Utility Maximization
ContinuedJuly 5, 2005
Graphical Understanding Normal
Indifference Curves
Downward Slope withbend toward origin
Graphical Non-normal
Indifference Curves
Y & X Perfect Substitutes
Graphical Non-normal
Only X Yields Utility
Graphical Non-normal
X & & are perfectcomplementary goods
Calculus caution
When dealing with non-normal utility functions the utility maximizing FOC that MRS = Px/Py will not hold
Then you would use other techniques, graphical or numerical, to check for corner solution.
Cobb-Douglas
Saturday Session we know that if U(X,Y) = XaY(1-a) then X* = am/Px
m: income or budget (I) Px: price of X a: share of income devoted to X Similarly for Y
Cobb-Douglas
How is the demand for X related to the price of X?
How is the demand for X related to income?
How is the demand for X related to the price of Y?
CES Example U(x,y) = (x.5+y.5)2
CES Demand
Eg: Y = IPx/Py(1/(Px+Py))
Let’s derive this in class
CES Demand | Px=5 I=100 & I = 150
I=150
I=100
CES | I = 100
Px=10
Px=5
For CES Demand
If the price of X goes up and the demand for Y goes up, how are X and Y related?
On exam could you show how the demand for Y changes as the price of X changes?
dY/dPx
When a price changes
Aside: when all prices change (including income) we should expect no real change. Homogeneous of degree zero.
When one prices changes there is an income effect and a substitution effect of the price change.
Changes in income
When income increases demand usually increase, this defines a normal good.
∂X/∂I > 0 If income increases and demand
decreases, this defines an inferior good.
Normal goods
As income increase (decreases) the demand for X increase (decreases)
Inferior good
As income increases the demandfor X decreases – so X is calledan inferior good
A change in Px
Here the price of X changes…thebudget line rotates about thevertical intercept, m/Py.
The change in Px
The change in the price of X yields two points on the Marshallian or ordinary demand function.
Almost always when Px increase the quantity demand of X decreases and vice versa.
So ∂X/∂Px < 0
But here, ∂X/∂Px > 0
This time the Marshallian or ordinarydemand function will have a positiveinstead of a negative slope. Note thatthis is similar to working with an inferior good.
Decomposition
We want to be able to decompose the effect of a change in price The income effect The substitution effect
We also will explore Giffen’s paradox – for goods exhibiting positively sloping Marshallian demand functions.
Decomposition
There are two demand functions The Marshallian, or ordinary, demand
function. The Hicksian, or income compensated
demand function.
Compensated Demand
A compensated demand function is designed to isolate the substitution effect of a price change.
It isolates this effect by holding utility constant.
X* = hx(Px, Py, U) X = dx(Px, Py, I)
The indirect utility function
When we solve the consumer optimization problem, we arrive at optimal values of X and Y | I, Px, and Py.
When we substitute these values of X and Y into the utility function, we obtain the indirect utility function.
The indirect utility function
This function is called a value function. It results from an optimization problem and tells us the highest level of utility than the consumer can reach.
For example if U = X1/2Y1/2 we know V = (.5I/Px).5(.5I/Py).5 = .5I/Px
.5Py.5
Indirect Utility
V = 1/2I / (Px1/2Py1/2) or I = 2VPx1/2Py1/2
This represents the amount of income required to achieve a level of utility, V, which is the highest level of utility that can be obtained.
I = 2VPx1/2Py1/2
Let’s derive the expenditure function, which is the “dual” of the utility max problem.
We will see the minimum level of expenditure required to reach a given level of utility.
Minimize
We want to minimize PxX + PyY
Subject to the utility constraint U = X1/2Y1/2
So we form L = PxX + PyY + λ(U- X1/2Y1/2)
Minimize Continued
Let’s do this in class… We will find E = 2UPx
1/2Py1/2
In other words the least amount of money that is required to reach U is the same as the highest level of U that can be reached given I.
Hicksian Demand
The compensated demand function is obtained by taking the derivative of the expenditure function wrt Px
∂E/∂Px = U(Py/Px)1/2
Let’s look at some simple examples
Ordinary & Compensated
State Px Py m Mx My U Hx
1 5 4 100 10 12.5 11.18033989 10
2 10 4 100 5 12.5 7.90569415 7.071067812
In this example our utility function is: U = X.5Y.5. We change the price of X from 5 to 10.