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

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Utility Maximization

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Utility maximization

Utility Maximization


July 5, 2005

Graphical understanding

Graphical Understanding

  • 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

Calculus caution

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 douglas1


  • 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?

Utility maximization


  • Example U(x,y) = (x.5+y.5)2

Ces demand

CES Demand

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

Let’s derive this in class

Ces demand px 5



CES Demand | Px=5

  • I=100 & I = 150

Ces i 100



CES | I = 100

For ces demand

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

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

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

Normal goods

As income increase (decreases) the

demand for X increase (decreases)

Inferior good

Inferior good

As income increases the demand

for X decreases – so X is called

an inferior good

A change in px

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 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

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.



  • 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.

Compensated demand

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

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 function1

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

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 2vpx 1 2 py 1 2

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.



  • 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

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

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

Ordinary & Compensated

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

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