According to an economic model, the budget b(t) at time t ≥ 0 in a household is chosen to maximise the lifetime utility U [b] = for d dt e-tu(c(t)), where u(c) ≥ 0 is the household utility function, ß> 0 is the discount rate, a constant, and c(t) is the household consumption satisfying the budgetary equation b'(t) = yb(t) + wc(t), b(0) = bo. In this equation, y> 0 is the bank interest rate, w> 0 is the household wage (both assumed constant in this model), and bo > 0 is the initial household budget b(0). Note that b(t) may be negative if the household is in debt, but c(t) > 0. The initial household consumption is c(0) = co > 0. The budget b(t) is subject to a No-Ponzi condition lim et b(t) 20 t-→∞ (which prevents the household financing current consumption through indefinitely borrowing and rolling over debt). Throughout this question you may assume that the usual theory of the calculus of variations is valid for this model on the infinite time interval t = [0, ∞).
According to an economic model, the budget b(t) at time t ≥ 0 in a household is chosen to maximise the lifetime utility U [b] = for d dt e-tu(c(t)), where u(c) ≥ 0 is the household utility function, ß> 0 is the discount rate, a constant, and c(t) is the household consumption satisfying the budgetary equation b'(t) = yb(t) + wc(t), b(0) = bo. In this equation, y> 0 is the bank interest rate, w> 0 is the household wage (both assumed constant in this model), and bo > 0 is the initial household budget b(0). Note that b(t) may be negative if the household is in debt, but c(t) > 0. The initial household consumption is c(0) = co > 0. The budget b(t) is subject to a No-Ponzi condition lim et b(t) 20 t-→∞ (which prevents the household financing current consumption through indefinitely borrowing and rolling over debt). Throughout this question you may assume that the usual theory of the calculus of variations is valid for this model on the infinite time interval t = [0, ∞).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:According
household
to an economic model, the budget b(t) at time t ≥ 0 in a
is chosen to maximise the lifetime utility
UM = [ die
₁
dt e-ßt u(c(t)),
where u(c) ≥ 0 is the household utility function, ß> 0 is the discount rate, a
constant, and c(t) is the household consumption satisfying the budgetary
equation
b'(t) = yb(t) + wc(t), b(0) = bo.
In this equation, y> 0 is the bank interest rate, w> 0 is the household wage
(both assumed constant in this model), and bo > 0 is the initial household
budget b(0). Note that b(t) may be negative if the household is in debt, but
c(t) > 0. The initial household consumption is c(0) = co > 0.
The budget b(t) is subject to a No-Ponzi condition
lim et b(t) ≥ 0
t-→∞
(which prevents the household financing current consumption through
indefinitely borrowing and rolling over debt).
Throughout this question you may assume that the usual theory of the
calculus of variations is valid for this model on the infinite time interval
t = [0, ∞).
Transcribed Image Text:(c) Hence show that the budget on the stationary path is given by
-) ert.
ekt
W
b(t) = | bo+ ~ +
Y
K
CO
W
Y
CO
K-Y
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