1.4.1 Indirect Utility 1.4.1 Indirect Utility

Lecture 2 Consumer Lecture 2 Consumer

theory (continued) theory (continued)

Topics 1.4 : Topics 1.4 :

Indirect Utility function and Indirect Utility function and

Expenditure function.

Expenditure function.

Relation between these two Relation between these two

functions.

functions.

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1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function Function

•• The level of utility when x(p,y) is The level of utility when x(p,y) is

chosen must be the highest one for chosen must be the highest one for

given prices and income.

given prices and income.

•• Different Prices and income will give Different Prices and income will give us the different level of utility.

us the different level of utility.

•• Let call the function that relate the Let call the function that relate the maximized utility to prices and

maximized utility to prices and income as the

income as the indirect utility functionindirect utility function

1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function Function

•• v(v(pp, y)= u(, y)= u(x(px(p,y)),y))

•• When u is continuous, x exist, When u is continuous, x exist, then v is well

then v is well--defined.defined.

•• From graph, v(p,y) gives the From graph, v(p,y) gives the value of utility of the highest value of utility of the highest

indifference curve that is indifference curve that is

tangent to the budget line.

tangent to the budget line.

u=v(p1,p2,y)

x1

x2

1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function: Properties Function: Properties

•• 1. Continuous at all p, y >01. Continuous at all p, y >0

•• 2. Homogenous of degree zero in 2. Homogenous of degree zero in (p, y)

(p, y)

•• 3. Strictly increasing in y3. Strictly increasing in y

•• 4. Decreasing in prices4. Decreasing in prices

•• 5. Quasiconvex in (p, y)5. Quasiconvex in (p, y)

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1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function: Properties Function: Properties

•• 6. Roy6. Roy’’s Identity:s Identity:

–– xx₁₁ (p, y) = -(p, y) = - ∂∂v/v/∂∂pp₁₁ / / ∂∂v/v/∂∂yy Proof

Proof

1.1. By the theorem of the MaximumBy the theorem of the Maximum 2.2. To show that v(To show that v(tptp,,tyty)=t)=t⁰⁰v(p,y). v(p,y).

Intuitively, the budget line is not Intuitively, the budget line is not shifted.

shifted.

3. 3. ∂∂v/v/∂∂y > 0. Using the envelop theorem, y > 0. Using the envelop theorem, we know this is equal to

we know this is equal to λλ^{* *} and > 0 .and > 0 . Intuitively, more income leads to higher Intuitively, more income leads to higher

utility.

utility.

1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function: Properties Function: Properties

4. 4. ∂∂v/v/∂∂p < 0. Use envelope p < 0. Use envelope theorem.

theorem.

5. By definition 5. By definition

v(pv(p^tt,,yy^tt) ) ≤≤ max[v(pmax[v(p¹¹,y,y¹¹ ), v(p), v(p²²,y,y²² )])]

Consumers prefers one of any two Consumers prefers one of any two

extreme budget sets to any extreme budget sets to any

average of the two.

average of the two.

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1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function: Properties Function: Properties

Or set {p: v(p,y)

Or set {p: v(p,y) ≤≤ k} or lower k} or lower contour set is a convex set.

contour set is a convex set.

p1

p2

1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function: Properties Function: Properties

v v(p,y)

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1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function: Properties Function: Properties

•• 6. Roy6. Roy’’s identity, froms identity, from

∂∂v/v/∂∂pp₁₁ = = -- λλ^{* *} xx₁₁^** And And

∂∂v/v/∂∂y = y = λλ^{* *}

1.4.1 Indirect Utility 1.4.1 Indirect Utility

Function: Properties Function: Properties

•• Let check all properties from Let check all properties from familiar utility.

familiar utility.

•• Using CD utility functionsUsing CD utility functions

•• v(p1,p2,y) = y / v(p1,p2,y) = y / pp₁₁^aapp₂₂^1-1-aa

•• Try CES function as shown in p. Try CES function as shown in p.

3131--32.32.

