Week 8: Consumer Theory Part 3 (Jehle and Reny,...

Week 8: Consumer Theory Part 3 (Jehle and Reny,Chapter 1)

Tsun-Feng Chiang*

*School of Economics, Henan University, Kaifeng, China

November 23, 2014

1 / 29

Consumer Theory Indirect Utility Function

1.4 Indirect Utility Function (Continued)

Example 1.3

Suppose the direct utility function is again the CES form, u(x1, x2) =(xρ1 + xρ2 )

1/ρ, where 0 6= ρ < 1. The consumer’s problem forminimization of expenditure function is,

minx1,x2p1x1 + p2x2 s.t. u = (xρ1 + xρ2 )1/ρ, x1 ≥ 0, x2 ≥ 0

and its Lagrangian,

L(x1, x2, λ) = p1x1 + p2x2 + λ[u − (xρ1 + xρ2 )1/ρ].

Assuming an interior solution in both goods, the first-order conditionsare as follows:

∂L∂x1

= p1 − λ(xρ1 + xρ2 )(1/ρ)−1xρ−1

1 = 0

2 / 29


Example 1.3 (Continued)

∂L∂x2

= p2 − λ(xρ1 + xρ2 )(1/ρ)−1xρ−1

2 = 0

∂L∂λ

= u − (xρ1 + xρ2 )1/ρ = 0

By rearranging these three equations and letting r ≡ ρ/(ρ− 1), weobtain the Hicksian demand functions,

xh1 (p1,p2,u) = u(pr

1 + pr2)

(1/r)−1pr−11 ,

xh2 (p1,p2,u) = u(pr

1 + pr2)

(1/r)−1pr−12 .

To form the expenditure function, substitute demand functions into theobjective function

e(p1,p2,u) = u(pr1 + pr

2)1/r

3 / 29


Relations between the Two

This section describes the relations between indirect utility andexpenditure functions, and the duality between Marshallian andHicksian demand functions.First, fix (p, y) and let u = v(p, y). The indirect utility function meansgiven the prices p, the utility level u can be attained by the incomelevel y . The expenditure function e(p,u), by the definition, says thatgiven the prices p, it is the smallest expenditure needed to attain alevel of utility at least u. Hence we must have e(p,u) ≤ y . Insertu = v(p, y) into e(p,u) ≤ y , we obtain

e(p, v(p, y)) ≤ y (eq.1)

Next fix (p,u) and let y = e(p,u). The expenditure function says thatgiven the prices p, y is the smallest income needed to attain the levelof utility u. If the consumer’s income were in fact y , then he couldattain at least the level of utility u. By definition, v(p, y) is the largest

4 / 29


utility level he can achieve given prices p and income y , this impliesv(p, y) ≥ u. Replace y in v(p, y) ≥ u with y = e(p,u), we obtain

v(p,e(p,u)) ≥ u (eq.2)

Under some familiar conditions, (eq.1) and (eq.2) must be held inequality.

Theorem 1.8 Relations Between Indirect Utility and ExpenditureFunctionsLet v(p, y) and e(p,u) be the indirect utility function and expenditurefunction for some consumer whose utility function is continuous andstrictly increasing. Then for all p� 0, y ≥ 0, and u ∈ U :

1 e(p, v(p, y)) = y2 v(p,e(p,u)) = u

This theorem allows us to derive both indirect utility and expenditurefunctions by solving either one of consumer’s maximization orminimization problem, but not both.

5 / 29


By solving a consumer’s minimization problem, we obtain e(p,u). ByTheorem 1.8,

e(p, v(p, y)) = y

Holding prices and viewing it as a function of utilty, it is possible toinvert the expenditure function by multiplying both sides by e−1,

v(p, y) = e−1(p : y).

Similarly, by solving a consumer’s maximization problem, we canobtain v(p, y). By Theorem 1.8, it is possible to derive the expenditurefunction using the

e(p,u) = v−1(p : u).

The maximization problem and minimization problem are conceptuallyjust opposite sides of the same coin. Mathematically, both the indirectutility function and the expenditure function are simply theappropriately chosen inverses of each other.

