chapter 2 duality and revealed preferencesflash.lakeheadu.ca/~kyu/e5113/nct2.pdf · Þnite...

Chapter 2 Duality and Revealed PreferencesEconomics 5113 Microeconomic Theory

Kam Yu

Winter 2019

Outline

1 Duality Theory: From Expenditure to UtilityExpenditure and Utility FunctionsConvexity and Monotonicity

2 Duality Between Direct and Indirect UtilityFrom Indirect to Direct Utility FunctionsInverse Demand Functions

3 Duality Between Demand and PreferencesImportant Characterization of the Demand FunctionThe Integrability Problem

4 Revealed PreferenceWeak AxiomStrong Axiom

Kam Yu (Lakehead) Chapter 2 Duality and Revealed Preferences Winter 2019 2 / 29

Duality Theory: From Expenditure to Utility Expenditure and Utility Functions

Recovering the Utility Function

Suppose that we are given a function E : Rn++ × R+ → R+ that

satisfies the seven properties of an expenditure function.

That is, E (p, u) is continuous, strictly increasing and unboundedabove in u, increasing, linearly homogeneous, concave, anddifferentiable in p.

Can we find an utility function U that gives E in an expenditureminimization problem?

The question is analogous to the Minkowski theorem in convexanalysis (see section 6.6 in Yu): A closed convex set is theintersection of its supporting half-spaces.



Another Maximization Problem

Consider the following constrained maximization problem:

maxu

u

subject to pTx ≥ E (p, u), for all p� 0,

u ≥ 0.

The control variable is u. The vector x is a parameter.

The vector p is a condition imposed on the maximization problemthat the first inequality constraint holds for all p ∈ R++.

Effectively we have infinitely many constraints, each p represents ahalf space containing the upper contour set %(x).

If a solution exists, the optimal choice u∗ is a function of x, that is,u∗ = U(x), which is also the value function.



Minkowski Theorem at Work

74 CHAPTER 2

Theorem 1.7, so that it is continuous, strictly increasing, and unbounded above in u, aswell as increasing, homogeneous of degree one, concave, and differentiable in p. Thus,E ‘looks like’ an expenditure function. We shall show that E must then be an expendi-ture function. Specifically, we shall show that there must exist a utility function on Rn

+whose expenditure function is precisely E. Indeed, we shall give an explicit procedure forconstructing this utility function.

To see how the construction works, choose (p0, u0) ∈ Rn++×R+, and evaluate E

there to obtain the number E(p0, u0). Now use this number to construct the (closed) ‘half-space’ in the consumption set,

A(p0, u0) ≡ {x ∈ Rn+ | p0 · x ≥ E(p0, u0)},

illustrated in Fig. 2.1(a). Notice that A(p0, u0) is a closed convex set containing all pointson and above the hyperplane, p0 · x = E(p0, u0). Now choose different prices p1, keep u0fixed, and construct the closed convex set,

A(p1, u0) ≡ {x ∈ Rn+ | p1 · x ≥ E(p1, u0)}.

Imagine proceeding like this for all prices p≫ 0 and forming the infinite intersection,

A(u0) ≡!

p≫0A(p, u0) = {x ∈ Rn

+ | p · x ≥ E(p, u0) for all p ≫ 0}. (2.1)

The shaded area in Fig. 2.1(b) illustrates the intersection of a finite number of theA(p, u0), and gives some intuition about what A(u0) will look like. It is easy to imaginethat as more and more prices are considered and more sets are added to the intersection, theshaded area will more closely resemble a superior set for some quasiconcave real-valuedfunction. One might suspect, therefore, that these sets can be used to construct something

x1

x2

x1

x2

A(p0, u0) !A(p, u0)

{x |p0 • x ! E(p0, u0)}

(a) (b)

Figure 2.1. (a) The closed half-space A(p0, u0). (b) The intersection of afinite collection of the sets A(p, u0).



Duality Theorem

We claim that if E satisfies the seven properties of an expenditurefunction, then the value function U(x) is increasing, unbounded above,and quasi-concave.

Sketch of the proof (see pages 74–76 in JR for details)

Existence of u∗: Since E is increasing and bounded above in u, thefeasible set,

G (x) ={u ≥ 0 : pTx ≥ E (p, u) ∀p� 0

},

is bounded. Since it is also closed, the solution exists by theWeierstrass theorem.

