Jane receives utility from days spent traveling on vacation domestically \((D)\) and days spent traveling on vacation in a foreign country ( \(F\) ), as given by the utility function \(U(D, F)=10 D F .\) In addition, the price of a day spent traveling domestically is \(\$ 100,\) the price of a day spent traveling in a foreign country is \(\$ 400,\) and Jane's annual travel budget is \(\$ 4000\). a. Illustrate the indifference curve associated with a utility of 800 and the indifference curve associated with a utility of 1200 b. Graph Jane's budget line on the same graph. c. Can Jane afford any of the bundles that give her a utility of \(800 ?\) What about a utility of \(1200 ?\) d. Find Jane's utility-maximizing choice of days spent traveling domestically and days spent in a foreign country.

Short Answer

Expert verified
a) The indifference curves are \(F = 80/D\) for U=800 and \(F = 120/D\) for U=1200. b) The budget line is \(F = 10 - 0.25D\). c) Jane can afford the bundles for U=800 if there are real positive solutions to the equation \(10 - 0.25D = 80/D\), and she can afford the bundles for U=1200 if there are real positive solutions to the equation \(10 - 0.25D = 120/D\). d) The utility-maximizing bundle is the solution to the equation \(-D/F = -0.25\), which will be the optimal choice.

Step by step solution

01

Illustrate the Indifference Curves

First, rearrange the utility function \(U(D, F)=10 D F\) to isolate \(F\), therefore \(F = U/(10D)\). For utility of 800, substitute \(U=800\) to get \(F = 80/D\). Similarly, for utility of 1200, substitute \(U=1200\) to get \(F = 120/D\). Plot these two equations to illustrate the indifference curves.
02

Draw the Budget Line

Jane's total budget is \$4000. Hence, the budget constraint equation will be \(100D + 400F = 4000\). We then isolate \(F\) to draw the budget line, resulting in \(F = 10 - 0.25D\).
03

Find the Affordable Bundles

Substitute the budget line \(F = 10 - 0.25D\) into each of the indifference curve equations \(F = 80/D\) or \(F = 120/D\) and find the solutions (values of \(D\) and \(F\)). For \(U=800\), if the solutions satisfy \(10 - 0.25D = 80/D\), then Jane can afford the bundles. If not, she can't afford them. For \(U=1200\), repeat the process with \(F = 120/D\).
04

Find the Utility-maximizing Choice

Set the marginal rate of substitution equal to the price ratio and solve the resulting equation under the budget constraint to find the utility maximizing choice. The marginal rate of substitution (slope of the indifference curve) is the negative ratio of their marginal utilities (\(-MU_D/ MU_F\)). Here, \(MU_D = 10F\), \(MU_F = 10D\), then their ratio is \(-D/F\). The price ratio (slope of the budget line) is \(-P_D/P_F = -100/400 = -0.25\). Set \(-D/F = -0.25\), from which we can find the optimal amount of \(D\) and \(F\).

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Most popular questions from this chapter

Suppose that Bridget and Erin spend their incomes on two goods, food \((F)\) and clothing (C). Bridget's preferences are represented by the utility function \(U(F, C)=10 F C,\) while Erin's preferences are represented by the utility function \(U(F, C)=.20 F^{2} C^{2}\) a. With food on the horizontal axis and clothing on the vertical axis, identify on a graph the set of points that give Bridget the same level of utility as the bundle \((10,5) .\) Do the same for Erin on a separate graph. b. On the same two graphs, identify the set of bundles that give Bridget and Erin the same level of utility as the bundle (15,8) c. Do you think Bridget and Erin have the same preferences or different preferences? Explain.

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