Tramontana.altervista.org Files Exercises (Consumer Theory)

Utility MaximizationChoice

Exercises of MicroeconomicsUtility Maximization - Choice (Ch. 7-8 Varian)

Fabio Tramontana (University of Pavia)

slides available at: http://tramontana.altervista.org/teaching.html

PhD in Economics at L.A.S.E.R.

Tramontana Exercises Micro


Outline

1 Utility Maximization

Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise



Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5

Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise 7.1

Consider preferences de�ned over the nonnegative orthant by

(x1,x2)� (y1,y2) if x1+ x2 < y1+ y2. Do these preferences exhibit

local nonsatiation?

If these are the only two consumption goods and the consumer

faces positive prices, will the consumer spend all of his income?

Explain.

What does �local nonsatiation� mean?

Local Nonsatiation

given any x in Xand any ε > 0, then there is some bundle y in X

with |x− y |< ε such that y � x .




Solution

In other words, there must always exist a better

bundle. Maybe it cannot be reached (it costs too much),

but it exists.

In our case the bundle (0,0) is the best one and no better bundle

exists.

So our consumer does not spend any amount of income.

It is not a real good what we are talking about.




Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise 7.2

A consumer has a utility function u(x1,x2) =max{x1,x2}. What is

the consumer's demand function for good 1? What is his indirect

utility function? What is his expenditure function?

The meaning of this kind of utility function is obvious. Only a good

is important, the one whose amount is the highest.

So, if prices di�er, the better choice for the consumer is to spend

all the income for the lowest priced good.

Otherwise, with the same prices, the better choice is the spend all

the income for only one good, randomly chosen.




Solution

So, the demand functions are

xi =

m/pi if pi < pj0 or m/pi if pi = pj0 if pi > pj

We can also build the indirect utility function:

v(p1,p2,m) =max{m/p1,m/p2}

The expenditure function relates income with utility and prices:

e(p,u). In our case:

e(p1,p2,u) = umin{p1,p2} .




Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise 7.3

A consumer has an indirect utility function of the form

v(p1,p2,m) =m

min{p1,p2}.

What is the form of the expediture function for this consumer?

What is the form of a (quasiconcave) utility function for this

consumer? What is the form of the demand function for good 1?

Let us start by writing down the indirect utility function in a

di�erent way:

v(p1,p2,m) =

m/p1 if p1 < p2m/p if p1 = p2m/p2 if p1 > p2




Solution

The best thing consists in starting from the demand function.

In fact, it is clear that the consumer will use all his income for the

good with the lowest price:

x1 =

m/p1 if p1 < p2any x1and x2such that p1x1+p2x2 =m if p1 = p20 if p1 > p2

and similarly for the demand function of the good 2.

This means that we have a �corner solution�, that is typical of a

linear utility function (or any monotonic transformation):

u(x1,x2) = x1+ x2




Solution

Finally, we must obtain the expenditure function, so a measure of

how much the consumer should spend in order to reach a certain

level of utility, given the goods' prices.

Our consumer spends:

xipi

where i denotes the good with the lowest price.

But the quantity of the good i is equal to m/pi that is also the

amount of utility reached.

In other words:

e(p1,p2,u) = umin{p1,p2} .




Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise 7.4

Consider the indirect utility function given by

v(p1,p2,m) =m

p1+p2.

(a) What are the demand functions? (b) What is the expenditure

function? (c) What is the direct utility function?

The standard way to obtain the (Marshallian) demand functions

given the indirect utility function is by using the Roy's identity:

Roy's identity

xi (p,m) =−∂v(p,m)

∂pi

∂v(p,m)∂m




Solution

So we need to calculate the partial derivatives of the indirect utility

function with respect to prices and income:

∂v∂p1

= ∂v∂p2

=− m(p1+p2)2

∂v∂m

= 1p1+p2

and by using the Roy's identity we obtain:

x1(m,p) = x2(m,p) =m

p1+p2

so the goods are equally consumed.




Solution

The expenditure function e(p,u) gives the information about how

much it costs to obtain a certain level of utility given the market

prices.

Costs substained by the consumer are generally of the following

form:

c = x1p1+ x2p2

We can use the demand functions to obtain:

c =m

p1+p2p1+

m

p1+p2p2




Solution

We know that the level of utility rechaed is exactly u = mp1+p2

, so

we have:

e(p1,p2,u) = (p1+p2)u

The direct utility function relates the level of utility with the

amount of goods consumed: u(x1,x2).We know that the consumer buys the same amount of the two

goods:

x1 = x2 =m

p1+p2

that also corresponds to the level of utility reached, given the

indirect utility function.




Solution

In other words we can write down:

u(x1,x2) =min{x1,x2} .




Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise 7.5

A consumer has a direct utility function of the form

U(x1,x2) = u(x1)+ x2.

Good 1 is a discrete good; the only possible levels of consumption

of good 1 are x1 = 0 and x1 = 1.

For convenience, assume that u(0) = 0 and p2 = 1.

(a) What kind of preferences does this consumer have?

These preferences are called �quasi-linear�, because they are

additive and linear in at least one good.




Exercise 7.5(b)

(b) The consumer will de�nitely choose x1 = 1 if p1 is strictly less

than what?

In order to answer to this question, let us consider the maximum

utility that can be reached with x1 = 0 and with x1 = 1.

If the consumer only consumes the good 2, given that its price is

equal to 1 and given that (from the utility function) there is 1-1

correspondence between level of utility and amount of good 2, we

have:

u|x1=0 =m




Solution

On the other hand, one unit of good 1 gives to the consumer an

utility of u(1).The consumer now can only buy an amount equal to m−p1 of the,

that is also the value of the utility brought by good 2.

So we have:

u|x1=1 = u(1)+m−p1

Now, in order to make the consumer preferer the situation with one

unity of good 1, this condition must be realized:

u|x1=0 < u|x1=1




Solution

That is:

p1 < u(1).




Exercise 7.5(c)

(c) What is the algebraic form of the indirect utility function

associated with this direct utility function?

It is a consequence of the previous point that the utility reached

will be the higher between the two considered, that is:

v(p1,p2,m) =max{m−p1+u(1),m} .



Exercise 8.5Exercise 8.6Exercise 8.7Exercise

Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise 8.5

Find the demand bundle for a consumer whose utility function is

u(x1,x2) = x32

1 x2 and her budget constraint is 3x1+4x2 = 100.

The Lagrangian function for this optimization problem is the

following:

L (x ,µ) = x32

1 x2−µ(3x1+4x2−100)

But it is more useful to see the utility function in logaritmic form

L (x ,λ ) =3

2lnx1+ lnx2−λ (3x1+4x2−100)




Solution

The �rst order conditions are obtained by di�erentiating the

Lagrangian function with respect to x1, x2 and λ . These derivatives

are equal to zero for the optimum bundle values:

32x1−3λ = 0

1x2−4λ = 0

3x1+4x2−100= 0

The �rst two equation can be seen as:

λ = 12x1

λ = 14x2

from which we obtain:

x1 = 2x2




Solution

By substituting x1 = 2x2 in the third f.o.c. we get:

6x2+4x2 = 100

⇓x∗2 = 10

and then:

x∗1 = 20; x∗2 = 10; λ ∗ = 140




Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise 8.6

Use the utility function u(x1,x2) = x12

1 x13

2 and the budget constraint

m = p1x1+p2x2 to calculate x(p,m), v(p,m), h(p,u) and e(p,u).

The Lagrangian function is the following:

L (x ,λ ) = x12

1 x13

2 −λ (p1x1+p2x2−m)

while the f.o.c. are:12x− 1

2

1 x13

2 = λp113x

12

1 x− 2

3

2 = λp2p1x1+p1x2 =m




Solution

by following a procedure similar to the one followed in the previous

exercise we obtain the Marshallian demand functions:

x1(p,m) = 35mp1

x2(p,m) = 25mp2

The indirect utility function can be obtained if we put the

Marshallian demand functions into the utility function:

v(p,m) =

(3

5

m

p1

) 12(2

5

m

p2

) 13

=(m5

) 56

(3

p1

) 12(

2

p2

) 13




Solution

It is now simple to obtain the expenditure function.

It is su�cient to take the indirect utility function and replace

�v(p,m)� with �u� and �m� with �e(p,u)�:

u =

(e(p,u)

5

) 56(

3

p1

) 12(

2

p2

) 13

and solve it for e(p,u):

e(p,u) = 5(p13

) 35(p22

) 25u

65




Solution

One of the properties of the expenditure function is the following:

Property of e(p,u)

If h(p,u) is the expenditure-minimizing bundle necessary to achieve

utility level u at prices p (Hicksian demand), then

hi (p,u) =∂e(p,u)

∂pifor i = 1, ...,k

so we need to calculate the partial derivatives of the expenditure

function, obtaining:

h1(p,u) =∂e(p,u)

∂p1=(p13

)− 25(p22

) 25 u

65

h2(p,u) =∂e(p,u)

∂p2=(p13

) 35(p22

)− 35 u

65

.




Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise 8.7

Extend the previous exercise to the case where

u(x1,x2) = (x1−α1)β1(x2−α2)

β2 and check the symmetry of the

matrix of substitution terms(

∂hj (p,u)∂pi

).