Note: Since E is linearly homogeneous in p, G (x) depends on therelative prices, not the absolute price of each good.



Monotonicity: Suppose that x1 ≥ x2. Then pTx1 ≥ pTx2 for allp� 0. Let u2 = U(x2). That is, u2 is the solution of themaximization problem for x2. Then pTx2 ≥ E (p, u2) and thereforepTx1 ≥ E (p, u2) for all p� 0. This means that u2 ∈ G (x1) and soU(x1) ≥ u2 = U(x2) as required.

Quasi-concavity: Suppose that U(x2) > U(x1). We want to showthat for any convex combination xα of the two bundles,

U(xα) ≥ min{U(x1),U(x2)

}= U(x1).

From the definition of G (x), we know that pTx1 ≥ E (p,U(x1)) andpTx2 ≥ E (p,U(x2)) for all p� 0. Therefore for 0 ≤ α ≤ 1,

αpTx1 + (1− α)pTx2 ≥ αE (p,U(x1)) + (1− α)E (p,U(x2)).

Since E is increasing in u, pTxα ≥ E (p,U(x1)). We conclude thatU(xα) ≥ U(x1).

The proof of unboundedness is left as an exercise.



Confirmation that U is a Utility FunctionNow we show that the value function U(x) in the above maximizationproblem is the utility function dual to expenditure function. That is,

E (p, u) = minx

{pTx : U(x) ≥ u, x ≥ 0

}.

Suppose that p0 � 0 and u0 ≥ 0. For any x ∈ Rn+ such that

U(x) ≥ u0, we have, by the definition of U and monotonicity of E inu,

pTx ≥ E (p,U(x)) ≥ E (p, u0)

for all p� 0.

Let p = p0. The above inequality can be stated asE (p0, u0) ≤ (p0)Tx for all x ∈ Rn

+ such that U(x) ≥ u0. This impliesthat

E (p0, u0) ≤ minx

{(p0)Tx : U(x) ≥ u0, x ≥ 0

}. (1)



Our goal is to show that (1) is an equality. To achieve this we shallfind an x0 ∈ Rn

+ such that U(x0) ≥ u0 and (p0)Tx0 ≤ E (p0, u0).

Since E is concave in p, by theorem 6.5 in Yu,

E (p, u0) ≤ E (p0, u0) +∇pE (p0, u0)T(p− p0)

= E (p0, u0) +∇pE (p0, u0)Tp−∇pE (p0, u0)Tp0

= E (p0, u0) +∇pE (p0, u0)Tp− E (p0, u0) (Euler’s thm)

= ∇pE (p0, u0)Tp.

Define x0 = ∇pE (p0, u0) so that we have E (p, u0) ≤ pTx0 for allp� 0. First, x0 ∈ Rn

+ because E is increasing in p. Second, by thedefinition of U, U(x0) ≥ u0.

Finally, applying Euler’s theorem to E at the point (p0, u0) gives

E (p0, u0) = ∇pE (p0, u0)Tp0 = (p0)Tx0

as required.


Duality Theory: From Expenditure to Utility Convexity and Monotonicity

Is U Generated by E Unique?

Recall that when we prove the existence and the seven properties ofE , we do not use the properties that U is quasi-concave andincreasing.

Now when we use E to generate an utility function by the aboveprocess, say,

W (x) = maxu

{u ≥ 0 : pTx ≥ E (p, u) ∀p� 0

},

are we sure that W = U if U is not increasing and quasi-concave?

The answer is, not necessary. The duality theorem we proved aboveguarantees that W is increasing and quasi-concave. So if U does notpossess these properties, then W 6= U.



General Relation Between W and U

In general, W (x) ≥ U(x) for all x ≥ 0.

Proof : By the definition of E , we have E (p,U(x)) ≤ pTx for allp� 0. This means that U(x) is in the feasible set of finding W .Therefore W (x) ≥ U(x).

Let %U and %W be the preference relations induced by U and Wrespectively. The above result implies that for any utility level v ,%U (v) ⊆ %W (v).



