1 endowments. 2 buying and selling trade involves exchange -- when something is bought something...

1

Endowments

2

Buying and Selling

Trade involves exchange -- when something is bought something else must be sold.

What will be bought? What will be sold? Who will be a buyer? Who will be a

seller?

3

Buying and Selling

And how are incomes generated? How does the value of income depend

upon commodity prices? How can we put all this together to explain

better how price changes affect demands?

4

Endowments

The list of resource units with which a consumer starts is his endowment.

A consumer’s endowment will be denoted by the vector (omega).

= endowment in good 1

= endowment in good 2

ω

1ω2ω

5

Endowments Example:

Let (1,2 This states that the consumer is endowed with 10 units of good 1 and 2 units of good 2.

If p1=2 and p2=3 What is the endowment’s value?

Endowment value is

p11 + p2 2 = This value can be exchanged for any consumption bundle costing no more than the endowment’s value.

6

Consumption bundles

The amount of goods that a consumer can choose to consume

A consumer consumption of good will be denoted by x

x1 = consumption of good 1

x2 = consumption of good 2

7

Budget Constraints Revisited

So, given p1 and p2, the budget constraint for a consumer with an endowment is

The budget set is

where x1 ≥ 0 and x2 ≥ 0

),( 21 ωω

p1x1 + p2x2 = p11 + p22

{(x1,x2) | p1x1 + p2x2 ≤ p11 + p22}

8

Budget Constraints Revisited

2

2211

pp+p ωω

x2

x1

Budget set

Budget constraint

2

1

pp

=slope -

{(x1,x2)|p1x1+ p2x2 ≤ p11+ p22}

p1x1 + p2x2 = p11 + p22

9

x2

x1

New constraint

Prices change from to and to

22112211 p′+p′=xp′+xp′ ωω

2

1

p

pslopeNew

-

1p 1p′ 2p′2p

New budget set

p1x1 + p2x2 = p11 + p22

2

2211

p′p′+p′ ωω

10

Prices change from to and to 1p 1p′ 2p′2p

The endowment point is always on the budget constraint.

So price changes pivot the constraint about the endowment point.

11

Net Demands

Definition: net demand is x –

for example

If x1 – 1 > 0

the consumer is a buyer of good 1

If x1 – 1 < 0

the consumer is a seller of good 1

12

The constraint

p1x1 + p2x2 = p11 + p22

can be written as

p1(x1 – 1) + p2(x2 – 2) = 0

That is, the sum of the values of a consumer’s net demands is zero.

Net Demands

13

Net Demands Suppose and p1=2, p2=3.

Then the constraint is

p1x1 + p2x2 = p11 + p22 = 26

If the consumer demands (x1*,x2*) = (7,4), Net demands are x1*- 1 = 7-10 = -3 and x2*- 2 = 4 - 2 = +2.

p1(x1 – 1) + p2(x2 – 2) =The purchase of 2 extra good 2 units at $3 each is funded by giving up 3 good 1 units at $2 each.

( , ) ( , ) 1 2 10 2

14

Net Demandsx2

x1

x2*

x1*

At prices (p1,p2), the consumer

p1(x1 – 1) + p2(x2 – 2) = 0

15

Net Demandsx2

x1

x2*

x1*

At prices (p1',p2'), the consumer

22112211 ppxpxp

16

Net Demandsx2

x1

x2*=

x1*=

At prices (p1'' p2''), the consumer

22112211 ppxpxp

p1(x1 – 1) + p2(x2 – 2) = 0

17

From Revealed Preference: A seller of good i who remains a seller of i

after price of i has decreased must be worse off

A buyer of good i must remain a buyer of i after price of i has decreased

Effect of a Price Decrease

18

Effect of a Price Decrease for a Seller

x2

x1

x2

x1

At prices (p1,p2), the consumer is a seller of good 1.

x1'

x2'

If after p1 decreases, the consumer remains the seller of good 1, he must be

U' U

UU'

19

Effect of a Price Decrease for a Buyer

x2

x1

x2

x1

At prices (p1,p2), the consumer is a seller of good 1.

x1'

x2'

The consumer MUST remain a buyer of good 1, He is

U' U

UU'

20

Net Demands and Price-Offer Curve

Price-offer curve represents bundles of goods that would be demanded at different prices.

It contains all the utility-maximizing gross demands for which the endowment can be exchanged such that the budget constraint is not violated i.e.

p1(x1 – 1) + p2(x2 – 2) = 0

21

Price-offer curve

good 1, good 2


x2

x1

p1(x1 – 1) + p2(x2 – 2) = 0

22

Price-offer curve

good 1, good 2


x2

x1

p1(x1 – 1) + p2(x2 – 2) = 0

23


Price-offer curve contains all theutility-maximizing gross demands

for which the endowment can be exchanged.

x2

x1

24

Slutsky’s Equation Revisited Slutsky: changes to demands caused by a

price change are the sum ofa pure substitution effect, andan income effect.

This assumed that income y did not change as prices changed. But

y = p11 + p22

does change with price. How does this modify Slutsky’s equation?

25

Slutsky’s Equation Revisited A change in p1 or p2 changes

y = p11 + p22, so there will be

an additional income effect, called the endowment income effect.

Slutsky’s decomposition will thus have three componentsa pure substitution effectan (ordinary) income effect, andan endowment income effect.

26

Slutsky’s Equation Revisited

Slutsky’s equation is now Total effect = Substitution Effect

+ Ordinary Income Effect

+ Endowment Income Effect

Suppose pi changes by ∆pi

The change in money income

∆m = ∆pi or ∆m =

i

i i∆pi

27

We can write Slutsky’s Identity as

∆xi ∆xis

∆xim xi(pi, m) ∆xi

m ∆m

∆pi ∆pi ∆m ∆m ∆pi

Endowment income effect ∆xim ∆m = ∆xi

m ωi

∆m ∆pi ∆m


Alternatively Slutsky’s equation is∆xi ∆xi

s (ωi – x i) ∆xi

m ∆pi ∆pi ∆m +

28


x1

2

1

x2 Initial prices are (p1, p2)

How is the change in demandfrom (x1, x2) to (x1', x2') explained?

x1

x2

x2'

x1'

Final prices are (p1', p2')

29


x1

2

1

x2 Pure substitution effect

30


x1

2

1

x2 Pure substitution effectOrdinary income effect

31


x1

2

1

x2 Pure substitution effectOrdinary income effectEndowment income effect

32

Slutsky’s Equation RevisitedOverall change in demand caused by a

change in price is the sum of:

(i) A pure substitution effectChange in demand at constant real income

(ii) An ordinary income effectChange in demand holding money income fixed

(iii) An endowment income effectChange in demand due to a change in endowment value

33

Labor Supply

A worker is endowed with $m of nonlabor income and R hours of time which can be used for labor or leisure. = (R,m).

Consumption good’s price is pc. w is the wage rate.

34

Labor Supply

The worker’s budget constraint is

where C, R denote gross demands for the consumption good and for leisure. That is

p C w R R mc ( )

p C wR wR mc

endowment value

expenditure

35

Labor Supply

p C w R R mc ( )

rearranges to

Cwp

Rm wR

pc c

.

36

Labor SupplyC

RR

endowment

m wRpc

Cwp

Rm wR

pc c

m

slope = , the ‘real wage rate’ wpc

37

Labor SupplyC

RR

endowment

Cwp

Rm wR

pc c

m

C*

R*

leisuredemanded

laborsupplied

1 endowments. 2 buying and selling trade involves exchange -- when something is bought something...

Documents