Intertemporal Choice Molly W. Dahl Georgetown University Econ 101 - PowerPoint PPT Presentation

Intertemporal Choice Molly W. Dahl Georgetown University Econ 101 – Spring 2009 1

The Intertemporal Choice Problem � Assume we have 2 periods � m 1 : endowment of money in period 1 � m 2 : endowment of money in period 2 � c 1 : consumption in period 1 � c 2 : consumption in period 2 2

The Intertemporal Budget Constraint � Suppose that the consumer chooses not to save or to borrow. � Q: What will be consumed in period 1? � A: c 1 = m 1 . � Q: What will be consumed in period 2? � A: c 2 = m 2 . 3

The Intertemporal Budget Constraint c 2 So (c 1 , c 2 ) = (m 1 , m 2 ) is the consumption bundle if the consumer chooses neither to save nor to borrow. m 2 0 c 1 m 1 0 4

The Intertemporal Budget Constraint � Now suppose that the consumer spends nothing on consumption in period 1; that is, c 1 = 0 and the consumer saves s 1 = m 1 - c 1 = m 1 � Let r be the interest rate. � What now will be period 2’s consumption level? 5

The Intertemporal Budget Constraint � Period 2 income is m 2 . � Savings plus interest from period 1 sum to (1 + r )m 1 . � So total income available in period 2 is m 2 + (1 + r )m 1 . � So period 2 consumption expenditure is = + + c m r m ( 1 ) 2 2 1 6

The Intertemporal Budget Constraint c 2 the future-value of the income + m 2 endowment + r m ( 1 ) 1 m 2 0 c 1 m 1 0 7

The Intertemporal Budget Constraint ( ) c 2 = + + c c m r m ( , ) 0 , ( 1 ) 1 2 2 1 + m 2 is the consumption bundle when all + r m ( 1 ) 1 period 1 income is saved. m 2 0 c 1 m 1 0 8

The Intertemporal Budget Constraint � Now suppose that the consumer spends everything possible on consumption in period 1, so c 2 = 0. � What is the most that the consumer can borrow in period 1 against her period 2 income of $m 2 ? � Let b 1 denote the amount borrowed in period 1. 9

The Intertemporal Budget Constraint � Only $m 2 will be available in period 2 to pay back $b 1 borrowed in period 1. � So b 1 (1 + r ) = m 2 . � That is, b 1 = m 2 / (1 + r ). � So the largest possible period 1 consumption level is m = + 2 c m 1 1 + r 1 10

The Intertemporal Budget Constraint ( ) c 2 = + + c c m r m ( , ) 0 , ( 1 ) 1 2 2 1 + m 2 is the consumption bundle when all + r m ( 1 ) 1 period 1 income is saved. the present-value of m 2 the income endowment 0 c 1 m 1 m 0 + 2 m 1 + r 1 11

The Intertemporal Budget Constraint ( ) c 2 = + + c c m r m ( , ) 0 , ( 1 ) 1 2 2 1 + m 2 is the consumption bundle when + r m ( 1 ) 1 period 1 saving is as large as possible. ⎛ ⎞ m = + 2 ⎜ ⎟ c c m ( , ) , 0 1 2 ⎝ 1 ⎠ + r 1 m 2 is the consumption bundle when period 1 borrowing is as big as possible. 0 c 1 m 1 m 0 + 2 m 1 + r 1 12

The Intertemporal Budget Constraint � Suppose that c 1 units are consumed in period 1. This costs $c 1 and leaves m 1 - c 1 saved. Period 2 consumption will then be = + + − c m r m c ( 1 )( ) 2 2 1 1 13

The Intertemporal Budget Constraint � Suppose that c 1 units are consumed in period 1. This costs $c 1 and leaves m 1 - c 1 saved. Period 2 consumption will then be = + + − c m r m c ( 1 )( ) 2 2 1 1 which is = − + + + + c r c m r m ( 1 ) ( 1 ) . 2 1 2 1 ⎪ ⎩ ⎧ ⎨ ⎩ ⎧ ⎪ ⎨ slope intercept 14

The Intertemporal Budget Constraint c 2 = − + + + + c r c m r m ( 1 ) ( 1 ) . 2 1 2 1 + m 2 + slope = -(1+r) ( r)m 1 1 m 2 0 c 1 m 1 m 0 + 2 m 1 + r 1 15

The Intertemporal Budget Constraint c 2 = − + + + + c r c m r m ( 1 ) ( 1 ) . 2 1 2 1 + m 2 + slope = -(1+r) ( r)m 1 Saving 1 Borrowing m 2 0 c 1 m 1 m 0 + 2 m 1 + r 1 16

The Intertemporal Budget Constraint + + = + + r c c r m m ( 1 ) ( 1 ) 1 2 1 2 is the “future-valued” form of the budget constraint since all terms are in period 2 values. This is equivalent to c m + = + 2 2 c m 1 1 + + r r 1 1 which is the “present-valued” form of the constraint since all terms are in period 1 values. 17

Comparative Statics � The slope of the budget constraint is − + ( 1 ) r � The constraint becomes flatter if the interest rate r falls. 18

19 c 1 ) r + 1 m 1 /p 1 Comparative Statics ( − slope = 0 m 2 /p 2 c 2 0

20 c 1 ) r + m 1 /p 1 1 Comparative Statics ( − slope = 0 m 2 /p 2 c 2 0

Comparative Statics c 2 − + ( 1 ) slope = r The consumer saves. m 2 /p 2 0 c 1 m 1 /p 1 0 21

Comparative Statics c 2 − + ( 1 ) slope = r The consumer saves. A decrease in the interest rate “flattens” the budget m 2 /p 2 constraint. 0 c 1 m 1 /p 1 0 22

Comparative Statics c 2 − + ( 1 ) slope = r If the consumer remains a saver then savings and welfare are reduced by a lower m 2 /p 2 interest rate. 0 c 1 m 1 /p 1 0 23

24 c 1 ) r m 1 /p 1 + Comparative Statics 1 ( − slope = 0 m 2 /p 2 c 2 0

25 c 1 ) r m 1 /p 1 Comparative Statics + 1 ( − slope = 0 m 2 /p 2 c 2 0

Comparative Statics c 2 − + slope = ( 1 ) r The consumer borrows. m 2 /p 2 0 c 1 m 1 /p 1 0 26

Comparative Statics c 2 − + slope = ( 1 ) r The consumer borrows. A a decrease in the interest rate “flattens” the budget constraint. m 2 /p 2 0 c 1 m 1 /p 1 0 27

Comparative Statics c 2 − + ( 1 ) slope = r If the consumer borrows then borrowing and welfare are increased by a lower interest rate. m 2 /p 2 0 c 1 m 1 /p 1 0 28