Consider a two-period, small open economy with endowments on tradable and nontradable goods. The representative household has lifetime utility U(CT1CN1, Cr2CN2) = log C₁₁ +log C1 + Blog Cr2+ Blog CN2 N1 Endowments are 21 22=5 and Qr₁=2.5, Qr2 = 7.5. interest rate is r* = 0.04 and the discount factor is 3=1/1.04 = 0.9615. = Initial NFA is zero. The world a. Compute equilibrium consumption of both goods, the trade balance, and the real exchange rate in both periods. b. Suppose that after the household chooses how much to borrow/save in period 0, the world interest rate rises to r=0.10. Recompute the equilibrium variables for period 2, and compute the difference between lifetime utility between this scenario and the scenario in part 1.
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- John 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?Exercise 2: Investment and Optimal Consumption Choice Kate has endowment E = (2775, 3000), i.e. she receives E1 = 2,775 in the first period and E2 = 3, 000 in the second period. Kate has access to a perfect capital market with interest rate r = 0.2 (i.e. 20%) per period. She also has access to a private investment opportunity. If she invests z in the private investment opportunity in the first period, it will return 180 · Vz – 100 if z > 100 R(z) = if z < 100 in the second period. Kate's preferences can be represented by the utility function u(x, y) = x · Y, where x is the number of dollars available to purchase goods in the first period and y is the number of dollars available to purchase goods in the second period. 1. What is the net present value, N PV (z), of the private investment opportunity? 2. What is Kate's budget constraint (including the investment opportunity)? 3. What is Kate's optimal consumption bundle (x*, y*)? 4. How will Kate finance this consumption plan (i.e. how…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…
- Consider a 2-period economy in which Yt = AtKt. The households start with aquantity a1 > 0 of initial assets and maximize lifetime utility given a per-periodutility function u(c) = ln (c). Assume that A1 = As1 = 1 −∆ and A2 = 1. What is rs1 equal to?a) A planner in a small, open economy has a utility function U = x1X2 and production takes place according to the production functions Y1 = 10L, – 0.5L Y2 = 10L2 – L The world prices of the goods are p1 10 and p2 5. The total amount of labour %| %3D available in the economy is 5 units. (i) Formulate the Lagrange equation. (ii) Write down the Karush-Khun-Tucker conditions. (iii) Determine a possible solution.For the given life time budget constraint; C, + C2 = W1 1 +r determine C1 and C2 to maximize the following utility function 1 C Ut 1- 0 + 1+p1 – 0 1 p = 1 r= 0,5 w = 220 220) ||
- Suppose the administration of a University that is located a high-crime urban area mustdecide on a policy to increase policing on its campus. The student population is 10,500 students but iscould increase if crime is reduced. The cost of added policing in terms of additional salary and equipmentis estimated to be $2.4 million per year. Suppose the administration plans to make their decision with thehelp of a benefit cost analysis of the policy. VSL is $11.8 million a. Explain how the concept of Value of a Statistical Life might factor into a benefit cost analysis of thispolicy. b. How many lives would need to be save to justify the added policing costs if only prevented deathswere considered as the policy benefit? c. Suppose it is believed that student enrollment will increase in the future if crime on campus is reduced.Write a formula to calculate the present value of benefits of policing if student enrollment increases by4% each year.Question 2An investor is to purchase one of three types of real estate, as illustrated inFigure below. The investor must decide among an apartment building, anoffice building, and a warehouse. The future states of nature that willdetermine how much profit the investor will make are good economicconditions and poor economic conditions. The profits that will result fromeach decision in the event of each state of nature are shown in Table below: Assume that it is now possible to estimate a probability of 0.60 that goodeconomic conditions will exist and a probability of .40 that poor economicconditions will exist. a) Determine the best decision by using expected opportunity loss. b) Develop a decision tree, with expected values at the probability nodes. c) Compute the expected value of perfect information.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 ???
- Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will beequal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their incomewill be y − L. The chance of a safe return is 50%.(i) Draw and carefully label the merchant’s endowment point, their expected income, and their certainty equivalent income in a 2-dimensional state-contingent consumption space. Use the diagram to illustrate and explain how the merchant would benefit from buying insurancein a competitive insurance market. At which point a risk-neutral insurance firm would maximisetheir profits by offering the merchant full insurance?Over their lifetime individuals typically prefer to have a reasonable _ consumption pattern. A) smooth B) volatile C) unpredictable D) very lowA 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.