Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter 2)economics-course.weebly.com/uploads/2/5/7/2/25725158/week_9.pdf · 2.1 Duality: A Closer Look The solutions to utility

Week 9: Topics in Consumer Theory (Jehle andReny, Chapter 2)

Tsun-Feng Chiang

*School of Economics, Henan University, Kaifeng, China

November 15, 2015

Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter 2)November 15, 2015 1 / 29

2.1 Duality: A Closer Look

The solutions to utility maximization problems and expenditureminimization problems are, in sense, the same. In this section, weshall explore further the connections among direct utility, indirect utility(from the maximization problem) and expenditure functions (from theminimization problem). The first duality is that every function of pricesand utility that has all the properties of an expenditure function is infact an expenditure function, i.e. there is a well-behaved utility functionthat generates it.

Consider any function of prices and utility, E(p,u), that may or may notbe an expenditure function. Suppose this function satisfies theexpenditure function properties 1 to 7 of Theorem 1.7, so that it iscontinuous, strictly increasing, and unbounded above in u, as well asincreasing in, homogeneous of degree one, concave, and differentiablein p. To show E is exactly an expenditure function, it must be shownthere must exist a utility function on Rn

+ whose expenditure function isprecisely E . Indeed we shall give an explict procedure for constructing


this utility function. To see how the construction works, choose(p0,u0) ∈ Rn

++ × R+, and evaluate E to obtain a number. Use thisnumber to construct the "half-space" in the consumption set,

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

illustrated in Figure 2.1(a) (see the next slide). Now choose differentprices p1, keep u0 fixed, and construct another set,

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

Imagine proceeding like this for all prices p� 0 and forming theinfinite intersection

A(u0) ≡ ∩p�0A(p,u0) = {x ∈ Rn+|p · x ≥ E(p,u0) for all p� 0}.(2.1)

The shaded area in Figure 2.1(b) illustrated the infinite intersection. Itis easy to imagine that as more and more prices are considered andmore sets are added to the intersection, the shaded area will be moreclosely resemble a superior set for some quasiconcave real-valuedfunction.


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


The following theorem says the the function constructed by this way isincreasing, unbounded above and quasiconcave, the properties like adirect utility fuunction.

Theorem 2.1 Constructing a Utility Function from an ExpenditureFunctionLet E : Rn

++ × R+ → R+ satisfy properties 1 through 7 of anexpenditure function given in Theorem 1.7. Let A(u) be as in (2.1).Then the function u : Rn

+ × R+ given by

u(x) ≡ max{u ≥ 0|x ∈ A(u)}

is increasing, unbounded above, and quasiconcave. (This function,u(x), represents convex, monotonic prefernces.)

If E(p,u) is really an expenditure function generated by some utilityfunction u(x). From the definition of an expenditure function,p · x ≥ E(p,u(x)) for all prices p� 0. Because E is strictly increasingin u, u(x) is the largest value of u such that p · x ≥ E(p,u) for all pricesp� 0. That is, u(x) is the largest value of u such that x ∈ A(u).


Theorem 2.1 tells us we can begin with an expenditure function anduse it to construct a direct utility function representing some convex,monotonic preferences.The following theorem says a functionsatisfying Theorem 1.7 is indeed an expenditure function.

Theorem 2.2 The Expenditure Function of Derived Utility, u, Is ELet E(p,u), defined on Rn

++ × Rn+, satisfy properties 1 to 7 of an

expenditure function given in Theorem 1.7 and let u(x) be derived fromE as in Theorem 2.1. Then for all nonnegative prices and utility,

E(p,u) = minx p · x s.t. u(x) ≥ u.

That is, E(p,u) is the expenditure function generated by derived utilityu(x).

Once we obtain the utitlity through an expenditure function, if theunderlying preferences are continuous and strictly increasing, we caninvert the function in u, obtain the associated indirect utility function,apply Roy’s identity, and derive the system of Marshallian demands aswell.


