4. Duality in Consumer Theory

EC 701, Fall 2005, Microeconomic Theory October 20, 2005 page 181

4. Duality in Consumer Theory

Definition 4.1. For any utility function U (x), thecorresponding indirect utility function is given by:

V (p,w) ≡ maxx{U (x) | x ≥ 0 , px ≤ w}

≡ maxx{U (x) | x ∈ Bp,w},

so that if x∗ is the solution to the UMP, then V (p,w) = U (x ∗).

Note that

V (p,w) ≡ maxx{U(x) |x ≥ 0, px ≤ w}

and

x(p,w) ≡ argmaxx

{U(x) |x ≥ 0, px ≤ w},so that

V (p,w) ≡ U(x(p,w)).


Example 4.1. Find the demand correspondence and theindirect utility function for the linear utility function

U = x + y .

•With the given utility function, x and y are perfect substitutesand the MU s are both 1 so the consumer will buy only thecheaper good.

• Let pm = min {px , py}. Demand for the cheaper good will bew/pm and demand for the more expensive good will be 0 .

• If px = py then demand for the goods can be any combination

such that expenditures add up to w .


• The consumer will always buy w/pm units of the cheaper good, so

his utility must also be w/pm. Therefore, the indirect utilityfunction is

v (px, py, w) =w

min {px, py}

2 1p p=

1Buy x1 2( , , )V p p w u=

2Buy x

1p

2p

0

increasing V

2 1p p=

1Buy x1 2( , , )V p p w u=

2Buy x

1p

2p

0

increasing V

• ¿¿Quasiconcave??


Proposition 4.1. Let p 00 = αp + (1 − α)p 0 andw 00 = αw + (1 − α)w 0 for α ∈ [0 , 1 ]. Then

Bp00,w00 ⊂ Bp,w ∪Bp0,w0.

(If a new price and wealth vector is a convex combination of

two price and wealth vectors, then the new budget set will be

contained with the union of the two original budget sets.)

1x

2x

,p wB ′ ′

,p wB ′′ ′′wpB ,

1x

2x

,p wB ′ ′

,p wB ′′ ′′wpB ,


Proof. We prove the contrapositive:

• If x /∈ Bp,w and x /∈ Bp0,w 0, then x /∈ Bp00,w 00.• But this must be true, because:

if px > w and p 0x > w 0

then [αp + (1 − α)p 0]x > αw + (1 − α)w 0.


Proposition 4.2. If U is continuous and locally nonsatiated

(lns), then V is:

i). Homogeneous of degree 0 .

ii). Strictly increasing in w and monotonically decreasing in p.

iii). Quasiconvex (no-better-than sets, B(p,w), are convex).

iv). Continuous in p and w .


Informal Proof.

i). Homogeneity: V doesn’t change if the budget set doesn’t

change.

ii). Strictly increasing in w ; decreasing in p:

• nonsatiated preferences =⇒ strictly increasing in w .

• decreasing in p, becauseincreases in p make the budget set smaller

new budget set is inside the old one.


1x

2x

,p wB ′ ′( , )x p w′ ′

( , )x p w,p wB ′′ ′′

wpB ,

( , )x p w′′ ′′

u

( , ) ( , )V p w V p w u′ ′= =( , )V p w u′′ ′′ ≤

1x

2x

,p wB ′ ′( , )x p w′ ′

( , )x p w,p wB ′′ ′′

wpB ,

( , )x p w′′ ′′

u

( , ) ( , )V p w V p w u′ ′= =( , )V p w u′′ ′′ ≤

iii). Quasiconvex: suppose V (p,w) = V (p 0,w 0) = u.

• Let p 00,w 00 be a convex combination of p,w and p 0,w 0.

• From previous proposition, we know:

if x ∈ Bp00,w 00 then it must be in either Bp,w or Bp0,w 0since u is the maximum utility available in those sets we

have V (p 00,w 00) ≤ V (p,w) = V (p 0,w 0).


iv). Continuity:

• Bp,w is “continuous” in p and wfor small changes in p and w , additional and excludedcommodity bundles are very close to the ones already there.

The continuity of U does the rest.

Yes, this is not really a proof, but the idea is the right one.