An alternative way to solve this problem consists in considering the

expenditure minimization problem, whose Lagrangian is:

L (x ,µ) = p1x1+p2x2−µ

[(x1−α1)

β1(x2−α2)β2−u

]from which we can derive the f.o.c.:

p1 = µβ1(x1−α1)β1−1(x2−α2)

β2

p2 = µβ2(x1−α1)β1(x2−α2)

β2−1

(x1−α1)β1(x2−α2)

β2 = u




Solution

If we divide the �rst equation by the second we obtain:

p1β2

p2β1=

x2−α2

x1−α1

The term x2−α2 can be obtained from the third equation:

x2−α2 =[(x1−α1)

−β1u] 1

β2

We can substitute it in the ratio between the �rst two equations

and solve it for x1, obtaining the Hicksian demand:

h1(p,u) = α1+

(p2β1

p1β2u

1β2

) β2β1+β2




Solution

The other demand is:

h2(p,u) = α2+

(p1β2

p2β1u

1β1

) β1β1+β2

The symmetry of the substitution matrix is proved by showing that:

∂h1(p,u)

∂p2=

∂h2(p,u)

∂p1=

[u

β1+β2

(β1

p1

)β2(

β2

p2

)β1] 1

β1+β2




Solution

The expenditure function is obtained by using the Hicksian

demands in p1x1+p2x2:

e(p,u)= p1

α1+

(p2β1

p1β2u

1β2

) β2β1+β2

+p2

α2+

(p1β2

p2β1u

1β1

) β1β1+β2

The following step consists in substituting �e(p,u)� with �m� and

�u� with �v(p,m)� in the expenditure function and solve it for

v(p,m) to obtain the indirect utility function:

v(p,m)=

[β1

β1+β2

(m−α2p2

p1−α1

)]β1[

β2

β1+β2

(m−α1p1

p2−α2

)]β2




Solution

Finally, we can use Roy's law to get the Marshallian demands:

x1(p,m) = 1β1+β2

(β1α2+β2

m−α1p1p2

)x2(p,m) = 1

β1+β2

(β2α1+β1

m−α2p2p1

) .




Outline


Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

2 Choice

Exercise 8.5

Exercise 8.6

Exercise 8.7

Exercise




Exercise (not from Varian)

Let there be two commodities, x1 and a composite commodity

called money M. assume a utility function

U = α log(x1)+βM

and income level Y . The price of x1 is p1 and the price of M is 1.

(a) Derive the Marshallian demand function for x1.

We can write down the constrained maximization problem:{maxx1,M

U = α log(x1)+βM

s.t. Y =M+p1x1




Solution

The Lagrangian is the following:

L (x1,M,λ ) = α log(x1)+βM−λ (M+p1x1−Y )

The f.o.c. are: ∂L∂x1

= α

x1−λp1 = 0

∂L∂M

= β −λ = 0∂L∂λ

= Y −M−p1x1 = 0

from the �rse f.o.c. we have that:

x1 =α

λp1




Solution

from the second we have that β = λ , so:

x1 =α

βp1.

Note that the demand for the good 1 is independent of income (has

zero income elasticity).




Exercise (b)

(b) Derive the Hicksian demand curve for x1.

To answer to this question it is better to write down the

constrained minimization problem:{minx1,M

M+p1x1

s.t. u = α log(x1)+βM

The Lagrangian is the following:

L (x1,M,µ) =M+p1x1+µ [α log(x1)+βM−u]




Solution

The f.o.c. are: ∂L∂x1

= p1−µα

x1= 0

∂L∂M

= 1−µβ = 0∂L∂ µ

= u−α log(x1)−βM = 0

from the �rst f.o.c. we have that:

x1 = µα

p1

given that, from the second f.o.c., we know that µ = 1/β :

x1 =α

βp1

that is the demand for good 1 is indipendent of the level of utility

reachable.Tramontana Exercises Micro



Exercise (c)

(c) What is the relationship between the Marshallian and the

Hicksian demands?

Is this a general relationship which holds for all utility functions

and, if not, why does it hold in this case?

Marshallian and Hicksian demands are the same;

In the Marshallian demand function, the quantity of good

demand depends upon its price and the income of the

consumer, while in the Hicksian demand function it depends

upon its price and the utility that the consumer wants to reach;




Solution

In our case the demand for the good 1 has zero elasticity (that

is, it's independent) both with respect to income and to

respect to utility. It only depends on the parameters and on

the price of the good. This is why the two demands are equal;

Obviously, this cannot be considered a general result, but only

what happens in this peculiar case.


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Tramontana.altervista.org Files Exercises (Consumer Theory)