Chapter4: Utility Maximization And Choice
Section: Chapter Questions
Problem 4.12P
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Review Appendix 2A of Griffin (on Lagrangian Method). Especially go over Box 2A.1. Using the utility function U = (x^1)(5w^1)(5), income = 40k, p_x = 1, find the demand function for water. (Follow the same steps.).
Transcribed Image Text:x0 +2-(-p) =0, and
40000- x-p w =0.
The first two equations can be combined to obtain optimal x in terms of w, yielding x= 30pw,
which can be substituted into the third necessary equation. Solving for w yields the water demand
function
%3D
40000
31p
which has the expected negative slope.
Transcribed Image Text:Interpretation of Lagrange Multipliers
The Lagrange multiplier is a "shadow price" associated with the constraint it premultiplies.
JE the constraint could be "relaxed" a little, the value of A tells us how much we will gain
in V. In some situations this type of information is very useful. While it is not especially
useful in this instance because utility has only ordinal significance, solving for à in the
boxed example tells us how much utility the consumer would gain if one had another
dollar of income. The result will be functionally dependent on water price.
Βox 2A.I
Example of demand obtained by utility maximization
A consumer has preferences between the total value of all other goods he consumes, x, and water,
w, that are given by the utility function U =x"w. The consumer has $40,000 of income to spend.
What is his demand for water?
Because of the simple way x is defined, its price is 1. (You can get another unit of the
good if you pay $1.) Because we are seeking a water demand function, the price of water
must be treated as variable. Let p be the price of water. The objective function is x"w with
decision variables x and w. The consumer's budget constraint is 1-x+p wS 40000 or
40000 - x-p w 20. The appropriate Lagrangian is
L(x, w, 2) = xw+2.(40000– x-p-w).
Necessary equations (2.48) and (2.49) are then applied:
30x w+2-(-1)=0,
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