utility maximization

Utility Maximization

A Utility Function Mathematically Representing Preferences

Utility functions have

Indifference curvesdescribe bundle-orderingpreferences

Y

U=U(x, y)

X

U

U(A) U(B) A B

U(A) U(B) A B

iff iff

A

B

U(A)

U(B)

• Assume that an individual has I dollars to allocate between good x and good y

pxx + pyy M

Consumption Opportunities: The Budget Constraint

x

y

The individual can affordto choose only combinationsof x and y in the shadedtriangle

y

M

p

x

M

p

The Budget Constraint•

• MC of consuming one more unit of x, the amount of y

that must be foregone.• The slope of the budget line is this MC.

x

y

pySlope

x p

x

y The slope is the changein y for a one unit increasein the consumption of x.

If Px = 10 and Py = 5, thenconsuming one more x meansconsuming two less y.

x

y y

pMy x

p p

y

M

p

x

M

p

• Keep buying x until the MB(x) = MC(x)• Interaction of…

– Preferences, diminishing MB because of diminishing MRS. MB = MRS

• MB in terms of y willing to be given up• In dollars, MB = py*MRS

– MC of x = px/py• MC in terms of Y given up• In dollars, MC = px.

Maximizing Utility

MB, MC

XX*

Not an indifference curve!

MC

MB

• To maximize utility, given a fixed amount of income, an individual will buy the goods and services:

– That exhaust total income

• Savings or borrowing is allowed (if we modify the budget

constraint to include a temporal component)

– So long as MB(x) ≥ MC(x), MB(y) ≥ MC(y), etc.– Or, until MB(x) = MC(x), MB(y) = MC(y)

Optimization Principle

• MRS is the maximum amount of y the person is willing to give up to consume more x; the definition of MB.

• px/py tells us the number of units of y that must be given up to consume one more x; the definition of MC.

Intuition

Px = 10

Py = 5

Slope of budget line = -2

y

x

A

B

At “B”, MRS<Px/Py, (MB < MC)

Utility and consumer surpluscan be increased by consumingless x.

At “A”, MRS>Px/Py (MB > MC),

You are willing to pay more than you have to, consumer surplus increases.

Utility and consumer surpluscan be increased by consuming more x.

U0

• At “C”, the MB = MC for the last unit of both goods consumed.

• That is, at “C”, MRS = px/py, or

Intuition

y

C

U1

x

A

B U0

x x

y y

yx

x y

U p

U p

UU

p p

• Unconstrained optimization is a lot easier to solve than constrained optimization.– Substitution: maximize the cross section of U

along the budget line– Lagrange method

Optimization

• This turns the constrained optimization into an unconstrained problem.

• Find the equation for the cross section of the U=U(x,y) above the budget line and maximize it -- i.e. find the top of the parabola

Substitution

x

y

U

x*

y*

• Substitute and maximize

Substitution

x y

1

1

1

1

; M=p x+p y

U

10

1

1

1

x x

y y y y

x

y y

x

y y

y

x y

x

U x y

p p xM Mxx x

p p p p

p xdU Mx

dx p p

p xMx

p p

Mpx

x p p

Mx

p

x y

y

y

y

M=p +p y1

p y1

1p y

1

y1 p

x

M

p

MM

M M

M

And substituteagain

Problem with this method

• It can get very mathematically complicated very quickly.

• Even U=xαyβ gets very tricky to solve.

LaGrange Method

• LaGrange knew that unconstrained optimization (like profit max) is relatively simple compared to constrained optimization.

• Taking what he knew unconstrained optimization he attempted to simplify the constrained maximization problem by making it mimic the unconstrained problem.

Unconstrained Optimization Examplemax ( , ) ( , )

FOC

( , ) ( , ) 0

( , ) ( , ) 0

Maximizing ( , ) ( , ) means

x x x x x

y y y y y

x x

y y

v f x y g x y

v f x y g x y f g

v f x y g x y f g

v f x y g x y

f g

f g

• Profit maximization is an example of this. We maximize the difference between two functions: π=R(q) – C(q).

• LaGrange wanted to find a way that was simpler than constrained optimization and more workable than simple substitution. – He wanted to make constrained optimization take the

form of the simpler unconstrained problem.

• First, let’s look at a simpler problem.

