Advanced Microeconomics (ES30025)

Advanced Microeconomics (ES30025)

Mathematics Review 2: The Lagrange Multiplier

Outline: I. Introduction

II. Duality Theory: Cobb Douglas Example III. Final Comments

I.

Introduction

The simplest method of solving a constrained maximization or constrained minimization problem is by the Lagrange multiplier method. In what follows we illustrate the method by considering, firstly, the consumer’s problems of maximizing (Cobb-Douglas) utility subject to a budget constraint, thereby yielding the consumer’s Marshallian demand function. We then, secondly, consider the consumer’s problem of minimizing expenditure subject to a (Cobb-Douglas) utility constraint, thereby yielding the consumer’s ( Hicks) Compensated demand function. We also allude to duality of the two problems and comment on the interpretation of the Lagrange Multiplier itself.

II.

Duality Theory: Cobb-Douglas Example

Marshallian Demand Functions

The Marshallian demand function is obtained by maximizing utility subject to a budget constraint. Thus: max x1,x2 Ax₁b_x 2 1−b st p₁x₁+ p₂x₂ =m (1) Solution: max x1,x2,λ { }L g _{= Ax} 1 b_x 2 1−b_{+λ m− p} 1x1− p2x2

(

)

(2) ∂Lg ∂x₁ =bAx1 b−1_x 2 1−b_−λp 1 =0 (3) ∂Lg ∂x₂ =

( )

1−b Ax1 b x₂−b_−λ p₂ =0 (4) ∂Lg ∂λ =m− p1x1− p2x2 =0 (5) Dividing (3) by (4):

bAx₁b−1_x 2 1−b 1−b

( )

Ax₁b_x 2 −b = λp₁ λp₂ ⇒ b 1−b

( )

⋅xx2₁ = p₁ p₂ ⇒ x₁= b 1−b

( )

⋅ pp2₁ x2 (6) Rearrange (5): x₂ = m p₂ − p₁ p₂ x1 (7) Substitute (7) in (6): x₁= b 1−b⋅ p₂ p₁ m p₂ − p₁ p₂ x1 ⎛ ⎝⎜ ⎞ ⎠⎟ ⇒ x₁=bm p₁ (8) Thus: x₂ = m p₂ − p₁ p₂ x1 ⇒ m p₂ − p₁ p₂ b m p₁ ⎛ ⎝⎜ ⎞ ⎠⎟ x₂ =

( )

1−b m p₂ (9)

The consumer’s Marshallian demand functions are thus:

um _{=u x} 1 m_,x 2 m

(

)

=A x

( )

₁m b _x 2 m

( )

1−b ⇒ um _{=A b}m p₁ ⎛ ⎝⎜ ⎞ ⎠⎟ b 1−b

( )

m p₂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ⇒ um _=Abb

( )

_1−b1−b m p₁b_p 2 1−b (11)

Note that the budget constraint is satisfied:

p₁x₁m_{+ p} 2x2 m_{= p} 1 b m p₁ ⎛ ⎝⎜ ⎞ ⎠⎟+p2

( )

1−b m p₂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=m (12)

Compensated Demand Function

The consumer’s Compensated demand function is obtained by minimizing expenditure subject to a utility constraint. Thus:

min x1,x2 p₁x₁+ p₂x₂ st Ax₁b_x 2 1−b _=u (13) Solution: min x₁,x₂,λ { }L g _{= p} 1x1+ p2x2 +θ u− Ax1 b_x 2 1−b

(

)

(14) ∂Lg ∂x₁ = p1−θbAx1 b−1_x 2 1−b _{=0 (15)} ∂Lg ∂x₂ = p2−θ

( )

1−b Ax1 b_x 2 −b _{=0 (16)} ∂Lg ∂θ =u −Ax1 b_x 2 1−b _{=0 (17)} Dividing (15) by (16) yields:

p₁ p₂ = θbAx₁b−1_x 2 1−b θ

( )

1−b Ax₁b_x 2 −b ⇒ p₁ p₂ = b 1−b

( )

⋅ xx2₁ ⇒ x₁= b 1−b⋅ p₂ p₁ x2 (18) Rearrange (17): x₂ = u Ax₁b ⎛ ⎝ ⎜ ⎞_⎠⎟ 1 1−b (19) Substitute in (19) in (18): x₁= b 1−b⋅ p₂ p₁ u Ax₁b ⎛ ⎝ ⎜ ⎞_⎠⎟ 1 1−b ( ) ⇒ x₁= b 1−b⋅ p₂ p₁ u A ⎛ ⎝⎜ ⎞ ⎠⎟ 1 1−b ( ) x₁ −b 1−b ( ) ⇒ x₁1+ b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ ₌ b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ u A ⎛ ⎝⎜ ⎞ ⎠⎟ 1 1−b ( ) ⇒ x₁ 1 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ ₌ b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ u A ⎛ ⎝⎜ ⎞ ⎠⎟ 1 1−b ( ) (20)