What if the utiltiy function is not strictly increasing and notquasiconcave? What would be the properties of derived utility functionconstructed from an expenditure function? Suppose only that u(x) iscontinuous. Let e(p,u) be the expenditure function generated by u(x).The continuity of u(x) is enough to guarantee that e(p,u) iswell-defined and continuous. Consider the utility function, call it w(x),generated by e(·) in the now familiar way, that is,

w(x) ≡ max{u ≥ 0|p · x ≥ e(p,u) p� 0}

By Theorem 2.1, w(x) is increasing and quasiconcave. Thus,regardless of whether or not u(x) is quasiconcave or increasing, w(x)will be both increasing and quasiconcave. Clearly, u(x) and w(x) neednot coincide.By the definition of e(p,u), that is, p · x ≥ e(p,u), and the defintion ofw(x), we can see w(x) ≥ u(x). For any number u ≥ 0, the level-usuperior set for u(x), say S(u), will be contained in the level-u superiorset for w(x), say T (u). Moreover, because w(x) is quasiconcave, T (u)is convex.


Figure 2.2. Duality between expenditure and utility.


In Figure 2.2 (a), u(x) is quasiconcave and increasing, then theboundary of S(u) yields the negatively sloped, convex indifferencecurve u(x) = u. Note that each point on the boundary is theexpenditure-minimizing bundle to achieve utility u at some price vectorp� 0. For any number u, by the definition of the expenditure functionit must be that u(x) ≥ u, which implies u(x) ≥ u ≥ w(x). But it hadbeen said w(x) ≥ u(x). Therefore, w(x) = u(x).

The case depicted in Figure 2.2 (b) is more interesting. There, u(x) isneither increasing nor quasiconcave. Note that some bundles on theindifference curve never minimize the expenditure required to obtainutility level u regardless of the price vector. The thick lines in Figure2.2(c) show those bundles that do minimize expenditure at somepositive price vector. For those bundle x on the thick linesegments inFigure 2.2(c), we therefore have as before that w(x) = u(x) = u. Butbecause w(x) is quasiconcave and increasing, the w(x) = uindifference curve must be as depicted in Figure 2.2(d). Thus, w(x)differs from u(x) only as much as is required to become strictlyincreasing and quasiconcave.


The implication for this case is that any observable demand behaviorthat can be generated by a nonincreasing, nonquasiconcave utilityfunction, like u(x), can also be generated by an increasing,quasiconcave utility function, like w(x). It is in the sense that theassumptions of monotonicity and convexity of preferences have noobservable implications for our theory of consumer demand.

Because the expenditure function and indirect utility are closely related(i.e. are inverses of each other), the duality between the expenditurefunction and the direct utility function implies there exists a dualitybetween the indirect utility function and direct utility function. Supposethat u(x) generates the indirect utility function v(p, y). Then bydefinition of the indirect utility function (see (1.12)), for every x ∈ Rn

+,v(p,p · x) ≥ u(x) holds for every p� 0. In addition, there will be besome price vector for which the equality holds. Evidently, we may write

u(x) = minp∈Rn++

v(p,p · x). (2.2)

(2.2) provides a mean for recovering the utility function u(x) fromknowledge of only the indirect utility function it generates.


Theorem 2.3 Duality Between Direct and Indirect UtilitySuppose that u(x) is quasiconcave and differentiable on Rn

++ withstrictly positive partial derivative there. Then for all x ∈ Rn

++, v(p,p · x),the indirect utility function generated by u(x), achieves a minimum in pon Rn

++, and

u(x) = minp∈Rn++

v(p,p · x). (T.1)

Beacuse v(p, y) is homogeneous of degree zero in (p, y), we havev(p,p · x) = v(p/(p · x),1) whenever p · x > 0. Consequently, letp · x = 1 we can rewrite (T.1) as

u(x) = minp∈Rn++

v(p,1). s.t. p · x = 1 (T.1’)

Whether we use (T.1) or (T.1’) to recover u(x) from v(p, y) does notmatter. Simply choose that which is more convenient. Onedisadvantage of (T.1) is that it always possess multiple solutionsbecuase of the homogeneity of v (i.e., if p∗ solves (T.1), then so doestp∗ for all t > 0).


Example 2.1

Convert the indirect utility function v(p, y) = y(pr1 + pr

2)−1/r to the

direct utility function. We use (T.1’) by setting y = 1, which yieldsv(p,1) = (pr

1 + pr2)−1/r . The direct utility function therefore will be the

minimum-value function

u(x1, x2) = minp1,p2(pr1 + pr

2)−1/r s.t. p1x1 + p2x2 = 1.