Definition 4.2. Given U (x), the expenditure minimizationproblem (EMP) is

minxpx

s.t. U(x) ≥ u

Definition 4.3. Given p, u, the expen-diture function e is defined by

e(p, u) = px∗,

where x∗ solves EMP.

• The expenditure function yields the minimum expenditure

required to reach utility u at prices p.

•More formally:e(p, u) = min

x{px |U(x) ≥ u}


Example 4.2. Find the expenditure function for the linearutility function U = x + y . How much do we have to spend to

get 100 units of utility if px = 5 and py = 7?

•We already know that the indirect utility function isv (px, py, w) =

w

min {px, py}.

• To find his expenditure function we setu =

w

min {px, py}and solve for w . We have

e (px, py, u) ≡ w = umin {px, py} .

• Expenditure to get u = 100 when px = 5 and py = 7 .

e (5, 7, 100) = 100min {5, 7} = 500.


Proposition 4.3. (Duality) Given U (x), continuous andlns and a constant vector of prices p À 0 , we have

i). If x∗ solves the UMP for w > 0 ,

• then x∗ solves the EMP when u is set to U (x∗)

• and e(p, v(p,w)) = w .

ii). If y∗ solves the EMP for u > U (0),

• then y∗ solves the UMP when w is set to py∗

• and v(p, e(p, u)) = u.


Informal Proof.

i). Given that x ∗ solves UMP for prices p and income w ,

• suppose that x ∗ does not solve the EMP for prices p andutility U (x∗).

• Then there is an x 0 that gives as much utility as x∗ but costsstrictly less (px 0 < px ∗),

• Thus, in the UMP, we can spend a little more than px 0 whilespending less than w .

• Therefore by nonsatiation, we can find x 00 with px 00 < w and

U (x 00) > U (x 0) ≥ U (x∗), a contradiction.• Therefore e(p, v(p,w)) = px ∗,

• and nonsatiation of U implies that px∗ = w .


ii). Given that y∗ solves EMP,

• we know that U (y∗) ≥ u.• Suppose that y∗ does not solve the UMP.• Then there is a y 0 with py 0 ≤ w (≡ py∗) such thatU (y 0) > U (y∗) ≥ u

• Therefore because U is continuous, we can choose y 00 < y 0

with U (y 00) > U (y∗) but py 00 < py 0 ≤ py∗, a contradiction(y∗ didn’t minimize expenditure as assumed).

• The continuity of U implies that U (y∗) = u, for otherwisemoney could be saved by allowing U (y∗) to fall withoutviolating the utility constraint of the EMP.


Proposition 4.4. For U continuous and nonsatiated, e(p, u)is

i). Homogeneous of degree 1 in p.

ii). Strictly increasing in u; increasing in p.

iii). Concave in p.

iv). Continuous in p, u.

Informal Proof.

i). Homogeneous in p:

e(αp, u) = minx{αpx |U (x) ≥ u}

= αminx{px |U (x) ≥ u}

= αe(p, u).


ii). Strictly increasing in u and increasing in p:

• For utility:By definition e(p, u) is the required expenditure to obtain u.

Suppose u 0 > u could be obtained by consuming x 0 withoutincreasing expenditures.

By continuity of u we could obtain u 00 > u even if we

consume a little less than x 0, that is at a lower expenditurethan e(p, u), a contradiction.

• For prices:Suppose that p 0 > p.

Then if x 0 solves the EMP at prices p 0 with U (x 0) ≥ u, wehave

e(p0, u) = p0x0 ≥ px0 ≥ e(p, u)

[Why not p 0x 0 > px 0 ?]


iii). Concave in p. Think of a consumer who normally consumes x 00

at prices p 00

• Suppose she spends a day at prices p and another day atprices p 0, where

p00 =1

2(p+ p0) .

• Could reach same utility at same average expense byconsuming x 00 both days.

• But can save money by adapting her choice of goods to thecurrent prices.

• By substituting cheap for expensive goods, you can get sameutility for less money at more extreme prices than at average

prices.


