estion 5 A utility function is given by u(x₁, x2)=-4x+80x18x+100x2-x1x2. Suppose that prices are p₁ = 5, and p2 = 1. Income is I. 1. Specify the Lagrangean using all three constraints. 2. Use the Lagrangean determine the range of values of income / for which the different constraints may bind. Note: these preferences are not monotone. As a consequence, the budget constraint itself may not bind.
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- Suppose a family (with 1 child) earns $50,000 per year and there is no publicly provided education. Assume the price of other goods is $2,000 and the price for education/ year is $5,000. Draw the family's budget constraint. Show how free public education worth $20,000 per student changes the budget constraint. Draw a set of indifference curves to show that the family increases education consumption after b. Show how a school voucher redeemable for $20,000 worth of education changes the family's budget constraint. What happens to the amount of education the family purchases for the child?Assuming the two good case. When a person is attempting to maximize utility and income increases, then: a. The budget constraint expands, shifting the budget line outward indicating more choices are now affordable and a higher utility level is now possible. b. The budget constraint contracts, shifting the budget line outward indicating more choices are now affordable and a higher utility level is now possible. c. The budget constraint expands, shifting the budget line inward indicating more choices are now affordable and a lower utility level is now possible. d. The budget constraint contracts, shifting the budget line outward indicating more choices are now affordable and a lower utility level is now possible.A possible explanation for the indecency might be the fact that the consumers are not all alive at the same time and therefore some mutually advantageous trades cannot occur. Consider an economy where consumer t receives an endowment of 1 unit of the single consumption good at time t and obtains utility only from consumption at times t and t + 1. All consumers meet at time 0 to trade. What is the equilibrium? Is exigency restored?
- In an environment in which all education is privately provided, suppose that a family with one child has $20,000 per year to spend on private goods and education. Draw this family’s budget constraint. Suppose now that an option of free public education with spending of $4,000 per pupil is introduced to this family. Draw three different indifference curves corresponding to the following three situations: (a) a free public education would increase the amount of money that is spent on the child’s education; (b) a free public education would decrease the amount of money that is spent on the child’s education; (c) a free public education would not affect the amount of money spent on the child’s education.Bob enjoys cookies (x) according to the utility function U(x)=20x- 2 tx , where t is a parameter that reflects how hungry he is. Cookies are costless in Bob’s world and so there is no income constraint. Using the envelope theorem, calculate how Bob’s maximum utility from eating cookies varies with t.2. Consider the two-good model of the utility maximization program subject to a budget constraint. The utility function U of a hypothetical rational consumer and his/her budget constraint are given, respectively, by: U = x1x2, (U) B = p1x1 + p2x2, (B) where xi = the consumer’s demand for consumption good i (i = 1, 2), pi = the price of consumption good i (i = 1, 2), and B = the (exogenously given) budget of the consumer. In this maximization program, assume the following data: B = 240, p1 = 10, p2 = 2. (a) Using the Lagrangian function L, derive the first-order (necessary) conditions for a (local) maximum of the utility function. (b) Compute the optimal values of all choice variables, i.e., x*1 , x*2, and λ* , in the program, where λ signifies the Lagrange multiplier. (c) Using the information of the bordered Hessian matrix H¯ , verify the second order (sufficient) condition for a (local) maximum of the utility function. Note:- Do not provide handwritten solution. Maintain accuracy…
- Consider a couple's decision about how many children to have. Assume that over a lifetime a couple has 100,000 hours of time to either work or raise children. The wage is $10 per hour. Raising a child takes 10,000 hours of time. a. Make a graph with the budget constraint showing the trade-off between lifetime consumption and number of children. (Ignore the fact that children come only in whole numbers!) Show indifference curves and an optimum choice. b. Suppose the wage increases to $15 per hour. Show how the budget constraint shifts. Using income and substitution effects, discuss the impact of the change on number of children and lifetime consumption. c. We observe that, as societies get richer and wages rise, people typically have fewer children. Is this fact consistent with this model? Explain.Assuming the two good case. When a person is attempting to maximize utility and the price of one of the two goods increases, then: a. The budget constraint will expand (rotate away from the origin), shifting the budget line outward indicating fewer choices are now affordable and a lower utility level is now possible. b. The budget constraint will contract (rotate towards the origin), shifting the budget line inward indicating more choices are now affordable and a lower utility level is now possible. c. The budget constraint will contract (rotate towards the origin), shifting the budget line outward indicating fewer more are now affordable and a higher utility level is now possible. d. The budget constraint will contract (rotate towards the origin), shifting the budget line inward indicating fewer choices are now affordable and a lower utility level is now possible.A young consumer has to decide how to spend his spare time. He can either watch movies x or play online video games y. Each round of the game cost 20 and each movie cost 20. He has 200 dollars to spend each week. In addition, he has time constraint. He can spend no more than 20 hours for entertainment each week. A movie last 2 hours and playing a game takes about 1 hour. His utility is given by √xy. Use the Kuhn-Tucker condition to find his optimal choice. (Assume that x and y can take any value, not necessarily integers.) Does the solution satisfy the necessary and sufficient condition for a maximum?
- A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle(8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consumingbundle (4, 4). She is indifferent between these two bundles. Assuming that the utilityfunction is Cobb-Douglas of the form U(x,y)=xα yβ, where α and β are positiveconstants, what is the utility function for this consumer?You have £20 per week to spend, and two possible uses for this money: telephoning friends back home, and drinking coffee. Each hour of phoning costs £2, and each cup of coffee costs £1. Your utility function is U(X,Y) = XY, where X is the hours of phoning you do, and Y the number of cups of coffee you drink. What are your optimal choices? What is the resulting utility level? You can use the standard result on the constrained maximization of such a function, but must state it clearly. Now suppose the price of telephone calls drops to £1 per hour. What are your optimal choices? What is the resulting utility level? How much income per week will enable you to achieve the same quantities at the new prices as the ones you chose before? What income will enable you to attain the same utility as you did before? Comment on your answer in the context of equivalent variation and compensating variation.If a competitive consumer chooses consumption program x=(20,5,7) at competitive prices p=(2,1,5), consumption program y=(15,10,6) at competitive prices p’=(3,1/2,5) then under the usual assumptions, A) violates the weak axiom of revealed preference (WARP) B) doesn’t violate WARP and exhibit direct revealed preference for x with respect to y. C) doesn’t violate WARP and exhibit direct revealed preference for y with respect to x. D) doesn’t violate WARP and doesn’t exhibit any direct revealed preference between x and y. E) NOPAC