Question

Suppose Paul’s utility is a function of her consumption of soda (X) and slices of pizza (Y) such that utility U = 10X0.5Y0.4. This means his marginal utility from consumption of sodas is MUX = 5X-0.5Y0.4 and her marginal utility from pizza is MUY = 4X0.5Y-0.6.

a.What is the value of Paul’s MRSXY when he buys 20 sodas (X) and 3 slices of pizza (Y)?

b.If the price of a soda is $1 and the price of slices of pizza is $4, show that Paul is not maximizing her utility with this consumption bundle.

c.Circle one: Paul should consume: more sodas and less pizza, or more pizza and less soda

d.For a budget of $54, what is Paul’s utility maximizing consumption levels of pizza and soda?e.What is his total utility at this level of consumption of soda and pizza? Round to the nearest tenth.

Answer 1

U = 10X^0.5Y^0.4

MUX = 5X^-0.5Y^0.4

MUY = 4X^0.5Y^-0.6

a.

MRSXY= MUX/MUY= 5X^-0.5Y^0.4 / 4X^0.5Y^-0.6 = 5Y/4X

when he buys 20 sodas (X) and 3 slices of pizza (Y):

MRSXY = 5(3)/4(20)= 3/16= 0.1875

b.

If the price of a soda is $1 and the price of slices of pizza is $4 that is

Ps= Price of soda = 1

Price of a slice of pizza= Pp= 4

For utility maximization the condition is:

MRSXY = Ps/Pp

Here Ps/Pp= 1/4= 0.25

So MRSXY is not equals to Ps/Pp which implies that Paul is not maximizing her utility with this consumption bundle.

c.

From above parts, we can observe that:

MRSXY < Ps/Pp

It implies that consumer is getting less utility by consuming an additional unit of Soda relative to utility from an additional unit of Pizza.

So Paul should consume:more pizza and less soda

d.

For a budget of $54:

Budget line: X+4Y= 54

Use optimal condition:

MRSXY= Ps/Pp

5Y/4X = 1/4

5Y= X Equation 1

Use this equation 1 in budget line:

5Y+4Y = 54

9Y= 54

Y= 6 Optimal quantity of Pizza

Use Y=6 in Equation 1:

X= 5(6)= 30 Optimal quantity of Soda

e. Total utility:

U = 10X^0.5Y^0.4

U = 10(10)^0.5(6)^0.4 = 64.75