1.Suppose that Chris's utility function is given by UC=QC1/2 RC1/2 , where QC and RC are his consumption of Q and R, respectively. Dana's utility function is given by UD=QD1/3 RD2/3, where QD and RD are her consumption of Q and R, respectively. Write an equation for the marginal rate of substitution (MRS) between Q and R for each of the two agents.
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1.Suppose that Chris's utility function is given by UC=QC1/2 RC1/2 , where QC and RC are his consumption of Q and R, respectively. Dana's utility function is given by UD=QD1/3 RD2/3, where QD and RD are her consumption of Q and R, respectively.
Write an equation for the marginal rate of substitution (MRS) between Q and R for each of the two agents.
2.Suppose that the price of good R is pR=1 and the price of good Q is pQ=2.
How much is Chris's and Dana's initial income, given his endowments and given these prices?
3.
At these prices, how many units of Q would Chris and Dana want to consume?
4.
At these prices, how many units of R would Chris and Dana want to consume?
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- Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…Smith and Jones are stranded on a desert island. Each has in her possession some slices of ham (H) and cheese (C). Smith prefers to consume ham and cheese in the fixed proportion of 2 slices of cheese to each slice of ham. Her utility function is given by Us = min(10H, 5C). Jones, on the other hand, regards ham and cheese as substitutes – she is always willing to trade 3 slices of ham for 4 slices of cheese, and her utility function is given by UJ = 4H + 3C. Total endowments are 100 slices of ham and 200 slices of cheese. a. Draw the Edgeworth Box diagram for all possible exchanges in this situation. What is the contract curve for this exchange economy? b. Suppose Smith’s initial endowment is 40 slices of ham and 80 slices of cheese (Jones has the remaining ham and cheese as her initial endowment). What mutually beneficial trades are possible in this economy and what utility levels will Smith and Jones enjoy from such trades? c. Now imagine a new endowment in which Smith has 60 slices…Consider the following utility function: U = 100x°.50,0.50 A consumer faces prices of P, = $2 and P, =$1 Assuming that graphically good X is on the horizontal axis and good Y is on the vertical axis, suppose the consumer chooses to consume 6 units of good X and 11 units of good Y. Then the marginal rate of substitution is equal to: MRS = (Enter your response rounded to two decimal places. Do not forget to include the negative sign.) Use absolute values. 15 stv МacВook Air 80 DI esc F1 F2 F3 F4 F5 F6 F7 F8 @ %23 2$ & 1 2 4 6 7 8 Q W T Y tab A S F H J K aps lock C V M control option command B H
- Economics Norton consumes only squash and tomatoes. His utility function is U(x, y) = x3. A canonical utility function. Consider the utility function u(c) 1-σ where c denotes consumption of some arbitrary good and ơ (Greek lowercase letter sigma") is known as the "curvature parameter" because its value governs how curved the utility function is. In the following, restrict your attention to the region c> (because "negative consumption" is an ill-defined concept). The parameter σ is treated as a constant. Plot the utility function for ơ-0. Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function? Plot the utility function for ơ marginal utility? Is marginal utility ever negative for this utility function? Consider instead the natural-log utility function u(c)=In(c). Does this utility function display diminishing marginal utility? Is marginal utility for this utility function? Determine the value of σ (if any value exists at all) that makes the general utility function presented above collapse to the…Consider the following utility function: U = 100x.50,0.50 A consumer faces prices of P, = $2 and Py =$1. Assuming that graphically good X is on the horizontal axis and good Y is on the vertical axis, suppose the consumer chooses to consume 6 units of good X an units of good Y. Then the marginal rate of substitution is equal to: MRS = 1.83. (Enter your response rounded to two decimal places. Do not forget to include the negative sign.) Use absolute values. The consumer should consume to maximize utility. more Y and less X the same amount of X and Y more X and less Y 15 tv MacBook Air 80 esc DII F1 F2 F3 F4 F5 F6 F7 F8 @ #3 % 2 7 8. Q W E Y tab s lock A S F G H J Z C V N M Onteel 8 : < CO T D4.13 CES indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by U(x, y) = (x° +y®)'/8 [in this function the elasticity of substitution o = 1/(1 – 6)]. a. Show that the indirect utility function for the utility function just given is V = I(p, + p,)¬/", where r = 8/(ò – 1) = 1 – 0. b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by E = V(p', + p,)''". f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.Fang likes playing badminton with her friends. Her utility function for playing badminton every week is given by U(t) = 11t – 2t2, where t is measured in hours. They play on a badminton court, which they can rent per hour. Suppose the current price to play on the badminton court is £2.50 per hour. How many hours should Fang play if she wishes to maximise her utility? Explain what we mean by the principle of diminishing marginal utility. Does the principle apply in Fang’s case? Explain why. In a diagram with income in pound sterling on the horizontal axis and quantity on the vertical axis, show the relationship between Fang’s budget and the number of hours that would maximise her consumer surplus.4. Assume Megan has preferences represented by utility function U (x₁, x2) = (x₁)¹/4 (x2)³/4 Good 1 is electricity (1) while good 2 is gasoline (x2). Suppose the consumer has an income level of m= 1000 dollars. In addition, the price of good 1 is p₁ = 5 and the price of good 2 is p2 = 5. (a) Compute the Marginal Rate of Substitution (MRS). (b) Find the optimal levels of electricity (x) and gasoline (x2) consumption for Megan. (c) Suppose the government imposes a per-unit tax of t = 5 dollars on gasoline. Find the optimal levels of electricity (x) and gasoline (x2) consumption for Megan with the tax.Mia is a registered nurse. She has 18 hours per day to devote to labor or leisure, and has $20 nonlabor income per day. She is paid $10 per hour for the first 8 hours of work and $15 per hour for overtime (for hours worked over 8 hours). Mia's preferences are represented by U(C, R) = CR utility function, where Cis the amount of dollars she spends on consumer goods and R be the number of hours of leisure that she chooses. Question 1 Part a Assuming she can work as many hours per day as she wishes, draw Mia's indif- ference curves, budget constraints and solve for her optimal consumption and leisure choices. Question 1 Part b Suppose that Mia's wage rate rises to $11 an hour for the first 8 hours. She is still paid $15 per hour for overtime. Again, find her optimal choice. Decompose the total change in demand due to a price change into a substitution effect, ordinary income effect and endowment income effect and graphically demonstrate each effect.Bunde's preferences are given by the utility function u(x1, x2) = x, + x2. For each of the following cases, decompose the price effects into the substitution and income effects using the Hicks-Allen decompositions. (For each part, use the given template to draw the constraints, and then use your completed graph to fill in the description. Enter any points in order from left to right as they would appear on the graph.) 20 15 10 5 10 15 20 (a) Suppose m = 120, P1 = 10, and p2 = 15. The price p1 then falls to 6, keeping p2 and m fixed. Given m = 120 and p, = 15, the budget B° is drawn for p, = 10. The utility-maximizing point on this budget is at A = given the linear preferences yielding utility y° = When p, falls to 6, the new budget is B" and the utility-maximizing point is C = ), yielding utility un = . The movement from A to C is the price effect of units of good 1. Removing income incrementally until Bunde can barely afford the old utility of at the new prices yields the line which…John and Belle consume only two goods, x and y. They have strictly convex preferences and no kinks in their indifference curves. At the initial endowment point, the ratio of John's marginal utility of x to his marginal utility of y is J and the ratio of Belle's marginal utility of x to her marginal utility of y is B, where J B. b. 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