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 lime-ou'(c) = ∞ (slope of the utility function becomes vertical as consumption approaches zero). The individual's lifetime utility is give by u(c₁) + 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 c₁ 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 max {u(c₁) + Bu(c₂)} C1,C2,81 subject to the first period budget constraint, qS1 + C₁ = y₁ along with the second period budget constraint, C2 = y2 + $1.
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 lime-ou'(c) = ∞ (slope of the utility function becomes vertical as consumption approaches zero). The individual's lifetime utility is give by u(c₁) + 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 c₁ 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 max {u(c₁) + Bu(c₂)} C1,C2,81 subject to the first period budget constraint, qS1 + C₁ = y₁ along with the second period budget constraint, C2 = y2 + $1.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Using this functional form in the intertemporal trade-off condition that you previ-
ously derived, do some algebra to obtain an equation that specifies how much the
individual would save as a function of q, B, y1 and y2.
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)
approaches zero). The individual's lifetime utility is give by u(cı) + 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 c1 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
= 0 (slope of the utility function becomes vertical as consumption
max {u(ci) + Bи(с2)}
C1,C2,81
subject to the first period budget constraint,
qsi + C1 = yY1
along with the second period budget constraint,
C2 = Y2 + $1.
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