Lagrange Method

max ( , ) ( , )v f x y g x y

max ( ) ( )v f x g x

• Maximize utility minus the cost of buying bundles. Think about a one-good world.

Maximize Utility - Expenditure

UU=U(x)

Expenditure = E = pxx

x*Problems: • Ux not measured in $ like E.• E is not constrained, we can spend as much as we like.

slope = Ex= px

slope = Ux

max ( ) ( )

max ( ) x

v f x g x

v U x p x

x

• First change the expenditure function by multiplying px by λ. Now call that function EU.

• We want λ to measure the marginal utility of $1.– In that case, units of x consumed would cost us utility and both U(x) and

EU(x) would be measured in the same units.

Maximizing Utility - Expenditure

x

UU=U(x)

EU=λpxx

x*

Slope = EUx= λpx

slope = Ux

max ( ) ( )

max ( )

U

x

v U x E x

v U x p x

• Problem, an infinite number of λ choices that will each solve this with a different x*

Maximizing Utility - Expenditure

UU=U(x)

x*

EUx = λ1 px

x* x*

EUx= λ2 px

EUx = λ3 px

x

• Now the slope of the expenditure function and expenditure are measured in utils, not dollars. But we are not constraining x yet.

• So first subtract λM from the expenditure function to get EL = λpxx - λM

LaGrange Method

UU=U(x)

EL = λpxx - λM

x*

slope = Ux

-λM

EU = λpxx

slope = ELx= λpx

x

• We know we want to find the x* such that that distance between U(x) and EL(x*) = U(x*). That is, where EL(x*) = 0

• So we maximize v = U(x) - 0• Substitute λpxx – λM = 0 in for 0, to constrain x* to our budget.

LaGrange Method

U U=U(x)

EL = λpxx - λM

x* slope = EL

x= λ px

slope = Ux

-λI

max ( ) ( )

max ( ) ( )

L

x

L U x E x

L U x p x MU=U(x*)-0

0x

• Our optimization becomes an unconstrained problem by including the requirement that λpxx = λM.

• λ is chosen along with x to maximize utility so that λ = the marginal utility of $1. That is, λpx = Ux.

LaGrange Method

U U=U(x)

EL = λ(pxx – M)

x* slope = ELx= λ px

slope = Ux

-λM

max ( ) ( ) xL U x p x M

U=U(x*)-0

0x

LaGrange Method

x

max ( ) ( ) or, equivalently

max ( ) ( )

FOC

: 0

: 0

L is the condition that when we

maximize ( ) - ( )

ensures that the solution satisfies ,

i.e. that ( *)

x

x

x x x

x

x x

x

L U x p x I

L U x I p x

L U p

L I p x

f g

v f x g x

L p x I

L x ( *)U x

• To maximize utility, maximize the height of the utility function above the plane

EL = λpxx + λpyy – λM• Such that

λpxx + λpyy – λM = 0

Two Goods: Lagrange’s Manufactured Plane

x

y

U U = U(x,y)

LaGrange PlaneEL=g(x,y)

EL= λpxx+ λpyy- λMg’x=EL

x= λ px

g’y=ELy= λ py

Whenx = 0 and

y = 0,U = - λM

ELx= λ px

ELy= λ py

Lagrange Method

U

UL=g(x,y) = 0λ(pxx+pyy – M) = 0x

y

U = U(x,y)max ( , ) 0, such that 0 = ( ) x yv U x y p x p y M

max ( , ) ( )

max ( , ) ( )

x y

x y

L U x y p x p y M

L U x y M p x p y

Basic Demand Analysis• Using Lagrangian to generate ordinary

(Marshallian) demand curves.– FOCs necessary– SOCs sufficient (check that they hold)– Ordinary (Marshallian) demand curves– Inverse demand curves– Meaning of λ– Indirect Utility– Expenditure Function– Comparative Statics General Results

Demand Functions using Lagrange’s Method

• Set up and maximize:x y

x x x x x

y y y y y

x y x y

x x

y y

yx

x y

L (x, y) U(x, y) (M p x p y)

L U p 0 U p

L U p 0 U p

L M p x p y 0 M p x p y

U p

U p

UU

p p

FOC: necessary conditions for a maximum

Solve to get two interesting results

, tangency condition

, bang for the buck the same for last unit

λ* chosen so that the constraint plane is parallel to the utility function.

