Complete utility maxamization problem using lagrangian method, ignore other question parts. Optimization with infinite horizonProblem 5.1. Consider a consumer who lives forever. She starts out with zero wealth ao = 0,receives an income stream yť and chooses a level of consumption c and savings at+1 every period.Her maximization problem is given by:subject toU = Σ B¹u(ct)t=0Ct+at+1 = Yt + (1 + r)at.• Set up the Lagrangean for this problem● Derive a relationship between u'(c+) and u'(ct+1)• Write the budget constrain in period t+1 and solve for at+1 as a function of Ct+1, Yt+1, at+2, r.Repeat the exercise for at+2 and plug both the expressions into the original budget constrain.• Use the pattern that emerges to re-write the budget constraint as a life-time budget constraint.This involves the (discounted) sum of income and consumption, but no assets.• Suppose u(c) = log(c). Use the FOC for consumption before to find a relationship betweenCt and Ct+1.• Combine the lifetime budget constraint with the first-order condition for consumption to solvefor consumption in period t as a function of income only

*As the question demands only the utility maximization problem we are solving only that , and…

5 Optimization with infinite horizon Problem 5.1. Consider a consumer who lives forever. She starts out with zero wealth ao = 0, receives an income stream yt and chooses a level of consumption ct and savings at+1 every period. Her maximization problem is given by: ∞ U = Σβtu(c) • t=0

5 Optimization with infinite horizon Problem 5.1. Consider a consumer who lives forever. She starts out with zero wealth ao = 0, receives an income stream yt and chooses a level of consumption ct and savings at+1 every period. Her maximization problem is given by: ∞ U = Σβtu(c) • t=0

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Chapter1: Making Economics Decisions

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Question

Complete utility maxamization problem using lagrangian method, ignore other question parts.

Optimization with infinite horizon
Problem 5.1. Consider a consumer who lives forever. She starts out with zero wealth ao = 0,
receives an income stream yť and chooses a level of consumption c and savings at+1 every period.
Her maximization problem is given by:
subject to
U = Σ B¹u(ct)
t=0
Ct+at+1 = Yt + (1 + r)at.
• Set up the Lagrangean for this problem
● Derive a relationship between u'(c+) and u'(ct+1)
• Write the budget constrain in period t+1 and solve for at+1 as a function of Ct+1, Yt+1, at+2, r.
Repeat the exercise for at+2 and plug both the expressions into the original budget constrain.
• Use the pattern that emerges to re-write the budget constraint as a life-time budget constraint.
This involves the (discounted) sum of income and consumption, but no assets.
• Suppose u(c) = log(c). Use the FOC for consumption before to find a relationship between
Ct and Ct+1.
• Combine the lifetime budget constraint with the first-order condition for consumption to solve
for consumption in period t as a function of income only

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