1. For each of the following three utility functions below, find the demand function by explicitly solving each utility maximization problem. (a) utx1,x2) = xx (b) u(x1, x2) = In.x +2ln.x2. (c) u(x1,x2) = x7x. 2. Answer the following: (a) Compare the three demands in Question (1), do you find the results sur- prising? Why or why not? (b) Make a serious attempt at coming up with a general statement that cap- tures these results.
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- 1.a Assume that a person’s utility function is given by the following function: ??=3?^(2/3)?^(1/3) Assume also that the price of X is £3, and the price of Y is £3 and that the budget is £45. What is the optimal amount of goods X and Y that should be purchased with this budget? b) Assume now that the price of good X is PD, while all other conditions remain the same. Find the optimal amount of good X that should be purchased for a generic price PD. In other words, find the individual demand function for good X. 1.b Assume now that the price of good X is PD, while all other conditions remain the same. Find the optimal amount of good X that should be purchased for a generic price PD. In other words, find the individual demand function for good X.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 preferencesa. Is the “more is better” assumption satisfied for both goods? Explain.b. What type of function is this? Explain.c. Provide expressions for the marginal utilities of x and y and does the marginal utility of x increase, remain constant, or diminish as the consumer buysmore units of x? Explain. Does the marginal utility of y increase, remain constant, or diminish as the consumer buysmore units of y? Explain.d. Provide an expression for the marginal rate of substitution of x for y.e. Is MRSx,y diminishing, constant, or increasing as the consumer substitutes more x for y alongan indifference curve? Explain.f. On a graph with x on the horizontal axis and y on the vertical axis, draw the indifferencecurves for U1 and U2 where U2 has a higher value than U1
- a. Is the “more is better” assumption satisfied for both goods? Explain.b. What type of function is this? Explain.c. Provide expressions for the marginal utilities of x and y and does the marginal utility of x increase, remain constant, or diminish as the consumer buysmore units of x? Explain. Does the marginal utility of y increase, remain constant, or diminish as the consumer buysmore units of y? Explain.d. Provide an expression for the marginal rate of substitution of x for y.e. Is MRSx,y diminishing, constant, or increasing as the consumer substitutes more x for y alongan indifference curve? Explain.f. On a graph with x on the horizontal axis and y on the vertical axis, draw the indifferencecurves for U1 and U2 where U2 has a higher value than U1 instructions: answer d,e, and f onlyThe preferences of a typical Californian can be represented by the following utility function: U (x1 , x2 ) = α ln(x1) + (1 − α) ln(x2) Here, x1 and x2 are the quantities of electricity and gasoline, respectively. The consumer faces prices given by p1 and p2 and has income m. Currently, the government has decided to impose a consumption restriction so that any person in the state is allowed to consume at most 50 units of electricity (x1 ≤ 50). Call this restriction a rationing constraint. (a) If α=0.25, m=100,and p1 =p2 =1, find the optimal consumption bundle of gasoline and electricity. Is the electricity rationing constraint binding (meaning does x1∗ = 50)? (b) Suppose that α = 0.75, but the other parameters are the same. What is the optimal consumption bundle? Is the rationing constraint on electricity consumption binding? (c) Now, assume that there is no rationing constraint. Assume m = 100 and p1 = p2 = 1, but α remains as a generic parameter. Solve for the optimal quantity…How does a consumer maximizes their utility given that they experience a budget constraint? Explain with graphical illustrations. Note:- Please avoid using ChatGPT and refrain from providing handwritten solutions; otherwise, I will definitely give a downvote. Also, be mindful of plagiarism. Answer completely and accurate answer. Rest assured, you will receive an upvote if the answer is accurate.
- Q2. Lucas likes lemon soda (X) and chips (Y). His utility function is given by: U (X, Y) = X0.2Y0.8 He earns $40 per week to spend on lemon soda (X) and chips (Y). The prices of lemon soda and chips are $2 and $4 respectively. Find out Lucas’s utility-maximizing bundle of lemon soda and chips (X*, Y*). Set up the utility-maximization problem and find out the price ratio of these two goods. Find out Lucas’s marginal utility of lemon soda (MUX) and marginal utility of chips (MUY). Calculate the MRSXY . Set up the optimal tangency condition and solve for Y in terms of X. Solve for Lucas’s optimal consumption bundle of lemon soda (X*) and chips (Y*). Draw the optimal consumption bundle on the budget constraint BC1 in Q1. Denote it as Bundle A. Make sure to indicate the optimal consumption of lemon soda (X*) and chips (Y*). Draw an indifference curve that is tangent to the budget constraint at Bundle A. Calculate the value of the MRSXY (the value not the formula) at the optimal…Q10. Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F. In Case 1, Total income is $100 , per unit prices of Food (F) are $2 , per unit prices of Cloth (C) are $10In Case 2, Total income is $100 , per unit prices of Food (F) are $2 , per unit prices of Cloth (C) are $15 Find the following for both cases, and contrast Case 2 with Case 1:a. What is the value of MRS at the optimal point and what does this value mean? b. What is the optimal consumption bundle i.e. (F*, C*)? c. Plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space (draw both case budget lines and point of optimality on one diagram)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.