•More formally:Suppose that for α ∈ (0 , 1), p 00 = αp + (1 − α)p 0

and suppose that x 00 solves the EMP with utility u, so thatU (x 00) ≥ u.Then px 00 ≥ e(p, u) and p 0x 00 ≥ e(p 0, u) [why?]Therefore

e(p 00, u) = p 00x 00 = (αp + (1 − α)p 0)x 00

= αpx 00 + (1 − α)p 0x 00

≥ αe(p, u) + (1 − α)e(p 0, u).

iv). Continuous in p, u. Follows from:

• continuity of the constraint set {x |U (x) ≥ u} as a functionof u

• and continuity of the objective function px in x and p.


Definition 4.4. Hicksian demand h(p, u) is a consumptionvector x ∗ that solves the EMP.

•We havee (p, u) = min

x{px |U(x) ≥ u}

and

h(p, u) = argminx

{px |U(x) ≥ u}

• Also, e(p, u) = ph(p, u).


Because x(p,w) solves the UMP and h(p, u) solves the EMP, theproposition on utility duality tells us:

Proposition 4.5. Given that U is continuous, u > U (0)and w > 0 , we have

i). x(p,w) = h(p, v(p,w)),

•Walrasian demand at wealth w = Hicksian demand at

utility level produced by w .

ii). h(p, u) = x(p, e(p, u)),

• Hicksian demand at utility u = Walrasian demand with

wealth required to reach u.


( ', )h p u

1x

( , )x p w

( ', )sx p x

u

2x u

( ', )x p w

Slutsky Compensation

HicksCompensation

PriceChange

( ', )h p u

1x

( , )x p w

( ', )sx p x

u

2x u

( ', )x p w

Slutsky Compensation

HicksCompensation

PriceChange

• Graph above shows the difference betweenSlutsky compensated demand xs(p

0, x)

and Hicksian demand h(p 0, u).


Suppose a consumer has consumption vector x(p,w) and utilityu = U (x(p,w)),

• and then prices change from p to p 0.

•We havexs(p

0, x) = x(p 0, p 0x(p,w))h(p 0, u) = x(p 0, e(p 0, u))

• Slutsky compensated demand = Walrasian demand when the

consumer is given sufficient wealth to buy his original consumption

vector x(p,w).

• Hicksian demand = Walrasian demand when the consumer is

given sufficient wealth to reach his original utility level,

u = U (x(p,w)).


•We know U (xs(p 0, x)) ≥ U (h(p 0, u)). [Why?]• As the price change p 0− p gets small, difference between Hicksiandemand and Slutsky demand becomes second-order small.

•We will show thatS(p,w) ≡ ∂xs(p

0, x)∂p0

¸p0=p≡ ∂h(p0, u)

∂p0

¸p0=p

• Both have the same derivatives at p 0 = p.

• Therefore, the Slutsky Equation is true for Hicksian compensateddemand.

• “Compensated demand” usually refers to Hicksian demand• Slutsky demand is rarely used.


Proposition 4.6. (M-C 3.E.4; Law of Demand) On average,

when prices rise, the substitution effect is negative. Moreformally:

• If U (x) is continuous and lns, and• h(p, u) is a function,• then for all p 00 and p 0

(p00 − p0)[h(p00, u)− h(p0, u)] ≤ 0.

Proof. We have

• (1) p 00h(p 00, u) ≤ p 00h(p 0, u) [why?]=⇒(2) p 00h(p 00, u)− p 00h(p 0, u) ≤ 0

• (3) p 0h(p 0, u) ≤ p 0h(p 00, u). [why?]=⇒(4) p 0h(p 0, u)− p 0h(p 00, u) ≤ 0

• Add (2) and (4) and factor the results.


4.1 The Envelope Theorem

• Suppose that a family of functions is described by f (x , y) fordifferent fixed parameters x and a variable y .

• At each point, we compare the values of all functions in thefamily, and choose the minimum value.

• This creates a new function g (y) ≡ minx f (x , y). The functiong(y) is called the lower envelope of f (x , y).

• In the figure, the family members and the lower envelope areplotted as functions of y .


y

1( , )f x y

)( yg

),(min)( yxfygx

=

2( , )f x y

3( , )f x y4( , )f x y

5( , )f x y

y

1( , )f x y

)( yg

),(min)( yxfygx

=

2( , )f x y

3( , )f x y4( , )f x y

5( , )f x y

• The theorem says that the slope of the envelope at any point is

the same as the slope of the member of the family that it touches.

