Chapter 4 · Chapter 4 UTILITY MAXIMIZATION AND CHOICE. 2 ... Tennis racket ? Cable TV? 4 What can...

1

Chapter 4

UTILITY MAXIMIZATION

AND CHOICE

2

Our Consumption Choices

Suppose that each month we have a

stipend of $1250.

What can we buy with this money?

3

What can we buy with this money?

Pay the rent, 750 $.

Food at WholeFoods, 300 $

Clothing 50 $

Gasoline, 40 $

Movies and Popcorn, 60 $

CD’s, 50 $

Tennis racket ?

Cable TV?

4

What can we buy with this money?

Pay the rent, 700 $.

Food at Ralph’s, 200 $

Clothing 40 $

Gasoline, 80 $

Movies and Popcorn, 60 $

CD’s, 20 $

Tennis racket 150 $

Cable TV 50 $

5

Which is Better?

If we like healthy food and our own bedroom, we will choose the first option

If we like crowded apartments and we absolutely need the Wilson raquet, we will choose the second option

The actual choice depend on our TASTE, on our PREFERENCES

6

Our Consumption Choices

Our decisions can be divided in two parts

We have to determine all available choices, given our income: BUDGET CONSTRAINT

Of all the available choices we choose the one that we prefer: PREFERENCES and INDIFFERENCE CURVES

7

UTILITY MAXIMIZATION PROBLEM

8

Basic Model• There are N goods.

• Consumer has income m.

• Consumer faces linear prices {p1,p2,…,pN}.

• Preferences satisfy completeness, transitivity

and continuity.

• Preferences usually satisfy monotonicity and

convexity.

• Consumer’s problem: Choose {x1,x2,…,xN} to

maximize utility u(x1,x2,…,xN) subject to budget

constraint and xi ≥ 0.

9

Budget Constraints• If an individual has m dollars to allocate

between good x1 and good x2

p1x1 + p2x2 m

Quantity of x1

Quantity of x2The individual can afford

to choose only combinations

of x1 and x2 in the shaded

triangle

If all income is spent

on x2, this is the amount

of x2 that can be purchased2p

m

If all income is spent

on x1, this is the amount

of x1 that can be purchased

1p

m

Slope=-p1/p2

10

Budget Constraints

• Suppose income increases

– Then budget line shifts out

• Suppose p1 increases

– Then budget line pivots around upper-left

corner, shifting inward.

11

Nonlinear Budget Constraints

• Example: Quantity Discounts.

• m=30

• p2=30

• p1=2 for x1<10

• p1=1 for x1>10

12

Consumer’s Problem (2 Goods)• Consumer chooses {x1,x2} to solve:

Max u(x1,x2) s.t. p1x1+p2x2 ≤ m

• Solution x*i(p1,p2,m) is Marshallian Demand.

Results:

1. Demand is homogenous of degree zero:

x*i(p1,p2,m)= x*i(kp1,kp2,km) for k>0

Idea: Inflation does not affect demand.

2. If utility is monotone then the budget binds.

13

UMP: The Picture

• We can add the individual’s preferences

to show the utility-maximization process

Quantity of x1

Quantity of x2

U1

A

The individual can do better than point A

by reallocating his budget

U3

C The individual cannot have point C

because income is not large enough

U2

B

Point B is the point of utility

maximization

14

Soln 1: Graphical Method• Utility is maximized where the indifference curve

is tangent to the budget constraint

• Equate bang-per-buck, MU1/p1 = MU2/p2

Quantity of x1

Quantity of x2

U2

B

constraintbudget of slope2

1

p

p

21constant 1

2 curve ceindifferen of slope

x

U

x

U

dx

dx

U

212

1

x

U

x

U

p

p

At optimum:

15

Soln2: Substitution Method• Rearrange budget constraint: x2 = (m-p1x1)/p2

• Turn into single variable problem:

Max u(x1,(m-p1x1)/p2)

• FOC(x1): MU1 + MU2(-p1/p2) = 0.

