Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y)=X³ Y UB(X,Y)=X*Y Therefore: • For consumer A: MUx= 3X2Y; MUY = X³ . For consumer B: MUX = Y; MUY = X The initial endowments are: A: X= 30; Y = 90 B: X= 50; Y 60 a) (16 points) Suppose the price of Y, Py = 1. Calculate the price of X, Px that will lead to a competitive equilibrium. b) (8 points) How much of each good does each consumer demand in equilibrium? Consumer A's demand for X: Consumer A's demand for Y: Consumer B's demand for X: Consumer B's demand for Y: c) (4 points) What is the marginal rate of substitution for consumer A at the competitive equilibrium?
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- 1. Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = X*Y UB(X,Y) = X*Y3 Therefore: • For consumer A: MUX = Y; MUY = X • For consumer B: MUX = Y3; MUY = 3XY2 The initial endowments are: A: X = 10; Y = 6 B: X = 14; Y = 19 show all work a) Suppose the price PY = 1. Calculate the price of X, PX that will lead to a competitive equilibrium. b) How much of each good does each consumer demand in equilibrium? Consumer A’s Demand for X: Consumer A’s Demand for Y: Consumer B’s demand for X: Consumer B’s demand for Y: c)What is the marginal rate of substitution for consumer A at the competitive equilibrium?1. Suppose that the representative consumer has a utility function defined over consumption over two dates of the form U(C₁, C₂) = c₁²c₂². The general form of the slope of the indifference curve for the representative consumer is - C₂/C₁. Moreover, remember that c₁ = y₁ − S and C₂ = y₂ + s(1 + r). a. Assume that the representative consumer has an endowment of consumption goods in the two periods of y₁ = 20 and y₂ = 10. Assuming an interest rate r = 1, compute the equilibrium allocation and the implied savings. b. Suppose that, because of an attack of pessimism, the representative consumer assumes that future income will drop so that y₂ = 0. What happens to the savings s in the first period? c. In the previous part, the interest rate remained at 1. Now, consider the savings function, that is, the relationship between the real rate of interest and the amount saved. The equilibrium interest rate is then determined as a market price in the Saving-Investment diagram. Given the typical shape…A 2015 report by the music industry estimated the revenue lost to the industry every yearfrom illegal downloading. In this problem we will derive some of the estimates that may havegone into their calculation (approximately).First, start with the individual consumer’s problem. Suppose a typical consumer has a yearlyentertainment budget of I that they can allocate between music downloads (D) and otherforms of entertainment (E). Consumer preferences are characterized by a utility functionU(D, E). a.) Write an expression for the consumer’s budget constraint as a function of their entertainment budget and the prices of music downloads (Pd ) and other entertainment (Pe). (b) Write the consumer’s constrained optimization problem in Lagrangian form. (Note: Youdo not need to solve it or derive first order conditions.)
- Zach's preferences are representable by the utility function u = 90.3927, where 9₁ and 92 denote his consumption of goods 1 and 2. (Answers to each of these questions are rounded, where required, to two decimal places.) Still assuming endowments of e₁ = 5 and e₂ = 9 and market prices p₁ = 20 and p2 = 30, what is the maximised value of Zach's utility? O Au= 6.74 O B. u = 5.39 O Cu = 5.65 O D.u = 7.56 O E. None of the aboveExercise 4 Consider an economy with two consumers, Alexia and Bart, who live two periods, t = 0 and t = 1. In each period they can consume one type of good and their preferences for consumption are given by U (co, ci) = c(ci)² _i = A, B. Alexia and Bart have the following endowment of good in each period M=1, M₁ = 1, MB = 2, MB = 2. In t = 0, Alexia and Bart can exchange a financial contract for the delivery of one unit of consumption good in t = 1 (a bond). Name p the price of the bond and b² the amount bought by agent i = =A, B. (a) Write down each agent's utility maximization and budget constraints assuming that he/she can trade the bond without restrictions. (b) Find each agent's optimal quantity b² as a function of the bond net return r. (c) Find the equilibrium value of r and the equilibrium demand/supply of each agent.ayesha derives utility from travelling and outdoor dinning o weekends as given utility function U(t,d)=TD.the price of a day spent travelling is $160{Pt=160} and price of dining outdoor $200{Pd=200}.ayesha annual budget for this is $8000. find ayesha's utility maximizing choice of days travelling and dining outside. and alsoo find uutility level from consuming that bundles .show findings graphically
