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(c1) + Bu(c2). In the first period of life, the individual has y1 units of income that can be either consumed or saved 0 (slope of the utility function becomes vertical as consumption In order to save the individual must purchase bonds at a price of

Microeconomic Theory
12th Edition
ISBN:9781337517942
Author:NICHOLSON
Publisher:NICHOLSON
Chapter17: Capital And Time
Section: Chapter Questions
Problem 17.1P
icon
Related questions
Question

Using the first-order conditions of this problem with respect to c1, c2 and s1, (i.e. the partial derivatives that have been set equal to zero) construct the optimal intertemporal consumption trade-off condition between c1 and c2. This trade-off is executed by variation in savings.

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(ci)+ Bu(c2).
In the first period of life, the individual has yı 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
= (slope of the utility function becomes vertical as consumption
max {u(c) + 8и(с2)}
C1,C2,81
subject to the first period budget constraint,
qs1+ C1 = y1
along with the second period budget constraint,
C2 = Y2 + S1.
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(ci)+ Bu(c2). In the first period of life, the individual has yı 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 = (slope of the utility function becomes vertical as consumption max {u(c) + 8и(с2)} C1,C2,81 subject to the first period budget constraint, qs1+ C1 = y1 along with the second period budget constraint, C2 = Y2 + S1.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
National Savings
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Microeconomic Theory
Microeconomic Theory
Economics
ISBN:
9781337517942
Author:
NICHOLSON
Publisher:
Cengage