6 / 29


Example 1.4Using the CES utility function in Example 1.1, in Example 1.2, we solvethe consumer’s maximization problem and obtain the indirect utilityfunction,

v(p, y) = y(pr1 + pr

2)−1/r

For the income level equal to e(p,u), therefore, we must have

v(p,e(p,u)) = e(p,u)(pr1 + pr

2)−1/r

From Theorem 1.8,

v(p,e(p,u)) = u

Combine the last two equations and solve it for e(p,u), we get theexpression

e(p,u) = u(pr1 + pr

2)1/r

7 / 29


Example 1.4 (Continued)This is the exact expenditure function we derive from Example 1.3. Onthe other hand, we can also derive the indirect utility function inExample 1.2 using the expenditure function in Example 1.3.

Except for the indirect utility functions and the expenditure functions,the solutions of the utility-maximization and expenditure-minimization,Marshallian demand functions and Hicksian demand functionsrespectively, also have a close relationship. The following theoremclarifies the links between Hicksian and Marshallian demands.Theorem 1.9 Duality Between Marshallian and Hicksian DemandFunctionsUnder Assumption 1.2 we have the following relations between theHicksian and Marshallian demand functions for p� 0, y ≥ 0, u ∈ U ,and i = 1, · · · ,n:

1. xi(p, y) = xhi (p, v(p, y)). 2. xh

i (p,u) = xi(p,e(p,u)).

8 / 29


The first relation says that the Marshallian demand at prices p andincome y is equal to the Hicksian demand function at prices p and theutility level that is the maximum that can be achieved at prices p andincome y . The second says that the Hicksian demand at given pricesp and utility level is equal to the Marshallian demand at the givenprices p and an income level equal to the minimum expenditure atthose prices to achieve that utility level.

Example 1.5From Example 1.2, the indirect utility function is

v(p, y) = y(pr1 + pr

2)−1/r .

From Example 1.3, the Hicksian demand function is

xhi (p,u) = u(pr

1 + pr2)

(1/r)−1pr−1i , i = 1,2

By Theorem 1.9,

9 / 29


Example 1.5 (Continued)

xhi (p, v(p, y)) = v(p, y)u(pr

1 + pr2)

(1/r)−1pr−1i

= y(pr1 + pr

2)−1/r (pr

1 + pr2)

(1/r)−1pr−1i

= ypr−1i (pr

1 + pr2)

−1

=ypr−1

ipr

1 + pr2, i = 1,2.

The final expression on the right-hand side is the Marshallian demand.Similarly we can derive Hicksian demand function from the Marshalliandemand function by Theorem 1.9

xi(p, y) =ypr−1

ipr

1 + pr2⇒ xi(p,e(p,u)) =

e(p,u)pr−1i

pr1 + pr

2= u(pr

1 + pr2)

1/r pr−1i

pr1 + pr

2

= u(pr1 + pr

2)(1/r)−1pr−1

i , i = 1,2.

10 / 29


The relations in Theorem 1.8 and Theorem 1.9 can be illustrated inFigure 1.18.Figure 1.18 Illustration of Theorems 1.8 and 1.19.

11 / 29


Fixing the prices p, given the income y , Figure (a) shows the consumercan achieve the maximum utility u(p, y) where the solutions are x∗1 andx∗2 . The corresponding Marshallian demand function of x1 is illustratedin Figure (b). On the other hand, to minimize the expenditure given theutility u, the consumer chooses x∗1 and x∗2 such that the expenditure isexactly as much as y . The corresponding Hicksian demand function ofxh

1 is illustrated in Figure (b). We can see that at the point x∗1 in Figure(b), the Marshallian and Hicksian demand functions intersect, whichmeans that given the prices, xh

1 (p, v(p, y)) is equal to x1(p, y) whenthe consumer minimizes his expenditure to achieve the highest level ofutility v(p, y), or x1(p,e(p, y)) is equal to xh

1 (p,u) when the consumermaximizes his utility with the smallest expenditure e(p, y).