- 1. A standard model of choice under risk is Expected Utility Theory (EUT) in which preferences over lotteries that pay monetary prizes (x₁, x2, ..., xs) with probabilities (P1, P2, ..., Ps) with Eps = 1 are represented by the function L S (a) What does it mean to say that a function represents the consumer's prefer- ences? Σpsu(xs) Choice 1 8=1 (b) State and briefly comment on the axioms required for the EUT representation. (c) Consider the following experiment of decision making under risk in which sub- jects are asked which lottery they prefer in each of the following two choices: Lottery B 0 with prob. 0.01 10 with prob. 0.89 50 with prob. 0.10 Lottery D Choice 2 Lottery A 0 with prob. 0 10 with prob. 1 50 with prob. 0 Lottery C 0 with prob. 0.90 10 with prob. 0 50 with prob. 0.10 Suppose that the modal responses are Lottery A in Choice 1 and Lottery D in Choice 2. Assume that utility of zero is equal to zero and illustrate why it is not possible to reconcile these experimental…Suppose Jimi has reference dependent preferences over guitars and money as in Tversky and Kahneman (1991). His utility functions are given below. Gains Gains 400 -2 -2 2 Guitars 2$ Losses Losses i-600 -2 What is the least amount of money Jimi is willing to accept to sell one of his guitars? (just enter a dollar amount, i.e., "10o0", not "$1000"Each day Amina, who is in third grade, eats lunch at school. She likes only Twinkies (t) and apple juice (a), and these provides her a utility of; utility = U(t,a) = vVta. If apple juice is graph on the vertical axis and Twinkies on horizontal axis. Amina has a dimisnishing MRS. %3D %3D True False
- Roxanne is considering chipping in to pay for some tree trimming in her neighborhood. If the trees are trimmed, everyone's view improves by the same amount. Roxanne's utility function is given by u(x,t)=3x+1200t, where x is the amount of money she has to spend and t is a variable that equals 1 if the trees are trimmed and 0 if they are not. 1st attempt See Hint Suppose Roxanne starts with $8000. Her utility of that amount of money without the trees being trimmed is 24000 (that is, u(8000,0)=24000). Calculate her reservation price for contributing to the tree-trimming project. $Suppose there are two consumers. A and B. The utility functions of each consumer are given by: U₁XX)-x²-y U₂XY)-x-² Therefore For consumer A: MUX-2XY: MU-X² For consumer B: MUx-Y²; MUy-2XY The initial endowments are: A:X-60; Y-150 B.X-45: Y-75 al Suppose the price of Y, Py-1. Calculate the price of X, P, that will lead to a competitive equilibrium. How much of each good does each consumer demand in equilibrium? Consumer A's demand for X: Consumer A's demand for Y Consumer B's demand for X: Consumer B's demand for Y d ts) What is the marginal rate of substitution for consumer A at the competitive equilibrium?In class discussions about uncertainty we assumed that the utility levels in each state of nature depends on c, which we might interpret as some aggregate con- sumption and we expressed utility as U(c). Now, let's extend this to a case where the utility level depends on consumption of two goods (this was the type of utility we used mainly in this course). Ben is a farmer who grows wheat and barley. However, his harvest is uncertain. If weather is good, he gets 200 lbs of wheat and 200 lbs of barley. If weather is bad, he gets only 100 lbs of wheat and 100 lbs of barley. His utility in each state of nature is U(w, b) = w¹/46³/4, where w and b represent his consumption of wheat and barley, respectively. Prices of wheat and barley are $1 in both state of nature. The probability of good weather is π. Question 3 Part a Express Ben's expected utility function. (Hint: find Ben's optimal consumption in each state of nature first) Question 3 Part b Let's assume π = 0.5. Knowing that bad weather…