Question

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) = 10DF. 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

a. Jane's indifference curve associated with a utility of 800 is shown as IC₁and 1C₂.

b. Jane's budget line on the same graph:

c. Yes, Jane can afford (29,2.75) and (11,7.25) gives her a utility of 800.

No, Jane can't afford any bundle that gives her a utility of 1200.

d. Jane's utility-maximizing choice of days spent traveling domestically and days spent in a foreign country is (11,7.25).

Step-by-step solution

Indifference curve for utilities 800 and 1200

Jane's utility function is:

U1(D, F) = 10DF = 800

U2(D, F) = 10DF = 1200

Jane's budget equation is:

100D +400F = 4000

Solving at utility =800

$$\begin{gathered} F=\frac{80}{D} \\ \mathrm{putting}\mathrm{in}\mathrm{the}\mathrm{budget}\mathrm{equation} \\ 100\mathrm{D}+\frac{400\times 80}{\mathrm{D}}=4000 \\ {\mathrm{D}}^2-40\mathrm{D}+320=0 \\ {\mathrm{D}}_1=28.94,29 \\ {\mathrm{D}}_2=11.05,11 \\ \mathrm{The}\mathrm{corresponding}\mathrm{values}\mathrm{for}\mathrm{days}\mathrm{spent} \\ \mathrm{traveling}\mathrm{on}\mathrm{vacation}\mathrm{in}\mathrm{a}\mathrm{foreign}\mathrm{country}\mathrm{will}\mathrm{be}: \\ {\mathrm{F}}_1=\frac{80}{{\mathrm{D}}_1}=2.75 \\ {\mathrm{F}}_2=\frac{80}{{\mathrm{D}}_2}=7.25 \end{gathered}$$

Jane's consumption bundles on the indifference curve are (29,2.75) or (11,7.25), which satisfies the utility condition (U=800) subject to budget constraint.

Similarly, Jane's travel utility of 1200 can be computed, which gives the following quadratic equation:

$$\begin{gathered} F=\frac{120}{D} \\ \mathrm{putting}\mathrm{in}\mathrm{the}\mathrm{budget}\mathrm{equation} \\ 100\mathrm{D}+\frac{400\times 120}{\mathrm{D}}=4000 \\ {\mathrm{D}}^2-40\mathrm{D}+480=0 \\ {\mathrm{D}}_1=20+9i \\ {\mathrm{D}}_2=20+9i \end{gathered}$$

The indifference curve for imaginary values of D can not be determined.

Jane's budget line and indifference curve

The optimal consumption bundle is the bundle of goods within the budget limit and has the highest utility (satisfaction)

The indifference curve (with optimal consumption bundle) and the budget line for Jane's utility of 800 are shown below:

Affordable bundles for utilities of 800 and 1200

Jane's indifference curve determined in step 1 shows that her travel consumption bundles for the utility of 800 are (29,2.75) or (11,7.25). In contrast, due to imaginary values computed for the utility of 1200, her indifference curve and the associated consumption bundle can’t be determined.

Utility maximizing choice

For the given utility function, Jane will maximize her utility when

$$\begin{gathered} \frac{M{U}_D}{P_D}=\frac{M{U}_F}{P_F} \\ \frac{10F}{100}=\frac{10D}{400} \\ D=4F \end{gathered}$$

Putting the relation in the utility function:

$$\begin{gathered} U(D,F)=10DF=40F^2 \\ \mathrm{for}\mathrm{F}=2.75 \\ {\mathrm{U}}_1=40\times 2.75\times 2.75=302.5 \\ \mathrm{for}\mathrm{F}=7.25 \\ {\mathrm{U}}_2=40\times 7.25\times 7.25=2,102.5 \end{gathered}$$

Thus, Jane's utility will maximize at (11,7.25).