3 Price elasticity and cross price elasticity of demand Suppose an individual who derives utility u(x, y) from consuming a units of good X and y units of good Y. Prices of goods X and Y are respectively Pa = 1 and py=p. Furthermore, the individual has a budget m. u(x, y) takes the following form: a> 0 and 3 > 0. (A) Derive the demands for goods X and Y. (B) Compute the price elasticity of demand for good Y. (C) Compute the cross price elasticity of good X for a change in p.
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- U(x, y) = xayb A consumer maximises utility subject to a budget constraint M = Pxx+Pyy Where px is the price of good x, py is the price of good y and M is the budget available. a. Derive an expression for the marginal utility of x. Under what condition is the marginal utility diminishing. b. Derive an expression for the marginal utility of y. Under what condition is the marginal utility diminishing.Question 7 A consumer has a utility function 100-(x₁-4)4 -(x2-3)4 and a budget constraint of 9x1+3x2 <200. How much of good 2 do they consume?1. An agent consumes quantity (X1,X2) of goods 1 and 2. Here is his utility function: U (x1,x2) = x13x2, his budget constraint is: p1x1+p2x2=m. (a) Calculate the agent's Marshallian demand (x*, x*2). (b) Derive the agent's expenditure function. (c) Roy's identity for good 1 states that: aV(pl,p2, m)/ х1* 3 дV (р1, р2, т) am Verify this equation (you need not verify it for good 2).
- 2 3 1. The consumer has Cobb-Douglas preferences for two goods x and y: u(x, y) = 10x¾yš Py> 0 and m > 0 is given. a) Set up the Expenditure Minimization Problem. b) Calculate MU and MUy. Ꮖ c) State the condition that explain why Cobb-Douglas preferences will guarantee an interior solu- tion. c) Give the two optimality conditions for x and y. i) ii) Px > 0, e) Solve for the compensated demand functions for good x and y.Pete consumes goods 1 and 2. Pete thinks that 3 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Pete's preferences? A B D E U (x₁,x₂) = 40x₁+30x - 10000 2 U(x₁,x₂) = 4x₁+3x₂+ 1000 2 U (x₁,x₂) = min {4x₁, 3x₂} U(x,x)=16x² +24xx₂+9x² More than one of the above does not represent Pete's preferences.Question 1 Consider a consumer with utility x₁2/3x21/3 and a budget constraint of 3x₁+6x2 ≤100. What is the maximum utility the consumer can achieve
- Mariah consumes music downloads (M) and concert tickets (C). Her utility function is giveh by U = 0.5M? + 2C². a. Using calculus, what is the marginal utility of music downloads? b. Using calculus, what is the marginal utility of concert tickets? Write the equation for the MRSMC (i.e., the MRS of C for M; M is increasing, C is С. decreasing). d. Would the bundles of (M = 4 and C = 1) and (M = 2 and C = 2) be on the same %3D %3D %3D %3D indifference curve? How would you know? e. Calculate the MRSMC when M = 4 and C =1 and when M = 2 and C = 2. Explain why the %3D %3D values of MRSMC differ across these two bundles. f. Based on your answers above, are Mariah's indifference curves convex? Explain using the values calculated. er thisDonald likes fishing (X1) and hanging out in his hammock (X2). His utility function for these two activities is u(x1, x2) = 3X12X24. (A) Calculate MU1, the marginal utility of fishing. (B) Calculate MU2, the marginal utility of hanging out in his hammock. (C) Calculate MRS, the rate at which he is willing to substitute hanging out in his hammock for fishing. (D)Last week, Donald fished 2 hours a day, and hung out in his hammock 4 hours a day. Using your formula for MRS from (c) find his MRS last week. (E) This week, Donald is fishing eight hours a day, and hanging out in his ham mock two hours a day. Calculate his MRS this week. Has his MRS increased or decreased? Explain why? (F) Is Donald happier or sadder this week compared to last week? Explain.A consumer with I dollars budget has the utility u(x,y) = x(y+1) over amounts of cake (x) and ice cream (y) she consumes. The prices are px , py respectively. (a) Derive her demand for cake (x), as a function of prices px , py and her budget I. (b) Looking at the demand function in (a), is cake a normal good or an inferior good? Are cake and ice cream complements or substitutes? (c) Calculate the (i) (own) price, (ii) income, (iii) cross- price elasticity of demand for cake at the point where I = 80, px =10, py = 20.
- Q50 Marginal utility analysis predicts a downward-sloping demand curve for good X because... a. As PX rises, the consumer increases purchases of X such that MUX/PX is equal to MU/P for all other products. b. As PX falls, the ratio MUX/PX becomes smaller, causing the consumer to purchase more of good X. c. As PX falls, the consumer increases purchases of X until MUX/PX is equal to MU/P for all other products. d. Utility-maximising consumers equate the marginal utility received for each product consumed. e. All demand curves are downward sloping, regardless of the behaviour of consumers.PROBLEM (7) A consumer with I dollars budget has the utility u(x,y) = x(y+1) over amounts of cake (x) and ice cream (y) she consumes. The prices are px, Py respectively. (a) Derive her demand for cake (x), as a function of prices px, py and her budget I. (b) Looking at the demand function in (a), is cake a normal good or an inferior good? Are cake and ice cream complements or substitutes? (c) Calculate the (i) (own) price, (ii) income, (iii) cross- price elasticity of demand for cake at the point where I = 80, px=10, py= 20..12) Leyla consumes goods X and Y. The price of good X is Px and the price of good Y isPy, Leyla’s income is I. If both prices and Leyla’s income increases by 50%, then theA) budget constraint will be unchanged.B) slope of the budget constraint will increase.C) slope of the budget constraint will decrease.D) budget constraint will shift outward in a parallel fashion.E) None of the above .