Any x* and y* that maximizes utilitywill also have to exhaust income.

• For utility to be maximized, it is necessary that the indifference curve is tangent to the budget constraint (as above).

• But it is not sufficient, we also need a diminishing MRS.

FOCs for an Optimum

y y

x x

FOCs satisfied

Utility Maximized y

x

FOCs satisfiedUtility Maximized

x x

y y

U p

U p

• Sufficient condition for a maximum to exist– If the MRS is non-increasing (utility function quasi concave) for all x, that is

sufficient for a maximum to exist – but it may not be unique.– If the MRS is diminishing (utility function strictly quasi concave) for all x, that is

sufficient for a unique maximum to exist. Need this to satisfy second order conditions for maximization.

SOCs for an Optimum

y y

x x

SOCs satisfied

Utility Maximized y

x

SOCs do not hold

Expenditure Minimization: SOC• The FOC ensure that the optimal consumption

bundle is at a tangency.• The SOC ensure that the tangency is a minimum, and

not a maximum by ensuring that away from the tangency, along the budget line, utility falls.

X

Y

U=U’

U=U*

U*>U’

• The second order conditions will hold if the utility function is strictly quasi-concave– A function is strictly quasi-concave if its bordered

Hessian is negative definite. That is:

• A function is strictly quasi-concave if:1. -UxUx < 0

2. 2UxUxyUy - Uy2Uxx - Ux

2Uyy > 0

Checking SOC:utility function strictly quasi-concave

00

0 and 0 x y

xx xx xy

x xxy yx yy

U UU

H H U U UU U

U U U

Checking SOC:Constrained Maximization

• The second order condition for constrained maximization will hold if the following bordered Hessian matrix is negative definite:

2 2 2

let ( , ) ( , ) ( , )

00

0 and 0

0, and 2 0

x y

x xx xy

y yx yy

x yx

x xx xyx xx

y yx yy

x x y xy x yy y xx

L x y U x y g x y

L L L

H L L L

L L L

p pp

H H p U Up U

p U U

p p p U p U p U

yxx y

UUp p,

Note:

So this Hessian andThe last only differ by

Ordinary Demand Curves• And from the FOC:

From these three equations and unknowns:

:

:

:

Solve to get the following:

* ( , , ), ordinary or Marshallian demand

* ( , , ), ordinary or Marshallian demand

* ( , , )

x x x

y y y

x y

x y

x y

x y

L U p

L U p

L M p x p y

x x p p M

y y p p M

p p M , = = , marginal utility of $1yx

x y

UU

p p

x x

y y

U pMRS

U p, solve for y to get income consumption curve

Inverse Demand Curves• Starting with the ordinary demand curves:

*

*

*

* *

Solve to get the following inverse demand equations:

( , , )

Recognize that at the optimal bundle

And

So the inverse demand curve tells us the

(willingness to give up y for a

x x y

x

y

x y

p p x p M

pMRS

p

p p MRS

ynother x) p .

I.e. the dollar value of the y that the individual is

willing to give up for an x.

Utility and Indirect Utility

• Maximum Utility, a function of quantities

• Indirect Utility a function of price and income

* * *

* *

*

*

,

Once we plug in

( , , ), ( , , ) and get

utility as a function of only prices and income

( , , ), ( , , )

, ,

x y x y

x y x y

x y

U U x y

x x p p M y y p p M

V V x p p M y p p M

V V p p M

Optimization :Expenditure Function

• Start with indirect utility

• Solve for M:

• This equation determines the expenditure needed to generate Ū, the expenditure function:

*

*

*

, ,

, ,

, ,

x y

x y

x y

V V p p M

M E p p U

E E p p U

Digression: Envelope Theorem• Say we know that y = f(x; ω)

– We find y is maximized at x* = x(ω)• So we know that y* = y(x*=x(ω),ω)).

• Now say we want to find out

• So when ω changes, the optimal x changes, which changes the y* function.

