Alberto's utility function is u(xã, YA) = min{¤Ã, YA}, where ï and yд are his consumptions of goods and y. Bella's utility function is u(XB,YB) = XBYB, where x and yß are her consumptions of goods and y. Alberto's endowment s 5 units of ï and no y. Bella's endowment is 11 units of y and no x. If x is the numeraire and p is the price of y in units of x, then in a competitive equilibrium
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- 5. Sheila and Bruce are taking a canoe trip. Sheila brought 10 boxes of peanuts (x) and 15 bags of chips (y). Sheila's utility function is U*(x,y) = lnvx*+ Invy°. Bruce also brought 20 boxes of peanuts and 5 bags of chips. Bruce's utility function is UB(x.y) = min[x', y'I. a) Illustrate the endowment point and draw a sample set of indifference curves through the endowment point. b) If Sheila and Bruce trade what will be the pattern of mutually beneficial trade? c) If the terms of trade are the number of bags of chips (y) per box of peanuts (x) then what is the largest value that these terms can be for a mutually beneficial trade in this economy? d) Find one mutually beneficial trade where the terms of trade are 1 bag of chips (y) per 2 boxes of peanuts (x). Suppose that Sheila and Bruce set up two competitive markets for peanuts and chips. Below you will show that if the price of peanuts (x) is $1 and the price of chips (y) is $2 then the markets for both peanuts and chips will clear.…Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (t) and soda (s), andthese provide him a utility ofutility = U (t s) =square root (ts).a. If Twinkies cost $0.10 each and soda costs $0.25 per cup, how should Paul spend the $1 hismother gives him in order to maximize his utility?b. If the school tries to discourage Twinkie consumption by raising the price to $0.40, by howmuch will Paul’s mother have to increase his lunch allowance to provide him with the same levelof utility he received in part (a)?3. Christina loves pizza and hates burger. Her utility function is U(p – b) = p – 6", where is the number of pizzas she consumes and b is the number of burgers she consumes. John likes both pizza and burger. His utility function is U(p, b) = p + 2\b Christina has an initial endowment of no pizzas and 8 burgers. John has an initial endowment of 16 pizzas and 8 burgers. a. Graph the initial endowment and label it E b. If Christina hates burgers and John likes them, how many burgers can Christina and John be consuming at a Pareto optimal allocation? What is John's marginal utility for pizzas and burgers? Mark the locus of Pareto optimal allocations of pizzas and burgers between Christina and John on the grapsh above.
- Exercise 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.Alex's utility is U (xA, YA) = min {TA, YA}, where zA, and y, are his consumptions of %3D goods a and y respectively. Becky's utility function is U (zB, YB ) = TBYB where ap and yp are her consumptions of goods a and y. Alex's initial endowment is 20 units of x and 12 units of y. Becky's initial endowment is 12 units of x and 20 units of y. At the Walrasian equilibrium, a) Both of them consume 16 units of good y each. b) Alex consumes 8 units of good y and Becky consumes 16 units of y. c) Both of them consumes 12 units of good x each. d) Alex consumes 12 units of good x and Becky consumes 20 units of x. e) None of the aboveFor problems 1, 2, 3, we consider an agent Ann who consumes goods and y and has a utility U (x, y) = x²y. In these problems, Good costs P and good y costs Py. (1) Ann has I to spend. (a) Write out Ann's utility maximization problem. (b) Find: MU, MUy, and MRSxy. (c) Write out Ann's tangency condition. What is the other equation needed to solve for Ann's demand? (d) Find Ann's demands for x and y as functions of Pr, Py, and I. (Partial credit if you do this only for I = 30, P₂ = 5 and Py = 2.)
- 4. Aaron and Burris have the following utility functions over two goods, x and y. Aaron’s utility function: UA(xA, yA) = min{xA/3, yA} Burris’s utility function: UB(xB, yB) = 9xB + 3yB Aaron’s endowment is eA = (2, 4). Burris’ endowment is eB = (10, 8). In an Edgeworth Box diagram, show which allocations are in the core. Solve for the set of Pareto optimal allocations (i.e. the contract curve) in the Edgeworth Box. Illustrate the contract curve in an Edgeworth Box diagram. Let good y be the numeraire (i.e. set py = 1 and let px = p). Solve for the Walrasian competitive equilibrium allocation and price ratio.Suppose that David and his friend Wilson derive utility from consuming two types of snacks: onion rings (9₁) and chips (9₂). The utility function for each individual is U (9₁, 92) = 9192. Their indifference curves for these two goods are assumed to have the usual (convex) shape. Suppose David has an initial endowment of 35 onion rings and 10 chips, and Wilson's initial endowment consists of 5 onion rings and 20 chips. (1) Draw an Edgeworth box and show the initial allocation of goods, to be labelled e. Indicate the initial quantities of each person's goods on the four axes.A and B consume only two goods, cider (C) and dumplings (D). A has an initial endowment of 10 bottles of C and 30 of D. Bob has an initial endowment of 50 bottles of cider and 50 dumplings. Alice’s utility function is uA(CA,DA) = 9ln(CA) + 10ln(DA), where CA and DA represent consumption of C and D, respectively. B’s utility function is uB(CB,DB) = CBXDB, where CB and DB denote B's consumption of C and D. a) Find the competitive equilibrium, i.e. the price ratio, of this exchange economy and the resulting equilibrium allocation. b) Find the expression of the contract curve for this economy and use your answer to check that the equilibrium allocation you found in (b) is indeed Pareto optimal.
- Please answer every part. 4. Consider an economy consisting of two individuals, Ann and Bob, and two goods, scotch and wine. Aun has 5 bottles of scoteh and 2 bottles of wine as her endowment, while Bob has 3 bottles of each. Suppose their preferences are described by the following utility functions uA(s, w) = sw and up(s, w) = s'u. Assume also that the prices of goods scotch and wine are represented by P,= 1 (scotch is the mumeraire), and P>0. a. Sketch the Edgeworth box of the economy with Ann at the lower left corner and Bob at the upper right corner; scotch on the horizontal axis, and wine on the vertical axis. Indicate the endowment point e in the box. b. Write the budget lines for Ann and Bob. e. Solve Ann's utility maximization problem. Expross Ann's optimal consumption bundle in terms of P. d. Solve Bob's utility maximization problem. Express Bob's optimal consumption bundle in terms of P. e. Define competitive equilibrium. Compute and plot the CE for this problem.2) Two consumers R and S share an endowment (x, y) of goods X and Y. The consumers' utilities from consumption can be written as UR = aln(XR) + bln(YR), and Us=aln(xs) + bln(ys), where XR, XS, YR, and ys are the quantities which each consumes, and a and b are parameters, for which a + b =1. Write a report in which you a) Define the concept of the contract curve, and show that for these endowments, it will be the upward diagonal of the Edgeworth box, which shows all possible divisions of the endowment. b) Assume that in the initial endowment, XRE = XE and ysE=yE. R begins with the endowment of X and S begins with the endowment of Y. By setting up a constrained maximisation problem for each consumer, obtain their offer curves, and hence obtain the Walrasian equilibrium. c) State the two welfare theorems, and explain how they apply in this case.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