•M-C has a more general version of the theorem: don’t worryabout it, because it is quite messy.


Proposition 4.7. (EnvelopeTheorem) Let g(y) = minx f (x , y), wheref (x , y) is differentiable. Then

g0(y) =∂f(x, y)

∂y

¸x=x(y)

where x(y) is the value of x thatminimizes f (x , y).

The intuition

• As y changes x also must change because x must alwaysminimizes f (x , y).

• If y changes by ∆y , the change ∆g(y) comes from two sources

directly from ∆y

and from ∆x (which is caused by ∆y).


• The envelope theorem says that if ∆y is small, the part of ∆gthat comes from ∆x (labeled ∆gx on the graph) is near 0 ,because...

The curves are flat at x(y), because x(y) minimizes f (x , y).

So at x = x(y), if y is held constant, ∆x produces a smallchange in f (x , y).

Almost all of the change in f (x , y) comes directly from ∆y .

x

2( , )f x y

2( )g y

1( )g y

1ˆ( )x y 2ˆ( )x y

),(min)( yxfygx

=

1( , )f x ygΔ

xΔ

xgΔ

ygΔ

x

2( , )f x y

2( )g y

1( )g y

1ˆ( )x y 2ˆ( )x y

),(min)( yxfygx

=

1( , )f x ygΔ

xΔ

xgΔ

ygΔ



Proof. Let x(y) be the solution of minx f (x , y). The f.o.c forx(y) is

∂f

dx

¸x=x(y)

= 0.

We can now write:

g(y) = f(x(y), y)),

so, by the chain rule,

g0(y) =∂f

dx

¸x=x(y)

x0(y) +∂f

∂y.

The first term is 0 .


Proposition 4.8. If U (x) is continuous and lns, and h(p, u)is a function, then

h(p, u) = ∇pe(p, u).

Proof.

•We know thate(p, u) = min

x{px |U(x) = u}

• Notice the equality constraint [why equality?]•We can write this as a saddle-point problem:

e(p, u) = maxλminx{px− λ[u− U(x)]}

• Envelope theorem says: in calculating ∂e/∂p, λ and x can betreated as constants at their optimal values.

• The only term that contains p explicitly is px .

• Thus ∇pe(p, u) ≡ ∂e/∂p = x∗ ≡ h(p, u).


Proposition 4.9. (M-C 3.G.2) For the Jacobian matrix

∂h(p, u)/∂p we have:

i). ∂h(p, u)/∂p = ∂2e(p, u)/∂p2

ii). ∂h(p, u)/∂p is negative semidefinite,

iii). ∂h(p, u)/∂p is symmetric, and

iv). [∂h(p, u)/∂p] · p = 0

Proof. We have:

i). 2nd derivative of e(p, u): Immediate fromh(p, u) = ∂e(p, u)/∂p

ii). Negative semidefinite: From concavity of expenditure function.


iii). Symmetric:

• The off-diagonal elements of ∂h(p, u)/∂p are thecross-partial derivatives of e(p, u).

• But well-behaved functions have symmetric cross-partialderivatives (i.e. ∂2f /∂x∂y = ∂2f /∂y∂x).

iv). [∂h(p, u)/∂p] · p = 0

• h(p, u) is homogeneous of degree 0 in p.• Result follows from Euler’s formula.


Proposition 4.10. (Slutsky equation for Hicksian demand.)Given U (x) strictly quasiconcave and well-behaved and thecorresponding indirect utility function V (p,w), we have

∂xi(p,w)

∂pj=∂hi(p, u)

∂pj− ∂xi(p,w)

∂wxj(p,w)

where u = V (p,w).

Proof. The proof depends on the previously-established identityh(p, u) ≡ x(p, e(p, u)).

• By chain rule:∂hi(p, u)

∂pj≡ ∂xi(p,w)

∂pj+∂xi(p,w)

∂w

∂e(p, u)

∂pj

• But∂e(p, u)

∂pj≡ hj(p, u) ≡ hj(p, V (p,w)) ≡ xj(p,w)

• Substitution completes the proof.