• Two methods to solve this…

* * * *

*

dy dy dy dx

d d dx d

*dy

d

Digression: Envelope Theorem• Start with: y = f(x; ω) and calculate x* = x(ω)• First option:

• y = f(x; ω), substitute in x* = x(ω) to get y* = y(x(ω); ω):

• Second option, turn it around:• First, take then substitute x* =

x(ω)

into yω(x ; ω) to get

•And these two answers are equivalent:

***dy x( ),dy

y ( )d d

f x,yy (x; )

***dy x( ),dy

y ( )d d

**y ( ) y ( )

• Plug the optimal values into the LaGrangian

x y x y x y x x y y x y

* * * * * **

x y x y x y

* * ** * *

x x y y

L* U x(p , p , M), y(p , p , M) (p , p , M) M p x(p , p , M) p y(p , p , M)

L x y x yU U 1 p p M p x * p y*

M M M M M M

L x yU p U p M

M M M

Differentiate with respect to M

*

x y

* * * **

x y

*

x y

*

p x * p y*M

L x y0 0 0

M M M MU

- p x - p y 0 (p , p , M)

L

M

M

And as we are at a maximum, the FOC get us:

As L= U (because M ),

In other words, when income rises by $1, you gain $1 worth of utility.

**U

M

Envelope Result, 1st option to get

Optimization: Envelope Result• Plug the optimal values into the LaGrangian

*x y x y x y x x y y x y

* * * * * ** * * *

x y x y x yx x x x x x

* ** *

x x yx x

L U x(p ,p ,M), y(p ,p ,M) (p ,p ,M) M p x(p ,p ,M) p y(p ,p ,M)

L x y x yU U p x p M p x p y

p p p p p p

L xU p U p

p p

xDifferentiate with respect to p

* ** * * *

y x yx x

* * * ** *

x x x x

* *

**

x

x xx y

x x x x

yx M p x

Lx

p yp p

L x y0 0 x 0

p p p p

U UL Ux x(p ,p ,M)

p p p p

p

And as we are at a maximum, the FOC get us:

As L= U and as , , means that

In other words, when the price of x rises by $1, you lose $1 worth of utility for every x bought.

** *

x

Ux

p

Optimization: Comparative Statics• If we have a specified utility function and we

derive the equations for the demand functions, the comparative statics are easy.– Take the derivatives to calculate the changes in x

and y when prices or income change.• However, what if all we know is U = U(x, y) and

we feel safe only assuming: Ux > 0 Uy > 0 Uxx < 0 Uyy < 0

• Can we get anything from that?

Optimization: Comparative Statics• Start with:

x y

x x x

y y y

x y

x y

x y

x y

L (x, y) U(x, y) (M p x p y)

L U (x, y) p 0

L U (x, y) p 0

L M p x p y 0

x* x(p , p , M)

y* y(p , p , M)

* (p , p , M)

FOC: necessary conditions for a maximum

And the equations for utility maximizing x, y,

Comparative Statics:Utility Maximizing x*, y*, λ*

x x y x y x y x

y x y x y x y y

x x y y x y

Substitute equations for x*, y* and * into the FOC

U x(p ,p ,M), y(p ,p ,M) (p ,p ,M)p 0

(1)

(2)

(3

U x(p ,p ,M), y(p ,p ,M) (p ,p ,M)p 0

M p x(p ,p ,M) p y(p ,p ,M) 0

Whatever happens to pri

)

c

es or income, consumption

will adjust to maximize utility.

xx xy x

yx yy y

x y

x y

x xx xy

y yx yy

x yU U p 0

M M Mx y

U U p 0M M M

x y1 p p 0

M MRearrange

x y0 p p 1

M M Mx y

p U U 0M M M

x yp U U 0

M M M

Tells us that if income increases by $1, so will

total expenditure.

Comparative Statics: Effect of a change in MDifferentiate (1), (2), (3) w.r.t. M

x yx y

p p 1M M

Side note:

Comparative Statics: Effect of a change in MPut in Matrix Notation

• Solve for

x y

x xx xy

y yx yy

2 2x y xy x yy y xx

y

x xy

y

?

y xy y xy yy

M0 p p 1x

p U U • 0M

0p U U y

M

H 2p p U p U p U 0

0 1 p

p 0 U

p 0 Ux0

M (

U U

Assuming

X could be either normal or inferior

)

p

H

.

p

x

M

Comparative Statics: Effect of a change in IPut in Matrix Notation

• Solve for

x y

x xx xy

y yx yy

2 2x y xy x yy y xx

x

x xx

y y

?

x x yyx xx

M0 p p 1x

p U U • 0M

0p U U y

M

H 2p p U p U p U 0

0 p 1

p U 0

p U 0y0

M (

Up p U

Assuming

Y could be either

)

normal or inferi

H

or.