The first order conditions for the Lagrangian require that the optimal p∗1and p∗2 satisfy

−(p∗r1 + p∗r2 )(−1/r)−1(p∗1)r−1 − λ∗x1 = 0,

−(p∗r1 + p∗r2 )(−1/r)−1(p∗2)r−1 − λ∗x2 = 0,

1− p∗1x1 − p∗2x2 = 0.

Solve the system of equations to obtain

p∗1 =x1/(r−1)

1

x r/(r−1)1 +x r/(r−1)

2

, p∗2 =x1/(r−1)

2

x r/(r−1)1 +x r/(r−1)

2


Example 2.1 (Continued)Substituting these into the objective function and forming u(x1, x2), weobtain

u(x1, x2) =

[x r/(r−1)

1 + x r/(r−1)2

(x r/(r−1)1 + x r/(r−1)

2 )r

]−1/r

=[(x r/(r−1)

1 + x r/(r−1)2 )1−r

]−1/r

= (x r/(r−1)1 + x r/(r−1)

2 )(r−1)/r .

Defining ρ ≡ r/(r − 1) yields

u(x1, x2) = (xρ1 + xρ2 )1/ρ

This is the CES direct utility function we started with in Example 1.2,as it should be.

The last duality result we take up concerns the consumer’s inversedemand function. The following theorem shows how to obtain theinverse demand from the direct utility function without solving themaximization problem.


Theorem 2.4 (Hoteling, Wold) Duality and the System of InverseDemandsLet u(x) be the consumer’s direct utility function and it differentiable.Then the inverse demand function for good i associated with incomey = 1 is given by

pi(x) =∂u(x)/∂xi∑n

j=1 xj(∂u(x)/∂xj)

Example 2.2Let’s take the case of the CES utility function once again. Ifu(x1, x2) = (xρ1 + xρ2 )

1/ρ, then

∂u(x)/∂xj = (xρ1 + xρ2 )(1/ρ)−1xρ−1

j

multiplying by xj , summing over j = 1,2, forming the required ratios,and invoking Theorem 2.4 gives the following system of inversefunctions when income y = 1:

p1 = xρ−11 (xρ1 + xρ2 )

−1; p2 = xρ−12 (xρ1 + xρ2 )

−1


2.2 Integrability

In Chapter 1.5, we showed that a utility maximizing consumer’sdemand function must satisfy homogeneity of degree zero in pricesand income, budget balancedness, symmetric, and negativesemidefiniteness, along with Cournot and Engel aggregation. But fromTheorem 1,17 the two aggregation results are derived from budgetbalancedness so they are redundant. There is another redundancy aswell. Of the remaining four conditions, only budget balancedness,symmetry, and negative semidefiniteness are truly independent:Homogeneity of degree zero is implied by budget balancedness andsymmetry.

Theorem 2.5 Budget Balancedness and Symmetry ImplyHomogeneityIf x(p, y) satisfies budget balancedness and its Slutsky matrix issymmetric, then it is homogeneous of degree zero in p and y .Proof: When budget balancedness holds,


Theorem 2.5 (Continued)y =

∑ni=1 pixi(p, y)

Take the first derivatives with respect to prices and income to obtainfor, i = 1, · · · ,n,

n∑j=1

pj∂xj(p, y)∂pi

= −xi(p, y), (P.1)

n∑j=1

pj∂xj(p, y)∂y

= 1. (P.2)

Fix p and y , then let fi(t) = xi(tp, ty) for all t > 0. Differentiating fi withrespect to t gives

f ′i (t) =n∑

j=1

∂xi(tp, ty)∂pj

pj +∂xi(tp, ty)

∂yy (P.3)


Theorem 2.5 (Continued)Now by budget balancedness, tp · x(tp, ty) = ty , so that dividing byt > 0, we may write:

y =∑n

j=1 pjxj(tp, ty) (P.4)

Substituting from (P.4) for y in (P.3) and rearranging yields

f ′i (t) =n∑

j=1

pj

[∂xi(tp, ty)

∂pj+∂xi(tp, ty)

∂yxj(tp, ty)

]

