Consider the two period consumption savings problem faced by an individual whose utility is defined on period consumption. This utility function u(c) has the properties that it is strictly increasing and concave, u'(c) > 0, u"(c) < 0 (where u'(c) denotes the first derivative while u"(c) represents the second derivative) and satisfies the Inada condition limc→0 = (slope of the utility function becomes vertical as consumption approaches zero). The individual's lifetime utility is give by u(ci)+ Bu(c2). In the first period of life, the individual has y1 units of income that can be either consumed or saved. In order to save, the individual must purchase bonds at a price of q units of the consumption good per bond. Each of these bonds returns a single unit of the consumption good in period 2. Total savings through bond purchases is s1 so that total expenditures on purchasing bonds is qs1. Let c denote the amount of consumption in period 1 chosen by the individual. In the second period of life, consumption in the amount c2 is financed out of the returns from savings and period 2 income, Y2. The problem of the individual is to maximize lifetime utility while respecting the budget constraints of periods 1 and 2 by choice of (c1, c2, 81). Formally, the individual solves the problem max {u(с) + Вu(с2)} C1,C2,81 subject to the first period budget constraint, qsi + C1 = Y1 along with the second period budget constraint, C2 = Y2 + $1. Answer the following: 1. Use the Method of Lagrange to solve this problem. To do so, construct the La-
Consider the two period consumption savings problem faced by an individual whose utility is defined on period consumption. This utility function u(c) has the properties that it is strictly increasing and concave, u'(c) > 0, u"(c) < 0 (where u'(c) denotes the first derivative while u"(c) represents the second derivative) and satisfies the Inada condition limc→0 = (slope of the utility function becomes vertical as consumption approaches zero). The individual's lifetime utility is give by u(ci)+ Bu(c2). In the first period of life, the individual has y1 units of income that can be either consumed or saved. In order to save, the individual must purchase bonds at a price of q units of the consumption good per bond. Each of these bonds returns a single unit of the consumption good in period 2. Total savings through bond purchases is s1 so that total expenditures on purchasing bonds is qs1. Let c denote the amount of consumption in period 1 chosen by the individual. In the second period of life, consumption in the amount c2 is financed out of the returns from savings and period 2 income, Y2. The problem of the individual is to maximize lifetime utility while respecting the budget constraints of periods 1 and 2 by choice of (c1, c2, 81). Formally, the individual solves the problem max {u(с) + Вu(с2)} C1,C2,81 subject to the first period budget constraint, qsi + C1 = Y1 along with the second period budget constraint, C2 = Y2 + $1. Answer the following: 1. Use the Method of Lagrange to solve this problem. To do so, construct the La-
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:Consider the two period consumption savings problem faced by an individual whose
utility is defined on period consumption. This utility function u(c) has the properties
that it is strictly increasing and concave, u'(c) > 0, u"(c) < 0 (where u'(c) denotes
the first derivative while u"(c) represents the second derivative) and satisfies the Inada
condition lim.→0 u'(c) = ∞ (slope of the utility function becomes vertical as consumption
approaches zero). The individual's lifetime utility is give by u(c)+ Bu(c2).
In the first period of life, the individual has y1 units of income that can be either
consumed or saved. In order to save, the individual must purchase bonds at a price of
q units of the consumption good per bond. Each of these bonds returns a single unit of
the consumption good in period 2. Total savings through bond purchases is sı so that
total expenditures on purchasing bonds is qsı. Let c̟ denote the amount of consumption
in period 1 chosen by the individual. In the second period of life, consumption in the
amount c2 is financed out of the returns from savings and period 2 income, Y2.
The problem of the individual is to maximize lifetime utility while respecting the
budget constraints of periods 1 and 2 by choice of (c1, C2, s1). Formally, the individual
solves the problem
со
max {u(c) + Bи(с2)}
C1,C2,81
subject to the first period budget constraint,
q$1 + C1 = Y1
along with the second period budget constraint,
C2 = Y2 + S1.
Answer the following:
1. Use the Method of Lagrange to solve this problem. To do so, construct the La-
grangean function for this problem. Use A1 as the Lagrange multiplier attached to
the period 1 budget constraint and A2 as the Lagrange multiplier attached to the
period 2 budget constraint.
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