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Math For Economics

This document introduces the concept of Lagrange multipliers and their marginal interpretation. It shows that the Lagrange multiplier λ represents the rate of change of the maximum value of the objective function with respect to variations in the constraint. This allows λ to be interpreted as the marginal value or shadow price of relaxing the constraint. The document applies this to consumer utility maximization and firm cost minimization, showing that the multiplier equals the marginal utility of income for consumers and the marginal cost of output for firms.

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90 views2 pages

Math For Economics

This document introduces the concept of Lagrange multipliers and their marginal interpretation. It shows that the Lagrange multiplier λ represents the rate of change of the maximum value of the objective function with respect to variations in the constraint. This allows λ to be interpreted as the marginal value or shadow price of relaxing the constraint. The document applies this to consumer utility maximization and firm cost minimization, showing that the multiplier equals the marginal utility of income for consumers and the marginal cost of output for firms.

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Math Nerd
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© © All Rights Reserved
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LECTURE 2: THE MEANING OF THE MULTIPLIERS

Reference: Pemberton & Rau Section 18.1

Consider the following problem:


maximise f ( x, y ) subject to g ( x, y )  0 (A)
Let the constrained maximum for this problem be at x  x * and y  y * .

Now consider the following variant with the same objective function:
maximise f ( x, y ) subject to g ( x, y )  b , (B)
where b is a constant.
The constrained maximum value of f ( x, y ) will now, in general, depend on b, and we
denote it by v(b) . We note that v(0)  f ( x*, y*) .
Define
H ( x, y )  f ( x, y )  v( g ( x, y )) .
Choose any point ( x, y ) and let b  g ( x, y ) . Then f ( x, y )  v(b) and so H ( x, y )  0 . But
g ( x*, y*)  0 and f ( x*, y*)  v(0) , so H ( x*, y*)  v(0)  v(0)  0 . Therefore H is
maximised at ( x*, y*) .
By the composite function rule,
H f g
  v( g ( x, y )) .
x x x
Hence, at the point ( x*, y*) we have
f g
 v(0) 0
x x
and similarly
f g
 v(0) 0.
y y
Setting   v(0) , we obtain the Lagrange multiplier rule for problem (A):
f g f g
  0,  0
x x y y
where  can be interpreted as the rate of response of the maximal value of f to
variations in the right-hand side of the constraint. The result generalises to problem (B) to
give:
Proposition For problem (B), let v(b) be the constrained maximum value of f, and let 
be the Lagrange multiplier at the optimum. Then   v(b) .

This proposition provides the marginal interpretation of the Lagrange multiplier.

Economic applications
Recall the utility-maximisation problem of Lecture 1:
maximise U ( x1 , x 2 ) subject to p1 x1  p 2 x 2  m.
The optimal levels of x1 , x 2 can be written

x1  f 1 ( p1 , p 2 , m), x 2  f 2 ( p1 , p 2 , m)
and the functions f 1 , f 2 are called the consumer's demand functions. If we substitute
these into the utility function, we get the maximum level of utility subject to this budget
constraint V ( p1 , p2 , m) . This is known as the indirect utility function. Then the
marginal interpretation of the Lagrange multiplier of the consumer’s problem is that
V
 .
m
Similarly, we may consider the firm’s cost minimisation problem
minimise w1 x1  w2 x2 subject to F ( x1 , x2 )  q.
The firm’s cost function C ( w1 , w2 , q) is defined to be the minimum level of cost subject to
the output constraint. Then, using the marginal interpretation of the multiplier, we get
C
 .
q

Exercises: Pemberton & Rau 18.1.1-18.1.3

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