12. Consider a pure 2-consumer 2-good exchange economy under a competitive price system. Consumers' A and B preferences are represented by utility functions Consider a pure 2-consumer 2-good exchange economy under a competitive price system. Consumers' A and B preferences are represented by utility functions uA(x1A,x2A)=x1A+2x2A, uB(x1B,x2B)=x1Bx2B, and their initial endowments are FA=(2,0) and (78=(0,2), respectively. Then, a competitive equilibrium allocation is: a) (x1A,x2A)=(0,1), (x1B.x2B)=(1,1) b) (x1A,x2A)=(0,2), (x1B,x2B)=(2,0) c) (x1A,x2A)=(1,1). (x1B,x2B)=(0,1) d) (x1A,x2A)=(1,1), (x1B,x2B)=(2,1) e) none of the previous statements a-d is correct.
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- 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…J 4 True or false? Be sure to explain your answer in detail. Suppose the utility function u(x1, x2) = 2x1x2 represents a consumer's preferences. Then the utility function v(x1,x2) = x1x2 also represents the same consumer's preferences3 Consider a pure exchange economy with 2 consumers and 2 goods. Consumer ? owns 8 units of good 1 and 1 unit of good 2, and his preference is represented by the following utility function: uA(x1, x2) = x1*x2. Consumer ? owns 2 units of good 1 and 4 units of good 2, and his preference is represented by the following utility function: uB(x1,x2)=x1+x2. Assume that the two consumers are allowed and able to trade with each other, and that good 1 is the numeraire. In this case, both consumers act as price-takers. Price takers: a market participant that is not able to dictate the prices in a market. Therefore, a price taker must accept the prevailing market price. Please find the competitive equilibrium of this pure exchange economy.
- A consumer has GH¢600 to spend on two commodities, A and B. Commodity A costs GH¢20 per unit and Commodity B costs GH¢30 per unit. Suppose that the utility derived by the consumer from x units of Commodity A, and y Commodity B is given by the Cobb-Douglas utility functionU (x, y) = 10x0.6y0.4a. How many units of each commodity should the consumer buy tomaximize utility?b. Is the budget constraint binding?1. Consider a consumer who chooses bundles consisting of two commodities, x and y. Suppose that, regardless of the prices px and py (which are always positive), the consumer chooses to consume x and y in a ratio of 2:1. This behavior is consistent with: a. A consumer having Cobb-Douglas utility function. b. The consumer’s utility function exhibiting perfect complements. c. The consumer’s utility function exhibiting perfect substitutes. d. Any of the above could be correct.You are choosing between two goods, X and Y, and your marginal utility from each is as shown in the table below. If your income is $9 and the prices of X and Y are $2 and $1, respectively, what quantities of each will you purchase to maximize utility? What total utility will you realize? Assume that, other things remaining unchanged, the price of X falls to $1. What quantities of X and Y will you now purchase? Using the two prices and quantities for X, derive a demand schedule (price–quantity-demanded table) for X.
- You are choosing between two goods, X and Y, and your marginal utility from each is as shown in the following table. If your income is $9 and the prices of X and Y are $2 and $1, respectively, what quantities of each will you purchase to maximize utility? What total utility will you realize? Assume that, other things remaining unchanged, the price of X falls to $1. What quantities of X and Y will you now purchase? Using the two prices and quantities for X, derive a demand schedule (a table showing prices and quantities demanded) for X.Suppose utility can be measured by "utils" and that Jane is consuming both lemons and cookies. The marginal utility from the last lemon consumed was 8 utils whereas the marginal utility from the last cookie consumed was 16 utils. Is it possible that Jane is maximizing total utility given the current combination of lemons and cookies consumed? Describe in detail what relationship would have to hold between the prices of lemons and cookies in order for Jane to be currently maximizing total utility.You are choosing between two goods, X and Y, and your marginal utility from each is shown in the following table. Units of X MUx Units of Y MUy 1 10 1 8 2 8 2 7 3 6 3 6 4 4 4 5 5 3 5 4 6 2 6 3 a. If your income is $9 and the prices of X and Y are $2 and $1, respectively, what quantities of each will you purchase to maximize utility? ______units of X and ______units of Y b. What total utility will you realize? ______utils c. Assume that, other things remaining unchanged, the price of X falls to $1. What quantities of X and Y will you now purchase? _____units of X and ______units of Y d. Using the two prices and quantities for X, complete the table to derive the demand schedule (a table showing prices and quantities demanded) for X. Instructions: Start with the highest price first Price of X Quantity Demanded of X $ $
- True or false with reasoning: 1) _______When we claim that utility can be ordinally measured, we assume that the consumer is able to measure the total and marginal utility received when one extra unit of a commodity is consumed. 2)_______If MRS between two goods is constant, then having more of one good without having more of the other does not increase utility. 3)_______Marginal Utility increases until total utility is at a maximum and then marginal utility decreases.Assume that Bob's utility function over beer x and pizza y is U(x,y) = 4x+12y. Which of the following statements is false? a) If the price of pizza is 12 and the price of beer is 3, then we can't determine how much pizza relative to beer Bob purchases. b) If the price of pizza is 16 and the price of beer is 5, then Bob only purchases pizza. c) If the price of pizza is 15 and the price of beer is 5, the own price elasticity of the demand for pizza as well as beer is infinite. d) It the price of pizza is 15 and the price of beer is 4, Bob purchases only beer e) Pizza and beer are perfect substitutes.Suppose that a fast-food junkie derives utility from three goods-soft drinks (x), hamburgers (y), and ice cream sundaes (z)− according to the Cobb-Douglas utility function U(x,y,z)=x0.5y0.5(1+z)0.5. Suppose also that the prices for these goods are given by px=1,py=4, and pz=8 and that this consumer's income is given by I=8.a. Show that, for z=0, maximization of utility results in the same optimal choices as in Example 4.1 . Show also that any choice that results in z>0 (even for a fractionalz ) reduces utility from this optimum.b. How do you explain the fact that z=0 is optimal here?c. How high would this individual's income have to be for any z to be purchased?