Bundle: Microeconomic Theory: Basic Principles and Extensions, 12th + MindTap Economics, 1 term (6 months) Printed Access Card
12th Edition
ISBN: 9781337198202
Author: NICHOLSON, Walter, Snyder, Christopher M.
Publisher: Cengage Learning
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Chapter 3, Problem 3.4P
a
To determine
To plot: Graphical representation of the given function and to show convexity of IC.
b)
To determine
To plot: Graphical representation of the given function and to show convexity of IC.
c)
To determine
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Consider the following utility functions:
(1) u(x₁, x₂) = x₁ + 2x2.
(2) u(x1, 1₂) = 2125 12.75
(3) u(x₁,1₂)=-1²-1₂.
(4) u(x1, 1₂) min{x1, 2x2).
For utility functions (1) to (3), give the equations of the indifference curves correspond-
ing to utility level k, where 22 is expressed as a function of 2₁ and k.
For utility functions (1) to (4), draw two indifference curves for each function. That
is, draw four graphs (one for each utility function) and two indifference curves on each
graph.
1.
Joan is on summer break and spends most of her time either playing video games
or browsing the internet. Her utility function is U (q1, 42)
399.24$, where qi represents the
0.2,0.8
%3D
hours spent browsing the internet and q2 hours playing video games. Joan does not have a budget
constraint but she has 15 hours to spend on both activities.
(a)
Write the equation for one of Joan's indifference curves when the level of utility is
equal to 100.
(b)
Write Joan's time constraint.
(c)
Find the amount of hours that Joan will spend on each activity that maximizes her
utility and satisfies her time constraint. Use either the substitution or Lagrangian method.
(d)
Can Joan afford any bundle that yields a utility of 50? Explain.
(e)
How does Joan's optimum amount of time spent on each activity change if her
utility function is U (q1, q2) = 5ln(3) + In(q1) + 4ln(q2)?
2. Now consider u(x1, 22) = 2 ln xỉ + ln x2
(a) Find expressions for MU1 and MU2.
(b) Find an expression for MRS12.
(c) Are the preferences represented by this function convex? Justify your answer using the
MRS or a graph of some indifference curves.
Chapter 3 Solutions
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- A consumer with income I=120 facing prices pX = 4 and pY = 8 for two goods X and Y (for each good she prefers more to less) chooses optimally to consume 12 units of X. If the prices change and now pX = 6 and pY = 4, what is the possible range for her optimal X consumption? (like, x >/ 7 or 10 >/ x >/7…etc. Use indifference curve analysis on a graph to reason about the possible locations of the new optimal bundle.)arrow_forward1. For each of the following scaling functions for a von Neumann-Morgenstern utility func- tion, determine the Marginal Rate of Substitution between X1 and X2 and the equation for an indifference curve through the consumption bundle (100,100) (solve for X2 on the left hand side of the equation). State 1 is the bad outcome that occurs with probability 0.2 and State 2 the good outcome that occurs with probability 0.8. Graph these indif- ference curves and comment on what you see. Is a consumer with these preferences risk averse, risk neutral, or risk loving? (a) V(X) = InX (b) V(X)= VX (c) V(X)= X (d) V(X)= X²arrow_forwardSuppose that one of your friends does not care if she consumes spaghetti (S) or noodle (N). She wants toconsume either two-dish spaghetti or two-dish noodle or any combination of both which adds up to twodishes.(i) Write the utility function of your friend to represent his preferences.(ii) Draw couple of indifference curves of your friend on graph with appropriate labels.(iii) Denote prices as Pn, Ps; income as M. If Pn < Ps, then determine the demand function of your friend foreach product as a function of prices and income.arrow_forward
- For each of the following utility functions draw the indifference curve that passes through (1,1). Label at least three points on each curve and indicate the direction of increased preference: a) u(x1, x2 ) = 3x1 b) u(x1, x2 ) = x1 + 2x2 c) u(x1, x2 ) = x1+ logx2 d) u(x1, x2 ) = min (2x1, x2) e) u(x1, х2) %3D max (1, x2)arrow_forwardDraw an indifference map for each of the following functions. U(x,y) = x2y3U (x, y) = 2x + 3yU (x, y) =x + ln(y)U (x, y) = min{2x, 3y}U (x, y) = max{3x, 2y}arrow_forwardWhat is the logarithmic transform of the utility function U=xα1xβ2xγ3U=x1αx2βx3γ given the budget constraint px1x1+px2x2+px3x3=Mpx1x1+px2x2+px3x3=M. Select one: a. ln U=ln xα1+ln xβ2+ln xγ3ln U=ln x1α+ln x2β+ln x3γ b. ln U=α ln x1−β ln x2−γ ln x3ln U=α ln x1−β ln x2−γ ln x3 c. ln U=α ln x1+β ln x2+γ ln x3ln U=α ln x1+β ln x2+γ ln x3 d. ln U=ln xγ1+ln xβ2+ln xα3arrow_forward
- Draw indifference curves for the following utility functions. a) u(x, y) = x+ 2y b) u(x, y) = min{x, 3y}arrow_forwardConsider a consumer with utility function u(x1, x2) = min{4 min{x1, x2}, x1 + x2} (a) Draw indifference curves passing through points (2, 2), (1, 2) and (4, 2). Make sure you correctly determine kink points. What properties of the preferences can you deduce from the shape of indifference curves? (b) When X = R2 +, does UMP have a solution when p1 = p2 = 0? What property of the preference relation did you use to get your answer? (c) Assume that prices are such that p1, p2 2 0 and that p` > 0 for some `e {1, 2} (i.e. the price of at least one good is positive). Derive Walrasian demand. What are the prices for which Walrasian demand is single-valued?arrow_forwardConsider a consumer with utility function u(x1, x2) = min{4 min{x1, x2}, x1 + x2} (a) Draw indifference curves passing through points (2, 2), (1, 2) and (4, 2). Make sure you correctly determine kink points. What properties of the preferences can you deduce from the shape of indifference curves? (b) When X = R 2 +, does UMP have a solution when p1 = p2 = 0? What property of the preference relation did you use to get your answer? (c) Assume that prices are such that p1, p2 > 0 and that p > 0 for some `e {1, 2} (i.e. the price of at least one good is positive). Derive Walrasian demand. What are the prices for which Walrasian demand is single-valued?arrow_forward
- 3.1 Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the MRS declines as x increases). a. U(x, y) = 3x + y. b. U(x, y) = √x. y. c. U(x, y) = √x + y. d. U(x, y) = √x² - y². xy e. U(x, y): x + yarrow_forwardSince the consumer is indifferent between all points on an indifference curve, no point is preferred. True Falsearrow_forwardSuppose that consumer has the following utility function: U(X, Y) = X@Y where 1 > a > 0 and 1 > b>0 are constants. Which of the following is correct? Preferences are convex and indifference curves are bowed inward towards the origin since Law of Diminishing Marginal Utility holds. Preferences are convex and indifference curves are bowed outward from the origin since Lavw of Diminishing Marginal Utility fails to hold. Preferences are concave and indifference curves are bowed outward from the origin since Law of Diminishing Marginal Rate of Substitution fails to hold. O Preferences are convex and indifference curves are bowed inward towards the origin since Law of Diminishing Marginal Rate of Substitution holds.arrow_forward
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