• Rearrange: MU1/MU2 = p1/p2

• Example: u(x1,x2) =x1x2

16

Soln3: Lagrange Method

• Maxmize the Lagrangian:

L = u(x1,x2) + [m-p1x1-p2x2]

• First order conditions:

MU1 - p1 = 0

MU2 - p2 = 0

• Rearranging: MU1/MU2 = p1/p2

• Example: u(x1,x2) =x1x2

17

LIMITATIONS OF FIRST ORDER

APPROACH

18

1. Second-Order Conditions

• The tangency rule is necessary but not

sufficient

• It is sufficient if preferences are convex.

– That is, MRS is decreasing in x1

• If preferences not convex, then we must

check second-order conditions to ensure

that we are at a maximum

19

1. Second Order Conditions

• Example in which the tangency condition is

satisfied but we are not at the optimal bundle.

Quantity of x1

Quantity of x2

B

A

A = local min

B = global max

C = local maxB

A

C

20

1. Second-Order Conditions

• Example: u(x1,x2) = x12 + x2

2

• FOCs from Lagrangian imply that

x1/x2 = p1/p2

• But SOC is not satisfied: L11 = L22 = 2 > 0

• Actual solution:

– x1 = 0 if p1>p2

– x2 = 0 if p1<p2

21

2. Corner Solutions• In some situations, individuals’ preferences

may be such that they can maximize utility

by choosing to consume only one of the

goods

Quantity of x

Quantity of yAt point A, the indifference curve

is not tangent to the budget constraintU2U1 U3

A

Utility is maximized at point A

22

2. Corner Solutions

• Boundary solution will occur if, for all (x1,x2)

MRS>p1/p2 or MRS<p1/p2

• That is bang-per-buck from one good is always

bigger than from the other good.

• Formally, we can introduce Lagrange multipliers

for boundaries.

• If preferences convex, then solve for optimal

(x1,x2). If find x1<0, then set x1*=0.

23

2. Corner Solutions

• Example: u(x1,x2) = x1 + x2

• MRS = - /

• Price slope = -p1/p2

• Which is bigger?

24

3. Non-differentiable ICs

• Suppose preferences convex.

• At kink MRS jumps down.

• Solution occurs at kink if

MRS- ≥ p1/p2 ≥ MRS+

25

3. Non-differentiable ICs

• Perfect complements: U(x1,x2) = min (x1, x2)

• Utility not differentiable at kink.

• Solution occurs at:

x1 = x2

26

GENERAL THEORY

27

The n-Good Case

• The individual’s objective is to maximize

utility = U(x1,x2,…,xn)

subject to the budget constraint

m = p1x1 + p2x2 +…+ pnxn

• Set up the Lagrangian:

L = U(x1,x2,…,xn) + (m - p1x1 - p2x2 -…- pnxn)

28

The n-Good Case• First-order conditions for an interior

maximum:

L/x1 = U/x1 - p1 = 0

L/x2 = U/x2 - p2 = 0•••

L/xn = U/xn - pn = 0

L/ = m - p1x1 - p2x2 - … - pnxn = 0

29

Implications of First-Order Conditions

• For any two goods,

j

i

j

i

p

p

xU

xU

/

/

• This implies that at the optimal

allocation of income

j

iji

p

pxxMRS ) for (

30

Indirect Utility Function• Substituting optimal consumption in the utility

function we find the indirect utility function

V(p1,p2,…,pn,m) = U(x*1(p1,…,pn,m),…,x*n(p1,…,pn,m))

• This measures how the utility of an individual

changes when prices/income varies

• Enables us to determine the effect of

government policies on utility of an individual.

31

Indirect Utility: Properties1. v(p1,..,pn,m) is homogenous of deg 0:

v(p1,..,pn,m) = v(kp1,..,kpn,km)

Idea: Inflation does not affect utility.

2. v(p1,..,pn,m) is increasing in income and

decreasing in prices.

3. Roy’s identity:

If p1 rises by $1, then income falls by x1×$1.

