Advanced MicroEconomics

AEB 6106: Problem Set 3

Due Oct. 10 at the start of class

1. In problem set 2, you worked with the following utility function:

𝑈 = 𝑥 4 5𝑦

1 5

You derived the Marshallian demand functions:

𝑥 = 4

5

𝐼

𝑃𝑥 , 𝑦 =

1

5

𝐼

𝑃𝑦

And the indirect utility function:

𝑉 = ( 4

𝑃𝑥 )

4 5

( 1

𝑃𝑦 )

1 5 𝐼

5

From this we can derive the expenditure function:

𝐸 = 5𝑈 ( 4

𝑃𝑥 )

− 4 5

( 1

𝑃𝑦 )

− 1 5

And compensated demand for x:

𝑥𝑐 = 𝑈 ( 4𝑃𝑦 𝑃𝑥

)

1 5

a. Derive the own-price Slutsky equation for this utility function for good x.

b. Explain what the terms in the Slutsky equation mean.

2. In problem set 2, you worked with the following utility function:

𝑈 = 3𝑥 + 10𝑦 You derived the Marshallian demand functions:

If 3

10 >

𝑃𝑥

𝑃𝑦 , then x =

𝐼

𝑃𝑥 , y = 0

If 3

10 <

𝑃𝑥

𝑃𝑦 , then y =

𝐼

𝑃𝑦 , x = 0

And the indirect utility function:

𝑉 = max {3 ( 𝐼

𝑃𝑥 ) , 10 (

𝐼

𝑃𝑦 )}

From this you could derive the expenditure function:

𝐸 = 𝑈 min { 𝑃𝑥 3

. 𝑃𝑦 10

}

And the compensated demand functions:

If 3

10 >

𝑃𝑥

𝑃𝑦 , then 𝑥𝑐 =

𝑈

3 , 𝑦 𝑐 = 0

If 3

10 <

𝑃𝑥

𝑃𝑦 , then 𝑦 𝑐 =

𝑈

10 , 𝑥𝑐 = 0

a. Derive the own-price Slutsky equation for this utility function for good x, assuming that 3

10 >

𝑃𝑥

𝑃𝑦

and that this remains the case after the price change.

b. Discuss the terms in the Slutsky equation for this utility function and the assumed scenario.

(Note- this discussion should vary considerably from your discussion in part a)

c. Derive the own-price Slutsky equation for this utility function for good x, assuming that

originally 3

10 >

𝑃𝑥

𝑃𝑦 but after the price change,

3

10 <

𝑃𝑥

𝑃𝑦 . Note that you will not be able to take partial

derivatives to derive the terms in the Slutsky equation, but will instead have to consider discrete

changes from the original price to the new price.

3. In problem set 2, you worked with the following utility function:

𝑈 = min{5𝑥, 6𝑦} You derived the Marshallian demand functions:

𝑥 = 6𝐼

6𝑃𝑥 + 5𝑃𝑦 , 𝑦 =

5𝐼

6𝑃𝑥 + 5𝑃𝑦

And the indirect utility function:

𝑉 = 30𝐼

6𝑃𝑥 + 5𝑃𝑦

From this, we could get the following expenditure function:

𝐸 = 𝑈(6𝑃𝑥 + 5𝑃𝑦)

30

And the compensated demand function for x:

𝑥𝑐 = 𝑈

5

a. Derive the own-price Slutsky equation for this utility function for good x.

b. Explain the terms in the Slutsky equation specifically for this utility function (Note- your

discussion should be different from the two previous discussions).

4. In problem set 2, you worked with the following utility function:

𝑈 = 4𝑥 1 4 + 3𝑦

You derived the Marshallian demand functions:

𝑥 = ( 𝑃𝑦

3𝑃𝑥 )

4 3

, 𝑦 = 𝐼

𝑃𝑦 − (

1

3 )

4 3

( 𝑃𝑦 𝑃𝑥

)

1 3

If you worked through the additional practice problems, you derived the indirect utility function:

𝑉 = 3 2 3 (

𝑃𝑦 𝑃𝑥

)

1 3

+ 3𝐼

𝑃𝑦

For this utility function, Hicksian and Marshallian demand for good x are the same:

𝑥𝑐 = ( 𝑃𝑦

3𝑃𝑥 )

4 3

Derive the own-price Slutsky equation for this utility function for good x.

5. For both goods x and y in each of the four problems above, indicate whether the goods are

normal or inferior goods.

6. For the utility functions in questions 1 – 4, determine if goods x and y are gross substitutes,

gross complements, or gross independent (Note- this requires 2 calculations) and determine if they

are net substitutes, net complements, or net indepent.

7. For the demand functions for good x contained in questions 1 – 4, derive the price, income, and

cross-price elasticities of demand.