Throughout this problem set, let X = R2, and suppose a consumer faces a price vector p = (P1, P2) and has an income m, where P1, P2, m > 0. In different questions, we will make different assumptions about what the consumer's utility u: X → R looks like. Throughout, let r* € R2 be the consumer's chosen bundle.

Throughout this problem set, let X = R2, and suppose a consumer faces a price vector p = (P1, P2) and has an income m, where P1, P2, m > 0. In different questions, we will make different assumptions about what the consumer's utility u: X → R looks like. Throughout, let r* € R2 be the consumer's chosen bundle.

Microeconomics A Contemporary Intro

10th Edition

ISBN:9781285635101

Author:MCEACHERN

Publisher:MCEACHERN

Chapter6: Consumer Choice And Demand

Section: Chapter Questions

Problem 13PAE

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1 Consider a consumer who consumes only two goods, 21 and z2. His utility function is u (21, 22) = In z1 + 2. where In represents the natural logarithm. 1.a Derive the consumer's MRS of good 1 for good 2 using calculus to calculate his marginal utility from z and his marginal utility from z2. 1.b If the price of z1 is $2 per unit and the price of z2 is $4 per unit, and the consumer's income is $100, What is the equation of this consumer's budget line? 1.c What are the optimal consumption choices, z and z5, for this consumer? Show your work.
Show whether or not the following utility functions are monotonic transformation of each other u = X,X2: u = lnX,lnX, %3! u = X, + X2: u = (X, + X,) u = X, + X2: u = (X1 + X2)? Show whether or not the following utility functions are convex U = x,°X, U = x, + X,2 U = X, + X,X2X
2. Consider an individual who not only consumes goods x and y, but also needs a mini- mum amount of x = xo to survive. Her (Stone-Geary) utility function is given by U(x, y) = (x — xo)¹/²y¹/² Let prices be pa, Py and income is I > Pro. (a) Show that this individual maximizes utility by spending (I-Pro) + Pro on good rand(I - Pro) on good y. Provide an economic interpretation for this result. [Hint: It might help to define z = x - xo and I = I - Pro. Then find the optimal z (and therefore x = z + xo) and y]
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We are told that Alfonso is a rational consumer, but we do not have much information about his preferences for Olives (good 1) and pickled onions (good 2). Suppose we observe that when the prices p1=p2=1 and his income is M=2, he chooses the bundle (1,1). Now we suppose we are told that p1 has decreased, that p2 has remained the same, and that Alfonso's income has been modified so that his initially chose bundle (1,1) is just affordable. Show that at his chosen bundle Alfonso consumes more Olives.
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1. Consider a consumer who consumes two goods. The utility function is given by u(x1, x₂) /2x1+1+√√√x₂, - where x > 0 denotes the amount of good i = the price of good 1 is 3, the price of good 2 is p2 > 0, and the income is 12. 1,2 consumed. Unless stated otherwise, (1) Calculate the marginal rate of substitution (measuring the value of good 1) at the consumption plan (x₁, x₂) = (3, 4). (2) Calculate the consumption plan that is optimal for the consumer. (3) Let v(p₂) denote the utility level at the optimal consumption plan. Calculate its derivative with respect to p2, i.e., v'(P2), at p2 = 3. (4) Suppose that p2 = 1 and the utility level is 10. Calculate the compensated demand for each good.
1. Suppose that U (f,c) = f + 8c² is a utility function that describes Amelia's preferences over two goods: fish (f) and custard (c). For the following, think of fish as the good graphed on the horizontal axis. Derive an expression for her marginal utility (Uf) from a small increase in f holding c fixed. Also find the marginal utility for custard (U.). b. What is Amelia's marginal rate of substitution (MRS)? Give a brief (2 sentences maximum) intuitive description of what MRS represents. If Amelia has 4 units of custard, holding her utility constant, how many units of custard would she be willing to give up in order to get one more unit of fish? c. Graph Amelia's indifference curve for a utility level of 40. Be sure to specify at least 3 bundles of goods on the indifference curve. d. Does the fact that Amelia's indifference curve intersects with the custard axis violate any of the 5 properties of indifference curves? Briefly support your answer. a. Give another utility function that…
Suppose a consumer’s utility function is u = x_1^(3/2) x_2^(3/2) . She spends her budget of £27 for two goods. The prices of both goods are p1 = 6 and p2 = 6 Now suppose that instead both goods are priced as follows: There is a discount of 50% on the price of good 1 on each additional unit in excess of 3 units, and there is a discount of 50% on the price of good 2 on each additional unit in excess of 3 units. Draw the new budget constraint and derive it analytically.
Let the following table represents the total utility of a given consumer, in the cardinal utility approach. Q 1 2 3 4 5 Tux 8 14 18 20 20 Tuy 6 10 13 15 16 Mux Muy Mux/px Muy/py D) Assuming the consumer has any amount of money (enough budget) how many of X and Y should the consumer buy, to maximize utility? E) What is the total utility of X and Y? F) Let now price of X is 4 birr per unit and price of Y is 2 birr per unit and budget of the consumer for consumption of X and Y is 20 birr. Given budget constraint how many of X and Y should the consumer buy to maximize utility? G) What are the total utility of X and Y
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