Agent also changes demand, but effect small

m

Vmppx

p

VNi

i

),,...,( 1

32

EXPENDITURE MINIMISATION

PROBLEM

33

Expenditure Minimization

• We can find the optimal decisions of our

consumer using a different approach.

• We can minimize her/his expenditure

subject to a minimum level of utility that

the consumer must obtain.

• This is important to separate income

and substitution effects.

34

Basic Model• There are N goods.

• Consumer has utility target u.

• Consumer faces linear prices {p1,p2,…,pN}.

• Preferences satisfy completeness, transitivity

and continuity.

• Preferences usually satisfy monotonicity and

convexity.

• Consumer’s problem: Choose {x1,x2,…,xN} to

minimise expenditure p1x1+…+pNxN subject to

u(x1,x2,…,xN) ≥ u and xi ≥ 0.

35

Consumer’s Problem (2 Goods)• Consumer chooses {x1,x2} to solve:

Min p1x1+p2x2 s.t. u(x1,x2) ≥ u

• Solution h*i(p1,p2,u) is Hicksian Demand.

• Expenditure function

e(p1,p2,u) = min [p1x1+p2x2] s.t. u(x1,x2) ≥ u

= p1h*1(p1,p2,u) +p2h*2(p1,p2,u)

• Problem is dual of UMP

36

Expenditure level E2 provides just enough to reach U

EMP: The Picture

Quantity of x1

Quantity of x2

U

Expenditure level E1 is too small to achieve U

Expenditure level E3 will allow the

individual to reach U but is not the

minimal expenditure required to do so

A

• Point A is the solution to the dual problem

37

Graphical Solution

• Slope of Indifference Curve = -MRS

• Slope of Iso-expenditure curves = -p1/p2

• At optimum

• Idea: equate bang-per-buck

• If p2=2p1 then MU2=2MU1 at optimum.

212

1

x

U

x

UMRS

p

p

38

Lagrangian Solution

• Minimize the Lagrangian:

L = p1x1+p2x2 + [u-u(x1,x2)]

• First order conditions:

p1 - MU1 = 0

p2 - MU2 = 0

• Rearranging: MU1/MU2 = p1/p2

• Example: u(x1,x2) =x1x2

39

EMP & UMP• UMP: Maximize utility given income m

– Demand is (x1,x2)

– Utility v(p1,p2,m)

• EMP: Minimize spending given target u

– Suppose choose target u = v(p1,p2,m).

– Then (h1,h2) = (x1,x2)

– Then e(p1,p2,u) = m

• Practically useful: Inverting Expenditure Fn

– Fixing prices, e(v(m)) = m, so v(m)=e-1(m)

– If u=x1x2 then e=2(up1p2)1/2. Inverting, v(m)=m2/(4p1p2)

40

Expenditure Function: Properties1. e(p1,p2,u) is homogenous of degree 1 in (p1,p2)

– If prices double constraint unchanged, so need

double expenditure.

2. e(p1,p2,u) is increasing in (p1,p2,u)

3. Shepard’s Lemma:

– If p1 rises by ∆p, then e(.) rises by ∆p×h1(.)

– Demand also changes, but effect second order.

4. e(p1,p2,u) is concave in (p1,p2)

),,(),,( 2121 upphuppep

i

i

41

e(p1,…)

But the consumer can do

better by reallocating

consumption to goods

that are less expensive.

Actual expenditures will

be less than esub

esub

If he continues to buy the

same set of goods as p*1

changes, his expenditure

function would be esub

Expenditure Fn: Concavity and Shepard’s Lemma

p1

e(p1,…)

At p*1, the person spends

e(p*1,…)=p*1x*1+ … + p*nx*n

e(p*1,…)

p*1

42

Hicksian Demand: Properties1. h(p1,p2,u) is homogenous of degree 0 in (p1,p2)

– If prices double constraint unchanged, so demand

unchanged.

2. Symmetry of cross derivatives

– Uses Shepard’s Lemma

3. Law of demand

– Uses Shepard’s Lemma and concavity of e(.)

2

12112

1

2

hp

epp

epp

hp

011

1

1

e

pph

p