1. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 1 MACROECONOMICS I UPF 2008-2009 LECTURE SLIDES SET 3 Professor Antonio Ciccone

2. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 2 II. ECONOMIC GROWTH WITH ENDOGENOUS SAVINGS

3. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 3 1. Household savings behavior

4. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 8 time HOUSEHOLD INCOME OF FARMER FIGURE 1: CONSUMPTION SMOOTHING: A VOLATILE INCOME PATH

9. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 9 time HOUSEHOLD INCOME OF FARMER HOUSEHOLD CONSUMPTION OF FARMER (“KEYNESIAN” theory) FIGURE 2: INCOME AND "KEYNESIAN CONSUMPTION"

10. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 10 time HOUSEHOLD INCOME OF FARMER HOUSEHOLD CONSUMPTION OF FARMER (EMPIRICAL OBSERVATION) FIGURE 3: CONSUMPTION SMOOTHING

11. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 19 C[0] C[1] Lw[0] Lw[1] GRAPHICALLY: INCOME LEVELS AND CONSUMTION

20. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 20 C[0] C[1] Lw[0] Lw[1] 1+r THE INTERTEMPORAL BUDGET CONSTRAINT

21. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 21 C[0] C[1] Lw[0] Lw[1] 1+r INTERTEMPORAL UTILITY MAXIMIZATION

22. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 22 C[0] C[1] Lw[0] Lw[1] 1+r C[0] C[1]

23. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 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. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 33 C[0] C[1] Lw[0] Lw[1] 1+r C[0] C[1] INCREASE IN INTEREST RATE HIGH INTEREST RATE LOW INTEREST RATE

34. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 34 AN EXAMPLE Take the following utility function: with FIRST-ORDER CONDITION BECOMES or

35. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 35 3. The case of 3 and more periods -- Timing -- Intertemporal budget constraint -- Optimality conditions -- Time consistency

36. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 36 TIMING t=0t=1 Q[0] w[0]Lw[1]Lw[2]L C[1] C[2] INITIAL WEALTH t=2 - interest r[0] - utility discount - interest r[1] - utility discount C[0] YOU ARE HERE

37. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 37 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

38. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 38 INTERTEMPORAL BUDGET CONSTRAINT

39. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 39 BUDGET CONTRAINT AND TIME EVOLUTION OF WEALTH t=0t=1t=2 Q[0]w[0]Lw[1]Lw[2]L C[1]C[2] C[3] C[0]

40. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 40 INTERTEMPORAL BUDGET CONSTRAINT

41. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 41 THE “PRESENT-VALUE BUDGET SURPLUS” = PERMANENT INCOME minus PRESENT VALUE CONSUMPTION = PRESENT VALUE OF END-OF-LIFE WEALTH

42. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 42 MAXIMIZE BETWEEN ADJACENT PERIODS OPTIMAL SOLUTION OF CONSUMPTION PROBLEM plus BUDGET CONSTRAINT WITH EQUALITY

43. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 43 INFINITE HORIZON =TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT (end of) PERIOD t

44. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 44 INTERTEMPORAL BUDGET CONSTRAINT NO-PONZI-GAME condition

45. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 45 TIME T  0 WHAT IF:

46. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 46 CAN INCREASE TIME-0 CONSUMPTION  CONSUMPTION PLAN NOT OPTIMAL! NECESSARY FOR OPTIMALITY:

47. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 47 TIME CONSISTENCY of HOUSOLD CONSUMPTION PLANS

48. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 48 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)

49. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 49 ***** 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)

50. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 50 3. Optimal consumption and savings in continuous time 1. Infinite horizon subject to = TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT TIME t

51. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 51 2. Intertemporal budget constraint Wealth in discrete time Wealth in continuous time

52. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 52 Intertemporal budget constraint in continuous time satisfied with equality if

53. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 53 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:

54. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 54 2) Hence the utility discount per unit of time is: 3) What is the limit as t  0? Hopital ’ s rule yields

55. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 55 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 ” )

56. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 56 TIME OPTIMAL CONSUMPTION PATH r =  C(t) C(0) CONSTANT CONSUMPTION IN TIME

57. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 57 TIME OPTIMAL CONSUMPTION PATH r >  C(t) C(0) INCREASING CONSUMPTION IN TIME

58. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 58 TIME OPTIMAL CONSUMPTION PATH r <  C(t) C(0) DEACREASING CONSUMPTION IN TIME

59. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 59 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)

60. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 60 THE INTERTEMPORAL BUDGET CONSTRAINT without initial wealth HENCE

61. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 61 6. Deriving the continuous time first-order condition MAXIMIZATION BETWEEN ANY TWO PERIODS SEPARATED BY TIME x subject to = TOTAL SPENDING IN TWO PERIODS

62. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 62 Take the following utility function: with

63. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 63 FIRST ORDER CONDITIONS FOR THE TWO PERIODS IN TIME making use of the utility function

64. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 64 REWRITING THIS CONDITIONS YIELDS subtracting 1 from both sides

65. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 65 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)?

66. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 66 Apply Hopital ’ s rule

67. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 67 HENCE as two periods become VERY CLOSE WHICH IS WHAT WE WANTED TO SHOW

68. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 68 SUMMARIZING QUESTION: What characterizes the optimal consumption PATH that solves subject to

69. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 69 ANSWER: and or

70. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 70 2. The Ramsey-Cass-Koopmans model

71. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 71 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

72. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 72 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)

73. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 73 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

74. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 74 CAPITAL MARKET EQUILIBRIUM E(4) E(5)

75. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 75 WHAT WE CANNOT KEEP IS INSTEAD: E(6) E(7) INTERTEMPORAL BUDGET CONSTRAINT where c[t] is CONSUMPTION per PERSON 2. Household behaviour

76. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 76 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

77. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 77 (E3) and (E4) imply recall that there is NO population growth and therefore E(9)

78. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 78 SO WE HAVE OUR TWO EQUATIONS: and

79. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 79 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

80. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 80 k c k-ISOCLINE: CAPITAL DOES NOT GROW k-ISOCLINE

81. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 81 k c k-ISOCLINE: CAPITAL DOES NOT GROW CHANGES IN k in PHASE DIAGRAM

82. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 82 Continue with the optimal consumption equation FIRST: Find ISOCLINE, which are the (c, k) combinations such that or

83. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 83 k c c-ISOCLINE: CONSUMPTION DOES NOT GROW k* is the k such that f’(k)=  c-ISOCLINE 0

84. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 84 k c c-ISOCLINE: CONSUMPTION DOES NOT GROW k* is the k such that f’(k)=  CHANGES IN c in PHASE DIAGRAM 0

85. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 85 k c c-ISOCLINE: CONSUMPTION DOES NOT GROW k* is the k such that f’(k)=  CHANGES IN c in PHASE DIAGRAM 0

86. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 86 k c k-ISOCLINE: CAPITAL DOES NOT GROW c-ISOCLINE: NO CONSUMPTION GROWTH k* PUTTING CHANGES in k and c TOGETHER 0

87. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 87 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* 0

88. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 88 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* 0

89. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 89 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

90. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 90 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

91. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 91 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

92. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 92 Q(t)=K(t) or q(t)=k(t) (1) Wealth=Capital (2) Intertemporal budget constraint with equality

93. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 93 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

94. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 94 time t NEGATIVE INTEREST RATE

95. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 95 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!!!!!!!

96. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 96 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) EQUILIBRIUM ( “ SADDLE ” ) PATH 0

97. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 97 SADDLE PATH SATISFIES INTERTEMPORAL BUDGET CONSTRAINT Capital market equilibrium: Income per worker=Labor income + Capital income: Hence:

98. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 98 Moreover: As: given that interest rates>0 for k<=k*

99. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 99 OPTIMALITY -- What would social planner do? - Social planner: dictator who decides allocation according to HH welfare subject to physical contraints

100. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 100 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)

101. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 101 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)

102. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 102 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* 0 ALL THE PATHS THAT SATISFY CONDITIONS (A) and (B)

103. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 103 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.

104. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 104 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

105. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 105 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

106. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 106 -- 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).

107. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 107 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

108. UPF, Macroeconomics I, 2008-09 SLIDE SET 3Slide 108 k c k-ISOCLINE: NO CAPITAL GROWTH c-ISOCLINE: NO CONSUMPTION GROWTH k* k(0) OPTIMAL AND EQUILIBRIUM ALLOCATION 0