANNO BE OPTIMAL!! Now suppose instead -- now the planner is a REAL MAGICIAN!! as the number of machines in the economy goes up by which is GREATER THAN machines built less depreciation (B)

105. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 105 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* 0 ALL THE PATHS THAT SATISFY CONDITIONS (A) and (B)

106. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 106 NOW NOTE: -- Starting the allocation by jumping ABOVE the SADDLE PATH CANNOT BE OPTIMAL because you end up violating the non-negativity constraint for capital -- Starting the allocation by jumping BELOW the SADDLE PATH CANNOT BE OPTIMAL either. The proof is to construct another path—that is clearly not optimal either— but that still is BETTER THAN the paths starting out below the saddle path. How to do that is explained on the next slides.

107. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 107 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) We are trying to show that the RED PATH CANNOT BE GLOBALLY OPTIMAL 0

108. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 108 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) CONSIDER THE ALTERNATIVE GREEN PATH, which: -- concides with RED PATH until k* is reached and then JUMPS UP to the green dot where is stay forever 0

109. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 109 -- The GREEN PATH CANNOT POSSIBLY BE OPTIMAL because consumption JUMPS and therefore the green path violates CONSUMPTION SMOOTHING, which was CONDITION A above. -- Still, the GREEN PATH is certaintly better than the RED PATH because it has the same consumption until k* and MORE consumption from there onwards!!! -- For all RED PATHS (that is, all paths starting below the saddle path), there is a GREEN PATH. So no paths starting below the saddle path can be optimal (despite the fact that it satisfies conditions A and B).

110. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 110 HENCE: The only path starting at k[0] that : -- satisfies CONDITIONS A and B, which are necessary for optimality -- satisfies non-negativity of capital -- satisfies that there is NO OTHER PATH we can construct that is better  IS THE SADDLE PATH  EQUILIBRIUM AND OPTIMAL ALLOCATIONS ARE EQUAL

111. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 111 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) OPTIMAL AND EQUILIBRIUM ALLOCATION 0

112. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 112 In the steady state: Savings rate is constant, just like in the Solow model But it is endogenous in the sense of depending on “fundamentals” like time preference etc. In the simplest case: S=I=  K Combined with: r+  =MPK and r= 

113. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 113 In the steady state with technological change: With technological change and population growth: S=I=(  +a)K Growth of consumption=  (r-  )  What is the relationship between the SS savings rate S/Y and the rate of technological change a ?

114. BGSE Growth and Development, 2008-09 SLIDE SET 3Slide 114 Comparative statics Greater impatience (discount rate)? (effects on income, capital, wages, interest rates) Capital income taxation? A temporary cut of lump-sum taxes?