yM

¶¶

Comparative Statics: Effect of a change in pxDifferentiate (1), (2), (3) w.r.t. px

xx xy xx x x

yx yy yx x x

x yx x

x yx x x

x xx xyx x x

y yx yyx x x

x yU U p 0

p p p

x yU U p 0

p p p

x yp x p 0

p p

Rearrange

x y0 p p x

p p p

x yp U U

p p p

x yp U U 0

p p p

Comparative Statics: Effect of a change in px

Put in Matrix Notation• Solve for

xx y

x xx xyx

y yx yy

x

2 2x y xy x yy y xx

y

x xy ?

y yy

x

2xy y yyx y

p0 p p xx

p U Up

0p U Uy

p

H 2p p U p U p U 0Assuming

X could be giffen

0 x p

p U

U p Upp 0 U px xx0

( )

.

p H

x

x

p

Comparative Statics: Effect of a change in px

Put in Matrix Notation… AGAIN• Solve for

xx y

x xx xyx

y yx yy

x

2 2x y xy x yy y xx

x

x xx ?

y yx

x

y yx yx x xx

Assuming

X and y could be compliments or substitute

p0 p p xx

p U Up

0p U Uy

p

H 2p p U p U p U 0

0 p x

p U

p U 0 p U pp p Uy x x0

)H

.

(

s

p

x

yp

¶¶

Comparative Statics:Preview of income and substitution effects

y x?

x

?x y

y

?

? ?

y x x

x

y

yyx

yx yx x

22 yxy y yyx y

y yx yx xx

xy yy yx x

xx

x

xpU p Up px xx

p ( ) ( )

p p xp U pp p U

Up U p

U pp U

U

y x xp ( ) ( )

x y0; 0

M ( ) I ( )

U Up p p p U

? ?

x y

x x

2y x p pp xx y

;

x

p ( ) p (

yM M

)

Rearrangethese

Sub inthese

Income effectmatters

Specific Utility Functions

• Cobb-Douglas• CES• Perfect Compliments

Cobb-Douglas: Utility Max

• Problem:

• Set up the LaGrangian

• FOC

x y

x y

1x x

1y y

x y

U(x, y) x y , s.t. M-p x-p y

L=x y + (M-p x-p y)

L : x y p 0

L : x y p 0

L : M-p x-p y=0

1x

1y

1

1

U x y

U x y

x y yMRS

x y x

Cobb-Douglas: Demand• FOC Imply, to maximize utility, these must hold.

• Plug into the budget constraint to get the ordinary (Marshallian) demand functions:

• Note, demand only a function of own price changes (one Cobb-Douglas weakness)

y x

x y

x y

yp xp x ; y

p p

M Mx* ; y*

p p

Cobb-Douglas: Demand• Preferences are homothetic (only a function of

the ratio of y:x). When income rises, optimal bundle along a ray from the origin.– Expenditure a constant proportion of income

– Income elasticities are = 1

x yp x* M; p y* M

Mx

x

dx M Me 1

dM x ( )pM

( )p

Cobb-Douglas: Indirect Utility

• Plug x* and y* into the utility function

x y

x y

U(x, y) x y

M Mx* ; y*

( )p ( )p

M MV

( )p ( )p

Cobb-Douglas: Expenditure Function

• Start with indirect utility function

• Solve for M, and then rename it Ex y

M MV

( )p ( )p

1

x y

( )VM

p p

1

x y

( )VE

p p

CES: Utility Max

• Problem:

• Set up the Lagrangian

• FOC

x y

x y

1x x

1y y

x y

U(x, y) x y , s.t. M - p x - p y

L x y (M - p x - p y)

L : x p 0

L : y p 0

L : M - p x - p y 0

1

1

x

y

xMRS

y

p x

p y

CES: Demand• FOC Imply

• Plug into the budget constraint and solve:

1 1

1 111

yx

y x

p xp yx ; y

p p

1 1y x

x yx y

M Mx ; y

p pp 1 p 1p p

CES: Indirect Utility• Plug x* and y* into the utility function

1 1y x

x yx y

1 1y x

x yx y

U(x, y) x y

M Mx ; y

p pp 1 p 1p p

M MV

p pp 1 p 1p p

CES: Expenditure Function• Solve for M, then rename E.

