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# Utility Maximization - PowerPoint PPT Presentation

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. X & & are perfect

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Utility Maximization

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## 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

X & & are perfect

complementary goods

• Non-normal

### 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

I=150

I=100

### CES Demand | Px=5

• I=100 & I = 150

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

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* = 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 = 2UPx1/2Py1/2

• 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

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