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

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter17: Capital And Time

Section: Chapter Questions

Problem 17.3P

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When analyzing how borrowing and lending affect the consumer's budget constraint, we measure spending in the current time period on the horizontal axis and spending in the future time period on the vertical axis. Assume that the interest rate at which the consumer can lend and borrow is 10%, income in period 1 is $1000 and income in period 2 is $1200. The point of maximum current consumption can be expressed as 1000+ 1200/1.1. 1000(1.1) + 1200. 1000+ 1200 + .1 1000/1.1 + 1200/1.1 + 1.
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2. Mr. A has the following utility function and budget constraints: Max 0.1Ln(C1) + 0.7Ln(C2) Subject to S1 + C1 = 100 C2 + S2 = (1 + r)S1 where C1 and C2 are consumption level at young and that at old respectively. Likewise, S1 and S2 are saving at young and saving at old respectively. a) Find out Mr. A’s optimal consumption levels (i.e. C1*, C2*) and optimal savings (i.e. S1*, S2*) in terms of interest rate r. b) Show clearly the results in part a) in a suitable diagram (with C1 as x-axis and C2 as y-axis). c) Is Mr. A a saver ? or a borrower ? d) If r is equal to 0 (i.e. saving gives no returns), will Mr. A still choose to save when he is young (i.e. is S1 still bigger than 0) ? Why ? e) Suppose that Mr. A is not allowed to save (i.e. S1 = 0). What are his optimal consumption levels ? Show his optimal consumption levels in the same diagram you prepare for part a) (with a suitable indifference curve). f) If r increases,…
Analysing Utility Function and Household Optimization Consider a household with the following utility function representing their preferences over consumption: with U = u(C) + Bu(C++1) u(C) = exp(-aC), BE (0,1), a>0 where Ct and Ct+1 represent consumption in the current and future periods, respectively. The household faces a two-period decision problem. They receive endowments of Yt and Yt+1 in the current and future periods, respectively. The real interest rate is denoted by rt. Notice: The utility function u(C) takes on negative values for all positive consumption levels. However, in economic models, the absolute value of utility is less important than how utility changes with consumption. A higher level of utility represents a more preferred outcome for the household. Solving for Current Consumption Demand Function Solve for the household's demand function for current consumption (Ct). Express Ct as a function of Yt, Yt+1, rt, and the parameters ẞ and a. Discuss what happens to Ct…
Malaysians are at their worst financial position in 2022 compared with the last five years, and thelooming recession may push them to the brink with dwindling savings coupled with theaftermath of the Covid-19 pandemic. A recent survey by Malaysian financial services websiteRinggitPlus revealed that 70 per cent of Malaysians saved less than RM500 per month (S$153)in 2022 or did not save at all. “This is compared with 52 per cent of Malaysians saving less thanRM500 monthly in 2021, the largest year-on-year increase since 2018,” the RinggitPlusMalaysian Financial Literacy Survey 2022 showed....Mr Patrick Tay, deals partner, economicsand policy at Malaysia PwC, is not surprised by the decline in savings in 2022. “One reason forthe depleted savings is low wages and higher inflation. But another key reason is that Malaysiansare spending more post-pandemic on holidays or other leisure expenses because some may feelthat they saved during the pandemic, and it is fine to now spend this year…
Patience has a utility function 1/2 1/2 U(cl, c2) = c? + 0.80c/? where c is her consumption in period 1 and c is her consumption in period 2. Her income in period 1 is 3 times as large as her income in period 2. At what interest rate will she choose to consume the same amount in period 1 as in period 2? a. 2 b. 0.13 c. 0.25 d. 0 е. О.38 20% 13% 25% 0%
[**] Simon has current wealth of $36, including $20 in cash. With probabil- ity Simon's money will be stolen (contingency 1), leaving him with only $16 to spend on consumption, and with probability his cash will not be stolen (contingency 2) so he can spend the entire $36 on consumption. Simon has utility function u (0₁.0₂) = √6 + 2√ where c is consumption spending in contingency i. Show Transcribed Text 1. What is the expected value of Simons contingent consumption? 2. What is the expected utility of Simons contingent consumption? 3. What is Simons MRS at his current contingent consumption bundle 4. An insurance company offers to fully insure Simon against the risk of theft for a premium (price) of $P. That is, if Simon pays $P to the insurance company, then the insurer will replace Simons $20 in contingency 1. If Simon buys the insurance his contingent consumption bundle is therefore (36 P; 36 P). i. What is Simons MRS at the fully insured contingent consumption bundle? ii. At what…
Assume that someone has inherited 2,000 bottles of wine from a rich uncle. He or she intends to drink these bottles over the next 40 years. Suppose that this person’s utility function for wine is given by u(c(t)) = (c(t))0.5, where c(t) is each instant t consumption of bottles. Assume also this person discounts future consumption at the rate δ = 0.05. Hence this person’s goal is to maximize 0ʃ40 e–0.05tu(c(t))dt = 0ʃ40 e–0.05t(c(t))0.5dt. Let x(t) represent the number of bottle of wine remaining at time t, constrained by x(0) = 2,000, x(40) = 0 and dx(t)/dt = – c(t): the stock of remaining bottles at each instant t is decreased by the consumption of bottles at instant t. The current value Hamiltonian expression yields: H = e–0.05t(c(t))0.5 + λ(– c(t)) + x(t)(dλ/dt). This person’s wine consumption decreases at a continuous rate of ??? percent per year. The number of bottles being consumed in the 30th year is approximately ???