12 / 29

Consumer Theory Properties of Consumer Demand

1.5 Properties of Consumer Demand

Economists generally prefer to measure important variables in real,rather than monetary, terms. Relative prices and real income are twosuch real measures.By the relative price of some good, we mean the number of units ofsome other good that must be forgone to acquire 1 unit of the good inquestion. If pi and pj are the prices of good i and j , respectively, therelative price of good i in terms of good j is

pi

pj=

y /unit iy /unit j

=y

unit i· unit j

y=

unit of junit of i

The price ratio pi/pj means how many units of j acquire when forgoinga unit of good i .By real income, we mean the maximum number of units of somecommodity the consumer could acquire if he spend his entire moneyincome. The real income measured by good i , for example, is

13 / 29


ypj

=y

y /unit j= units of j .

One of the properties of the indirect utility function is homogeneity ofdegree zero in prices and income. This implies a consumer has nomoney illusion, that is, his utility and consumption bundle do notchange as both the prices and income change by the same proportionso his real income is unchanged. The following theorem formalizes thedescription above:

Theorem 1.10 Homogeneity and Budget Balancedness

Under Assumption 1.2, the consumer demand function xi(p,y),i = 1,2, · · · ,n, is homogeneous of degree zero in all prices andincome, and it satisfies budget balancedness, p · x(p, y) = y for all(p, y).Proof: We had proved the indirect utility is homogeneous of degreezero in prices and income, so that

14 / 29


Theorem 1.10 (Continued)

v(p, y) = v(tp, ty) for all t > 0.

This is equivalent to the statement

u(x∗(p, y)) = u(x∗(tp, ty)) for all t > 0.

where x∗ is the demand that maximizes the utility. Because the budgetsets at (p, y) and (tp, ty) are the same, each of x∗(p, y) and x∗(tp, ty)was feasible when the other was chosen. By Assumption 1.2, the utilityfunction u is strict quasiconcave, the previous equality implies that

x∗(p, y) = x∗(tp, ty) for all t > 0.

Thus we prove the demand is homogeneous of degree zero in pricesand income.

15 / 29


Income and Substitution Effects

When the price of a good declines, the quantity demanded of thatgood is uncertain. See Figure 1.19 where the price of good 1 declines.The quantity demand of good 1 increases in (a); does not change in(b); and decreases in (c).Figure 1.19 Response of quantity demanded to a change in price.

16 / 29


When the price of a good declines, there are at least two conceptuallyseparate reasons why we expect some change in the quantitydemand. First, that good becomes relatively cheaper compared toother goods, so we expect the consumer to substitute the relativelycheaper good for the now relatively more expensive ones. This is thesubstitution effect (SE). At the same time, however, when the price ofa good declines, the real income measured by that good increases.The effect of increase of the purchasing power on quantity demands iscalled the income effect (IE).As the price change occurs, we can only observe the quantitydemands before and after the price changes. The observabledifference is called the total effect (TE). However, the Hicksiandecomposition can distinguish the quantity demand due to the SE andIE. The decomposition formalizes the intuitive notion of the SE as thefollowing: The SE is that hypothetical change in consumption thatwould occur if relative prices were to change to their new level but themaximum utility the consumer can achieve were kept the same as

17 / 29


before the price change. The TE is then defined as whatever is left ofthe total effect after the SE. In other words, the TE is the sum of the SEand the IE.Figure 1.20(a) (see the next slide) gives a clear example. When theprices change from (p0

1,p02) to (p1

1,p02) (p1 declines, and p2 does not

change), the utility level increases from u0 to u1 and the quantitieschange from (x0

1 , x02 ) to (x1

1 , x12 ), which is the TE. Suppose there is no

change in utility, the change in prices leads the quantity demandsmove from (x0

1 , x02 ) to a hypothetical level (xs

1 , xs2 ), which is the SE.

Then the change from (xs1 , x

s2 ) to (x1

1 , x12 ) is the IE.