1 1y x

x yx y

1 1y x

x yx y

M MV

p pp 1 p 1p p

1 1V

p pp 1 p 1p p

2M

CES: Expenditure Function (cont)• Solve for M, then rename E.

1 1y x

x yx y

1

1 1y x

x yx y

VM

1 12

p pp 1 p 1p p

VE

1 12

p pp 1 pp p

1

1

Perfect Compliments: Utility Max• Problem:

• No Lagrangian, just exhaust income such that:

• So plug this condition into the budget equation– Essentially, we substitute the expansion path into the

budget line equation.

x yU(x, y) min( x, y), s.t. M-p x-p y

xy

Perfect Compliments: Demand• Demand equations

x y

x y

x y

yM=p +p y

M= p +p y

My

p +p

x y

x y

x y

xM=p x+p

M= p + p x

Mx

p + p

Perfect Compliments: Indirect Utility

• Since utility from x = utility from y at utility max:

x y

x yx y

x y x y

x yx y

M MV(p ,p ,M) min ,

p +pp +p

M MV(p ,p ,M) or V(p ,p ,M)

p +pp +p

Perfect Compliments: Expenditure Function

• Since the utility from consumption of each must be equal,

x y x yp +p p +p

E V, or E V

Bonus Topics

• Money Metric Utility Function• Homogeneity• Corner Solutions (Kuhn-Tucker)• Lump-Sum Principle• MRS and MRT (Marginal Rate of

Transformation – slope of PPF)

Money Metric Utility Function• Start with an expenditure function and replace

with Ū with U=U(x,y)

• Now we know the minimum expenditure to get the same utility as the bundle x’, y’.

• That is, for an x’, y’, this function tells us the cost of being at the tangency on the same indifference curve. As all bundles on the same curve get the same min. expenditure and prefernece ordering is preserved, this is a utility function.

x yE e p ,p , U x yE e p ,p , U(x, y)

E Ex yU U p ,p , x, y

Evaluating Housing Policy• Assume you find the poor generally expend

1/3 of income on housing.• The government wants to double the quality

of housing the poor consume at the same 1/3 their income.

• How to evaluate?

Pre-Public Housing

H

YIf expenditure on housing is generally 1/3 of income, assume U=h1/3y2/3

ph=$1px=$1M=$1,000

333

667

667x y

M 2Mh* ; y*

3p 3p

Public Housing

H

Y ph=$1px=$1M=$1,000Qualified citizens get 667 housing units for 1/3 of income ($333)

333

667

667

2133

x y

1 23 3

E1 2

3 3

VE

1 23p 3p

h yU

1 23 3

Public Housing: Money Metric Utility

H

Y ph=$1px=$1M=$1,000Qualified citizens get 667 housing units for 1/3 of income ($333)

333

667

667

2133

x y

1 23 3

E1 2

3 3

VE

1 23p 3p

h yU

1 23 3

UE=1,000

UEPH=1,261

The extra housing has a value to the poor of $261. Depending on the cost to thegovernment of providing the housing, the program can be evaluated.

Homogeneity• If all prices and income were doubled, the

optimal quantities demanded will not change– the budget constraint is unchanged

xi* = xi(p1,p2,…,pn,M) = xi(tp1,tp2,…,tpn,tM)

• Individual demand functions are homogeneous of degree zero in all prices and income

• To test, replace all prices and income with t*p and t*M. The quantity demanded should be unaffected (all the “t” should cancel out).

Corner Solutions

X

YNon-corner solutions: Optimal bundle will be where x > 0, y > 0 and MRS = px/py

Two corner solutions: Optimal bundle will be where x =0

Corner Solutions

• At “A”, MRS = px/py, but the optimal quantity of X = 0

X

Y

A

Corner Solutions

• At “B”, the tangency condition holds where x* < 0. Given the budget line, the optimal feasible x is where x = 0 and MRS < px/py

X

Y

B

Corner Solution

• To develop these conditions as part of a Lagrangian equation, we add non-negativity constraints: (we could add if we wanted to be really thorough).

• Lagrangian:

– Requiring x = s2 is simply a way of ensuring that x ≥ 0.

