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

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 p 1 =2 and p 2 =3 What is the endowment’s value? Endowment value is p 1  1 + p 2  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 x 1 = consumption of good 1 x 2 = consumption of good 2

7 Budget Constraints Revisited So, given p 1 and p 2, the budget constraint for a consumer with an endowment is The budget set is where x 1 ≥ 0 and x 2 ≥ 0 p 1 x 1 + p 2 x 2 = p 1  1 + p 2  2 {(x 1,x 2 ) | p 1 x 1 + p 2 x 2 ≤ p 1  1 + p 2  2 }

8 Budget Constraints Revisited x2x2 x1x1   Budget set Budget constraint {(x 1,x 2 )|p 1 x 1 + p 2 x 2 ≤ p 1  1 + p 2  2 } p 1 x 1 + p 2 x 2 = p 1  1 + p 2  2

9 x2x2 x1x1   New constraint Prices change from to and to New budget set p 1 x 1 + p 2 x 2 = p 1  1 + p 2  2

10 Prices change from to and to 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 x 1 –  1 > 0 the consumer is a buyer of good 1 If x 1 –  1 < 0 the consumer is a seller of good 1

12 The constraint p 1 x 1 + p 2 x 2 = p 1  1 + p 2  2 can be written as p 1 (x 1 –  1 ) + p 2 (x 2 –  2 ) = 0 That is, the sum of the values of a consumer’s net demands is zero. Net Demands

13 Net Demands Suppose and p 1 =2, p 2 =3. Then the constraint is p 1 x 1 + p 2 x 2 = p 1  1 + p 2  2 = 26 If the consumer demands (x 1 *,x 2 *) = (7,4), Net demands are x 1 *-  1 = 7-10 = -3 and x 2 *-  2 = = +2. p 1 (x 1 –  1 ) + p 2 (x 2 –  2 ) = The purchase of 2 extra good 2 units at $3 each is funded by giving up 3 good 1 units at $2 each.

14 Net Demands x2x2 x1x1   x2*x2* x1*x1* At prices (p 1,p 2 ), the consumer p 1 (x 1 –  1 ) + p 2 (x 2 –  2 ) = 0

15 Net Demands x2x2 x1x1   x2*x2* x1*x1* At prices (p 1 ',p 2 '), the consumer

16 Net Demands x2x2 x1x1 x 2 *=   x 1 *=   At prices (p 1 '' p 2 ''), the consumer p 1 (x 1 –  1 ) + p 2 (x 2 –  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 x2x2 x1x1   x2x2 x1x1 At prices (p 1,p 2 ), the consumer is a seller of good 1. x1'x1' x2'x2' If after p 1 decreases, the consumer remains the seller of good 1, he must be U' U U U'

19 Effect of a Price Decrease for a Buyer x2x2 x1x1   x2x2 x1x1 At prices (p 1,p 2 ), the consumer is a seller of good 1. x1'x1' x2'x2' The consumer MUST remain a buyer of good 1, He is U' U U U'

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. p 1 (x 1 –  1 ) + p 2 (x 2 –  2 ) = 0

21 Price-offer curve good 1, good 2 Net Demands and Price-Offer Curve x2x2 x1x1   p 1 (x 1 –  1 ) + p 2 (x 2 –  2 ) = 0

22 Price-offer curve good 1, good 2 Net Demands and Price-Offer Curve x2x2 x1x1   p 1 (x 1 –  1 ) + p 2 (x 2 –  2 ) = 0

23 Net Demands and Price-Offer Curve Price-offer curve contains all the utility-maximizing gross demands for which the endowment can be exchanged. x2x2 x1x1  

24 Slutsky’s Equation Revisited Slutsky: changes to demands caused by a price change are the sum of  a pure substitution effect, and  an income effect. This assumed that income y did not change as prices changed. But y = p 1  1 + p 2  2 does change with price. How does this modify Slutsky’s equation?

25 Slutsky’s Equation Revisited A change in p 1 or p 2 changes y = p 1  1 + p 2  2, so there will be an additional income effect, called the endowment income effect. Slutsky’s decomposition will thus have three components  a pure substitution effect  an (ordinary) income effect, and  an endowment income effect.

26 Slutsky’s Equation Revisited Slutsky’s equation is now Total effect = Substitution Effect + Ordinary Income Effect + Endowment Income Effect Suppose p i changes by ∆p i The change in money income ∆m = ∆p i or ∆m = ∆p i

27 We can write Slutsky’s Identity as ∆x i ∆x i s ∆x i m x i (p i, m) ∆x i m ∆m ∆p i ∆p i ∆m ∆m ∆p i Endowment income effect ∆x i m ∆m = ∆x i m ω i ∆m ∆p i ∆m Slutsky’s Equation Revisited Alternatively Slutsky’s equation is ∆x i ∆x i s (ω i – x i ) ∆x i m ∆p i ∆p i ∆m +

28 Slutsky’s Equation Revisited x1x1 22 11 x2x2 Initial prices are (p 1, p 2 ) How is the change in demand from (x 1, x 2 ) to (x 1 ', x 2 ') explained? x1x1 x2x2 x2'x2' x1'x1' Final prices are (p 1 ', p 2 ')

29 Slutsky’s Equation Revisited x1x1 22 11 x2x2 Pure substitution effect 

30 Slutsky’s Equation Revisited x1x1 22 11 x2x2 Pure substitution effect  Ordinary income effect 

31 Slutsky’s Equation Revisited x1x1 22 11 x2x2 Pure substitution effect  Ordinary income effect  Endowment income effect 

32 Slutsky’s Equation Revisited Overall change in demand caused by a change in price is the sum of: (i)A pure substitution effect Change in demand at constant real income (ii)An ordinary income effect Change in demand holding money income fixed (iii) An endowment income effect Change 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 p c. 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     endowment value expenditure

35 Labor Supply  rearranges to 

36 Labor Supply C R R  endowment   m slope =, the ‘real wage rate’

37 Labor Supply C R R  endowment  m C* R* leisure demanded labor supplied