The term in bracket is the ij − th element of the Slutsky matrix. Whensymmetry of the Slutsky matrix holds, the ij − th element is equal tothe ji − th element. Consequently we may interchange i and j withinthose brackets and maintain equality. Therefore,

f ′i (t) =n∑

j=1

pj

[∂xj(tp, ty)

∂pi+∂xj(tp, ty)

∂yxi(tp, ty)

]


Theorem 2.5 (Continued)

=

n∑j=1

pj∂xj(tp, ty)

∂pi

+ xi(tp, ty)

n∑j=1

pj∂xj(tp, ty)

∂y

=

1t

n∑j=1

tpj∂xj(tp, ty)

∂pi

+ xi(tp, ty)1t

n∑j=1

tpj∂xj(tp, ty)

∂y

=

1t[−xi(tp, ty)](by (P.1)) + xi(tp, ty)

1t[1](by (P.2))

= 0.

Since the demand fi(t) = xi(tp, ty) does not change in t , it ishomogeneous of degree zero.

In summary, for a utility-maximizer’s system, the demand function mustsatisfies the three properties,


Budget Balancedness: p · x(p, y) = yNegative Semidefiniteness: The associated Slutsky matrix s(p, y)must be negative semidefinite.Symmetry: s(p, y) must be symmetric.

On the other hand, it can be proved that demand behavior isconsistent with the theory of utility maximization if and only if it satisfiesbudget balancedness, negative semidefiniteness, and symmetry. Thisimpressive result warrants a formal statement.

Theorem 2.6 Integrability Theorem

A continuously differentiable function x : Rn+1++ → Rn

++ is the demandfunction generated by some increasing, quasiconcave utility function if(and only if, when utility is continuous, strictly increasing, and strictlyquasiconcave) it satisfies budget balancedness, symmetry, andnegative semidefiniteness.


In practice, this theorem allows us to recover the utility function giventhe demand functions satisfying the three properties.

Example 2.3Suppose there are three goods and that a consumer’s demandbehavior is summarized by the functions

xi(p1,p2,p3, y) =αiypi, i = 1,2,3,

where αi > 0, and α1 + α2 + α3 = 1. It is straightforward to check thatthe demand functions satisfies budget balancedness, and the Slutskyequation is symmetric and negative semidefinite. Consequently, byTheorem 2.6, x(p, y) must be utility-generated.The first step is to derive the expenditure function. Remember theShephard’d lemma

∂e(p1,p2,p3,u)∂pi

= xhi (p1,p2,p3,u)


Example 2.3 (Continued)By duality between Hicksian demand functions and Marshalliandemand function,

xhi (p1,p2,p3,u) = xi(p1,p2,p3,e(p1,p2,p3,u)) =

αie(p1,p2,p3,u)pi

Therefore,

∂e(p1,p2,p3,u)∂pi

=αie(p1,p2,p3,u)

pi, i = 1,2,3

This is a partial differential equation. Move e(p,u) to the left hand side,∂pi to the right hand side, and take the integral on the both side∫

∂e(p,u)e(p,u)

= αi

∫∂pi

pi, i = 1,2,3

You would have no trouble deducing that


Example 2.3 (Continued)ln(e(p,u)) = αi ln(pi) + constant, i = 1,2,3

The only additional element to keep in mind is that when partiallydifferentiating with respect to, say, pi , all the other variables, p1,p3, andu, are treated as constants. With this in mind, it is easy to see thethree equations above imply the following three:

ln(e(p,u)) = α1ln(p1) + c1(p2,p3,u),ln(e(p,u)) = α2ln(p2) + c2(p1,p3,u),ln(e(p,u)) = α3ln(p3) + c3(p1,p2,u),

where the ci(·) functions are the constant terms. Because the threeequations should hold simultaneously, the only possibility is

ln(e(p,u)) = α1ln(p1) + α2ln(p2) + α3ln(p3) + c(u),

where c(·) is some function of u. But this means that


Example 2.3 (Continued)e(p,u) = c(u)′pα1

1 pα22 pα3

3

Because e(p,u) is strictly increasing in u, we can choose any c(u)′

that is strictly increasing. It does not matter what strictly increasingfunctions we choose because any monotonic transform can keep theorder of utility. We may choose c(u)′ = u, so that

e(p,u) = upα11 pα2

2 pα33

We can then know the utility function.


