Utility Maximization Steps

ECON 6500

The MRS and the Cobb-Douglas

Consider a two-good world, x and 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 = p x x + py y
Unless there is a Corner Solution, the solution will occur where the highest indifference curve is tangent to
the budget constraint. Equivalent to that is the statement: The Marginal Rate of Substitution equals the
price ratio, or
px
M RS =
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 M RS. As it turns out, every
utility function has its own M RS, 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 M RS without
calculus.
For example, if the utility function is
U = xy
then
y
M RS =
x
This is a special case of the "Cobb-Douglas" utility function, which has the form:
U = xa y b
where a and b are two constants. In this case the marginal rate of substitution for the Cobb-Douglas utility
function is ³a´ ³y ´
M RS =
b x
regardless of the values of a and b.

Solving the utility max problem

Consider our earlier example of "Skippy" where
U = xy
y
M RS =
x
Suppose Skippy’s budget information is as follows: B = 100, px = 1, py = 1. Her budget constraint is
B = px x + py y
100 = x + y

Step 1 Set MRS equal to price ratio

px
M RS =
py
y 1
=
x 1
y = x
this relationship must hold at the utility maximizing point.

1
Step 2 Substitute step 1 into budget constraint
Since y = x, the budget constraint becomes

100 = x + y
= x+x
= 2x

Solving for x yields

100
x= = 50
2
Therefore
y = 50
and
u = (50)(50) = 2500

Change the price of x

Now suppose the price of x falls to 0.5 or 1/2. Re-do steps 1 and 2,
px
M RS =
py
y 0.5 1
= =
x 1 2
1
y = x
2
Substitute this new relationship into the budget constraint

100 = x + y
1
100 = x + x
2
100 = 1.5x
100
x = = 66.7
1.5
y = 33.3

General Solution to Cobb-Douglas Utility

Using the general form of the Cobb-Douglas
U = xa y b
where
ay
M RS =
bx
and the budget constraint in the form
B = p x x + py y
where the price ratio is px /py , the first rule of utility maximization yields
px
M RS =
py
ay px
=
bx py
b px
y = x
a py

2
Substituting into the budget constraint yields
µ ¶
b px
B = px x + py x
a py
b
B = px x + px x
¡ a+b ¢ a
B = a px x (see footnote for algebra)
³ ´B
x∗ = a
a+b p
x

Similarly, we can find y by the same method, which gives us

µ ¶
∗ b B
y =
a + b py

The solutions for x and y are called the consumer’s DEMAND FUNCTIONS.
Note that in our first example where U = xy, the values of a and b are a = b = 1 substituting into x∗
and y ∗ we get
³ ´B
x∗ = 1
1+1 px
B
x∗ =
2px
and
B
y∗ =
2py
Use the values of px , py , and B to test to see if these equations give you the solutions in example One.
If we substitute the answers back into the utility function, we get
µ ¶µ ¶
B B
U = xy =
2px 2py
2
B
U =
4px py

This gives you the utility number directly from the budget and prices. If you re-arrange this expression
to get B by itself, you get p
B = 4px py U
You can use this equation to calculate the amount of budget is needed if you know prices AND the desired
utility number (Helpful for CV and EV)

0 The trick used here is as follows:

b a b
x+ x = x+ x
a a a
 
a b
= + x
a a
a+b
= x
a