Proposition 4.11. (Roy’s Identity) Given U (x) strictlyquasiconcave and well-behaved and the corresponding indirect

utility function V (p,w), we have

xj(p,w) = −∂V (p,w)

∂pj

Á∂V (p,w)

∂w.

Proof.

• First, the intuition:

∂V (p,w)

∂pj

Á∂V (p,w)

∂w$∆u

∆pj

Á∆u

∆w

=∆w

∆pj= −xj (p,w)

We overlooked some little details:

for example, xj (p,w) changes when p changes,

but xj (p,w) is a utility maximizer, so the envelope theoremtells us that we can ignore this change.


• Formal proof:Let u = V (p,w), so that w = e(p, u)

We have V (p, e(p, u)) ≡ uHold u constant. By the implicit-function theorem, we have:

∂e(p, u)

∂pj≡ − ∂V (p,w)

∂pj

Á∂V (p,w)

∂w

but

∂e(p, u)

∂pj≡ hj (p, u) = hj (p,V (p,w))

≡ xj (p,w).


• The chart below summarizes the duality between the UMP andthe EMP.

• It is taken (with editorial errors corrected) from M-C, p. 75.

( , )x p w

( , ( , ))u V p e p u=

EquationSlutsky

( , )V p w ( , )e p u

( , )h p u

EMPUMP

( , ( , ))w e p V p w=

( , ) ( , ( , ))x p w h p V p w= ( , ) ( , ( , ))h p u x p e p u=

'R oy sIdentity

( , )( , )p

h p ue p u

=∇

( , )x p w

( , ( , ))u V p e p u=

EquationSlutsky

( , )V p w ( , )e p u

( , )h p u

EMPUMP

( , ( , ))w e p V p w=

( , ) ( , ( , ))x p w h p V p w= ( , ) ( , ( , ))h p u x p e p u=

'R oy sIdentity

( , )( , )p

h p ue p u

=∇


.


5. Welfare Analysis

• Changes in price (and incomes) lead to changes in level of utilitythat consumers can obtain.

•Many economic policies affect prices:competition policy

foreign trade policy

tax law

business regulations

• It would be useful to be able to measure the effect of pricechanges on utility.


• Can we use the indirect utility function?If prices and incomes change from p0 ,w0 to p1 ,w1 ,

then utility increases by ∆U = v(p1 ,w1)− v(p0 ,w0).But not very useful for evaluating policy:

◦ Utility is not an observable economic variable.◦ Psychologists have done little to create tools for measuringutility.

◦Most economists don’t trust the measurements ofpsychologists.

◦Most economists consider utility to be only an ordinal ranking.


• All is not lost.• Economists can count money.• Traditional (Marshall’s) monetary measure of utility change fromadditional goods: willingness-to-pay (WPT).

• Suppose the consumer is willing to pay (at most) $100 for 40kilograms of rice. Then he is indifferent between:

(40 kilograms of rice and $100 less)

and no change

• Conclusion: WTP is a measure of utility gain from goods.


Definition 5.1. The willingness to pay WTP(x) for acommodity vector x is the maximum amount the consumer

would voluntarily pay for x .

Definition 5.2. Consumer surplus (CS) gained from x is

given by WTP(x)− px .

•WTP can be measured in the market.

•WTP $ area under demand curve


x

0( , )x p w

( , )x p w

p

0p

WTP

Marginalp

0 0( , )p x p w

CS

x

0( , )x p w

( , )x p w

p

0p

WTP

Marginalp

0 0( , )p x p w

CS


•More precisely:Let p(x ,w) be the demand-price function

◦ inverse of demand function◦ prices p at which the consumer would demand xWe have

WTP(x) $Z x

0

p(x,w)dx

or, equivalently

the willingness to pay for the marginal unit of a good at demand

x(p,w) is given by

MWTP(x) = p(x,w).

¿¿¿Why???


Consumer does not want to buy unit x when p > p(x ,w),

but DOES buy it when p = p(x ,w).

• But the measurement is not exact.Suppose x is selling at prices p0

We would like to know the MWTP(x) at prices p0 ,

but p(x ,w) gives us the MWTP(x) at prices p = p(x ,w).

Why are they different?

Because of the income effect.


•We can find other monetary equivalents of the utility changescaused by price changes.

