14.451 Lecture Notes Economic Growth

14.451 Lecture Notes
• Combining the above, we can rewrite the resource constraint as
and the time constraint as
with
3.1.3 The Ramsey Problem
ct + kt+1 ≤ F (kt,lt)+(1− δ)kt,
zt =1− lt,
ct ≥ 0, lt ∈ [0, 1], kt ≥ 0.
• The social planner chooses a plan {ct,lt,kt+1} ∞ t=0 so as to maximize utility subject to
the resource constraint of the economy, taking initial k0 as given:
3.1.4 Optimal Control
max U0 =
∞X
β t U(ct, 1 − lt)
t=0
ct + kt+1 ≤ (1 − δ)kt + F (kt,lt), ∀t ≥ 0,
ct ≥ 0, lt ∈ [0, 1], kt+1 ≥ 0., ∀t ≥ 0,
k0 > 0 given.
• Let µ t denote the Lagrange multiplier for the resource constraint. The Lagrangian of
the social planner’s problem is
L0 =
∞X
β t U(ct, 1 − lt)+
t=0
∞X
t=0
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µ t [(1 − δ)kt + F (kt,lt) − kt+1 − ct]