Utility Maximization Steps
ECON 6500
The MRS and the Cobb-Douglas
Consider a two -go od world, xand y. Our consumer, Skippy, wishes to maximize utility, denoted U(x, y).
Her problem is then to Maximize:
U=U(x, y)
subject to the constraint
B=pxx+pyy
Unless there is a Corner Solution, the solution will occur where the highest indiļ¬erence curve is tangent to
the budget constraint. Equivalent to that is the statement: The Marginal Rate of Substitution equals the
price ratio,or
MRS =px
py
This rule, combined with the budget constraint, give us a two-step procedure for finding the solution to the
utility maximization problem.
First, in order to solve the problem, we need more information about the MRS. As it turns out, every
utility function has its own MRS, which can easily be found using calculus. However, if we restrict ourselves
to some of the more common utility functions, we can adopt some short-cuts to arrive at the MRS without
calculus.
For example, if the utility function is
U=xy
then
MRS =y
x
This is a special case of the "Cobb-Douglas" utility function, which has the form:
U=xayb
where aand bare two constants. In this case the marginal rate of substitution for the Cobb-Douglas utility
function is
MRS =Ā³a
bĀ“Ā³y
xĀ“
regardless of the values of aand b.
Solving the utility max problem
Consider our earlier example of "Skippy" where
U=xy
MRS =y
x
Suppose Skippyās budget information is as follows: B= 100,p
x=1,p
y=1. Her budget constraint is
B=pxx+pyy
100 = x+y
Step 1 Set MRS equal to price ratio
MRS =px
py
y
x=1
1
y=x
this relationship must hold at the utility maximizing point.
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