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Lecture 6: Duality

Advanced Microeconomics I

Yosuke YASUDA

Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp

October 21, 2014

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Connection between UMP and EMP

There is a strong link between the utility maximization problem (UMP) and the expenditure minimization problem (EMP). Let us first consider the following practice question.

✞

✝

☎

Q A consumer has the following indirect utility function:✆ v(p₁, p₂, ω) = ^ω

2

2p1^p2

.

1 What is the consumer’s Marshallian demand for good 1?

2 What is the expenditure function?

3 What is the consumer’s Hicksian demand for good 1?

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Answers

1 Using Roy’s identity, we obtain

x₁(p, ω) = −

∂v(p,ω)

∂p₁

∂v(p,ω)

∂ω

= −

− ^ω2

2p²₁p₂ ω p₁p₂

= ^ω

2p₁^.

2 By duality (explained formally later) and the indirect utility function, the indirect utility function can be translated into the following expenditure function:

u= ^{e(p, u)}

2

2p₁p₂ ^{⇐⇒ e(p, u) =p2p1p2u.}

3 Using Shephard’s lemma, we obtain x^h₁(p, u) = ^{∂e(p, u)}

∂p₁ ⁼ r p₂u

2p1

.

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Dual Problem - Example (1)

✞

✝

☎

Ex A dual turtle problem.✆

1 The maximal distance a turtle can travel in 1 day is 1 km.

2 The minimal time it takes a turtle to travel 1 km is 1 day. Let M (t) be the maximal distance the turtle can travel in time t.

✞

✝

☎

Q What kinds of conditions (on M (t)) are needed for the above✆ statements to be equivalent?

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Dual Problem - Example (2)

Theorem 1

Assume M is “strictly increasing” and “continuous.” Then, The maximal distance a turtle can travel in t^∗ is x^∗

is equivalent to

The minimal time it takes a turtle to travel x^∗ is t^∗ Proof.

(⇒): If the maximal distance that the turtle can pass within t^∗ is x^∗, and if the minimal time to cover the distance x^∗ is strictly less than t^∗, then by strict monotonicity the turtle would cover a distance strictly larger than x^∗.

(⇐): If it takes t^∗ to cover the distance x^∗ and if the turtle passes a larger distance than x^∗ in t^∗, then by continuity the turtle will be beyond the distance with strictly less time than t^∗.

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Dual Problem - Theory (1)

Applying the duality idea to the consumer problem, we can establish the close relationship between the indirect utility and expenditure functions, and between the Marshallian and Hicksian demand functions.

Let v(p, ω) and e(p, u) be the indirect utility function and expenditure function. Then, by definition, the following property must hold:

e(p, v(p, ω)) ≤ ω for all (p, ω) ≫ 0. v(p, e(p, u)) ≥ u for all (p, u) ∈ Rⁿ₊₊^×R.

The next theorem demonstrates that under certain conditions on preferences, both of these inequalities must be equalities.

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Dual Problem - Theory (2)

Theorem 2

Suppose the consumer’s preference satisfy continuity and monotonicity. Then for all p ≫ 0, ω ≥0 and u ∈ R:

e(p, v(p, ω)) = ω (1)

and

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

Holding prices in both functions constant, we can invert the indirect utility function in its income variable. Applying the inverse function, denoted by v⁻¹(p : ·), to both sides of (2), we obtain

e(p, u) = v⁻¹(p : u).

Similarly, applying the inverse function of the expenditure function, denoted by e⁻¹(p : ·), to both sides of (1), we obtain

v(p, ω) = e⁻¹(p : ω).

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Dual problem - Theory (3)

Theorem 3

Suppose the consumer’s preference is continuous, monotone and strictly convex. Then, we have the following relations between the Hicksian and Marshallian demand functions for p ≫ 0, ω ≥0 and u ∈ R, and i = 1, 2, ..., n:

x_i(p, ω) = x^h_i(p, v(p, ω)) and

x^h_i(p, u) = xi(p, e(p, u)).

1 Marshallian demand at prices p and income ω is equal to the Hicksian demand at those prices and the maximum utility level that can be achieved at those prices and income ω.

