B.3 Marie has preferences over two goods, cake q₁ and bread q2. She chooses quantities to consume soas to best satisfy these preferences subject to the budget constraint p1q1 + P292 y where p₁ andP2 are prices and y is total budget.Suppose that Marie's preferences are represented by utility function u(q1, 92) = 91 + In (bq1 +92)where b≥ 0 is a preference parameter. Assume that p1/bp2 ≥ y/ (P1 - bp2) ≥ 1.(a) Show that Marie's indifference curves are downward sloping and that her weakly preferred setsare convex for all possible values of b.(b) Show that her Marshallian demand for cake gi isYfi (y, P1, P2)1P1-bp2and find her Marshallian demand for bread, f2 (y, P1, P2). Discuss the shape of Engel curvesfor the two goods.(c) Explain why Marshallian demand curves for normal goods slope down. Are there any values ofb for which either cake or bread could be a Giffen good for Marie? Discuss.(d) Find the form of the indirect utility function and expenditure function and hence establishexpressions for the Hicksian demands for the two goods.

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Answered: B.3 Marie has preferences over two…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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6. Consider a consumer with the utility function u(x,, x2) = In(x,) + x2 and the budget constraint p, x1 + P2X2 = m. Derive the consumer's demand functions for x, and x2.
5. Suppose a preference relation is represented by a utility function u(x₁, x₂) = x₁ + x₂. Explain why the function -u(x₁, x₂) = −x₁ - x₂ does not represent the same preference relation. It is enough to explain using two appropriate bundles.
Amy consumes x and y and her preferences can be represented by the following utility function U(x,y) = 4x + y. 1. Are Amy's preferences transitive? 2. Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain. 6- 3. Can you use (2) to answer whether Amy's preferences are strictly monotonic? 4. What is MRS.y ? Is MRSx.y diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? 5. On a graph with x on the horizontal axis and y on the vertical axis, draw a typical indifference curve. Also indicate on your graph whether the indifference curve will intersect either or both axes. Label the curve U1. 6. Are Amy's preferences convex? Are they strictly convex? Explain
2. Anne's preference can be represented by the following utility function: U(x, y) = x® y". a) Find the optimal bundle for Anne that minimizes her expenditure. b) Write down the expenditure function. c) In a graph, show how her optimal bundle would change if the price of x increases (assume price of y remains the same).
A consumer has a budget of £12 to split between two goods: good 1 has a price of 2, good 2 has a price of 3. Write the consumer’s budget constraint algebraically. Convert this budget constraint into the formula for the budget line. Show this line on a suitably labelled graph. The consumer’s budget increases to £24. Show the effect of this change graphically. A consumer dislikes good 1, and dislikes good 2. Show these preferences on a suitably labelled graph with an indifference curve. Label the graph: which areas would be preferred to those on the line?
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…
Anthony seeks to maximize the following utility function u(x, y) = x'/3y2/3 subject to the budget constraint Pæa + PyY = I 1 where pr, Py, x, y, I > 0. a) Find Anthony's utility-maximizing bundle (x*, y*) as a function of pæ, Py, and I. b) Show that y* is decreasing in py and increasing in I (hint: use partial derivatives). c) What share of Anthony's income is spent on x? What share is spent on y? In other words, calculate Pa and Pu. Are these shares a function of prices? Pyy* Note: The above utility function is Cobb-Douglas, and all Cobb-Douglas functions have these share formulas for any values for the exponents. d) What is the impact of a change in pr on Anthony's utility?
Suppose you are given the following information for a particular individualconsuming two goods, a and b: Pa = $5, Pb = $6, MUa = 100, MUb = 200, and income (m) = $200.a) Sketch the budget set. What is the slope of the Budget Line? What are maximal possibleconsumptions of a and b?b) What is the MRSab for the two goods?c) Is this person maximizing her utility? How can you tell?d) Should she consume more of good a or of b? Explain.e) Why can’t you tell what her optimal bundle is? Explain.
Consider a consumer with M = $120 to spend (income) and faces prices PX = $15 and PY =$5 for goods X and Y. Her utility function is U(X, Y) = X1/2 + Y1/2.a Carefully. express the consumer’s choice problem, using the given information (this is whereyou write out the max operator, the choice variables, the objective function, and the budget constraint). b. Compute the absolute value of the consumer’s marginal rate of substitution, and inspect it todetermine the shape of the consumer’s indifference curves: C-shaped, linear, )-shaped, or some other shape.To show your work, neatly use the arrow argument, increasing X (↑) and decreasing Y (↓) to see whether|MRS(X, Y)| is diminishing along an indifference curve. c. If the indifference curves are C-shaped write out the budget line and the equal slopes conditionthat characterize an interior solution to the consumer’s choice problem. Use the particulars for the givenconsumer. Solve these conditions to find the interior solution. On the other…
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I. You only consume two goods and your preferences are represented by utility function 2 U(x₁, x₂) = (x1.5 + x2.5) for x₁ > 2 and x₂ > 0.

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