Brenda wants to buy a new car and has a budget of 25,000 ....

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?

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?

Key Concepts

Trade-Off Analysis

Trade-off analysis in consumer choice involves understanding the opportunity cost associated with increasing consumption of one good at the expense of another. It encapsulates the idea that, due to limited resources, consuming more of one item requires sacrificing consumption of another. By analyzing the incremental costs and benefits, consumers can make optimized decisions that maximize their overall satisfaction.

Utility Maximization

Utility maximization is the process by which a consumer selects the combination of goods that maximizes their satisfaction, subject to the constraint of their limited budget. This involves choosing a point where the highest possible indifference curve is tangent to the budget constraint, meaning the marginal rate of substitution equals the ratio of the goods' prices. In this condition, any move from the optimal bundle would result in a lower level of utility.

Marginal Rate of Substitution (MRS)

The marginal rate of substitution is a key concept that measures the rate at which a consumer is willing to give up units of one good in exchange for an additional unit of another good while maintaining the same level of utility. It is defined as the ratio of the marginal utilities of the two goods, and it is used to understand consumer preferences and the trade-offs they are willing to make in their consumption choices.

Indifference Curves

Indifference curves represent a consumer's preferences by depicting all combinations of two goods that provide the same level of satisfaction or utility. Each curve corresponds to a different utility level, and consumers are indifferent among the combinations lying on the same curve. These curves are typically downward sloping and convex to the origin, reflecting the trade-offs consumers are willing to make between goods.

Budget Constraint

The budget constraint is a fundamental concept in consumer theory that represents all combinations of goods and services a consumer can afford given their income and the prices of those goods. It is typically depicted as a line or curve in a graph where the slope shows the rate at which one good must be sacrificed to consume more of the other. This constraint sets the feasible set of choices from which a consumer can select to maximize their satisfaction.

Transcript

00:01 So brenda wants to buy a new car and she has a budget of $25 ,000.

00:10 And she has just found a magazine that signs each year an index for stalling and an index for gas mileage.

00:17 So each index runs from 1 to 10, with 10 representing either the most styling or the best mileage.

00:24 And while looking at the list of cars, she observes that on average, as the style index increases by 1 unit, the price of so as the style index increases by one unit the price of the car increases by $5 ,000 and she also observes that as the gas mileage index rises by one units the price increase by $2 ,500 so as gas mileage index increases by one unit price increases by 2 ,500 and the index runs from 1 to 10.

01:29 So now the first question says illustrates the various combination of style s, s being the style index and g being the gas mileage that brenda could select with a $25 ,000 budget and place the gas mileage under a horizontal axis.

01:50 So here we have style s and style g.

01:55 So to answer the first question, for every style that brenda, so for every style index that brenda has to spend $5 ,000 on, so for every style index, right, brenda has to spend $5 ,000, $5 ,000.

02:11 So if brenda spends the amount of $25 ,000 on style, she can achieve the style index of five, because five times on $5 ,000 is $25 ,000.

02:21 So with our budget of $25 ,000 she can afford a style index of $5 ,000 each so a gas mileage index rises by one unit when she spends $2 ,500 right so she can achieve a gas index or index of 10 because when the $25 ,000 is spent on guys mileage because $2 ,500 multiplied by $10 is $25 ,000.

03:05 So with that budget of $25 ,000, she can afford a gas index of 10.

03:35 So now the slope of the budget line will be represented by minus ps divided by pg is equal to minus 2 ,500 divided by 5 ,000 and then this will give you a slope of minus 1 over 2.

04:02 So the various combinations of gas mileage, bng and stal bns, that brenner can select with that 2 ,500, 25 ,500 will be shown in a table that i'll be drawing right now.

04:17 So various combinations with $25 ,000 budgets.

04:32 So we'll have the gas mileage over here, then the stow over here, then the gas mileage of 5 ,000, i'm sorry, $2 ,500, and the stow is $5 ,000, and the total price.

05:15 So the gas mileage is 10, 8, 6, 4, 2, and 0, and the style is 0, 1 .1.

05:42 Two three four and five so the available combination for the gas mileage of two thousand five hundred would be with gas mileage of ten would be twenty five thousand and for this one will be twenty thousand and this will be fifteen thousand then ten thousand then five thousand and then zero for the stow was zero thousand, 10 ,000, 15 ,000, 20 ,000, and 25 ,000.

06:26 And the total price would be 25 ,000, 25 ,000, 25 ,000, 25 ,000 and 25 ,000.

06:38 So this is the available combination for the budgets that we have of $25 ,000.

06:44 So moving on to the next question.

06:54 So 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...

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