2.3 Revealed Preference

We had discussed about the properties of demand functions derivedfrom the consumer’s preference or utility. The problem is thatpreference is not observable so we cannot directly examine thedemand functions. The theory of revealed preference says thatvirtually every prediction ordinary consumer theory makes for aconsumer’s observable market can also be derived from a few simpleand sensible assumption about the consumer’s observable choicesthemselves, rather than about his unobservable preferences.

The basic idea is simple: If the consumer buys one bundle instead ofanother affordable bundle, then the first bundle is considered to berevealed preferred to the second. Thus his tastes had been revealedby his choices. Instead of laying down axioms on a person’spreferences as well we did before, we instead make assumptionsabout the consistency of the choices that are made. Formally,


Definition 2.1 Weak Axiom of Revealed Preference (WARP)A consumer’s choice behavior satisfies WARP if for every distinct pairof bundles x0, x1 with x0 chosen at prices p0 and x1 chosen at pricesp1,

p0 · x1 ≤ p0 · x0 =⇒ p1 · x0 > p1 · x1

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

To better understand the implications of this definition, look at Figure2.3.(see the next slide). In both graphs, the consumer facing p0

chooses x0, and facing p1 chooses x1. In (a), when the price is at p0,the consumer chooses x0 although x1 is affordable to him (x1 is insidethe budget constraint). When the prices change from p0 to p1, hechooses x1 because x0 becomes not affordable to him. Therefore, hisbehavior does not violate WARP.


In (b), at price p0, both x0 and x1 are affordable to him. His choice ofx0 means he prefers x0 to x1. When the prices change from p0 to p1,he chooses x1 instead of x0 although x0 is affordable to him.

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


A Numerical Example: WARP

In the table above, first bundle is chosen at first prices, second bundleat second prices and third bundle at third prices. So the diagonalcorresponds to actual choices.From the first row, because we see the consumer chooses (10,1)although (5,5) and (5,4) are affordable. So we know (10,1) % (5,5)and (10,1) % (5,4). From the second row, we know (5,5) % (5,4).From the third row, we know (5,5) % (10,1) which contradicts(10,1) % (5,5). His behavior violates WARP.


Now suppose a consumer’s choice behavior satisfies WARP. Letx(p, y) denote the choice function (note well this is not a demandfunction because we have not mentioned utility or utility maximization),the quantities the consumer chooses facing prices p and income y .For p� 0, we assume the choice function x(p, y) satisfies budgetbalancedness, i.e. p · x(p, y) = y . The first consequence of WARP andbudget balancedness is that the choice function x(p, y) must behomogeneous of degree zero in (p, y).

Next consequence of WARP is the negative semidefiniteness of theSlutsky equation. Then it can also be proved the Slutsky equation issymmetric by budget balancedness and homogeneity of degree zero.However, symmetry only applies in the case of two goods. When thereare only two goods, the choice function x(p, y) satisfies both negativedefiniteness and symmetry given the assumptions of WARP andbudget balancedness. Then the choice function is actually a demandfunction because we would then be able to construct a utility functiongenerating it. On the other hand, we also can use the choice functionto recover the utility function using integrability.


For more than two goods, WARP and budget balancedness does notnecessarily imply symmetry of the Slutsky matrix because of the lackof transitivity. To have a choice function to be equivalent to the demandfunctions, we need a stronger axiom than WARP imposed on theconsumer’s behavior, that is Strong Axiom of Revealed Preference(SARP).

Definition: Strong Axiom of Revealed Preference (Mas-Colell et.al, 1995)The choice function x(p, y) satisfies the strong axiom of revealedpreference if for any list,

(p1, y1), (p2, y2), · · ·, (pN , yN)

with xn+1(pn+1, yn+1) 6= xn(pn, yn) for all n ≤ N − 1, we havepN · x1(p1, y1) > yN whenever pn · xn+1(pn+1, yn+1) ≤ yn for alln ≤ N − 1.

In words, if x1(p1, y1) is directly or indirectly preferred to xN(pN , yN),then xN(pN , yN) cannot be revealed preferred to x1(p1, y1).


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Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter 2)economics-course.weebly.com/uploads/2/5/7/2/25725158/week_9.pdf · 2.1 Duality: A Closer Look The solutions to utility