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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

Advanced Engineering Mathematics
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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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: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) =
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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