Intertemporal Choice

1. Intertemporal Choice Lecture 13

2. Topics to be Discussed • We discuss and examine the decision to save or consume over time • Modify earlier model and allow savings and lending to be part of it

3. The Model • Let C1 = Consumption in period 1 • Let C2 = Consumption in period 2 • M1 = money income in period 1 • M2 = money income in period 2 • First, let P1 = P2 = 1

4. The Model C2 (Period 2) • C1 ≤ m1 : Cannot consume more than income in period 1 or this consumer Can’t borrow in period 1 • Can save in first period to consume more than m2 in second period • C2 = m2 + (m1 – C1) Slope = -1 m2 Endowment m1 C1 (Period 1) Savings in period 1 Income

5. The Model: Adding “r” • If there are CREDIT MARKETS, then possibility exists for saving and borrowing • Let “r” be the interest rate (in decimal form) • An individual can potentially save and borrow in period 1 and consumption pattern in period 2 will follow

6. If Individual SAVES in Period 1 • C1 < m1 • m1 – C1 > 0  Savings • r*(m1 – C1)  Interest earned • C2 = m2 + (1 + r)(m1 – C1) • Consumption in period 2 is income in period 2 plus savings and interest earned in period 1

7. If Individual BORROWS in Period 1 • C1 > m1 • C1 – m1> 0  Borrowings • r*(C1 – m1)  Interest on borrowings • C2 = m2 – (1 + r)(C1– m1) • Consumption in period 2 is income in period 2 less borrowings and interest on borrowings in period 1

8. Intertemporal Budget Constraint • Either of C2 equations could be re-arranged to get the intertemporal budget constraint • C2 = m2 – (1 + r)(C1– m1) • Total consumption = total income • C1(1 + r) + C2 = (1 + r)m1 + m2 • C1 + C2 / (1 + r) = m1 + m2 / (1 + r) FV Budget Constraint PV Budget Constraint

9. Exercise • r = 10% • Rs100 today is worth how much tomorrow? • Rs100 tomorrow is worth how much today? • FV = 100 ( 1 + 0.1) = 110 • PV = 100 / ( 1 + 0.1) = 90.9

10. Budget Constraint C2 (Period 2) • Regardless of whether we use PV or FV Budget Constraint, the graph looks like this • FV Budget Constraint • C1(1 + r) + C2 = (1 + r)m1 + m2 • Set C1 = 0, C2 = (1 + r)m1 + m2 • Set C2 = 0, C1 = m1 + m2 / (1 + r) • Slope = ΔC2/ΔC1 • – (1+r) OR – P1/P2 Slope: -(1 + r) m2 Endowment m1 C1 (Period 1)

11. The Model: Adding Preferences C2 (Period 2) • If preferences were like Perfect Compliments, then the individual would like to consume in a certain proportion over the two periods • If U = min ( C1, C2 ), then the individual wants to consume the same amount in each period C1 = C2 C2* C1* C1 (Period 1)

12. The Model: Adding Preferences C2 (Period 2) • However, we normally assume we have well behaved preferences such as: • An average level of consumption is preferred for both periods, but the individual is willing to substitute … ? C1 (Period 1)

13. The Model: Adding Preferences • Since first period consumption is higher than the income, this individual is a borrower • Net Buyer of C1 C2 (Period 2) Endowment m2 C2* C1* m1 C1 (Period 1)

14. The Model: Adding Preferences C2 (Period 2) • Since first period consumption is lower than the income, this individual is a lender • Net Seller of C1 C2* Endowment m2 m1 C1* C1 (Period 1)

15. Effect of changing interest rate, ‘r’ • Suppose, r rises to r1 • Rise in r makes first period consumption more expensive (in terms of forgone period 2 consumption) C2 (Period 2) Slope: -(1 + r1) m2 Endowment Slope: -(1 + r) m1 C1 (Period 1)