Figure 1.20(b) shows that because the SE measures how the quantitydemands changes along the indifference curve as the price of a goodchanges, the SE is just the quantity change along the Hicksiandemand function. The Marshallian demand curve picks up the TE ofthe price change. The two diverge from one another is equal to the IE.The relationship between the TE, SE and IE are summarized in theSlutsky equation.

18 / 29


Figure 1.20 The Hicksian decomposition of a price change (p1 declines).

19 / 29


Theorem 1.11 The Slutsky Equation

Let x(p, y) be the consumer’s Marshallian demand system. Let u∗ bethe level of utility the consumer achieves at prices p and income y .Then,

∂xi(p, y)∂pj︸︷︷︸TE

=∂xh

i (p,u∗)

∂pj︸︷︷︸SE

−xj(p, y)∂xi(p, y)∂y︸︷︷︸

IE

, i , j = 1, · · · ,n.

For the own price change, the equation becomes

∂xi(p, y)∂pi

=∂xh

i (p,u∗)

∂pi− xi(p, y)

∂xi(p, y)∂y

(eq.3)

The term on the left is the slope of the Marshallian demand curve forgood i , the response of quantity demanded to a change in own price.The sign of this term is determined by the other two terms on the right.

20 / 29


The first right-handed term in (eq.3) is the slope of the Hicksiandemand curve whose sign is prove by the theorem:

Theorem 1.12 Negative Own-Substitution Terms

Let xhi (p,u) be the Hicksian demand for good i . Then

∂xhi (p,u)∂pi

≤ 0, i = 1, · · · ,n.

Proof: By Shephard’s lemma (Theorem 1.7),

∂e(p,u)∂pi

= xhi (p,u)

Because the expenditure function is concave in prices, the secondderivative is nonpositive,

∂e2(p,u)∂p2

i=∂xh

i (p,u)∂pi

≤ 0

21 / 29


For the sign of the second right-handed term, it is uncertain. It couldbe ∂xi(p, y)/∂y > 0 for the good i is normal, or ∂xi(p, y)/∂y < 0 forthe good i is inferior. For the normal good i , the left-handed term of(eq.3) ∂xi(p, y)/pi < 0 or the Law of Demand holds.

Theorem 1.13 the Law of DemandA decrease in the own price of normal good will cause quantitydemanded to increase. If an own price decrease causes a decrease inquantity demanded, the good must be inferior.

To move beyond our current knowledge about Marshallian andHicksian demands, we have to probe the system of substitution termsa bit deeper. We first establish that "cross-substitution terms" aresymmetric.

Theorem 1.14 Symmetric Substitution Terms

Let xh(x,u) be the consumer’s system of Hicksian demands andsuppose that e(·) is twice continuously differentiable. Then,

22 / 29


Theorem 1.14 Symmetric Substitution Terms

∂xhi (p,u)∂pj

=∂xh

j (p,u)∂pi

, i , j = 1, · · · ,n

If we arrange all substitute terms in Theorem 1.14 into a matrix withthe own-substitution terms on the diagonal and cross-substitution termoff-diagonal, then the matrix is negative semidefinite.

Theorem 1.15 Negative Semidefinite Substitution Matrix

Let xh(p,u) be the consumer’s system of Hicksian demands, and let

σ(p,u) ≡

∂xh

1 (p,u)∂p1

· · · ∂xh1 (p,u)∂p1

.... . .

...∂xh

n (p,u)∂p1

· · · ∂xhn (p,u)∂pn

Called the substitution matrix, contain all the Hicksian substitution terms.Then the matrix σ(p,u) is negative semidefinite.

23 / 29


The Hicksian demand function and the substitution matrix areunobservable. Using the Slutsky equation, we can transform thesubstitution matrix into an observable matrix with the same properties.

