Expenditure Minimization

Expenditure Minimization

• Set up optimization problemx y

x y

x x x x x

y y y y y

x x

y y

E p x p y, U U(x, y)

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

FOC

L : p U 0 p U

L : p U 0 p U

L : U U(x, y) 0 U U(x, y)

With the result that

p U

p U

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 indifference curve, expenditure rises.

X

Y

E=E*

E=E’E*<E’

Expenditure Minimization: SOC• The second order condition for constrained minimization will hold

if the following bordered Hessian matrix is positive definite:

2 2 2

let ( , ) ( , )

00

0 and 0

0, and 2 0

x y

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 p x p y U U x y

L L L

H L L L

L L L

U UU

H H U U UU U

U U U

U U U U U U U U

Will hold if the Hessian of the Lagrangian is is Positive Definite

Note, -(-Ux )2 =-Ux 2

< 0 and (so long as μ > 0), 2UxUxy Uy -Uy2Uxx-Ux

2Uyy > 0, so theseconditions are equivalent to checking that the utility function is strictly quasi-concave.

Expenditure Minimization

• Solve FOC to get:

x x

y y

c* cx y

c* cx y

* cx y

p U

p U

U U(x, y)

And get the compensated, Hicksian, demand curves

x x p ,p ,U

y y p ,p ,U

p ,p ,U

Expenditure Minimization

• Back into the expenditure function determine minimum expenditure:

• Solve for Ū to get the indirect utility function:

* c cx x y y x yE p x p ,p ,U p y p ,p ,U

* * *x y x yV V p ,p ,E V p ,p ,M

Interpreting μ: Envelope Result• Start with L*

c c

c

* c c c cx x y y x y x y x y x y

*c* c* * * c* c* c* c* * * c* c*

x U y U U U U Ux y

*c* * c* c* *U x U yx

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

L Up x p y U U x U y U x , y )

U U

Lx p U y p U

U

Differentiate with respect to U

c

c c

c* * * * c* c*U Uy

*c* * c* c* * c* * c* * c* c*U x U y Ux y

* *c* c* * c* *U U U

** c* c*

x y

U U x , y

Lx p U y p U U U x , y

U

L Lx 0 y 0 0

U U

Ex , y 0 E* p ,p , U

U

,

Because U-U , L*= and

In other words, if you want to*

increase utility by 1 util, you need to

increase expenditure by .

c* cx y

c* cx y

*x y

x x p ,p ,U

y y p ,p ,U

p ,p ,U

Finding : Envelope Result• Start with L*.

c cx x x x x

cx

* c c c cx x y y x y x y x y x y

x

*c* c* c* * * c* * c* * * c* c*

x p y p p p px yx

** * c* *

x p yxx

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

p

Lp x x p y U x U y U U x , y

p

Lp U x p U

p

Differentiate with respect to

cx x x

x x x x

* c* c* c* * c* c*p p py

* *c* c* c* * c*p p p p

x x

*c* c* c*

x

y x U U x , y

L L0 x 0 y x 0 x

p p

EU - U x , y 0 E x

p

x

x

,

Because , L= ,

In other words, if p increases by $1, you need to increase expenditure by c*.

c* cx y

c* cx y

*x y

x x p ,p ,U

y y p ,p ,U

p ,p ,U

*

x

E

p

Expenditure Minimization• Comparative Statics

c* cx y

c* cx y

c* cx y

* c * c cx x x y x x y x y

* c * c cy y x y y x y x y

* * c cx y x y

Plug the optimization functions into the FOC

x x p ,p ,U

y y p ,p ,U

p ,p ,U

FOC

L : p p ,p ,U U x p ,p ,U , y p ,p ,U 0

L : p p ,p ,U U x p ,p ,U , y p ,p ,U 0

L : U U x p ,p ,U , y p ,p ,U 0

Comparative Staticsx

c* c* ** *

xx xy xx x x

c* c* *c* c

yx yy yx x x

c* c*

x yx x

c* c*

x yx x

* c**

x xxx x

Differentiate with respect to p

x y1 U U U 0

p p p

x y0 U U U 0

p p p

x y0 U U 0

p p

Rearrange

x y U U 0

p p

xU U

p p

c**

xyx

c* c* c**

y yx yyx x x

yU 1

p

x yU U U 0

p p p

Comparative Statics: Effect of a change in px

Put in Matrix Notation• Solve for

xx y c

x xx xyx

y yx yy c

x

2 2x y xy x yy y xx

y

x xy

2c*y yy y

x

p0 U U 0

xU U U 1

p0U U U

yp

Assuming SOC are satisfied H 2 U U U U U U U 0

0 0 U

U 1 U

U 0 U Ux0

p ( )H

Compe

nsated demand curves must be downward sloping.

