14.451 Lecture Notes Economic Growth
14.451 Lecture Notes Economic Growth
14.451 Lecture Notes Economic Growth
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• Define λt ≡ β t µ t and<br />
George-Marios Angeletos<br />
Ht ≡ H(kt,kt+1,ct,lt,λt) ≡<br />
≡ U(ct, 1 − lt)+λt [(1 − δ)kt + F (kt,lt) − kt+1 − ct]<br />
H is called the Hamiltonian of the problem.<br />
• We can rewrite the Lagrangian as<br />
∞X<br />
L0 =<br />
=<br />
β<br />
t=0<br />
t {U(ct, 1 − lt)+λt [(1 − δ)kt + F (kt,lt) − kt+1 − ct]} =<br />
∞X<br />
t=0<br />
or, in recursive form<br />
β t Ht<br />
Lt = Ht + βLt+1.<br />
• Given kt, ct and lt enter only the period t utility and resource constraint; (ct,lt) thus<br />
appears only in Ht. Similarly, kt,enter only the period t and t +1utility and resource<br />
constraints; they thus appear only in Ht and Ht+1. Therefore,<br />
Lemma 9 If {ct,lt,kt+1} ∞ t=0 is the optimum and {λt} ∞ t=0 the associated multipliers, then<br />
taking (kt,kt+1) as given, and<br />
kt+1 =argmax<br />
k 0<br />
taking (kt,kt+2) as given.<br />
(ct,lt) =argmax<br />
c,l<br />
Ht<br />
z }| {<br />
H(kt,kt+1,c,l,λt)<br />
Ht + βHt+1<br />
z }| {<br />
H(kt,k 0 ,ct,lt,λt)+βH(k 0 ,kt+2,ct+1,lt+1,λt+1)<br />
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