predictions of utility theory about the nature of demand suplementary references: layard, p &...

Predictions of Utility Theory About the Nature of Demand

Suplementary References:

Layard, P & A.Walters:

Microeconomic Theory p135-137

Deaton A., & J. Muellbauer, Economics and Consumer Behavior p14-16

p43-46

Predictions of Utility Theory About the Nature of Demand

• Given the axioms (or properties) 1-7 we have assumed about the utility function, these imply certain things about the demand function.

• We can now want to derive a set of core properties we would expect any demand function we estimated to exhibit

Testable Predictions and the Theory

• We can then test the demand functions we derived and ask if they exhibit these properties.

• If they don’t then we either have a problem with our data, or we have used the wrong functions or estimation method

• OR (more seriously),

• with our theory!

PROPERTIES OF DEMAND FUNCTION

1. The Adding-Up Condition

The effect of a change in income on the demand for all goods

Pxx + Pyy = m

Pxdx + Pydy = dm

1dm

dyP

dm

dxP yx

1dm

dy

y

m

m

yP

dm

dx

x

m

m

xP yx

(dm)

x

m

m

x

y

m

m

y

1dm

dyP

dm

dxP yx

1dm

dy

y

m

m

yP

dm

dx

x

m

m

xP yx

1dm

dy

y

m

m

yP

dm

dx

x

ms yx




Pxx + Pyy = m

Pxdx + Pydy = dm

1dm

dyP

dm

dxP yx

1dm

dy

y

m

m

yP

dm

dx

x

m

m

xP yx

1dm

dy

y

m

m

yPs yxx




Pxx + Pyy = m

Pxdx + Pydy = dm

1dm

dyP

dm

dxP yx

1dm

dy

y

m

m

yP

dm

dx

x

m

m

xP yx

Property 1: Sx x + Sy y = 1 (adding-up condition)




Pxx + Pyy = m

Pxdx + Pydy = dm

Testable property 2.

Homogeneity

Now ’ing PRICES AND INCOME

If demand is unaffected by an equi-proportional change in all prices and income then there is an absence of money illusion

That is, if I choose the bundle (x, y) with prices Px, Py and income m, then I will choose the same bundle with 2 Px, 2Py and 2m.

Formal Statement

Formally, a function is homogeneous of degree t, if when all prices and income change by

x = f ( Px, Py, m) = t f ( Px, Py,m)

Claim: Demand functions are homogeneous of degree zero in prices and income, that is

x = f ( Px, Py, m)

= t f ( Px, Py,m) = 0 f ( Px, Py,m)

= f (Px, Py,m)

Other examples of homogeneous Functions

Production Function: Q = ƒ (L, K)

What happens if we scale up all inputs by a factor of

ƒ(L, K) = ?

Homogeneity of degree t implies ƒ(L, K) = t ƒ(L, K)

What is t ?

If we have CRS, that is, if the production function is homogeneous of degree 1, then t=1 and

= 1 ƒ(L, K) = ƒ(L, K)

e.g. ƒ(2L, 2K) = 2 ƒ(L, K)=2Q

dmm

xdP

P

xdP

P

xdx y

yx

x

dmm

m

xm

xdP

P

P

xP

xdP

P

P

xP

x

x

dxy

y

y

yx

x

x

x

111

Homogeneity of degree zero in prices and income seems a reasonable property, after all it simply implies the absence of money illusion.

What does homogeneity imply about our demand functions in general?

Taking the total derivative of x= x(Px, Py, m)we get:

dmm

xdP

P

xdP

P

xdx y

yx

x

dmm

m

xm

xdP

P

P

xP

xdP

P

P

xP

x

x

dxy

y

y

yx

x

x

x

111



Taking the total derivative of x= x(Px, Py, M)we get:

dmm

xdP

P

xdP

P

xdx y

yx

x

dmm

m

xm

xdP

P

P

xP

xdP

P

P

xP

x

x

dxy

y

y

yx

x

x

x

111



Taking the total derivative of x= x(Px, Py, m)we get:

m

dm

P

dP

P

dP

y

y

x

x

Ox

dx

m

dm

x

m

m

x

P

dP

x

P

P

x

P

dP

x

P

P

x

x

dx

y

yy

yx

xx

x

Will imply no change in the demand for x

If the demand function is homogeneous of degree zero in prices and income then changing Px, Py,and m in the same proportion:

dmm

m

xm

xdP

P

P

xP

xdP

P

P

xP

x

x

dxy

y

y

yx

x

x

x

111

x

x

x

xy

yx

xx

x P

dP

x

m

m

x

P

dP

x

P

P

x

P

dP

x

P

P

x

x

dx

Cournot Condition:

m

dm

x

m

m

x

P

dP

x

P

P

x

P

dP

x

P

P

x

x

dx

y

yy

yx

xx

x

x

x

x

xy

yx

xx

x P

dP

x

m

m

x

P

dP

x

P

P

x

P

dP

x

P

P

x

0

x

xx

x

xxp

x

xxp P

dP

P

dP

P

dPyx

0

xxpxp yx 0

0 xxpxp yx

Property 2: The Cournot Condition

Homogeneity of the demand function for x requires

Similarly for the demand function for y,

homogeneity requires

0 xxpxp yx

0 yypyp xy

predictions of utility theory about the nature of demand suplementary references: layard, p &...

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