Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

Uxy( , )measures the total satisfaction or utility the consumer receives from having x

units of the first commodity and y units of the second.) (Answer: xy=18, = 8 )

5.2 The Lagrange Multiplier
In some problems, we need to compute the Lagrange multiplierλ since it has the
following useful interpretation.

Suppose M is the maximum (or minimum) value off(xy, )subject to the

constraintgxy K( , )=. The Lagrange multiplier λis the rate of change of M

with respect to K. That is,

dM

dK

λ=

Hence,

λ Change in M resulting from a 1-unit increase in K.

Example 4
Suppose the editor in Example 2 is allotted $60,200 instead of $ 60,000 to spend
development and promotion of the new book. Estimate how the additional $ 200 will
affect the maximum sales level.

Solution
In Example 2, you solved the three Lagrange equations
1/ 2
3/2

30

20

60

xy

x

xy

λ

λ

=

=

+=

to conclude that the maximum value M off(xy, )subject to the constrain

occurred when and. To find

xy+= 60

x= 36 y= 24 λsubstitute these values of x and y into the

first or second Lagrange equation. Using the second equation, you get

(^) ()
3/2
λ==20 36 4, 320
The goal is to estimate the changeΔMin the maximal sales that will result from an
increase ofΔ=K 0.2(thousand dollars) in the available funds.
Since
dM
dK
λ= , the one-variable approximation formula, gives

4, 320(0.2) 864

dM

MKK

dK

Δ ≈ Δ=Δ=λ =

That is, maximal sales of the book will increase by approximately 864 copies if the
budget is increased from $ 60,000 to $60,200 and the money is allocated optimally.

Example 5
Suppose the consumer in example 3 has $601 instead of $600 to spend on the two
commodities. Estimate how the additional $1 will affect the maximum utility.
(Answer: ΔM0.22)