Raising every term by the power (1-b) yields:

x₁= b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ( ) u A ⎛ ⎝⎜ ⎞ ⎠⎟ (21)

x₂ = u Ax₁b ⎛ ⎝ ⎜ ⎞_⎠⎟ 1 1−b ( ) = u A ⎛ ⎝⎜ ⎞ ⎠⎟ 1 1−b ( ) x₁ −b 1−b ( ) ⇒ x₂ = u A ⎛ ⎝⎜ ⎞ ⎠⎟ 1 1−b ( ) b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ( ) u A ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ −b 1−b ( ) = b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ( ) ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ −b 1−b ( ) _u A ⎛ ⎝⎜ ⎞ ⎠⎟ ⇒ x₂ = b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ −b u A ⎛ ⎝⎜ ⎞ ⎠⎟ ⇒ x₂ = 1−b b ⎛ ⎝⎜ ⎞ ⎠⎟ p₁ p₂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ b u A ⎛ ⎝⎜ ⎞ ⎠⎟ (22)

The consumer’s ( Hicks) Compensated demand functions are thus:

x₁h ₌ b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ( ) u A ⎛ ⎝⎜ ⎞ ⎠⎟ x₂h ₌ 1−b b ⎛ ⎝⎜ ⎞ ⎠⎟ p₁ p₂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ b u A ⎛ ⎝⎜ ⎞ ⎠⎟ (23)

Recall that the ( Hicks) Compensated demand functions measure the change in demand following a change in price holding the price of the other good and the consumer’s real income (vis. His ability to enjoy a particular level of utility) constant. Thus, the (Hicks) Compensated demand function measures the own-price substitution effect.

Note that the utility constraint is satisfied:

A x

( )

₁h b _x 2 h

( )

1−b = A b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ( ) u A ⎛ ⎝⎜ ⎞ ⎠⎟ ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ b 1−b b ⎛ ⎝⎜ ⎞ ⎠⎟ p₁ p₂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ b u A ⎛ ⎝⎜ ⎞ ⎠⎟ ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ 1−b ⇒ Ax₁b_x 2 1−b _=u (24)

Note that minimized expenditure is given by:

mh _{= p} 1x1 h_{+ p} 2x2 h ⇒ mh _{= p} 1 b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ( ) u A ⎛ ⎝⎜ ⎞ ⎠⎟+ p2 1−b b ⎛ ⎝⎜ ⎞ ⎠⎟ p₁ p₂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ b u A ⎛ ⎝⎜ ⎞ ⎠⎟ (25) Thus:

mh _{= p} 1 b_p 2 1−b u A ⎛ ⎝⎜ ⎞ ⎠⎟ b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ 1−b ( ) + 1−b b ⎛ ⎝⎜ ⎞ ⎠⎟ b ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⇒ mh _{= p} 1 b_p 2 1−b u A ⎛ ⎝⎜ ⎞ ⎠⎟ b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ −b 1+ b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⇒ mh _{= p} 1 b_p 2 1−b u A ⎛ ⎝⎜ ⎞ ⎠⎟ b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ −b 1 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⇒ mh _{= p} 1 b_p 2 1−b u A ⎛ ⎝⎜ ⎞ ⎠⎟ 1 b ⎛ ⎝⎜ ⎞ ⎠⎟ b 1 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ 1−b ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (26)

III. Final Comments

Interpretation of the Lagrange Multiplier

The Lagrange multiplier measures the marginal effect of a relaxation in the constraint on the consumer’s objective function. Thus, the Lagrange multiplier in the first problem, λin the first problem measures the marginal effect on utility of a relaxation in the budget constraint and may therefore be interpreted as the consumer’s marginal utility of income. The Lagrange multiplier in the second problem, θ, measures the marginal effect on expenditure on a relaxation in the utility constraint.

Second-Order Conditions

Note that we have not explored the second-order conditions to confirm that we have indeed found the utility maximizing and cost minimizing choices for the consumer. We will generally assume that second-order conditions are satisfied in Lagrange multiplier constrained optimization problems.

Duality

Note the duality of the problem. Set the constraint level of utility _u in the consumer’s (Hicks) Compensated demand function (23) equal to the Marshallian maximized utility _um _in

x₁= b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ( ) um A ⎛ ⎝⎜ ⎞ ⎠⎟ ⇒ x₁= b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1−b ( ) 1 A ⎛ ⎝⎜ ⎞ ⎠⎟ Ab b

( )

_1−b 1−b m p₁b_p 2 1−b ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⇒ x₁=b1−b+b

( )

_1−b −( )1−b+( )1−b _p 2 1−b ( )−( )1−b p₁−( )1−b−bm ⇒ x₁=bm p₁ (27)

This is the level of demand determined by the Marshallian demand function (10). A similar result occurs if we set the budget constraint in the consumer’s Marshallian demand function (10) equal to the (Hicks) Compensated minimized expenditure, _mh _{in Equation (26):}

x₁=bm h p₁ = b p₁ p1 b_p 2 1−b u A ⎛ ⎝⎜ ⎞ ⎠⎟ 1 b ⎛ ⎝⎜ ⎞ ⎠⎟ b 1 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ 1−b ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ ⇒ x₁= b 1 b ⎛ ⎝⎜ ⎞ ⎠⎟ b 1 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ 1−b ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ p₂ p₁ ⎛ ⎝⎜ ⎞ ⎠⎟ 1−b u A ⎛ ⎝⎜ ⎞ ⎠⎟ ⇒ x₁= b1−b 1 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ 1−b ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ p₂ p₁ ⎛ ⎝⎜ ⎞ ⎠⎟ 1−b u A ⎛ ⎝⎜ ⎞ ⎠⎟ ⇒ x₁= b 1−b ⎛ ⎝⎜ ⎞ ⎠⎟ p₂ p₁ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1−b u A ⎛ ⎝⎜ ⎞ ⎠⎟ (28)