c*

x

x

p

Expenditure Minimization: Example

.5x y x y

.5x y

.5x x

y y .5

.5

yx x

y x y

E p x p y, U xy ,M 60,p 1,p 2

L p x p y U xy

FOC

L : p y

xL : p

2y

L U xy

With the result that

2p yp p x2yx and y

p x p 2p

Expenditure Minimization• Combining with

.5

2/31/32y* * x

x y

L U xy

yields the Hicksian Demand Functions

2p U p Ux and y

p 2p

Expenditure Minimization

• Expenditure Function

• And solving this for U would yield U* = V *(px,py,M)

* c* c*x y

2/31/32y* x

x yx y

E p x p y

2p U p UE p p

p 2p

• Homogeneity– a doubling of all prices will precisely double the value

of required expenditures• homogeneous of degree one

• Nondecreasing in prices– E*/pi 0 for every good, i

• Concave in prices– When the price of one good rises, consumers respond

by consuming less of that good and more of other goods. Therefore, expenditure will not rise proportionally with the price of one good.

Properties of Expenditure Functions

E(p1,…)

Since the consumption pattern will likely change, actual expenditures will be less than portrayed Ef such as E(px,py,U*). At the px where the quantity demanded of a good becomes 0, the expenditure function will flatten and have a slope of 0.

Ef

If the consumer continues to buy a fixed bundle as p1’ changes (e.g. goods are perfect compliments), the expenditure function would be Ef

Concavity of Expenditure Function

px

E(px,py,U*)

E(px’,py…U*)

px’

Max and Min RelationshipsUtility MaxL = U(x) + λ(M-g(x))x* = x(px, M)

Indirect UtilityU* = U *(x*)V * = V *(px, M)

Expenditure FunctionSolve V * for M (M=E)E * = E *(px, U)

Expenditure MinL = g(x) + μ(U-U(x)))xc* = xc (px, U)

Expenditure FunctionE* = E *(xc*)E * = E *(px, U)

Indirect UtilitySolve E * for U (E=M)U * = V *(px, M)

xE g(x) p x

Shephards Lemma and Roy’s Identity

• Two envelope theorem results allow:– Derivation of ordinary demand curves from the

expenditure function– Derivation of compensated demand curves from

the indirect utility function

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

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 we get the identity

***dy x( ),dy

y ( )d d

f x,yy (x; )

***dy x( ),dy

y ( )d d

**y ( ) y ( )

This is the basis for…• Roy’s Identity

– Allows us to generate ordinary (Marshallian) demand curves from the indirect utility function.

• Shephard’s Lemma– Allows us to generate compensated (Hicksian)

demand curves from the expenditure function.

Roy’s Identity: Envelope Theorem 1

x y

* * *x y x y x y

* * * * * * *x y

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

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

L U x, y I p x p y

x x p ,p ,M y y p ,p ,M p ,p ,M

L U x , y M p x p y

L V (x p ,p ,M , y p ,p ,M ) p ,p ,M I p x p ,p ,M p y p ,p ,M

L V p ,p ,M p ,p ,M I p x p ,p ,M p y p ,p ,M

Derive

; ;

Substitute

x x y y x y

* *x y x y

x

y

x

* *x

I p x p ,p ,M p y p ,

L p ,p ,M V p

p ,M 0

L V p ,p

,p ,M

,M

p p

Note that

So

Option 1: Plug the optimal choice variable equations into the Lagrangian and THEN differentiate

* * *x y x y x y

*x

x y

x y

x

* *x

y

y

yx

x y

x

x x p ,p ,M y y p ,p ,M p ,p ,M

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

L p ,p ,Mp ,p ,M

L(x, y,p ,p , I, )x

p

x x p ,

x pp

p ,M p ,p ,M

Assume we already have derived

, ,

Start with

NOW plug in , and to get

* * *x

x y

* *x y x y

x yx

y* *U x , y V p ,

,p ,M

L p ,p ,M V p ,p ,Mx p ,p ,M

p M

p ,M

M M

Remember that = i.e. =

Option 2: Differentiate the Lagrangian and THEN plug the optimal choice variable into the derivative equation

Roy’s Identity: Envelope Theorem 2

* *x y x y

x yx

* *x y x y

x x

*

*

*x y

x y

x

x y

x

*x y

xy *

x y

L (

L (p ,p , I) V (p ,p , I)

p p

V (p ,p , I)

p

V (p ,p , I

p ,p , I) V (p ,p , I)x(p ,p , I)

p I

V (p ,p , I)x(p ,p , I)

