2. Suppose that two products, i= 1, 2, are located at the extreme ends of the [0,1] interval. Let li denote the location of the product and x the consumer locations. Consumers buy one unit and are uniformly distributed on the unit interval; their indirect utility is given by r - pit |li - xl, where r is the willingness to pay, pi the price set by firm i, and t a parameter that measures the disutility from not consuming their ideal product. Marginal cost of production is constant and equal to c. Solve for the Stackelberg equilibrium in prices where the firm located at zero is the leader firm. Is there an advantage for being a price leader, that is, is there a first-mover advantage? Why? Explain. Compare the results with simultaneous choice of prices.

2. Suppose that two products, i= 1, 2, are located at the extreme ends of the [0,1] interval. Let li denote the location of the product and x the consumer locations. Consumers buy one unit and are uniformly distributed on the unit interval; their indirect utility is given by r - pit |li - xl, where r is the willingness to pay, pi the price set by firm i, and t a parameter that measures the disutility from not consuming their ideal product. Marginal cost of production is constant and equal to c. Solve for the Stackelberg equilibrium in prices where the firm located at zero is the leader firm. Is there an advantage for being a price leader, that is, is there a first-mover advantage? Why? Explain. Compare the results with simultaneous choice of prices.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Q2. Suppose a consumer seeks to maximize the utility function U (x, y) = (x + 2) (y + 1), where x and y represent the quantities of the two goods consumed. The prices of the two goods and the consumer's income are pa, py, and I. (c) Show the bordered Hessian matrix, H for this problem. What does the second order condition require for this problem? Show if it is satisfied.
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Questions d) to f)
Assume that the preferences of a consumer can be expressed by the utility function 0.75 utx), 4 + x (i) Show that based on this utility function, the demand functions are for the benefits provided by 81 p x = 4 3 81 р; т and x P2 P der 1 p is the price per unit 1 x, 2 p is the price per unit 2 x and m are the income of the consumer. (ii) Explain what is meant by the cross-price elasticity of a good. Calculate the cross-price elasticity of the goodigiven in question (i), and give an interpretation of the result.
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2. Suppose John's utility function is UX,Y) = 4/XY , where X is consumption of beer and Y is consumption of pizza. For this utility function, ( Calculus Kahuna say: MUx = 2\Y/X and MUy = 2\X/Y ) a) Find three different bundles: (X,Y) = (2, 4½), (2, 3), (4, 24) and find the utility they provide John. Plot the three bundles and connect them with an indifference curve. J-L. IIII -+-+- - -4-L 7-T-T-r הרד - דר -LJ-J-L- 1-L -LJ-J-I ---
Guerdon always puts half a sliced banana, q, , on his bowl of cereal, q, The two 14 goods are perfect complements. What is his utility function? Derive his demand curve for bananas graphically and mathematically. 12- 10- e, Guerdon's utility function (as a function of q, and q2) is 6- VA U= min/2q1.92} 4- O B. U=q, +92- 2- 0+ OC. U=29, + q92- 10 12 14 16 18 20 6 22 Bananas O D. U=0.5q1 + q2- OE. U=q, xq2- 5.00- 4.50- Guerdon's demand function for bananas (as a function of the price of bananas, P1. the price of a bowl of cereal, p2, and income, Y) is 4,00- 3.50- 91 = (p1 + 2p2) (Properly format your expression using the tools in the E 3.00 palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the_ character.) 2.50- * 2.00- P. 1.50- 1.00- 0.50- 0.00+ 10 12 14 Bananas tv 80 DI DD F1 F3 F4 F5 F6 F7 F8 F10 @ %23 $ & 1 3 4 7 Q W E Y U S F H. J く C V M option command command optic ーの * CO Cereal (bowls) p. S per banana B * * N A
Q 15. suppose the consumer's utility function is given by U(x,y) =x^1/4 y^3/4 where the equation for this consumer's demand curve for x is?
. Answer all parts (a) (c) of this question. (a) Consider an agent whose preferences over any couple (x1, x2), where 2₁ ER+ and x2 € R+, e.g., apples and oranges, is such that she prefers the bundle that is closest to having the same number of apples and oranges. Write a utility function u: R² → R+ which represent these preferences. A politician remarks "Our recent increases in the wage rates of teachers has been a total suc- cess! The shortage of teachers has been reduced drastically. Another, similar wage increase should eliminate this shortage entirely" (b) Explain and illustrate in a diagram what is meant by "income effects" and "substitution effects" of a wage rate change. (c) Explain and illustrate how you would model the labour supply decision of a potential teacher. Do you agree that the wage increase will increase the labour supply in this case? Carefully outline the assumptions underlying your argument.
Exercise 4. In the lectures we have been working the the CRRA utility function, u : R+ which we defined, for any n≥0, as → R, Prove that u(x): -{ 21--1 1-n if ŋE [0,1) U (1,∞) In a if n = 1. 2--1 lim == Inz. →1 1-7 (7) Hint: Use L'Hôpital's Rule. Exercise 5. Consider the following two-period optimization problem we studied in class: max u (x1)+ẞu (x2), s.t. x1+px2=y, (1.22)ER² (8) where u is given by (7), x; represents consumption of good i = {1,2}, ẞ € [0, 1] is the discount factor, p = R++ is the relative price of good 2 in terms of good 1, and y = R++ is income expressed in terms of good 1. For the following two questions, consider an arbitrary n > 0. 1. Find the bundle (1, 2) that solves (8). 2. Notice that the solution (1, 2) depends on the parameters (n,3,p,y), which the con- sumer takes as given. Derive the expressions for ǝr/Op and ǝx/Əp.