Economics 326: Marshallian Demand and Comparative Statics

Economics 326: Marshallian Demand and Comparative

Statics

Ethan Kaplan

September 17, 2012

Outline 1. Utility Maximization: General Formulation 2. Marshallian Demand 3. Homogeneity of Degree Zero of Marshallian Demand 4. Engel Curves, Normal Goods, Luxury Goods, Gi?en Goods 5. Corner solutions 6. Indirect Utility

1 Utility Maximization: General Formulation

General maximization problem with two goods:

max

CX ;CY

U

(CX ;

CY

)

s:t: PCX CX + PCY CY I

Lagrangian:

L = Uh(CX; CY ) +

i

I PCX CX PCY CY

FOCS:

@L = @U (CX; CY )

@CX

@CX

@L = @U (CX; CY )

@CY

@CY

PCX = 0 (1) PCY = 0 (2)

@L

=I @

PCX CX

PCY Y = 0

Solve for and

using equations (1) and (2)

@U (CX;CY )

= @CX PCX

@U (CX;CY )

= @CY PCY

Equation the two equations for ; we get

@U (CX;CY ) @U (CX;CY )

@CX

=

@CY

PCX

PCY

Interpretation of the above equation? Equate marginal utility per dollar accross goods.

Another formulation:

M RS =

@U (CX;CY )

@CX @U (CX;CY )

=

@CX

PCX PCY

Interpretation? Equating the MRS with the negative of the price ratio or the ratio of the marginal utilities with the price ratio. Intuitively, if the price for a good is high, then the marginal utility must also be high because its an expensive good.

2 Marshallian Demand

We now go back to our previous example from the previous lecture:

max

C X ;CY

L

(CX; CY

;

)

=

CX0:5CY0:5 +

h

i

I PCX CX + PCY CY

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