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Utility Maximization

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Utility Maximization

Continued

July 5, 2005

- Normal Indifference Curves

Downward Slope with

bend toward origin

- Non-normal Indifference Curves

Y & X Perfect Substitutes

- Non-normal

Only X Yields Utility

X & & are perfect

complementary goods

- Non-normal

- 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.

- 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

- 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?

- Example U(x,y) = (x.5+y.5)2

Eg: Y = IPx/Py(1/(Px+Py))

Let’s derive this in class

I=150

I=100

- I=100 & I = 150

Px=10

Px=5

- 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

- 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.

- 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.

As income increase (decreases) the

demand for X increase (decreases)

As income increases the demand

for X decreases – so X is called

an inferior good

Here the price of X changes…the

budget line rotates about the

vertical intercept, m/Py.

- 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

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.

- 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.

- There are two demand functions
- The Marshallian, or ordinary, demand function.
- The Hicksian, or income compensated demand function.

- 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)

- 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.

- 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

- 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.

- 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.

- 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)

- 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.

- 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

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