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 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?

Short answer

a. Brenda can select combinations of style and gas mileage that satisfy the equation S = 5 - 0.5*G. b. With her preferences, Brenda will choose a car with more style than gas mileage. c. If Brenda's MRS is S/(4G), she would like to have a car with style index 8 and gas mileage index 2. d. If Brenda's MRS is (3S)/G, she would like to have a car with style index 2 and gas mileage index 3.

Step-by-step solution

Identify the budget constraint

Given the information, Brenda has a budget of $25,000. Each unit increase in the style index costs $5000 and each unit increase in the gas mileage index costs $2500. The budget constraint equation can be represented as: $25,000 = $5000*S + $2500*G. Divide the equation by 2500 to simplify: 10 = 2S + G.

Solve for a in the budget constraint equation

To visualize on a graph with gas mileage on the horizontal axis, solve for S: S = 5 - 0.5*G. This means that for each possible gas mileage index value, we can calculate the corresponding style index value Brenda can afford.

Interpret Brenda's preferences

Given that Brenda gets three times as much satisfaction from an extra unit of styling as she does from gas mileage, it means the marginal rate of substitution (MRS) of gas mileage for styling is 1/3. This indicates Brenda is willing to sacrifice 3 units of gas mileage for one more unit of style.

Find the optimal combination with Brenda's marginal rate of substitution (MRS)

Suppose Brenda's MRS (of gas mileage for styling) is equal to S/(4G). This means the optimal combination will satisfy the equation S/G = 4. Substituting S = 4G into the budget constraint equation from Step 2 (2S+G=10), we get G = 2 and S=8.

Find the optimal combination with a new MRS

Suppose Brenda's MRS is now equal to 3S/G. This means the optimal combination will satisfy the equation S/G = 1/3. Replacing S = G/3 into the budget constraint equation from Step 2, we get G = 3 and S=2.

Key concepts

Marginal Rate of Substitution

The marginal rate of substitution (MRS) is a cornerstone concept in microeconomics, particularly when explaining consumer behavior in the context of trade-offs. It refers to the rate at which a consumer is willing to substitute one good for another, while maintaining the same level of overall satisfaction or utility.

Think of it as the exchange rate between apples and oranges in a fruit market of consumer preferences. If you are equally happy with either fruit, you might be willing to trade one apple for one orange. But what if you love apples a bit more? You might need two oranges to compensate for giving up an apple. This is your personal MRS between apples and oranges.

In Brenda's case, her MRS indicates how many extra miles per gallon in gas mileage she is willing to forego for an additional unit of style in her new car purchase. Initially, Brenda's MRS of 3 to 1 suggests that style is much more important to her than gas mileage. Later, the specific MRS expressions given in the scenarios, such as \( S/(4G) \) and \( (3S)/G \), provide us with a mathematical reflection of Brenda's preferences, dictating her optimal choice between styling and gas mileage.

Consumer Preferences

Consumer preferences reveal how individuals prioritize different attributes of a product or service. Each person has their own unique set of preferences based on their tastes, experiences, and values, which guide their decisions in the market.

In the example of Brenda's car purchase, preferences are what make her value styling more than gas mileage. If Brenda always receives three times as much satisfaction from an extra unit of styling as she does from gas mileage, she exhibits a strong preference for style over fuel efficiency.

It's not just about liking one attribute more than another; it's about how much more. Her specific preference ratio helps us predict her behavior — which cars will catch her eye and which ones won't make the cut. By translating these preferences into tangible numbers, economists can model and foresee consumer choices within the confines of budget constraints.

Economic Trade-offs

Economic trade-offs are at the heart of decision-making. Every choice has an opportunity cost, meaning that by choosing one option, you're inherently giving up the opportunity to choose others.

For Brenda, this means every dollar she spends on styling is a dollar she can't allocate towards gas mileage. The budget constraint equation \(25,000 = 5,000S + 2,500G\) that was provided in her car search scenario, lays out the trade-offs Brenda must consider given her limited budget of \$25,000.

Understanding these trade-offs is crucial because they reflect the scarcity of resources, in this case, Brenda’s budget. She must balance the allure of a stylish car against the practicality of fuel efficiency, within the bounds of what she can afford. Her decisions will mirror her effort to maximize satisfaction from this purchase, a concept known to economists as utility maximization.