Corner Solutions

• FOC

x x x

y y y

x y

2

s

L U p 0

L U p 0

L M p x p y 0

L x s 0

L 2 s 0

Corner Solutions

• Kuhn-Tucker Condition

• This tell us that either:• μ =0 (the optimum is at a tangency)• s = 0 (the optimum is where x = 0)• μ =0 and s = 0 (the optimum is at a tangency where x

= 0)

sL 2 s 0

Corner Solutions

• The usual assumption is that the optimal bundle will be where x>0, y>0 and

X

Y

x x x

y y y

x x

y y

L U p 0

L U p 0

U p

U p

If s > 0 and μ = 0

x

y

pMRS

p

Corner Solutions

• At “A”, , but the optimal quantity of x = 0

X

Y

A

If s = 0 and μ = 0

x x x

y y y

x x

y y

L U p 0

L U p 0

U p

U p

x

y

pMRS

p

Corner Solutions

• At B’, the tangency condition holds where x* < 0. • The optimum is where x=0 and

X

Y

B’

x x x

y y y

x x

y y

x x x

y y y

x

y

L U p 0

L U p 0

U pU p

U U pMRS

U U p

pMRS

p, at B

If s = 0 and μ > 0

x

y

pMRS

p

B

Kuhn-Tucker Example• Utility: U=xy+20y, M = 40, px = $4 and py = 1.

x y

L xy 20y (40 4x y)

L : y 4 ; L : x 20

L : 40 4x y

yGets 4x 80 y, x 20

4Solve x 5, y 60

Oops.

Looks Like• Tangency where x=-5, y=60

X

Y yMRS = 4

x 20

How about a feasible optimum?• Tangency where x=-5, y=60

X

Y

40MRS = 2

0 20

x

y

p60 4MRS =

5 20 p 1

x

y

p 4Slope =

p 1

Kuhn-Tucker Set-up

• Utility: U=xy+20y, M = 40, px = $4 and py = 12

x

y

2

L xy 20y (40 4x y) (x s )

L : y 4 0

L : x 20 0

L : 40 4x y 0

L : x s 0

Ls : 2 s 0 Kuhn-Tucker Condition

Kuhn-Tucker Result

xx y

y

x

y

x

y

Kuhn-Tucker: 2 s 0, so or s or both = 0

pyUse L and L :

x 20 p

pyIf μ=0, optimum is a tangency where

x 20 p

If s 0, optimal x 0.

If 0, and s 0 optimum at corner and:

p y yMRS

p x 20 x 20

If

0, and s 0 optimum at corner but tangency where x 0.

If 0, and s 0 optimum at interior.

Kuhn-Tucker Result• In this example, at x = 0, y = 40, 0 :

x

yAt corner, this condition holds: =4

x 20So 4x 80 y and 40

y 40At the corner, MRS = 2

x 20 0 20If p 2, then the optimum is at x 0, y 40,

0 and s 0.

How about a feasible optimum?• Optimum where x=0, y=40

X

Y

x

y

x

y

x

y

At optimum:

p y yMRS =

p x 20 x 20

p 40 40 40MRS =

p 20 20

p y y4 MRS = 2

p x 20 x 20

Solving Kuhn-Tucker

• If you find that the optimal bundle is not on the budget constraint, check all corners for a maximum utility.

Lump Sum Tax• Taxing a good vs taxing income

– Tax on x only

– Lump sum tax (income tax)

x y

x

y y

x y

x y

x

y y

p x p y M

px

p p

x * y*

p x * p y* M x *

p x p y M x *

pM x *y x

p p

Mwhich is y= -

and are optimal bundles so

, and R* =

Lump Sum Tax• Difference in the budget lines: sales tax

x

y y

* x

y y y

* x x

y y y y y

*

y

p x *y*

p p

p x * x *y

p p p

p x * p x *M x *

p p p p p

x *

p

Without a tax, M

With the unit tax on x, at any x*,M

MAt any x*, y*-y

y*-y

Lump Sum Principle

• A tax on x rotates the budget line to have:

X

Y

x

y

pslope

p

y

M

p

x

M

px

M

p

y*

x*

y

x *

p

yτ*

Lump Sum Tax• Difference in the budget lines: income tax

x

y y

* xR

y y y

* x xR

y y y y y

*R

y

p x *y*

p p

x *

p x *M x *y

p p p

p x * p x *M x *

p p p p p

x *

p

Without a tax, M

With an income tax R*=

MAt any x*, y*-y

y*-y

Lump Sum Principle• Tax paid = x*τ. Alternatively, an income tax of that same

amount would shift the budget line so that the consumer can just afford the same bundle they chose under the tax on x.

