Question

I need help with Marginal rate of substitution

Brenda wants to buy a new car and has a budget of S25,000. 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 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 $5,000. She also observed that as the gas-mileage index rises by one unit, the price of the car increases by $3,125. 10 Illustrate the various combinations of style (S) and gas mileage (G) that Brenda could select with her $25,000 budget. Using the line drawing tool, illustrate Brenda's budget line. Label this line Budget Line 田 Carefully follow the instructions above, and only draw the required object. Suppose Brenda's preferences are such that she always receives five times as much satisfaction from an extra unit of styling as she does from gas mileage. What type of car will Brenda choose? Brenda will select a car with 5 units of styling and 0 units of gas mileage. (Enter your response rounded to two decimal places.) Gas mileage (index units) Suppose that Brenda's marginal rate of substitution (of gas mileage for styling) is equal to 3G.What value of each index would she like to have in her car? Brenda will select a car withunits of styling andunits of gas mileage. (Enter your response rounded to two decimal places.)

Answer 1

Let S be the style of the car and G be the gas mileage.

According to the information given, P_S = 5000 and P_G = 3125

Budget constraint is given by the following equation:

P_SS + P_GG = 25000 => 5000S + 3125G = 25000

Simplifying we get: 8S + 5G =40

Also given that Brenda's preferences are such that she receives five much times as extra satisfaction from styling as she does from the gas mileage i.e.

$\frac{MU_S}{MU_G} = 5 \Rightarrow U = 5S + G$ type utility function

For an interior solution of utility maximization problem -

$\frac{MU_S}{MU_G} = \frac{P_S}{P_G} = MRS_{S,G}$ which doesn't hold true in our case. Hence we have a corner solution.

If S = 0, then G = 8 by budget constraint, U = 8

If G = 0, then S = 5 and U = 25

Thus utility is maximized at the combination (S,G) = (5,0)

Now if

$\frac{MU_S}{MU_G} = MRS_{S,G} = \frac{1}{MRS_{G,S}}$

$\Rightarrow \frac{1}{\frac{S}{3G}} = \frac{3G}{S} = 5 \Rightarrow S = \frac{3G}{5}$

Substituting in budget constraint, we get:

8S+ 5G = 40

$\Rightarrow \frac{24G}{5} + 5G = 40 \Rightarrow \frac{24G+25G}{5} = 40$

$\Rightarrow49G = 200 \Rightarrow G = \frac{200}{49} = 4.082$

and

$S = \frac{3G}{5} = \frac{3*4.082}{5} = 2.45$