When U is Not Increasing and Not Quasi-Concave

80 CHAPTER 2

x1

x2

(a)

S(u) ! T(u)

u(x) ! ux1

x2

(b)

S(u)

u(x) ! u

0

x1

x2

(c)

u(x) ! u

0x1

x2

(d)

w(x) ! u

u(x) ! u

0

x1

x2

(e)

w(x) ! u

u(x) ! u

0

Budget Line

x*

y/p1

Figure 2.2. Duality between expenditure and utility.

80 CHAPTER 2

x1

x2

(a)

S(u) ! T(u)

u(x) ! ux1

x2

(b)

S(u)

u(x) ! u

0

x1

x2

(c)

u(x) ! u

0x1

x2

(d)

w(x) ! u

u(x) ! u

0

x1

x2

(e)

w(x) ! u

u(x) ! u

0

Budget Line

x*

y/p1

Figure 2.2. Duality between expenditure and utility.



When is U = W ?

If U is increasing and quasi-concave, then W = U.

Proof : Suppose that x0 ≥ 0 and let U(x0) = u. By monotonicity andquasi-concavity there exists a price vector p0 such that

E (p0, u) = (p0)Tx0,

which is a supporting hyperplane of the convex set %(x0). By definition,W (x0) satisfies the constraint pTx0 ≥ E (p,W (x0)) for all p� 0. Whenp = p0, we have (p0)Tx0 ≥ E (p0,W (x0)), which means

E (p0, u) ≥ E (p0,W (x0)).

Since E is increasing in u, U(x0) = u ≥W (x0).


Duality Between Direct and Indirect Utility From Indirect to Direct Utility Functions

Just Another Minimization Problem

Recall that in the utility maximization problem, given anyconsumption bundle x, and imagine that y = pTx, we have, for anyp� 0,

V (p,pTx) ≥ U(x). (2)

Is there some prices p that turns the inequality into an equality?

The answer is yes. In fact if % is strictly convex, p is unique. Thebudget constraint pTz = y is a supporting hyperplane for %(x) at thebundle x.

Inequality (2) implies that

U(x) = minp

{V (p,pTx) : p� 0

}.

This provides a mean to recover the utility function from the indirectutility function.


Duality Between Direct and Indirect Utility From Indirect to Direct Utility Functions

From V to U

Suppose that the utility function U of a consumer is quasi-concave andthe marginal utilities of all goods are positive, that is, ∇U(x)� 0. LetV (p, y) be the indirect utility function generated by U. Then for allconsumption bundle x,

U(x) = minp

{V (p,pTx) : p� 0

}. (T.1)

Proof : Given any x ≥ 0, we have, for all p� 0,

V (p,pTx) ≥ U(x).

Let p = ∇U(x), λ = 1, and y = pTx. Then we have

∇U(x) = λp and pTx = y ,

which are the sufficient conditions for the utility maximization problem.


Duality Between Direct and Indirect Utility Inverse Demand Functions

Variations

Since V is homogeneous of degree zero in (p, y), the duality relationcan be written as

U(x) = minp

{V (p, 1) : pTx = 1,p� 0

}. (T.1’)

In practice, the problem in (T.1’) may be analytically more convenientthen that in (T.1).

Suppose that y = 1. Then the inverse demand function is

p = d−1(x) =∇U(x)

∇U(x)Tx.

Proof : In the UMP, the Lagrange multiplier theorem implies that∇U(x) = λp. Together with the budget constraint pTx = 1, we have

∇U(x)

∇U(x)Tx=

λp

λpTx=

p

pTx= p.


Duality Between Demand and Preferences Important Characterization of the Demand Function

Can We Get U from d?

Since consumer behaviours are observable, we can in principleestimate the ordinary demand function d .

An important question is, what properties of d must possess so thatwe can derive the utility function from it?

So far we have derived the following properties of the demandfunction:

1 Budget balancedness: pTd(p, y) = y .2 The Slutsky matrix S(p, y) is symmetric.3 The Slutsky matrix is negative semidefinite.4 Homogeneity of degree zero in (p, y).5 Cournot and Engel aggregations.

We have shown that property 1 implies 5. Are there any moreredundancy?



Budget Balancedness and Symmetry Imply Homogeneity

Properties 1 and 2 imply property 4.

Proof : Differentiate the budget constraint with respect to pi and y , weget, for i = 1, . . . , n,

n∑j=1

pj∂dj(p, y)

∂pi= −di (p, y), (3)

n∑j=1

pj∂dj(p, y)

∂y= 1. (4)

For any p and y , define a single variable function fi (t) = di (tp, ty) fort > 0. Our goal is to show that fi (t) = di (p, y), which means that fi isindependent of t, or f ′i (t) = 0 for all t > 0.