• Theoretically more sound, [Robert Willig argued otherwise]• but more difficult to measure and use.• Suppose a consumer faces prices p0 and has wealth w .• Think of a possible price change from the price level p0 to the

level p1 .

• The price change would change utility from u0 = v(p0 ,w) tou1 = v(p1 ,w).


Definition 5.3. The compensatingvariation [in wealth], CV (p0 , p1 ,w) isthe amount of money that a consumer

would have to pay after a price change

from p0 to p1 in order to revert to her

original level of utility.

•Mathematically:v(p1, w − CV (p0, p1, w)) = v(p0, w) = u0


Definition 5.4. The equivalentvariation [in wealth], EV (p0 , p1 ,w) isthe amount of money a consumer would

have to receive in place of a price change

from p0 to p1 in order to reach the level

of utility that the price change would have

created.

•Mathematically:v(p0, w + EV (p0, p1, w)) = v(p1, w) = u1

• Both compensating and equivalent variation are positive for aprice change that increases utility.


•With CV we are increasing wealth as we increase prices in order

to keep the consumer at her original level of utility, u0 .

•With EV we are decreasing wealth instead of increasing pricesin order to keep the consumer at the new lower level of utility, u1 ,

that would have been created by the price increase.

• Both measures determine the change in wealth that is preciselyequivalent to a change in prices,

• but at different levels of utility.


Proposition 5.1. Both CV and EV can be expressed by use

of the expenditure function as follows:

i). CV (p0 , p1 ,w) = e(p1 , u1)− e(p1 , u0), andii). EV (p0 , p1 ,w) = e(p0 , u1)− e(p0 , u0),

where u0 = v(p0 ,w) and u1 = v(p1 ,w).


Proof. Remember that v and e are inverses. We have:

i).

e(p1 , u1)− e(p1 , u0)= e(p1 , v(p1 ,w))

−e(p1 , v(p1 ,w −CV (p0 , p1 ,w)))= w − w +CV (p0 , p1 ,w)

= CV (p0 , p1 ,w)

ii).

e(p0 , u1)− e(p0 , u0)= e(p0 , v(p0 ,w + EV (p0 , p1 ,w)))

−e(p0 , v(p0 ,w))= (w + EV (p0 , p1 ,w))− w= EV (p0 , p1 ,w)


• Keep in mind thate(p1, u1) ≡ e(p0, u0) ≡ w [why?]

so that we can write

i). CV (p0 , p1 ,w) = e(p0 , u0)− e(p1 , u0), andii). EV (p0 , p1 ,w) = e(p0 , u1)− e(p1 , u1).

• Because h(p, u) ≡ ∇pe(p, u), we know:


Proposition 5.2. CV and EV are given by the

path-independent line integrals

CV (p0, p1, w) =

Z p0

p1

h(p, u0) · dp,

and

EV (p0, p1, w) =

Z p0

p1

h(p, u1) · dp.where p1 and p2 are price vectors (not scalers) and h(p, u) · dp(a scalar) is the inner product of the vectors h(p, u) and dp asp moves along a path from p1 to p0 in price space.

• The expression h(p, u) · dp ≡ de(p, u) is a scalar quantity thatrepresents the change in expenditure in all markets as the price

vector is continuously changed.

•We have assumed that e is such that the integral is notpath-dependent, which means it doesn’t matter how we get from

p1 to p0 . So if p1 = (2 , 4) and p0 = (6 , 7) we could integrate aswe go from (2 , 4) to (6 , 4) to (6 , 7), or we could integrate as wego from (2 , 4) to (2 , 7) to (6 , 7), and the value of the integralwould be thes same.


• The following graph illustrates the idea when all but one price iskept constant.

0( , )h p u

x1( , )x p w0( , )x p w

( , )x p w

1( , )h p u

p

0p

CV

1p

0( , )h p u

x1( , )x p w0( , )x p w

( , )x p w

1( , )h p u

p

0p

CV

1p

0( , )h p u

x1( , )x p w0( , )x p w

( , )x p w

1( , )h p u

p

0p

EV

1p

0( , )h p u

x1( , )x p w0( , )x p w

( , )x p w

1( , )h p u

p

0p

EV

1p


• Suppose we cut price $1 at a time.• The wealth released by each price cut is equal to the currentamount we are buying.

