Each individual's lifetime budget constraint is given by C1 + C2/(1 + r) = Individuals choose consumption in each period by maximizing lifetime ut this lifetime budget constraint. (a) What is the individual's optimal consumption in each period? How he or she do in the first period? (b) The government has decided to set up a social security system. Thi $10 from each individual in the first period, put it in the bank, and person with interest in the second period. Write out the new constraint. How does the system affect the amount of private savin system affect national savings (total savings in society)? What is the of social security system?
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- A consumer's income in the current period is y = 100 and income in the future period is y'= 120. He or she pays lump-sum taxes t=20 in the current period and t'= 10 in the future period. The real interest rate is 0.1, or 10%, per period. (a) Determine what the consumer's optimal current-period and future-period consump tions are, and what optimal saving is, and show this in a diagram with the consumer's budget constraint and indifference curves. Is the consumer a lender or a borrower? (b) Now suppose that instead of y = 100 the consumer has y = 140 Again, determine optimal consumption in the current and future periods and optimal saving, and show this in a dia gram. Is the consumer a lender or a borrower? (c) Explain the differences in your results be tween parts (a) and (b).Assume a consumer has current-period income y = 200, future-period income y′ = 150, current and future taxes t = 40 and t′ = 50, respectively, and faces a market real interest rate of r = 0.05, or 5% per period. The consumer would like to consume according to the following utility function: U (c, c′ ) = ln(c) + ln(c′ ). Show mathematically the lifetime budget constraint for this consumer. Find the optimal consumption in the current and future periods and optimal saving. Suppose that instead of r = 0.05 the interest rate is r = 0.1. Repeat parts (a) and (b). Does the substitution effect or the income effect dominate?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, β, y1 and y2.
- Assume an intertemporal budget constraint that shows how consumption can be traded off between two periods, t and t+1. Assume the consumer can save and borrow at the same interest rate of 10%. Assume the consumer collects income of $100 in each period. To gain an extra $10 dollars in period t+1, what must the consumer give up in period t? 11 10 1 10John and Peter are two representative consumers/investors who maximize the utility of consumption. John's utility of consumption is characterized as ln(x) + 2ln(y) while Peter puts more weight on the current consumption level and has a utility function of 2ln(x) + ln(y). John has a wealth of ($10, $20) thousand, while Peter has a wealth of ($20, $15) thousand now and next year, respectively. (a) What are the optimal consumption plans forJohn and Peter,respectively,if the interest rate is 5% per annum? (b) If John and Peter are the only investors/consumers, what is the equilibrium interest rate? (c) Further to part (b), how much do they borrow or lend to each other?(50 points) Consider the two-period economy with investment discussed in class. Forthe sake of simplicity, assume that the utility is time separable and that the per-periodutility is equal to:u(C) = log CAssume also that the technology to produce the consumption goods has constant returns to scale. Moreover, capital is the only input to production. Hence the currentperiod technology can be represented as:Y = zK1where Y , K, and z stand for output, capital and total factor productivity in the currentperiod. The future period technology is akin to the current period technology. There isno government expenditure in the current and future period so that G = 0 and G0 = 0.Finally there is full capital depreciation, namely δ = 1.(a) (10 points) Write down the social planner’s problem. State the variables withrespect to which the social planner maximises.(b) (10 points) Derive the first order conditions to the planner problem. Rearrangethere condition to find the conditions characterising the…
- Consider an economy where individuals live for two periods only. Their utility function over consumption in periods 1 and 2 is given by U = 2 log(C1) + 2 log(C2), where C1 and C2 are period 1 and period 2 consumption levels respectively. They have labor income of $100 in period 1 and labor income of $50 in period 2. They can save as much of their income in period 1 as they like in bank accounts, earning interest rate of 5 percent per period. They have no bequest motive, so they spend all their income before the end of period 2. a. What is each individual’s lifetime budget constraint? If they choose consumption in each period so as to maximize their lifetime utility subject to their lifetime budget constraint, what is the optimal consumption in each period? How much do the consumers save in the first period? b. Suppose that the government introduces a social security system that will take $10 from each individual in period 1, put it in a bank account, and transfer it back to…Consider the intertemporal consumption problem of Mr Cronus between two periods, say this yearand next year. His utility function takes the form U (c1; c2) = pc1 +0:97pc2, where c1 and c2 arehis consumption this and next year respectively. It can be shown (and you do not have to) thatthis utility function satis es diminishing marginal rate of substitution.His yearly income is stable at 100 unit (let say a unit is ten-thousand). He faces di¤erent interestrates between borrowing and saving. Speci cally, the saving interest rate is 0:02, whereas theborrowing interest rate is 0:04.(a) Describe the budget set facing Mr Cronus.(b) Is Mr Cronus a borrower? Explain your answer.(c) Is Mr Cronus a saver? Explain your answer.Clare is contemplating her possible consumption patter for this year and next. She know that she will have income of $50,000 this year and $55,000 next yea. Her plan is to consume $40,000 this year (t=0). She is also going to invest 30,000. This investment has a positive NPV of $450. She decides to take the investment; in addition, the return on the investment is 9.62%. What consumption she can expect at t=1? (show a detailed procedure)
- Suppose that a consumer/investor has an initial endowment only for the current period, which is Eo =450. She may consume today or in the next period only (two-period model). The interest rate for borrowing and lending in the capital market is 5% (a)Depict the budget constraint for the investor in an inter-temporal consumption diagram! What is the maximum amount the consumer is able to consume in the next period? (b)The consumption preferences of the consumer/investor are best described by a square root function, defined over current and future consumption. What is his optimal consumption plan? Show your calculations! Depict the results in appropriate diagram. Which amount is invested in the capital market?Suppose that an economy's net investment flow is I(t) = 2t + 10t1/2.Letting K(0) is the initial investment, use the definite integralto find the level of capital: a.Next 5 yearsb. Initial investment, K(0) = 100Eva lives for two periods and wants to have equal consumption in both. Her first period income is $Y subscript 1 and her second period income is $Y subscript 2. The interest rate is 7%. If Y subscript 1 = 81713, and Y subscript Training 2 = 80175, what is her optimal saving?