Econ 102 SY 2008 2009

Lecture 4

Demand Curve

July, 2008

Econ 102 SY 2008 2009

Key concepts

Demand functions vs. correspondences Market demand Income and price elasticities Demand curve estimation Consumer surplus

Econ 102 SY 2008 2009

Continuity of the demand function

Continuity of Demand Function. Suppose preference set is continuous and weakly convex, and (p;m) > 0. Then, the demand curve x(p;m) is a upper hemi-continuous convex-valued correspondence. Furthermore, if the weak convexity is replaced by the strict convexity, x(p;m) is a continuous single-valued function.

Econ 102 SY 2008 2009

Weakly convex preferences

Recall: CONVEXITY. Given x; x0 in X such that x0 > x, then

it follows that tx +(1- t)x0 > x for all 0 < t < 1.

STRICT CONVEXITY. Given x; x0 in X such that x0

> or~x, then it follows that tx +(1- t)x0 > x for all 0 < t < 1.

WEAK CONVEXITY. Given x; x0 in X such that x0 > or~x, then it follows that tx +(1- t)x0 > or~ x for all 0 < t < 1.

Econ 102 SY 2008 2009

Illustrations Discontinuous demand due to non convex preferences

y

x

a

b

c

d

x

px

pxa

pxb

pxd

xa xb xc xdxa xb xc xd

a

b

c

d

Econ 102 SY 2008 2009

Inverse demand function x=f(p;m) is the demand function p=p(x) is the inverse demand function

i =

i =-

∀

∂∂∂∂

∀∂∂

∂

∂

∑

∑

∀

∀

ii

i

ii

ii

ii

i

xxuxu

m)m;x(p

0pm

xxu

x

u

mxpxx

u

0mxpL

0px

uL

)mxp(u(x)L

ii

ii

ii

ii

i

ii

x

ii

i

i

= =

=-=

i =-=

-+=

∑∑

∑

∑

∀∀

∀

∀

∂

∂

∀∂

∂

Max

Econ 102 SY 2008 2009

y

-a)m1(p

x

amp

0pm

U

x

aUm

U

myy

-a)U1(x

x

aU

0mxpL

0py

-a)U1(L

0px

aUL

)mxp(yxL

y

x

x

ii

i

yy

xx

ii

i-a)1(a

=

=

=-

=

=+

=-=

=-=

=-=

-+=

∑

∑

∀

∀

Max

Econ 102 SY 2008 2009

Illustrationx

p

p

x

Inverse demandDemand function

Econ 102 SY 2008 2009

Integrability problem Given a system of demand functions x(p;m). Is there

necessarily a utility function from which these demand functions can be derived? --- integrability problem.

Yes. if the system of demand functions satisfy these properties:

1. Nonnegativity: x(p;m) >= 0. 2. Homogeneity: x(tp; tm) = x(p;m). 3. Budget Balancedness: px(p;m) = m. 4. Symmetry: The Slutsky matrix is symmetric. 5. Negative Semi-definite: The Slutsky matrix S is

negative semi-definite.

Econ 102 SY 2008 2009

Revealed Preference

Criticism of preference theory: “too strong on the grounds that individuals are unlikely to make choices through conscious use of a preference relation.

Samuelson developed revealed preference theory as alternative based on weaker set of hypotheses.

Econ 102 SY 2008 2009

Principle behind revealed preference

preference statements are constructed only from observable decisions, that is, from actual choice made by a consumer.

An individual preference relation, even if it exists, can never be directly observed in the market.

Consumers make actual choices given prices and income.

Revealed preference is recovering the preference behavior from actual choices.

Econ 102 SY 2008 2009

Assumptions made in revealed preference theory

In general there are no restrictions: anything is possible.

Need to rule out this trivial case. The underlying utility function to be locally

non-satiated

Econ 102 SY 2008 2009

Definitions Direct Revealed Preference: If ptxt => ptx, then u(xt) =>

u(x). We will say that xt is directly revealed preferred to x, i.e. xtRDx. Assuming that the data were generated by utility maximization, we can conclude that if xtRDx implies u(xt) => u(x).

Strictly Direct Revealed Preference: If ptxt > ptx, then u(xt) > u(x) or xt is strictly direct revealed preferred to x, i.e. xtPDx.

Revealed Preference: xt is said to be revealed preferred to x if there exists a finite number of bundles xt ,x1; x2,…, xn such that xtRDx1; x1RDx2,…, xn-1RDxn; Or xtRx.

Transitive closure of the relation RD. xtRx implies u(xt) => u(x).

Econ 102 SY 2008 2009

GARP: Generalized Axiom of Revealed Preference

If xt R xs then xs cannot be strictly directly revealed preferred to xt.

xt R xs implies not xs PD xt. In other words, xt R xs implies psxs <= psxt.

Two standard conditions of GARP: WARP and SARP.

Econ 102 SY 2008 2009

WARP: Weak Axiom of Revealed Preference

If xtRD xs and xt is not equal to xs, then it is not the case that xs RD xt, i.e., pt xt => pt xs implies ps xt => ps xs.

Requires: the demand function is single valued.

Econ 102 SY 2008 2009

SARP: Strong Axiom of Revealed Preference

If xtR xs and xt is not equal to xs, then it is not the case that xs R xt, i.e., pt xt > pt xs implies ps xt > ps xs.

Requires: the demand function is single valued.

Econ 102 SY 2008 2009

Recoverability of preference relation

Boundaries. RP revealed preferred, RW revealed worse

Relative to xoxs

x’

Econ 102 SY 2008 2009

Recoverability of preference relation

Boundaries. RP revealed preferred, RW revealed worse

With many observed points the NRP and NRW becomes tighter.

Econ 102 SY 2008 2009

Income – Leisure Model

Slutsky’s equation

ji

j

i

j

i

jji

j

i

j

i

ji

j

i

j

i

ji

j

i

j

i

xdm

)x(p; pdx

p

)x(p; ph

dp

)x(p; pdx

)xx(dm

)x(p; pdx+

p

)x(p; ph

dp

)x(p; pdx

xdm

)x(p; pdx

p

)x(p; ph

p

)x(p; px

xdm

)x(p; pdx+

p

)x(p; px

dp

)x(p; pdx

-=

endowment, zero if

-=

-=

but

=

constant income real

constant income real

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

Econ 102 SY 2008 2009

Illustration: price x increase

x

y

UA

UB

vvB

A

A’

UA’

xAxA’xB

Econ 102 SY 2008 2009

Income – Leisure model

x is consumption basket w is wage L is Leisure Tbar is endowment of time H or hours of work is (Tbar-L)

Tx + wL = wpsuch that

L),u(xmax

x

x;L

Econ 102 SY 2008 2009

Income-Leisure Model

Income effect is positive; assuming Leisure is normal good Possible that income dominates the substitution effect And thus the demand for Leisure will have portion where it is like

that of Giffen good Supply of hours of work is the negative of the demand of Leisure Upward sloping supply of hours of work

)LT()Tw(d

)Tw; w,dL(p

w

)Tw; w,(ph

dw

)Tw; w,dL(p

ttanconsU

i -+=

∂

∂

Econ 102 SY 2008 2009

Demand for Leisure

L

w

Econ 102 SY 2008 2009

Individual Hours of work supply

H=Tbar-L Backward

bending supply function of Hours of Work

H=Tbar-L

w

Econ 102 SY 2008 2009

End of Lecture 4

Demand Curve