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 $2000. Suppose F is on the horizontal axis and D is on the vertical axis. Her optimal consumption point is given by(partial units are allowed)

1. F = 0 and D = 100

2. F = 5 and D = 20

3. F = 2.5 and D = 10

4. F = 2.5 and D = 20

Answer 1

Annual budget = $2,000

Price of a day spent travelling domestically (Pd) = $100

Price of a day spent travelling in a foriegn country (Pf) = $400

The budget equation is as follows -

[Price of day spent travelling domestically * Number of days travelling on vacation domestically] + [Price of day spent travelling in foreign country * Number of days travelling on vacation in foreign country] = Annual budget

[Pd * D] + [Pf * F] = Annual budget

100D + 400F = 2,000

The utility function is as follows -

U = 10DF

Calculate MU_D -

MU_D = dU/dD = d(10DF)/dD = 10F

Calculate MU_F -

MU_F = dU/dF = d(10DF)/dF = 10D

The condition for optimal consumption bundle is as follows -

MU_D/MU_F = P_D/P_F

10F/10D = 100/400

F/D = 1/4

4F = D

or,

D = 4F

Putting the value of D in budget equation,

100D + 400F = 2,000

[100 * 4F] + 400F = 2,000

400F + 400F = 2,000

800F = 2,000

F = 2,000/800

F = 2.5

D = 4F = 4 * 2.5 = 10

So,

Her optimal consumption bundle is as follows -

F = 2.5 and D = 10

Hence, the correct answer is the option (3)