14.451 Lecture Notes Economic Growth

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• Define λt ≡ β t µ t and George-Marios Angeletos Ht ≡ H(kt,kt+1,ct,lt,λt) ≡ ≡ U(ct, 1 − lt)+λt [(1 − δ)kt + F (kt,lt) − kt+1 − ct] H is called the Hamiltonian of the problem. • We can rewrite the Lagrangian as ∞X L0 = = β t=0 t {U(ct, 1 − lt)+λt [(1 − δ)kt + F (kt,lt) − kt+1 − ct]} = ∞X t=0 or, in recursive form β t Ht Lt = Ht + βLt+1. • Given kt, ct and lt enter only the period t utility and resource constraint; (ct,lt) thus appears only in Ht. Similarly, kt,enter only the period t and t +1utility and resource constraints; they thus appear only in Ht and Ht+1. Therefore, Lemma 9 If {ct,lt,kt+1} ∞ t=0 is the optimum and {λt} ∞ t=0 the associated multipliers, then taking (kt,kt+1) as given, and kt+1 =argmax k 0 taking (kt,kt+2) as given. (ct,lt) =argmax c,l Ht z }| { H(kt,kt+1,c,l,λt) Ht + βHt+1 z }| { H(kt,k 0 ,ct,lt,λt)+βH(k 0 ,kt+2,ct+1,lt+1,λt+1) 46
Equivalently, taking (kt,kt+2) as given. 14.451 Lecture Notes (ct,lt,kt+1,ct+1,lt+1, ) = arg max c,l,k0 ,c0 ,l0[U(c, l)+βU(c0 ,l 0 )] s.t. c + k 0 ≤ (1 − δ)kt + F (kt,l) c 0 + kt+2 ≤ (1 − δ)k 0 + F (k 0 ,l 0 ) • We henceforth assume an interior solution. As long as kt > 0, interior solution is indeed ensured by the Inada conditions on F and U. • The FOC with respect to ct gives The FOC with respect to lt gives ∂L0 ∂ct = β t ∂Ht ∂ct =0⇔ ∂Ht ∂ct =0⇔ Uc(ct,zt) =λt ∂L0 ∂lt t ∂Ht = β ∂lt =0⇔ ∂Ht ∂lt =0⇔ Uz(ct,zt) =λtFL(kt,lt) Finally, the FOC with respect to kt+1 gives ∂L0 = β ∂kt+1 t · ∂Ht + β ∂kt+1 ∂Ht+1 ¸ =0⇔ ∂kt+1 −λt + β ∂Ht+1 =0⇔ ∂kt+1 λt = β [1 − δ + FK(kt+1,lt+1)] λt+1 47
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14.451 Lecture Notes Economic Growt
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Contents Preface ix 1 Introduction
Page 5 and 6: 14.451 Lecture Notes 3.1.4 OptimalC
Page 7 and 8: 14.451 Lecture Notes 5.2 SomeExampl
Page 9 and 10: Preface This is the set of lecture
Page 11 and 12: Chapter 1 IntroductionandGrowthFact
Page 13 and 14: 14.451 Lecture Notes 1.2 The World
Page 15 and 16: 14.451 Lecture Notes • Figure 4 3
Page 17 and 18: 14.451 Lecture Notes 9. Openness: I
Page 19 and 20: Chapter 2 The Solow Growth Model (a
Page 21 and 22: 14.451 Lecture Notes • We say tha
Page 23 and 24: 14.451 Lecture Notes 2.1.3 The Reso
Page 25 and 26: whereas consumption is given by 14.
Page 27 and 28: 14.451 Lecture Notes where the latt
Page 29 and 30: 14.451 Lecture Notes 2.2 Decentrali
Page 31 and 32: 14.451 Lecture Notes • Let K m t
Page 33 and 34: Equivalently, 14.451 Lecture Notes
Page 35 and 36: 14.451 Lecture Notes where Kt ≡ R
Page 37 and 38: 14.451 Lecture Notes Proof. Follows
Page 39 and 40: 14.451 Lecture Notes • Combining
Page 41 and 42: 14.451 Lecture Notes We would expec
Page 43 and 44: • Straightforward algebra gives W
Page 45 and 46: 14.451 Lecture Notes • Suppose th
Page 47 and 48: 2.6 Miscellaneous 14.451 Lecture No
Page 49 and 50: 14.451 Lecture Notes 2.6.3 Introduc
Page 51 and 52: Chapter 3 The Neoclassical Growth M
Page 53 and 54: 14.451 Lecture Notes ut+1 : There i
Page 55: 14.451 Lecture Notes • Combining
Page 59 and 60: equivalently 14.451 Lecture Notes
Page 61 and 62: 14.451 Lecture Notes • Let B be t
Page 63 and 64: 14.451 Lecture Notes 3.2 Decentrali
Page 65 and 66: 14.451 Lecture Notes only the total
Page 67 and 68: 14.451 Lecture Notes • Finally, i
Page 69 and 70: 14.451 Lecture Notes The constraint
Page 71 and 72: 14.451 Lecture Notes where f(k) ≡
Page 73 and 74: 14.451 Lecture Notes Proposition 15
Page 75 and 76: 14.451 Lecture Notes individual’s
Page 77 and 78: 14.451 Lecture Notes 3.3 Steady Sta
Page 79 and 80: 3.3.2 Transitional Dynamics 14.451
Page 81 and 82: 14.451 Lecture Notes Proof. The pol
Page 83 and 84: 14.451 Lecture Notes • The ˙c =0
Page 85 and 86: 14.451 Lecture Notes 3.5 Comparativ
Page 87 and 88: 14.451 Lecture Notes • We conclud
Page 89 and 90: where At denotes TFP. Note that 14.
Page 91 and 92: 14.451 Lecture Notes • Note that
Page 93 and 94: 14.451 Lecture Notes 3.6.2 Impulse
Page 95 and 96: Chapter 4 Applications 4.1 Arrow-De
Page 97 and 98: 14.451 Lecture Notes bles. We can t
Page 99 and 100: 14.451 Lecture Notes • Then, the
Page 101 and 102: 14.451 Lecture Notes • More gener
Page 103 and 104: 14.451 Lecture Notes • Consider n
Page 105 and 106: 14.451 Lecture Notes aggregate path
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14.451 Lecture Notes notes to be co
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Chapter 5 Overlapping Generations M
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14.451 Lecture Notes Life-cycle con
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Note that Recall that ∂m ∂R 14.
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where the scalar ζ>0 is given by 1
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14.451 Lecture Notes • Finally, n
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Chapter 6 Endogenous Growth I: AK,
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14.451 Lecture Notes so that consum
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14.451 Lecture Notes • Finally, t
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14.451 Lecture Notes • Combining
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14.451 Lecture Notes Like in the si
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14.451 Lecture Notes Combining the
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14.451 Lecture Notes • Households
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14.451 Lecture Notes Proposition 25
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Chapter 7 Endogenous Growth II: R&D
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14.451 Lecture Notes More generally
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14.451 Lecture Notes you can think
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14.451 Lecture Notes Free entry in
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14.451 Lecture Notes 7.1.8 Pareto A
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14.451 Lecture Notes 7.1.9 Introduc
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7.2.2 The Value of Innovation 14.45
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14.451 Lecture Notes It follows tha
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14.451 Lecture Notes • Compare no
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