Utility Maximization Continued July 5, 2005

Graphical Understanding Normal Indifference Curves Downward Slope with bend toward origin

Graphical Non-normal Indifference Curves Y & X Perfect Substitutes

Graphical Non-normal Only X Yields Utility

Graphical Non-normal X & & are perfect complementary 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) = X a Y (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 demand for X decreases – so X is called an inferior good

A change in Px Here the price of X changes…the budget line rotates about the vertical 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 ordinary demand function will have a positive instead of a negative slope. Note that this 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* = h x (P x, P y, U) X = d x (P x, P y, I)

The indirect utility function When we solve the consumer optimization problem, we arrive at optimal values of X and Y | I, P x, and P y. 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 = X 1/2 Y 1/2 we know V = (.5I/Px).5 (.5I/Py).5 =.5I/P x.5 P y.5

Indirect Utility V = 1/2I / (Px 1/2 Py 1/2 ) or I = 2VPx 1/2 Py 1/2 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 = 2VPx 1/2 Py 1/2 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 P x X + P y Y Subject to the utility constraint U = X 1/2 Y 1/2 So we form L = P x X + P y Y + λ(U- X 1/2 Y 1/2 )

Minimize Continued Let’s do this in class… We will find E = 2UP x 1/2 P y 1/2 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 P x ∂E/∂P x = U(Py/Px) 1/2 Let’s look at some simple examples

Ordinary & Compensated StatePxPymMxMyUHx In this example our utility function is: U = X.5 Y.5. We change the price of X from 5 to 10.