Question

Jane’s utility function over jelly, J, and peanut butter, N, isU = JN. The price of a jar of jelly is $5. The price of a jar of peanut butter is $10. Jane has a budget of $100 to allocate to these two items. If Jane maximizes her utility, how much of each good does she consume?

1. J* = 5 jars, N* = 5 jars.

2. J* = 10 jars, N* = 10 jars.

3. J* = 10 jars, N* = 0 jars.

4. J* = 5 jars, N* = 10 jars.

5. J* = 10 jars, N* = 5 jars.

Answer 1

The utility maximizing combination of goods will be such that Jane spends all her income and attains highest possible utility.

Her utility function is given as,

U = J × N

Here J = jars of jelly

N = Peanut butter.

The utility maximizing combination of two goods is calculated using the following equation,

MUj/MUN = Pj/PN --------(1)

Pj = price of jelly

PN = price of peanut butter

The budget constraint of Jane is equal to,

Pj × J + PN × N = M

Here M = income of Jane

J = quantity of jelly

N = quantity of peanut butter.

We know the income of Jane is equal to $100. Putting M = $100 and Pj = 5 and PN = 10 we get budget constraint as,

5J + 10N = 100 -------(1)

Here MUj = marginal utility of jelly which will be calculated as,

MUj = U/J

MUj = (J×N)/J

MUj = N

And similarly MUN = marginal utility of peanut butter which is calculated as

MUN = U/N

MUN = (J×N)/N

MUN = J

And the price of jelly = $5

Price of peanut butter = $10

Putting all these values in equation (1) we get,

N*/J*= 5/10

N*/J* = 1/2

J* = 2N* ------------- (3)

Now putting J = 2N from equation (3) in equation (2) we get,

5×(2N*) + 10×N* = 100

10N* + 10N* = 100

20N* = 100

N* = 100/20

N* = 5.

Now putting N = 5 in equation (3) we get,

J* = 2N*

J* = 2×5

J* = 10

So the utility maximizing combination of jelly and peanut butter for Jane is equal to,

J* = 10 jars, N* = 5 jars.

So as you can see that correct answer is option E.