Duality Theory

1
Duality Theory

• AEDE 711
• October 10, 2005

2
Duality

• Need a practical way to link utility maximization
  to what we observe in markets.
• Duality provides link between ordinary
  (uncompensated) and compensated demand
• Primal Utility Maximization
• Dual Expenditure Minimization.

3
Dual to Utility Maximization Expenditure
Minimization

• We have already seen how we can derive ordinary
  demand functions from utility maximization.
• But it is difficult to derive compensated demand
  functions from utility maximization.
• A different, but related problem would be to
  minimize expenditure, subject to a minimum level
  of utility (U from utility max?)

4
Expenditure Function

• The dual problem is to allocate income so as to
  reach a given utility level with the least
  expenditure.
• Reverse the problem.

Dual
Primal
Solution gives us Maximum utility (U) with
Income constrained to
Solution gives us Minimum Expenditure (I)
with Utility constrained to
5
Definition Indirect Utility Function

• Use utility maximization to derive optimal values
  of X as a function of price and income.

• Now substitute X into the utility function to
  get maximum utility.
• U(X1,X2,,Xn)

• V is the indirect utility function maximum
  utility as a function of prices and income.
• Can in some cases be estimated directly.

6
Definition Expenditure Function

• Minimize expenditure, subject to UU().

Derive compensated Demand curves

• Optimal solution expressed as

• E is the expenditure function minimum
  expenditure utility as a function of prices and
  utility.

7
Graphically

• U in primal U in dual
• I in primal I in dual
• (X,Y) in primal
• (Xc,Yc) in dual
• This holds only at optimum, when relative prices
  change,
• (X,Y) ? (Xc,Yc)
• Primal leads to ordinary demand functions
• Dual leads to compensated demand functions.

8
Compensated Demand Functions
Compensated demand functions allow us to
assess the change in income needed to stay on
the same indifference curve (hold
utility constant) when prices change.
B
A
C
9
What can we do with this information?

• Derive the optimal subsidy
• Derive the Slutsky equation
• Calculate Substitution and Income Effects

10
Example

• Derive compensated demand functions.
• Compare ordinary (uncompensated) demand to
  compensated demand functions.
• Determine optimal subsidy to return an individual
  to initial utility level when one of the prices
  has increased.
• Show Slutsky Equation
• Calculate substitution and income effects

11
Example

• Derive compensated demand curves from dual problem

(1) Set up Lagrangian and solve FOCs
12
(2) Use first two FOCs to solve for X and Y
(3) Derive compensated demand for Xc and Yc.
13
(4) Derive Expenditure Function (Minimum
expenditure necessary to maintain constant
utility, U)
A
(5) Indirect Utility function is (at A)
14
Compare to Ordinary (uncompensated) Demand
15
(1) Use first two FOCs to solve for X and Y
(2) Demand for X, X, is found by substituting Y
into 3rd FOC and solving for X
(3) Substitute X into Y, and get Y
16
Compare
Ordinary
Compensated
Demand (uncomp.)
Comp. Demand
Expenditure Function
Indirect Utility
17
What compensation do you give for an increase in
PX?

• Calculate compensation required to maintain
  utility at U when price changes.

(1) Calculate indirect utility at optimal point
(2) Insert compensated demand functions into
objective of dual holding utility at V and
varying PX I PXX PYY
(3) Have I from minimization problem, now just
need to calculate the subsidy/compensation
as S I - I
18
Show compensation for Example
(1) Compensation, S, is (I - I) I is
expenditure function evaluated at new price PX,
and U I is expenditure function evaluated at
original PX, PY, and U (2) Obtain compensated
demand with new PX
19
(2 continued) Substitute compensated demand
functions into I
(3) Calculate subsidy S I - I
20
Numerical Example

• I200 PY 2 PX 1

Compensated Demand?
How much demanded?
21
Optimal SubsidyS I - I
If PX remains constant gt
If PX doubles to 2 gt
22
The Optimal Subsidy is the same as estimating CS
from the compensated demand curve
Px
Consumers Surplus Area A B

Px1
A
B
Px0
Xc(Px Py, I)
X0
X1
X
23
Show how CS from Xc Optimal Subsidy
Hotellings Lemma (derivative of the expenditure
function is compensated demand)
Re-arrange
Re-write
Integrate
24
Consumers Surplus S Area (A B)
Px
Px1
A
B
Px0
Xc(Px Py, I)
X0
X1
X
25
Derive the Slutsky Equation
Primal and dual give us same demand at the
optimal bundle
Differentiate both sides
Re-arrange
26
Note that I PXX PYY gt
Therefore
Which is the Slutsky equation we discussed
earlier..
Slope of Ordinary Demand Function Slope of
Compensated demand (Slope of Engel Curve)(X)
27
Numerical Ex. Slutsky Equation

• I200 PY 2 PX 1

28
Calculate Substitution and Income Effects

• What is the Substitution Effect?
• Xc(P1U0) - Xu(P0U0)
• What is income Effect?
• Xu(P1 U1) - Xc(P1U0)

Y
B
I1
A
C
U0
U1
X
P1
P0
S
I
29
PX rises from 1 -gt 2

• I200 PY 2 PX 1

Substitution Effect Xc(P1U0) - Xu(P0U0)
Since Xu(P0U0) 100 gt Substitution Effect
70.7 - 100 - 29.3
Income Effect Xu(P1 U1) - Xc(P1U0)
Since Xc(P1U0) 70.7gt Income Effect 50
70.7 - 20.7
Total Effect Substitution Effect Income
Effect -29.3 (-20.7) - 50 Alternatively,
total effect Xu(PX2U1) - Xu(PX1U0) 50 -
100 -50
30
Substitution 70.7 100 - 29.3
Income 50 -70.7 - 20.7
31
Summary Whats the point?

• Ordinary demand (Xu)
• Derived from Utility Maximization
• Compensated demand (Xc)
• Derived from Expenditure Minimization
• Relationship between Ordinary and compensated
  demand
• Slutsky Equation
• Slope of Ordinary Demand Function
• Slope of Compensated demand (Slope of Engel
  Curve)(X)
• Slope is same if income effect 0 (dx/dI 0)
• Normal Good (dX/dI gt0) gt
• Inferior Good (dX/dI lt0) gt
• Giffen Good (dX/dI lt0) gt slope of ordinary
  demand curve is

32
PX
Normal Good For any change in price, you get a
bigger shift along the ordinary demand curve than
along the compensated demand curve. Income
effect reinforces substitution effect
Xu
Xc
x
PX
Inferior Good For any change in price, you get a
smaller shift along the ordinary demand
curve than along the compensated demand
curve. Income effect opposes substitution
effect
Xc
Xu
x
33
Summary

• Price changes have two effects
• Change the optimal bundle
• Change income
• When measuring change in CS, need to account for
  both effects
• Marshallian CS
• Hicksian CS
• They will differ, depending on income effects
• Difference is typically small for small
  (marginal) price changes