X

Y

y

M

p

x

M

px

M

p

y*

x*

y

M x *

p

x

M x *

p

y

x *

p

y

x *

p

Lump Sum Principle

• When

• Indirect Utility function is:

.5

x y

U xy

2M Mx y

3p 3p

;

.5

x yx y

2M MV(p ,p ,M)

3p 3p

Lump Sum Principle

• Set, I=60, py = 2, px=1

• Utility is:

x y

2M Mx 40; y 10

3p 3p

.5

x yx y

2M MV(p ,p ,M) 126.49

3p 3p

Lump Sum Principle

• With a $1 tax on x,

• Utility is:

• And tax revenue is $20.

x y

2M Mx 20; y 10

3 p 1 3p

.5

*x y

x x y

2M MV V(p ,p ,M)

3 p 1.

3p4

P63 2

Lump Sum Principle

• With a $20 income tax,

• Utility is:

• 68.85 > 63.24

* *

x y

2(M 20) M 20x =26.667; y 6.667

3P 3P

.5

*x y

x y

2(M 20) M 20V V(p ,p ,M) 68.85

3P 3P

• If the utility function is U = Min(x,4y) then plug x=4y into the budget equation to get:

• The indirect utility function is then:

The Lump Sum Principle: Perfect Compliments

*

x y

Mx

p 0.25p

*

x y

My

4p p

x yx y x y

M MV(p ,p ,M) Min ,4

p 0.25p 4p p

• Set, I=60, py = 2, px=1

• Utility is:

The Lump Sum Principle: Perfect Compliments

*

x y

Mx 40

p 0.25p

*

x y

My 10

4p p

x yx y x y

M MV(p ,p ,M) Min ,4 40

p 0.25p 4p p

• With a $1 tax on x,

• Utility is:

The Lump Sum Principle: Perfect Compliments

*

x y

Mx 24

(p 1) 0.25p

*

x y

My 6

4(p 1) p

x yx y x y

M MV(p ,p , I) Min ,4 24

(p 1) 0.25p 4(p 1) p

• With a $24 income tax,

• Utility is:

• Since preferences do not allow substitution, the consumer makes the same choice either way and the tax does not affect utility.

The Lump Sum Principle: Perfect Compliments

*

x y

M - 24x 24

p 0.25p

*

x y

M - 24y 6

4p p

x yx y x y

M - 24 M - 24V(p ,p ,M) Min ,4 24

p 0.25p 4p p

MRS=MRT• Using Varian’s example about milk and butter

– B* and M* may be different for all consumers.– However, the MRS at the tangency is the same for ALL

consumers, no matter the income or preferences.

Butter

Milk

Pb=3; Pm=1

MRS=3 MRS=3

MRS=3

Marginal Rate of Transformation(a.k.a Rate of Product Transformation)

• Varian is sort of implicitly assuming the PPF is linear. So there is a constant trade of in producing butter or milk as resources are reallocated.

Milk

MRT=3

Butter

MRS=3

Social Indifference curve

Marginal Rate of Transformation• With a more realistic PPF, the MRT rises as

more butter and less milk is produced.

Milk

MRT=3

Butter

b m

m b

b b

m m

b m

b m

C qMRT

C q

C pMRT

C p

C C

p p

Assuming competative firms producing b and m

or , marginal cost p the same at

the margin for all goods.

MRS=MRT• In the long run (π=0), the cost of producing

butter must be 3 times the cost of producing milk. That is, the tradeoffs in consumption = the tradeoff in production

Milk

px/py=3

Butter

b b b

m m m

C p UMRT MRS

C p U

Back to Varian’s treatment• A new technology that allows you to produce

butter with 4 gallons of milk is not going to be a winner as everyone would choose the cheaper butter previously offered.

Butter

Milk

MRT=4Pb=4; Pm=1

• However, a new technology that allows you to produce 1 pound of butter with 2 gallons of milk IS going to be a winner!

Butter

Milk

MRS=MRT=2

Pb=2; Pm=1

MRS=MRT• Here is what improved butter making

technology does with a more standard PPF.

Milk

px/py=3

Butter

px/py=2