A decision maker allocates an endowment of W > 0 dollars across two periodst = 1, 2. He discounts the future by β ∈ (0, 1) while facing a gross interest rateof R > 1. His utility is the same as studied in class. Solve for the intertemporalchoice problem. Show that the optimal consumption is decreasing over time ifβR < 1, constant over time if βR = 1, and increasing over time if βR > 1.
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A decision maker allocates an endowment of W > 0 dollars across two periods
t = 1, 2. He discounts the future by β ∈ (0, 1) while facing a gross interest rate
of R > 1. His utility is the same as studied in class. Solve for the intertemporal
choice problem. Show that the optimal consumption is decreasing over time if
βR < 1, constant over time if βR = 1, and increasing over time if βR > 1.
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- H3. An investor with an initial endowment of $ 16,000 is confronted with the following productivity curve: C1= 240 (16,000 − C0)0.5 where C0 denotes consumption at present, and C1 consumption in the future. Assume the interest rate (for borrowing and lending) is 20%. The investor's utility function, from which it is possible to derive his indifference curves, is defined as: U(C0, C1) =C0C1 . What is the NPV of the investment chosen by the investor? Show proper step by step calculationConsider an individual who lives for two periods and has utility of lifetime consumption U = log(C1) + 1/1+δ log(C2), where C1 and C2 are the consumption levels in the first and second period respectively, and δ, 0 1 > 0 in the first period and no income in the second period, so Y2 = 0. He can transfer some income to the second period at a before-tax rate of return of r, so saving $S in the first period gives $[1 + r]S in the second period. The government levies a capital tax at rate τ on capital income received in the second period. The tax proceeds are paid as a lump-sum transfer to the following generation. The present generation does not care about the next one. a. What is the lifetime consumption profile of this individual? What is his lifetime indirect utility function expressed as a function of Y1 and b. Evaluate the change in initial income Y1 that is required to compensate the individual for the welfare loss due to the capital income tax τ. c. What is…Consider the two-period endowment economy we discussed in class. In this economy, a household lives for two periods and can save s. The return on saving is the real interest rate,r. Income is exogenous and given as y and y'. Households have utility overconsumption,c, in each period by U(c)=In(c)+B In(c') where future utility is discounted with rate B. Households choose consumption in both periods,c and c'. 1-Write down the current period and future period budget constraints. Derive the lifetime budget constraints. 2-What is the slope of the lifetime budget constraint? What are the intercepts? What is the significance of the endowment point? 3-Write down the expression for the consumer's optimization problem. What is the tangency condition? Provide an intuitive explanation for this condition. 4-Solve for c,c', and s. Given that B=1 and r=0, how would you modify the answers for c,c; and s? Explain.
- 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?An investor with an initial endowment of $ 16,000 is confronted with the following productivity curve: C1= 240 (16,000 - C0)^0.5 where C0 denotes consumption at present, and C1 consumption in the future. Assume the interest rate (for borrowing and lending) is 20%. The investor's utility function, from which it is possible to derive his indifference curves, is defined as: U(C0,C1) =C0C1 How much will the investor invest in production? A)$16,000 B)$0 C)$10,000 D)$12,000John 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?
- In the two-period Fisher model of consumption, suppose that the first period income is $5,000 and the second period income is $5,000 for both Matt and Paola. The interest rate is 10 percent. Matt’s lifetime utility function is C1 + C2 while Paola’s lifetime utility function is C1 + 0.8C2. If there is a borrowing constraint, whose consumption is affected by that?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?If Samantha's income is reduced to zero after she loses her job, her consumption will be ________ and her saving will be ________. Group of answer choices greater than zero; less than zero less than zero; less than zero greater than zero; greater than zero less than zero; greater than zero
- 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…Assume Marco is initially borrowing and investing 100, with a return on investment of 50% and an interest rate on borrowing at 10%. The return on investment falls to 5%. Which statements are correct? Select one or more: A. Marco’s decision to continue to invest will depend on his preference between consumption today and consumption in the future. B. Marco will wish to invest and borrow, but he will be worse off than when the return to investment was 50%. C. If he continues to invest and borrow, the dashed line representing his new frontier will start at 105 on the y axis and be shallower than the solid red line, so he’ll continue to invest and borrow D. In the remaining questions, assume the central bank now cuts interest rates so that the real interest rate falls to zero. Marco will still not wish to invest and borrow. E. If he just invests his money in the bank instead, his frontier will cross the x axis at 100 and be steeper than the frontier if he invests.…Over a three-year period, an individual exhibits the following consumption behavior: Px Py x y Year 1 3 3 7 4 Year 2 4 2 6 6 Year 3 5 1 7 3 Is this behavior consistent with the axioms of revealed preference?