1. Expenditure Minimization

2. Expenditure Minimization
Set up optimization problem

3. 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. Y
E*<E’
E=E’
E=E* X

4. Expenditure Minimization: SOC
The second order condition for constrained minimization will hold if the following bordered Hessian matrix is positive definite:
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-Ux2Uyy > 0, so these
conditions are equivalent to checking that the utility function is strictly quasi-concave.

5. Expenditure Minimization
Solve FOC to get:

6. Expenditure Minimization
Back into the expenditure function determine minimum expenditure:
Solve for Ū to get the indirect utility function:

7. Interpreting μ: Envelope Result
Start with L*

8. Finding : Envelope Result
Start with L*.

9. Expenditure Minimization
Comparative Statics

10. Comparative Statics

11. Comparative Statics: Effect of a change in px Put in Matrix Notation
Solve for

12. Expenditure Minimization: Example

13. Expenditure Minimization
Combining with

14. Expenditure Minimization
Expenditure Function
And solving this for U would yield U* = V *(px,py,M)

15. Properties of Expenditure Functions
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.

16. Concavity of Expenditure Function
If the consumer continues to buy a fixed bundle as p1’ changes (e.g. goods are perfect compliments), the expenditure function would be Ef Ef
E(px,py,U*)
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.
E(p1,…)
E(px’,py…U*) px
px’

17. Max and Min Relationships
Utility Max
L = U(x) + λ(M-g(x))
x* = x(px, M)
Expenditure Min
L = g(x) + μ(U-U(x)))
xc* = xc (px, U)
Indirect Utility
U* = U *(x*)
V * = V *(px, M)
Expenditure Function
E* = E *(xc*)
E * = E *(px, U)
Expenditure Function
Solve V * for M (M=E)
E * = E *(px, U)
Indirect Utility
Solve E * for U (E=M)
U * = V *(px, M)

18. 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

19. 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…

20. 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

21. This is the basis for… Roy’s Identity Shephard’s Lemma
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.

22. Roy’s Identity: Envelope Theorem 1
Option 1: Plug the optimal choice variable equations into the Lagrangian and THEN differentiate

23. Roy’s Identity: Envelope Theorem 2
Option 2: Differentiate the Lagrangian and THEN plug the optimal choice variable into the derivative equation

24. Envelope Theorem and Roy’s Identity

25. Shephard’s Lemma: Envelope Theorem 1
Option 1: Plug the optimal choice variable equations into the Lagrangian and THEN differentiate

26. Shephard’s Lemma: Envelope Theorem 2
Differentiate the Lagrangian and THEN plug the optimal choice variable into the derivative equation

27. Shephard’s Lemma
Bring results of Option 1 and Option 2 together:

28. The Relationships When E* = M’ And U* = Ū Primal Dual
Max U(x), s.t. M = px
L=U(x)-λ(p•x-M)
Marshallian Demand
x* = x(p,M’)
λ=UM
Min E=p•x, s.t. Ū=U(x)
L=px-μ(Ū=U(x))
Hicksian Demand
x*=xc(p, Ū)
μ=EU
x(p,M’) = x* = xc (p,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))
Indirect Utility Function
U* = U(x*)
U* = U(x*=x(p,M’))
U* = V(p, M’)
Expenditure Function
E* = 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)

29. Indirect Utility Function
The Relationships
Primal
Dual
Indirect Utility Function
U* = V*(p, M)
Expenditure Function
E* = E*(p, U)
Roy’s
Identity
Shephard’s
Lemma
xi* = xi(p,M)= -
xi* = xci (p,U) =
∂V*(p,M)
∂pi ∂M
∂E*(p,U)
∂pi

30. Ordinary (Marshallian) Demand
Slope of budget line from px/py to steeper px’/py
Income is fixed at M’, but
utility falls y
px/py Ū U2
px’/py
x*=x(px,py,M’)
px/py x xb xa xb xa x
Qd falls from xa to xb
Qd falls from xa to xb

31. Compensated (Hicksian) Demand
Slope of budget line from px/py to steeper px’/py
Utility is fixed at Ū, but
expenditure rises y
px/py
x*=xc(px,py, Ū)
px’/py
x(px,py,M’)=xc(px,py,Ū) U1
px/py x xc xa xc xa x
Qd falls from xa to xc
Qd falls from xa to xc

32. Ordinary (Marshallian) Demand
Slope of budget line from px/py to flatter px’’/py
Income is fixed at M’, but
utility rises y
px/py U0 Ū
px/py
x*=x(px,py,M’)
px’’/py xa x xb xa xb x
Qd falls from xa to xb
Qd falls from xa to xb

33. Compensated (Hicksian) Demand
Slope of budget line from px/py to flatter px’’/py
Utility is fixed at Ū, but
expenditure falls y
px/py
x*=xc(px,py,Ū) Ū
px/py
x(px,py,Ī)=xc(px,py,Ū)
px’’/py xa xc x xa xc x
Qd rises from xa to xc
Qd rises from xa to xc

34. 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.