Use the budgetary equation to express the functional U[b] in terms of b and b', then calculate the Euler-Lagrange equation, and show that it may be written as a differential equation in c(t): u" (c(t)) c'(t) = (3-y) u'(c(t)). Alitu Gu
Use the budgetary equation to express the functional U[b] in terms of b and b', then calculate the Euler-Lagrange equation, and show that it may be written as a differential equation in c(t): u" (c(t)) c'(t) = (3-y) u'(c(t)). Alitu Gu
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.EA: Extended Application Managing Renewable Resources
Problem 2EA
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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, ∞).
![(
Use the budgetary equation to express the functional U[b] in terms of b
and b', then calculate the Euler-Lagrange equation, and show that it
may be written as a differential equation in c(t):
u" (c(t)) c' (t) = (B - y) u' (c(t)).
For the remainder of this question, the utility function u(c) is given, for
c> 0, by
c¹-0
1-0¹
where 0 <0 <1 is a constant and 3 (1 - 0)y.
u(c)
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F6c285032-8a53-4812-b342-c0c0d41973c6%2Fjzowzy8_processed.png&w=3840&q=75)
Transcribed Image Text:(
Use the budgetary equation to express the functional U[b] in terms of b
and b', then calculate the Euler-Lagrange equation, and show that it
may be written as a differential equation in c(t):
u" (c(t)) c' (t) = (B - y) u' (c(t)).
For the remainder of this question, the utility function u(c) is given, for
c> 0, by
c¹-0
1-0¹
where 0 <0 <1 is a constant and 3 (1 - 0)y.
u(c)
=
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