Utility Maximization

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.