Question

Brenda wants to buy a new car and has a budget of \(25,000. She has just found a magazine that assigns each car an index for styling and an index for gas mileage. Each index runs from 1 to 10, with 10 representing either the most styling or the best gas mileage. While looking at the list of cars, Brenda observes that on average, as the style index increases by one unit, the price of the car increases by \)5000. She also observes that as the gas-mileage index rises by one unit, the price of the car increases by \(2500.

a. Illustrate the various combinations of style (S) and gas mileage (G) that Brenda could select with her \)25,000 budget. Place gas mileage on the horizontal axis.

b. Suppose Brenda’s preferences are such that she always receives three times as much satisfaction from an extra unit of styling as she does from gas mileage. What type of car will Brenda choose?

c. Suppose that Brenda’s marginal rate of substitution (of the gas mileage for styling) is equal to S/(4G). What value of each index would she like to have in her car?

d. Suppose that Brenda’s marginal rate of substitution (of the gas mileage for styling) is equal to (3S)/G. What value of each index would she like to have in her car?

Short answer

a. Brenda’s optimal consumption bundle of style and gas mileage are P(2,4), P’(4,3), & P’’(6,2).

b. When Brenda chooses a car whose styling level is 6 and gas mileage level 2, she will receive three times as much satisfaction from an extra styling unit as she does from gas mileage.

c. The index value Brenda will have in the car is (1,4), level 1 for styling and 4 for gas mileage.

d. The value of the index Brenda will have in the car is (3,1) level 3 for styling and 1 for gas mileage.

Step-by-step solution

Brenda’s index for styling and gas mileage

The index table below shows that Brenda, with a budget of $25,000, will choose either (2,4), (4,3), or (6,2).

The following diagram shows Brenda’s budget line and optimal consumption bundles at P, P’, & P’’.

The intersection of the budget line and the different IC, gives different optimal bundles, as shown above.

When Brenda’s satisfaction for styling is thrice that of gas mileage

Numeric value to the quality of goods signifies the satisfaction level of a consumer.

The consumption bundle P’’(6,2) in the index table implies that Brenda chooses styling level 6 and gas mileage level 2; her satisfaction level for style is 6, and gas mileage is 2.

Thus, the consumption bundle (6,2) gives Brenda three times satisfaction for styling to gas mileage.

When Brenda’s MRS is S/4G

The marginal rate of substitution is the ratio of marginal utilities of goods X and Y.

For MRS=S/4G, Brenda will choose the styling of level 1 and gas mileage of level 4, for which the corresponding prices are $2,500 and $20,000, respectively.

The cost of the car at this MRS will be $22,500, which satisfies Brenda’s budget constraint of $25,000.

When Brenda’s MRS is 3S/G

For MRS=3S/G, Brenda will choose styling level 3 and gas mileage level 1, for which the corresponding prices are $7,500 and $5,000, respectively.

The cost of the car at this MRS will be $12,500, which satisfies Brenda’s budget constraint of $25,000.