Consider a two-period, small open economy with endowments on tradable and nontradable goods. The representative household has lifetime utility U(C₁₂C₁₂C₁₂₂CN₂) = log C₁₁+ log C₁₁ + b log C2 + blog C2 NI Endowments are QN₁QN₂=5 and Qr₁= 2.5, Qr₂ = 7.5. interest rate is r = 0.04 and the discount factor is b = 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.
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- 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…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?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) ||
- A planner in a small, open economy has a utility function U = x1x2 and production takes place according to the production functions Y1 = 10L1 – 0.5L{ Y2 = 10L2 – L3 The world prices of the goods are p, = 10 and p, = 5. The total amount of labour %3D available in the economy is 5 units. i) Formulate the Lagrange equation and determine a possible solution.Q2: Consider a person who is thinking about whether to engage in a life of crime. He knows that, if he gets caught, he will be in jail and his consumption will be low, xº, but if he does not get caught, he will be able to consume an amount x₁ that is considerably above Χρ· (a) Suppose that x₁ = 20; x₁ = 80 (where both are expressed in thousands of pounds) and suppose the probability of getting caught is 8 = 0.5. What is the expected consumption level if the life of crime is chosen? (b) Suppose the potential criminal's tastes over gambles can be expressed using the following utility function u(x) = In (x). Calculate the person's expected utility from a life of crime. How does it compare with the utility of the expected value of consumption? Based on your answer, explain this individual's attitude towards risk and draw the consumption/utility relationship. (c) Consider the level of consumption this person could attain by not engaging in a life of crime. What level of consumption from an…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…
- Angelina lives on an island where she produces two goods, apples (x) and bananas (V), according to the production possibility frontier 200 = x² +y², and she consumes all the %3D goods herself. Her utility function is U (x, y) = xy³. Find her utility maximizing x and y as well as the value of 2 (Lagrange multiplier).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 – L3 The world prices of the goods are p1 10 and P2 5. The total amount of labour 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. b) Let aggregate national income be given by Y; = C; + It + G; Where C, I and G are consumption, investment and government expenditure. Assume that consumption is given by C = mY; %3D Investment is given by I; = i + «(Y¢-1– Y;-2) %3D And government expenditure declines over time according to G¢ = Go(1 – 8)*, 0 < 8 < 1 Where 8 is the exogenous rate of decline of government expenditure (i) Derive the linear second order difference equation. (ii) Solve the difference equation.(c) Consider a two-period small open endowment-economy populated by a large number of identical households with preferences described by the lifetime utility function (C1)*" (C2)2 whe C1 and C2 denote consumption in periods 1 and 2, and a1 and az are the weights on consumption. Suppose that households receive exogenous endowments of goods given by Q1 = 10 and Q2 = 11 in periods 1 and 2. Every household enters period 1 with some debt (B5) inherited from the past. Let B be equal to -5. The interest rate on these liabilities, denoted ro, is 20 percent. Finally, suppose that the country enjoys free capital mobility and that the world interest rate on assets held between periods 1 and 2, denoted r*, is 10 percent. 1. Calculate the value of the household's lifetime wealth, Ý. 2. Assume that a1 = a2. Compute the equilibrium levels of consumption in periods 1 and 2. 3. Now, compute the equilibrium levels of the trade balance and the current account in period 1. 4. Assume now that the endowment…
- considere a pure exchange economy with two connumeren, indexed by i-1.2, and two goods, indexed by l = 1.2. Consumers are endowed with goods. we = (40) fore connumere 1 and W₁ = (0.1) fare consumere 2. Preferences of each Consumer are represented by the utility functions. A U₂ (xss X₂₂) = ln Rgs + ln £₁2 U₂ (×₂₁, X₂=) = ln X₂₁ + 1/ ln X₂₂ 1 write down the budget comtraint fore connumeres in this problem. In your comtraint, let ty Are the price of good I and fo be the price of good 2. 2. Find the demand functions for both goods for comumer 1 and Consumen 2. - (2) - (4) 3. Explain why the allocation (1₁, ₁2) = (0.5, 0.5) is Not farce to efficient.f) i) Find the degree of homogeneity of the following functions and comment on their returns to scale i. Q = K² + 5KL – 4L² ii. Q = 13KSL ii. A consumer has the following utility function a. U = Ax“yf Find the MU, and MU, iii. Given the following consumption function C = 70 + 0.85Y Find the corresponding savings function and marginal propensity to saveA 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.