Using the product rule,

f ′i (t) =n∑

j=1

pj∂di (tp, ty)

∂pj+ y

∂di (tp, ty)

∂y. (5)

Budget balancedness means that tpTd(tp, ty) = ty , or simplypTd(tp, ty) = y , which can be written as y =

∑nj=1 pjdj(tp, ty).

Substitute this y into equation (5), we get

f ′i (t) =n∑

j=1

pj

[∂di (tp, ty)

∂pj+∂di (tp, ty)

∂ydj(tp, ty)

]. (6)

Notice that the expression inside the square bracket is the i-j term of theSlutsky matrix.



Since the Slutsky matrix is symmetric, we can switch the i and the j insidethe bracket in equation (6):

f ′i (t) =n∑

j=1

pj

[∂dj(tp, ty)

∂pi+∂dj(tp, ty)

∂ydi (tp, ty)

]

=

n∑j=1

pj∂dj(tp, ty)

∂pi

+ di (tp, ty)

n∑j=1

pj∂dj(tp, ty)

∂y

=

1

t

n∑j=1

tpj∂dj(tp, ty)

∂pi

+ di (tp, ty)1

t

n∑j=1

tpj∂dj(tp, ty)

∂y

= −1

tdi (tp, ty) +

1

tdi (tp, ty) (by equations (3) and (4))

= 0.


Duality Between Demand and Preferences The Integrability Problem

We Only Need Three Properties for d

Now we are back to the original question. Given the observableordinary demand function that satisfies budget balancedness, with asymmetric and negative semidefinite Slutsky matrix, can we recoverthe utility function?

The answer is yes, and it is called the integrability problem.

We do this by showing that given the ordinary demand function, theexpenditure function exists. With the expenditure function, the utilityfunction can be derived.

The process is called integrability probably because the solutioninvolves a system of partial differential equations, hence the need forintegrations.

A sketch of the theory for the n = 2 case is discussed here.

For details, see “The Integrability Problem” by K.C. Border.


http://people.hss.caltech.edu/~kcb/Notes/Demand4-Integrability.pdf


The Integrability Problem

Suppose that a demand function d for two goods satisfies budgetbalancedness, with a symmetric and negative semidefinite Slutskymatrix.

Using the homogeneity property we can normalize the price of good 2,that is, we can set p2 = 1.

For any price-income point (p1, 1, y), the demand function isd(p1, 1, y), with utility u∗ = V (p1, 1, y). The expenditure functioncan be expressed as E (p1, 1, u

∗) = e(p1).

The function e is the solution of the following first-order ordinarydifferential equation:

de(p1)

dp1= d1(p1, 1, e(p1)), (7)

with the initial condition e(p01) = y0.



Existence of Solution

We invoke the Picard Theorem for the sufficient condition forexistence of a unique solution for e: If d1(p1, y) and ∂d1/∂y arecontinuous on a neighbourhood B of the initial point (p01 , y

0), thenthere exists an interval I centred at p01 and a unique function e(p1)that solve the differential equation.

Given that d satisfies the three conditions, the solution e(p1) has allthe properties of an expenditure function:

1 Continuous as a solution to an ODE.2 Increasing in p1, since d1(p1, 1, e(p1)) ≥ 0.3 Concave in p1:

d2e(p1)

dp21=

∂d1(p1, 1, e(p1))

∂p1+∂d1(p1, 1, e(p1))

∂yd1(p1, 1, e(p1))

= s11 ≤ 0.


http://mathworld.wolfram.com/PicardsExistenceTheorem.html


Example of a First-Order Ordinary Differential Equation

Suppose that y = f (t) and satisfies the equation

dy

dt= ky , (8)

where k is a given constant. We also know that at time t = 0, y = y0.

We know from calculus that if y = cekt , then the derivative satisfiesequation (8), where c is an arbitrary constant.

Substitute the initial condition in the solution, we have

y0 = cek×0,

which gives c = y0.

Therefore the solution for this ODE is

y = y0ekt .



General n-Commodity Case

The problem becomes a system of first-order partial differentialequations:

∇e(p) = d(p, e(p)),

with initial condition e(p0) = y0.