• But we are holding utility constant as we cut price,• by taking this money away from the consumer as it is released.

• So, as the price changes, the amount purchased remains on thecompensated demand curve h(p, u).

• The total amount of money taken away is the compensatingvariation.

• Compensating variation accumulates only in those markets inwhich price is changed. For a small price change, it is proportional

to the amount of the purchase in the corresponding market at the

current price.


•We can also use changes in consumer surplus as a monetarymeasure of a welfare change caused by a price change.

•More specifically, think of ∆CS the amount of additional moneymade available to the consumer as the price is gradually lowered.

• As with CV, ∆CS accumulates only in those markets in whichprice is changed. For a small price change, it is proportional to the

amount of the purchase in the corresponding market at the

current price..

• But the difference between CV and ∆CS , is that with CV wetake away the funds released as the price falls, and with ∆CS welet the consumer keep them.


•We have:

CS(p0) =

Z x(p0 ,w)

0

(p(x ,w)− p0)dx

CS(p1) =

Z x(p1 ,w)

0

(p(x ,w)− p1)dxso that

∆CS = CS(p1)−CS(p0)=

Z p0

p1

x(p,w)dp


• For normal goods, the various measures of welfare change arerelated as follows:

CV (p0, p1, w) < ∆CS < EV (p0, p1, w)

0( , )h p u

x1( , )x p w0( , )x p w

( , )x p w

1( , )h p u

p

0p

CSΔ

1p

0( , )h p u

x1( , )x p w0( , )x p w

( , )x p w

1( , )h p u

p

0p

CSΔ

1p


• For inferior goods the inequalities are reversed. See below.

0( , )h p u

x1( , )x p w0( , )x p w

( , )x p w

1( , )h p u

p

0p

CSΔ

1p0( , )h p u

x1( , )x p w0( , )x p w

( , )x p w

1( , )h p u

p

0p

CSΔ

1p

• Are the differences large?


Example 5.1. For the linear utility function U = x + y findthe demand correspondence, the indirect utility function, the

expenditure function, and the Hicksian compensated demand.

Then, for py = 2 and w = 60 find the compensating variation,

the equivalent variation and the change in consumer surplus if

px changes from 3 to 1 .

•With the given utility function, x and y are perfect substitutesand the MU s are both 1 so the consumer will buy only thecheaper good.

• Let pm = min {px , py}. Demand for the cheaper good will bew/pm and demand for the more expensive good will be 0 .

• If px = py then demand for the goods can be any combination

such that expenditures add up to w .


• The consumer will always buy w/pm units of the goods, so his

utility must also be w/pm. Therefore, the indirect utility functionis

v (px, py, w) =w

min {px, py}

• To find his expenditure function we setu =

w

min {px, py}and solve for w . We have

e (px, py, u) ≡ w = umin {px, py} .•We have

hx (px, py, u) =∂e

∂px=

(u for px < py

0 for px > py;

likewise for hy .

• Note that ∂e/∂px is undefined at px = py, but in that case it is

clear that hx and hy ∈ [0 , u] and hx + hy = u.


• Before the price change, when px = 3 , py = 2 and w = 60 , we

have

u0 =w

min {px, py}= 30,

and after px changes to 1 , we have

u1 =w

min {px, py}= 60.

•e (p0 , u0) ≡ e (3 , 2 , 30) = 30 · 2 = 60

e (p1 , u0) ≡ e (1 , 2 , 30) = 30 · 1 = 30

CV ≡ e (p0 , u0)− e (p1 , u0) = 30

•e (p0 , u1) ≡ e (3 , 2 , 60) = 60 · 2 = 120

e (p1 , u1) ≡ e (1 , 2 , 60) = 60 · 1 = 60

EV ≡ e (p0 , u1)− e (p1 , u1) = 60


• Consumer surplus changes as px changes but only while px < py,

that is as px goes from 2 to 1 . In that range we havex (px , py,w) = w/px so

∆CS =

Z 2

1

60

pxdpx = 60 [log 2 − log 1 ]

= 60 log 2 = 41 .6 .

Note that 41 .6 is between 30 and 60 .

4. Duality in Consumer Theory

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Transcript of 4. Duality in Consumer Theory