•• v(p1,p2,y) = y / (v(p1,p2,y) = y / (pp₁₁^{r r} + p+ p₂₂^rr))^--1/r1/r

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1.4.2 Expenditure 1.4.2 Expenditure

Function Function

•• What is the minimum level of What is the minimum level of money or expenditure needed money or expenditure needed for given prices to get a given for given prices to get a given

level of utility?

level of utility?

•• minmin_xx px stpx st. u(x) . u(x) ≥≥ u.u.

•• We specify the attainable level We specify the attainable level of u, then ask for the minimum of u, then ask for the minimum

money of achieving it.

money of achieving it.

x1 x2

u Slope=-p1/p2 Isoexpenditure:

e=p1x1+p2x2

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1.4.2 Expenditure 1.4.2 Expenditure

Function Function

•• minmin_xx px stpx st. u(x) . u(x) ≥≥ u. u.

•• Let Let xx^hh(p,u) solves the problem(p,u) solves the problem

•• The lowest expenditure to get u The lowest expenditure to get u is equal to

is equal to pxpx^hh(p,u) (p,u)

•• Let e(p,u) Let e(p,u) ≡≡ pxpx^hh(p,u) (p,u)

•• Called the Expenditure function.Called the Expenditure function.

1.4.2 Expenditure 1.4.2 Expenditure

Function Function

•• xx^hh(p,u) is called Hicksian (p,u) is called Hicksian Demand or Compensated Demand or Compensated

demand function.

demand function.

•• We can draw the Hicksian We can draw the Hicksian

demand for good1 by letting the demand for good1 by letting the price of x1 to vary, as we did for price of x1 to vary, as we did for

the Marshallian demand.

the Marshallian demand.

x1 x2 Let p

changes from p

to p

Before: slope= - p₁⁰ / p₂⁰

After: slope= - p₁¹ / p₂⁰

a

b

u

x1 p1

x1(p1, p

,u) p

p

We assume p2 and u

constant: Hicksian

demand for good 1

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1.4.2 Expenditure 1.4.2 Expenditure

Function Function

•• Hicksian demand tells us what Hicksian demand tells us what consumption bundles achieves a consumption bundles achieves a

targeted utility and minimizes total targeted utility and minimizes total

expenditure.

expenditure.

•• Compensated term means Compensated term means ““as price of as price of good 1 varies (goes up), you lose some good 1 varies (goes up), you lose some

utility, so to keep u constant, you utility, so to keep u constant, you

needed to be compensated to be as needed to be compensated to be as

happy as before.

happy as before.

•• Hicksian demand is not directly Hicksian demand is not directly

observable since it depends on utility, observable since it depends on utility,

which is not observable.

which is not observable.

1.4.2 Expenditure 1.4.2 Expenditure

Function: properties Function: properties

•• Theorem 1.7Theorem 1.7

•• 1. Continuous in p1. Continuous in p

•• 2. Strictly increasing in u2. Strictly increasing in u

•• 3. Increasing in p3. Increasing in p

•• 4. Homogenous of degree 1 in p4. Homogenous of degree 1 in p

•• 5. Concave in p5. Concave in p

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1.4.2 Expenditure 1.4.2 Expenditure

Function: properties Function: properties

•• 6. Shepard6. Shepard’’s Lemmas Lemma

xx₁₁ ^hh(p,u) = (p,u) = ∂∂e(p,u)/e(p,u)/∂∂pp₁₁ Proof.

Proof.

(try it by yourself, same ideas as (try it by yourself, same ideas as

in the indirect utility) in the indirect utility)

1.4.2 Expenditure 1.4.2 Expenditure

Function: properties Function: properties

•• ExampleExample

–– Min p1x1 + p2x2Min p1x1 + p2x2 –– St. u(x1, x2) =uSt. u(x1, x2) =u L(x1,x2,

L(x1,x2,λλ) = p1x1 + p2x2 ) = p1x1 + p2x2 –– [u(x1, x2) [u(x1, x2) ––u]u]

