utility maximazation
Utility Maximization Steps
MPP 801
Fall, 2007
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 x This is a special case of the "Cobb-Douglas" utility function, which has the form:
M RS =
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 = x M RS
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
1
Step 1 Set MRS equal to price ratio px py
1
=
1
= x
M RS
=
y x y
this relationship must hold at the utility maximizing point.
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 x= 100
= 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,
M RS
=
y x =
y
=
px py 0.5
MPP 801
Fall, 2007
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 x This is a special case of the "Cobb-Douglas" utility function, which has the form:
M RS =
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 = x M RS
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
1
Step 1 Set MRS equal to price ratio px py
1
=
1
= x
M RS
=
y x y
this relationship must hold at the utility maximizing point.
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 x= 100
= 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,
M RS
=
y x =
y
=
px py 0.5