fi = rn(n)n + с — F (Z)
(3.3c)
and the No-Ponzi-Game condition limt→∞ ne-rnt ≥ O, together with the initial stocks of
durable consumption and net debt: a(O) = no > O, n(O) = no > O. The current-value
Hamiltonian for this problem corresponds to
where μ now corresponds to the shadow value of international assets. As in the closed-
economy framework, consumers make their choices taking the aggregate levels of durable
goods and their stocks as given. Furthermore, agents make their consumption∕savings
decision holding the interest rate on bonds rn(n) constant. This implies that the Hrst
order conditions are equal to the following expressions
Cc + a Y
c + a's(cm )]
+ V(Z) + φ (c — δa) + μ [rn(n)n + c — F(Z)]
(3.4)
и и , .1 U4 [c + a, s (г)] s' (1) , , _ .
uc [c + a,s (г)] + =μ - φ, (3.5a)
O + Ji
V '(Z) = —μF '(Z) (3.5b)
φ = (β + Φ)φ — Uc [c + α,s (г)] - U [c + "'ɪ(/)] s'(1), (3.5c)
O + √/
μ = μ [β — rn(n)] (3.5d)
The optimality conditions (3.5a)-(3.5c) for the open economy have an interpretation simi-
lar to their counterparts (2.1Oa)-(2.1Oc) for the closed economy. The exception is equation
(3.5d), which describes the optimal path of the shadow value μ if the stock of net debt
is chosen optimally. Our speciHcation of preferences in equations (2.3a)-(2.3b) guaran-
tees that the Hamiltonian (3.4) is jointly concave in the control variables c and Z and
the state variables a and n. This implies that if the limiting transversality conditions
limt→∞ aφe-∣βt = limt→∞ nμe-βt = O hold, then necessary conditions (3.5a)-(3.5d) are
17
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