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 gas mileage for styling) is equal to \(S /(4 G)\) What value of each index would she like to have in her car? d. Suppose that Brenda's marginal rate of substitution (of gas mileage for styling) is equal to \((3 S) / G\). What value of each index would she like to have in her car?

Step by step solution

01

Determine the budget in terms of styling and gas mileage

Given the price increases by $5000 for each additional unit of style and by $2500 for each additional unit of gas mileage, we can derive the following equation: \( 5000S + 2500G = 25000 \). This equation represents Brenda's budget in terms of style and gas mileage.

02

Manipulate the budget equation to put gas mileage on the horizontal axis

The budget equation is manipulated to express it in terms of gas mileage. Therefore, \( G = 10 - 2S \). This gives the combinations of gas mileage \( G \) and styling \( S \) Brenda could have with her $25000 budget. When styling increases by a unit, the gas mileage capacity decreases by 2, and vice versa.

03

Find the type of car Brenda would choose given her preferences

Given that Brenda receives three times as much satisfaction from an extra unit of styling as she does from gas mileage, Brenda would prefer a car with more style than gas mileage. Until the trade-off becomes 1 unit of style for 3 units of gas mileage (due to price constraints), Brenda will select cars with more style.

04

Determine value of each index for \(S /(4 G)\)

Marginal rate of substitution (MRS) is the amount of one good that a consumer is willing to give up for an additional unit of another good. Here we have \( MRS = S /(4 G) \). From the budget constraint \( G = 10 - 2S \), substituting \( G \) in this equation, we get: \( S = 4S / (40 - 8S) \). Solving this equation, we get: \( S = 4 \) and \( G = 10-2*4 = 2 \).

05

Determine value of each index for \((3 S) / G\)

Using the same method as in Step 4, we now have a different MRS ratio, \( MRS = (3 S) / G \). By substituting our budget constraint of \( G = 10 - 2S \) in this equation, we get: \( 3S / (10 - 2S) = 3 S \). Solving the equation gives \( S = 1.5 \) and \( G = 7 \).

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