The problem is as if we try to recover the expenditure function fromthe demand function using Shephard’s lemma.

Frobenius theorem: A necessary and sufficient condition for thesolution to exist is the symmetry of the Slutsky matrix.

Negative semi-definiteness of the Slutsky matrix implies that e hasthe properties of an expenditure function.


Revealed Preference Weak Axiom

Consumer Theory without Axioms

Paul Samuelson suggested that we could derive a consumer theoryfrom observing their behaviours.

Weak axiom of revealed preference (WARP): Suppose that twodistinct bundles x0 and x1 are chosen by a consumer when prices arep0 and p1 respectively. Then

(p0)Tx1 ≤ (p0)Tx0 (9)

implies that(p1)Tx0 > (p1)Tx1. (10)

When prices are p0, the consumer chooses bundle x0. Inequality (9)means that bundle x1 is feasible but not chosen. It is natural toconclude that x0 % x1.

When prices are p1, the consumer chooses bundle x1. Inequality (10)means that bundle x0 is not feasible, otherwise it would be chosen bythe consumer.



WARP and Not WARP

92 CHAPTER 2

The basic idea is simple: if the consumer buys one bundle instead of another afford-able bundle, then the first bundle is considered to be revealed preferred to the second. Thepresumption is that by actually choosing one bundle over another, the consumer conveysimportant information about his tastes. Instead of laying down axioms on a person’s pref-erences as we did before, we make assumptions about the consistency of the choices thatare made. We make this all a bit more formal in the following.

DEFINITION 2.1 Weak Axiom of Revealed Preference (WARP)

A consumer’s choice behaviour satisfies WARP if for every distinct pair of bundles x0, x1with x0 chosen at prices p0 and x1 chosen at prices p1,

p0 · x1 ≤ p0 · x0 "⇒ p1 · x0 > p1 · x1.

In other words, WARP holds if whenever x0 is revealed preferred to x1, x1 is never revealedpreferred to x0.

To better understand the implications of this definition, look at Fig. 2.3. In both parts,the consumer facing p0 chooses x0, and facing p1 chooses x1. In Fig. 2.3(a), the consumer’schoices satisfyWARP. There, x0 is chosen when x1 could have been, but was not, and whenx1 is chosen, the consumer could not have afforded x0. By contrast, in Fig. 2.3(b), x0 isagain chosen when x1 could have been, yet when x1 is chosen, the consumer could havechosen x0, but did not, violating WARP.

Now, suppose a consumer’s choice behaviour satisfies WARP. Let x(p, y) denote thechoice made by this consumer when faced with prices p and income y. Note well that thisis not a demand function because we have not mentioned utility or utility maximisation –it just denotes the quantities the consumer chooses facing p and y. To keep this point clearin our minds, we refer to x(p, y) as a choice function. In addition to WARP, we make one

x1

x2

p1p0

x0x1

(a)

x1

x2

p1p0

x0

x1

(b)

Figure 2.3. The Weak Axiom of Revealed Preference (WARP).



Two Observations

1 The weak axiom can be alternatively stated as follows. If theLaspeyres quantity index QL between period 0 and period 1 is lessthan or equal to 1, then the Paasche qauntity index QP is less than 1,where

QL =(p0)Tx1

(p0)Tx0, QP =

(p1)Tx1

(p1)Tx0.

2 A continuous, increasing, and quasi-concave utility function satisfiesthe weak axiom.

Question: Is the converse of statement 2 also true? If it is, then the weakaxiom of revealed preference characterizes a well-defined utility function.


Revealed Preference Strong Axiom

Revealed Preference and Integrability

If the revealed preferences of a consumer satisfy the weak axiom andbudget balancedness, then the resulting choice function (see pages93–95 in JR)

1 is homogeneous of degree 0 in (p, y),2 has a negative semidefinite Slutsky matrix,3 (when n = 2) has a symmetric Slutsky matrix.

Therefore, in the case of n = 2, we can invoke integrability to recoverthe utility function.

For the general case where n > 2, we need the strong axiom ofrevealed preference: Suppose that a sequence of distinct bundlesx0, x1, . . . , xk , where xi−1 is revealed preferred to xi (in the sense ofthe weak axiom) for all i = 1, . . . , k. Then xk cannot be preferred tox0.


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