Question

Anne has a job that requires her to travel three out of every four weeks. She has an annual travel budget and can travel either by train or by plane. The airline on which she typically flies has a frequent-traveler program that reduces the cost of her tickets according to the number of miles she has flown in a given year. When she reaches 25,000 miles, the airline will reduce the price of her tickets by 25 percent for the remainder of the year. When she reaches 50,000 miles, the airline will reduce the price by 50 percent for the remainder of the year. Graph Anne’s budget line, with train miles on the vertical axis and plane miles on the horizontal axis.

Short answer

Anne’s budget lines are given by MM₁ and MM₂. The budget line rotates outward when her real income increases due to discounted prices on airplane tickets when she traveled 25,000 and 50,000 miles.

Step-by-step solution

Anne’s budget equation

Anne’s budget equation is given by:

$C_{T}T+C_{P}P\le M$

where C_t= Cost of train ticket

C_P= Cost of Plane ticket

T & P = Train and Plane miles

M = Anne's Income

Anne’s budget equation after price reductions 

The budget line rotates due to the change in real income. Taking the change in the price of goods at the x-axis, the budget line rotates outward on the horizontal axis when real income increases due to a price decrease and rotates inward when real income decreases due to the rise in the price level.

When Anne gets a 25% discount on plane tickets for traveling 25,000 miles, the budget equation will be:

$$\begin{gathered} C_{T}T+\frac{25\times C_P}{100}\times 25000\le M \\ {C}_{T}T+\frac{C_P}{4}\times 25000\le M \end{gathered}$$

When Anne gets a 50% discount on plane tickets for traveling 50,000 miles, the new budget equation will be:

$$\begin{gathered} C_{T}T+\frac{50\times C_P}{100}\times 50000\le M \\ {C}_{T}T+\frac{C_P}{2}\times 50000\le M \end{gathered}$$

Due to the discount on plane tickets, Anne’s relative income changes. So, the budget line shifts from MM₁ to MM₂.

The figure shows distances traveled by train and airplane as T₁and P₁initially (when Anne travels 25,000 by plane) and T₁and P₂finally (when Anne travels 50,000 by plane). The shift in the budget line is due to the change in real income at a discounted price.