Suppose u(x1, x2) =x

Suppose u(x1, x2) =x₁₁^aaxx₂₂^1-1-aa answer

answer

•• xx₁₁ ^hh(p,u) = a(p(p,u) = a(p₂₂ / p/ p₁₁))^1-1-aa uu

•• xx₂₂ ^hh(p,u) = (1(p,u) = (1--a)(pa)(p₁₁ / p/ p₂₂))^aa uu

•• And e(p,u) = pAnd e(p,u) = p₁₁^aapp₂₂^1-1-aauu

•• Check ShepardCheck Shepard’’s lemmas lemma

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1.4.3 Relations between 1.4.3 Relations between

Expenditure and Indirect utility Expenditure and Indirect utility

•• If you did my assignment, you know If you did my assignment, you know that

that

•• v(p1,p2,y) = y / v(p1,p2,y) = y / pp₁₁^aapp₂₂^1-1-aa

•• If we compare this with the If we compare this with the expenditure function

expenditure function

•• e(p,u) = pe(p,u) = p₁₁^aapp₂₂^11-a-auu

•• We see that the expenditure function We see that the expenditure function is the inverse of indirect utility

is the inverse of indirect utility

function. We can inverse v function to function. We can inverse v function to

see y that relates to u and p.

see y that relates to u and p.

Theorem 1.8 Duality between Theorem 1.8 Duality between indirect utility and expenditure indirect utility and expenditure

•• If we replace y with e, and v with u in If we replace y with e, and v with u in the indirect utility function we will get the indirect utility function we will get

the expenditure function, and vice the expenditure function, and vice

versa.

versa.

•• 1. e(p, v(p,y) ) = y 1. e(p, v(p,y) ) = y

•• The minimum expenditure necessary The minimum expenditure necessary to get utility v(p,y) is y.

to get utility v(p,y) is y.

•• 2. v(p, e(p,u) ) = u2. v(p, e(p,u) ) = u

•• The maximum utility from income The maximum utility from income

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Theorem 1.8 Duality between Theorem 1.8 Duality between indirect utility and expenditure indirect utility and expenditure

•• This duality lets us work with one This duality lets us work with one problem and invert them to get

problem and invert them to get solution of another problem.

solution of another problem.

x2

x1 u(x

)=u

y/p2

(x

,x

)

Maximized utility and

minimized expenditure

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Theorem 1.9 Duality between Theorem 1.9 Duality between

Marshallian and Hicksian Demand Marshallian and Hicksian Demand

•• If x* solves UPM, let u=u(x*), If x* solves UPM, let u=u(x*), then x* solves EM.

then x* solves EM.

•• Where UPM: Where UPM: maxmax_xx u(x) u(x) st pxst px ≤≤ yy

•• And EM: minAnd EM: min_xx px stpx st u(x) u(x) ≥≥ u.u.

•• Similarly, If x* solves EM, let y Similarly, If x* solves EM, let y

==pxpx*, then x* solves UPM.*, then x* solves UPM.

Theorem 1.9 Duality between Theorem 1.9 Duality between

Marshallian and Hicksian Demand Marshallian and Hicksian Demand

•• 1. x (p, y) = 1. x (p, y) = xx^hh(p, v(p,y) )(p, v(p,y) )

•• The Marshallian demand at The Marshallian demand at income y is the same as

income y is the same as

Hicksian demand at utility v(p,y) Hicksian demand at utility v(p,y)

•• 2. 2. xx^hh(p,u) = x (p, e(p,u) )(p,u) = x (p, e(p,u) )

•• Hicksian demand at utility u is Hicksian demand at utility u is the same as the Marshallian

the same as the Marshallian demand at income e(p,u).

demand at income e(p,u).

p1

x1 x

x

x

=x

(p,u)= x

(p,v(p,y))=x

(p,y)

UMP x(p,y)

v(p,y)

EM

x

(p,u)

e(p,u)

xx₁₁^hh(p,u) = (p,u) =

∂∂e(p,u)/e(p,u)/∂∂pp₁₁

xx₁₁ (p, y) = (p, y) = --

∂∂v/v/∂∂pp₁₁ / / ∂∂v/v/∂∂yy