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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- Using a diagram, draw the outcome of optimal choice using an indi↵erence curve and the individual’s lifetime budget constraint.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 > 1.2. 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,…
- Rodrigo is taking a year between high school and college to work and save up. His utility from consumption each year is U(c) = discounts future utility by B. Rodrigo is going to make $I his year of working, and whatever he doesn't consume from that income will a savings account which will earn return r before he consumes next year. He has to pay for school expenses E in year two, before he consumes (but after return has been realized). 1-o and he go intoConsider a household with the following utility function representing their preferences over consumption: U = u(Ct) + Bu(C++1) with =- u(C) = exp(-aC), BE (0,1), a > 0 where C and C++1 represent consumption in the current and future periods, respectively. The household faces a two-period decision problem. They receive endowments of Y, 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. Question: Formulate the household's budget constraints for the current and future periods. Com- bine them to derive the household's intertemporal budget constraint. Write down the household's optimization problem (objective function) that they seek to maximize.…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 ???
- Merge the two budget constraints into a single lifetime budget constraint by eliminating savings from the system of two constraints.only typed solutionAnalysing 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…
- The utility maximization problem is given by 0-1 Ө-1\0-1 max uf = Cit,C2t,St 0 + a, (c2t) subject to Cit + St = Wt + e C2t = (1+ rt+1)s, By solving the maximization problem, characterize the saving function depending on the value of 0, i.e., there are three cases.Fred is planning his consumption over two time periods. Fred's preferences for consumption in period and two can be represented by the following utility function: U(c,,c,) = C +(1+p) C" , where pis the subjective discount rate, and c;,c, is consumption in the first and second period. Fred's income in the first period is y, and grows by g % from the first period to the second period. Fred has access to perfect financial markets. The rate of interest is r>0. (a) Derive Fred's demand functions for consumption in the two periods as functions of p,r , y and g. (b) Derive Fred's demand for borrowing/saving as a function of p,r, y and g. (c) Give a condition involving the relationship between r and g for when Fred will borrow and when he will save.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?