I

x(p ,p , I)V (p

x,p , I)

)

p

I

Option 1 yields:

Option 2 yields:

Envelope Theorem and Roy’s Identity

Shephard’s Lemma: Envelope Theorem 1

x y

* c * c *x y x y x y

* c c * c cx x y y x y x y x y x y

c* c*

* *x y

*x

L p x p y U U x, y

x x p ,p ,U , y y p ,p ,U , p ,p ,U

L p x p ,p ,U p y p ,p ,U p ,p ,U U U x p ,p ,U , y p ,p ,U

U U x , y 0

E E p ,p ,U

L p

Minimize

Derive

Plug them in to get

Since this reduces to

*y x y

x x

,p ,U E p ,p ,U

p p

Option 1: Plug the optimal choice variable equations into the Lagrangian and THEN differentiate

Shephard’s Lemma: Envelope Theorem 2

* c * c *x y x y x y

x y

x

*x y

y

x

*x y

x

c

cy

x

x x p ,p ,U y p ,p ,U p ,p ,U

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

L x, y,p ,p ,M,x

p

x x p ,p ,U

L p ,p ,Ux p ,p ,U

p

Having already derived

, y ,

Now start with

Take the derivative:

NOW plug in to get:Differentiate the Lagrangian and THEN plug the optimal choice variable into the derivative equation

Shephard’s Lemma

• Bring results of Option 1 and Option 2 together:

*x y c

x

* *x y x y

x x

*x

yx

cy

y

xx

L p ,p ,Ux p ,p ,U

p

x p ,p

L p ,p ,

,U

U E p ,p ,U

p p

E p ,p ,U

p

The Relationships

Primal Dual

Max U(x), s.t. M = pxL=U(x)-λ(p•x-M)

Marshallian Demandx* = x(p,M’)

λ=UM

Min E=p•x, s.t. Ū=U(x)L=px-μ(Ū=U(x))

Hicksian Demandx*=xc(p, Ū)

μ=EU

Indirect Utility FunctionU* = U(x*)

U* = U(x*=x(p,M’))U* = V(p, M’)

Expenditure FunctionE* = p•x*

where x*=xc(p, Ū)M’=E* = E(p, Ū)

U* =V(p,M’) when solved for M’ is E*= E(p, Ū)

x(p,M’) = x* = xc (p,Ū)when

E* = M’ and U* = Ū

When E* = M’And U* = Ū

x* = x(p,M)x*=xc(p,E(P,U))

x* = xc (p,U)x*=x(p,V(p,M))

The RelationshipsPrimal Dual

Indirect Utility FunctionU* = V*(p, M)

Expenditure FunctionE* = E*(p, U)

xi* = xi(p,M)= - xi* = xci (p,U) =

∂V*(p,M)∂pi

∂V*(p,M)∂M

∂E*(p,U)∂pi

Roy’sIdentity

Shephard’sLemma

Ordinary (Marshallian) Demand

y

xb xa xb xa

x*=x(px,py,M’)

ŪU2

x x

px/py

px/py

px’/py

Slope of budget line from px/py to steeper px’/py

Qd falls from xa to xbQd falls from xa to xb

Income is fixed at M’, but utility falls

Compensated (Hicksian) Demand

y

xc xa xa

x*=xc(px,py, Ū)

U1

x x

px/py

px/py

px’/py

Slope of budget line from px/py to steeper px’/py

Qd falls from xa to xcQd falls from xa to xc

xc

Utility is fixed at Ū, butexpenditure rises

x(px,py,M’)=xc(px,py,Ū)

Ordinary (Marshallian) Demand

y

xbxa

x*=x(px,py,M’)

ŪU0

x

px/py

px/py

px’’/py

Slope of budget line from px/py to flatter px’’/py

Qd falls from xa to xbQd falls from xa to xb

Income is fixed at M’, but utility rises

xbxa x

Compensated (Hicksian) Demand

y

Ū

px/py

px/py

px’’/py

Slope of budget line from px/py to flatter px’’/py

Qd rises from xa to xcQd rises from xa to xc

x(px,py,Ī)=xc(px,py,Ū)

x*=xc(px,py,Ū)

Utility is fixed at Ū, butexpenditure falls

xcxa x xcxa x

Ordinary and Compensated• If price changes and Qd changes along the

ordinary demand curve, then utility changes and you jump to a new compensated demand curve.

• If price changes and Qd changes along the compensated demand curve, then expenditure needed changes and you jump to a new compensated demand curve.

• Which curve is more or less elastic depends on whether the good is normal or inferior.