Theorem 1.16 Symmetric and Negative Semidefinite Slutsky Matrix

Let x(p, y) be the Marshallian demand system. Define the ij th Slutskyterms as

∂xi(p, y)∂pj

+ xj(p, y)∂xi(p, y)∂y

and form the entire n × n Slutsky matrix of price and incomeresponses as follows:

s(p, y) =

∂x1(p,y)∂p1

+ x1(p, y)∂x1(p,y)∂y · · · ∂x1(p,y)

∂pn+ xn(p, y)

∂x1(p,y)∂y

.... . .

...∂xn(p,y)∂p1

+ x1(p, y)∂xn(p,y)∂y · · · ∂xn(p,y)

∂pn+ xn(p, y)

∂xn(p,y)∂y

Then s(p, y) is symmetric and negative semidefinite.

24 / 29

Consumer Theory Some Elasticity Relations

Some Elasticity Relations

The consumer’s Marshallian demand should satisfy the budgetbalance, which says that the budget constraint must hold with equalityat every set of prices and income, or that

y =n∑

i=1

pixi(p, y) = p · x(p, y).

From this equality, we can derive some properties that can be testedempirically. These properties are expressed in terms of price andincome elasticities of demand.

Definition 1.6 Demand Elasticities and Income SharesLet xi(p, y) be the consumer’s Marshallian demand for good i . Then let

ηi ≡∂xi(p, y)∂y

yxi(p, y)

, εij ≡∂xi(p, y)∂pj

pj

xi(p, y)

be income elasticity and price elasticity, respectively.25 / 29


Definition 1.6 (Continued)and let

si ≡pixi(p, y)

y, so that si ≥ 0 and

n∑i=1

si = 1.

be income share, or proportion of consumer’s income, spent onpurchase of good i .

The income elasticity of demand for good i measures the percentagechange in the quantity of i demanded per 1 percent change in income.The price elasticity of demand for good i measures the percentagechange in the quantity of i demanded per 1 percent change in the pricepj . If j = i , εii is called the own-price elasticity of demand for good i .If j 6= i , εij is called the cross-price elasticity of demand for good iwith respect to pj .

26 / 29


Theorem 1.17 Aggregation in Consumer Demand

Let x(p, y) be the consumer’s Marshallian demand system. Let ηi , εij ,and si , for i , j = 1, · · · ,n, be as defined before. Then the followingrelations must hold among income shares, price and incomeelasticities of demand:1. Engel aggregation :

∑ni=1 siηi = 1.

2. Cournot aggregation :∑n

i=1 siεij = −sj , j = 1, · · · ,n.

Proof: We begin by recalling that the budget constraint requires

y =∑n

i=1 pixi(p, y)

For all p and y . Take the derivative on both sides with respect to y ,

1 =∑n

i=1 pi∂xi∂y

Multiply the right hand side by yxi

xiy = 1. Use the definition of the

income elasticity, then Engel aggregation is proved.

27 / 29


Theorem 1.17 (Continued)To prove Cournot aggregation, first rewrite the budget constraint:

y =∑n

i 6=j pixi(p, y) + pjxj(p, y), for an specific j ,

Take the first derivative with respective to pj on both sides, we obtain

0 =

n∑i 6=j

pi∂xi

∂pj

+ xj + pj∂xj

∂pj,

Move the term −xj to the left hand side and multiply both sides bypj/y . We then obtain

−pjxj

y=

n∑i=1

pi

y∂xi

∂pjpj

Multiply and divide by xi on the right hand side, and use the definitionof the price elasticity to complete the proof of Cournot aggregation.

28 / 29


Announcement

Homework 2Due on Monday, December 1st, 2014 in class.

Second Midterm ExamDate: Monday, December 15th, 2014Time: 9:00 am ∼ 11:30 amLocation: To be AnnouncedCoverage: Jehle and Reny, Chapter 1 and 2

Others: Closed Book, Closed Notes. No scratch paper (will beprovided by the proctor), calculator, cellphone and other electronicdevices are allowed. Pens, correction tools and basic requirements oflife (ex. water, tissue paper, and medicine) are allowed.

29 / 29

Week 8: Consumer Theory Part 3 (Jehle and Reny,...

Documents

Transcript of Week 8: Consumer Theory Part 3 (Jehle and Reny,...