2 Hicksian demand at any prices p and utility level u is the same as the Marshallian demand at those prices and an income level equal to the minimum expenditure necessary at those prices to achieve utility level u. 8 / 14

Slutsky Equation (1)

When the price of a good declines, there are two conceptually separate reactions: The consumer is expected to substitute the relatively cheaper good for the now relatively more expensive good (= substitution effect), and to arrange her purchases of all goods due to the expansion of her effective income, i.e., the budget set (= income effect).

Theorem 4

Suppose the consumer’s preference is continuous, monotone and strictly convex, and all the relevant functions are differentiable. Let u^∗ be the level of utility the consumer achieves at prices p and income ω^∗. Then, for i, j= 1, ..., n,

∂xi(p, ω^∗)

∂pj TE

= ^∂x

hi^{(p, u∗)}

∂pj SE

−xj(p, ω^∗)^{∂xi(p, ω∗)}

IE ∂ω

.

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Slutsky Equation (2)

By duality, x^h_i(p, u^∗) = xi(p, e(p, u^∗)).

Since this equality holds for all p ≫ 0, differentiating both sides with respect to p_j preserves the equality.

∂x^h_i(p, u^∗)

∂p_j ⁼

∂x_i(p, e(p, u^∗))

∂p_j ⁺

∂x_i(p, e(p, u^∗))

∂ω

∂e(p, u^∗)

∂p_j ^. By duality and Shephard’s lemma,

e(p, u^∗) = e(p, v(p, ω^∗)) = ω^∗

∂e(p, u^∗)

∂pj

= x^h_j(p, u^∗) = x^h_j(p, v(p, ω^∗)) = xj(p, ω^∗). Substituting these relations into the second equation,

∂x^h_i(p, u^∗)

∂p_j ⁼

∂xi(p, ω^∗)

∂p_j ⁺

∂xi(p, ω^∗)

∂ω ^{xj(p, ω}

∗_).

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Slutsky Equation (3)

The above equation is called the Slutsky equation (sometimes called the “Fundamental Equation of Demand Theory”), which provides neat analytical expressions for substitution and income effects. When j = i, the Slutsky equation shows the response of the Marshallian demand to a change in own price.

∂x_i(p, ω)

∂pi

= ^∂x

hi^{(p, u∗)}

∂pi

−x_i(p, ω)^{∂xi(p, ω)}

∂ω ^.

Although substitution effects are not observable, demand theory can provide some strong properties on own-price effects and cross-substitution effects. The first claim says that own-price effects can never be positive.

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Substitution Effects (1)

Theorem 5

Suppose e(p, u) is twice continuously differentiable in p. Then, for i= 1, ..., n, ^∂x

hi^{(p, u)}

∂pi

≤0. Proof.

By Shephard’s lemma, x^h_i(p, u) = ^{∂e(p, u)}

∂p_i ^.

Differentiating again with respect to p_i, we obtain

∂x^h_i(p, u)

∂pi

= ^∂

2_{e(p, u)}

∂p²_i ^.

The right hand side must be non-positive since the expenditure function is a concave function of p.

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Substitution Effects (2)

The non-positive own-price effects give us some implication to the response of the Marshallian demand as well.

A good is called normal (resp. inferior) if consumption of it increases (resp. declines) as income increases, holding prices constant.

A decrease in the own price of a normal good will cause quantity demanded to increase. If an own price decrease causes a decline in quantity demanded (known as Giffen’s paradox), the good must be inferior.

The next theorem says that “cross-substitution effects” are symmetric.

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Substitution Effects (3)

Theorem 6

Suppose e(p, u) is twice continuously differentiable in p. Then, for i, j= 1, ..., n, ^∂x

hi^{(p, u)}

∂p_j ⁼

∂x^h_j(p, u)

∂p_i ^. Proof.

By Shephard’s lemma,

∂x^h_i(p, u)

∂p_j ⁼

∂

∂p_j⁽

∂e(p, u)

∂p_i ^{) =}

∂²e(p, u)

∂p_j∂p_i ^{, and}

∂x^h_j(p, u)

∂pi

= ^∂

∂pi

(^{∂e(p, u)}

∂pj

) = ^∂

2_{e(p, u)}

∂pi∂pj

. By Yong’s theorem, we complete the proof:

∂²e(p, u)

∂p_j∂p_i ⁼

∂²e(p, u)

∂p_i∂p_j ^.

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