1. Use the Method of Lagrange to solve this problem. To do so, construct the La- grangean function for this problem. Use λ1 as the Lagrange multiplier attached to the period 1 budget constraint and λ2 as the Lagrange multiplier attached to the period 2 budget constraint. Consider the two period consumption savings problem faced by an individual whoseutility is defined on period consumption. This utility function u(c) has the propertiesthat it is strictly increasing and concave, u'(c) > 0, u"(c) < 0 (where u'(c) denotesthe first derivative while u"(c) represents the second derivative) and satisfies the Inadacondition 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 eitherconsumed or saved. In order to save, the individual must purchase bonds at a price ofq units of the consumption good per bond. Each of these bonds returns a single unit ofthe consumption good in period 2. Total savings through bond purchases is s1 so thattotal expenditures on purchasing bonds is qs1. Let c1 denote the amount of consumptionin period 1 chosen by the individual. In the second period of life, consumption in theamount 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 thebudget constraints of periods 1 and 2 by choice of (c1, C2, s1). Formally, the individualsolves the problem= (slope of the utility function becomes vertical as consumptionmax {u(c) + 8и(с2)}C1,C2,81subject to the first period budget constraint,qs1+ C1 = y1along with the second period budget constraint,C2 = Y2 + S1.

The saving function shows the positive relationship between the saving rate and the output of…

1. Use the Method of Lagrange to solve this problem. To do so, construct the La- grangean function for this problem. Use λ1 as the Lagrange multiplier attached to the period 1 budget constraint and λ2 as the Lagrange multiplier attached to the period 2 budget constraint.

1. Use the Method of Lagrange to solve this problem. To do so, construct the La- grangean function for this problem. Use λ1 as the Lagrange multiplier attached to the period 1 budget constraint and λ2 as the Lagrange multiplier attached to the period 2 budget constraint.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter17: Capital And Time

Section: Chapter Questions

Problem 17.1P

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