16. Effect of changing interest rate, ‘r’ • Suppose, r falls to r2 • Fall in r reduces cost of period 1 consumption (in terms of forgone period 2 consumption) C2 (Period 2) Slope: -(1 + r) m2 Endowment Slope: -(1 + r2) m1 C1 (Period 1)

17. Effect of changing interest rate, ‘r’ • FV Budget Constraint • C1(1 + r) + C2 = (1 + r)m1 + m2 • Set C1 = 0, C2 = (1 + r)m1 + m2 • Set C2 = 0, C1 = m1 + m2 / (1 + r) • If r ↑, C1 falls and C2 increases • If r ↓, C1 increases and C2 falls

18. Slutsky Equation • Effect of Price Change: • Substitution effect • Income effect • ΔC1/ΔP1 = ΔC1S/ΔP1 + ΔC1m/ΔM (Real M) • If price fell, ΔC1S/ΔP1 will always be positive and vice versa • If real income increases, ΔM is positive • But whether this positive change in income will cause ΔC1m to increase or fall, will depend on the good being an inferior, giffen or normal!

19. Slutsky Equation • Let P1 to be ‘r’; so a rise in P will be reflected in the rise in ‘r’ • Total effect of interest rate change = substitution effect + income effect • ΔC1/Δr = ΔC1S/Δr+ ΔC1m/ΔM * (m1 – C1)

20. ΔC1/Δr = ΔC1S/Δr+ ΔC1m/ΔM * (m1 – C1) • r increased: • ΔC1S/Δr  fall or will have a negative sign • IF BORROWER • m1 – C1 < 0 • ΔC1m/ΔM  Assuming Period 1 consumption is a normal good  fall and have a negative sign • ΔC1/Δr will fall overall or will be < 0 for the borrower

21. ΔC1/Δr = ΔC1S/Δr+ ΔC1m/ΔM * (m1 – C1) • r increased: • ΔC1S/Δr  fall or will have a negative sign • IF LENDER • m1 – C1 > 0 • ΔC1m/ΔM  Assuming Period 1 consumption is a normal good  fall and have a negative sign • ΔC1/Δr can be < 0 OR > 0 depending on the size of income and substitution effects!

22. Interest Rate Increased C2 (Period 2) • If this individual is a borrower…? • C1 falls while C2 can increase or fall • Rise in ‘r’ makes borrower worse-off m2 E B A C2* m1 C1* C1 (Period 1)

23. Interest Rate Increased • If this individual is a lender…? • C1 increases here while it could fall as well! • Rise in ‘r’ makes lender better-off C2 (Period 2) B C2* A m2 E m1 C1* C1 (Period 1)

24. Conclusion • Borrower • R falls, remain a borrower  Better-off • R increases, remain borrower  Worse-off • R increases, become lender  could be better off or worse off • Lender • R increases, remain a lender  Better-off • R falls, remain a lender  Worse-off • R falls, become a borrower  could be better off or worse off

25. Jennifer lives two periods. In the first period, her income is fixed at $10000; in the second, it is $20000. she can borrow and lend at the market rate of 7% • Sketch her intertemporal budget constraint • The ‘r’ increases to 9%. Sketch the new budget constraint. What effect do you expect this change to have on her saving? • Suppose that Jennifer is unable to borrow at any interest rate, although she can still lend at 9%. Sketch her intertemporal budget constraint.

26. A student entering college has been given $15000 by his parents. This is to be the student’s pocket money during all four years of college. Suppose further that he can borrow and lend freely at a market rate of 5% • Firstly, write down the equation of his intertemporal budget constraint • If the interest rate were higher than 5%, would the student be better or worse off? (Make sure you use your answer in part 1 to explain)

27. Exercise • Suppose a Pakistani consumer is very uncertain about future increase in price level and hence, prefers consuming today rather than saving. Whereas, Japanese consumer prefers saving over consumption in the current time period. • Draw appropriate Intertemporal Choice models for the two consumers. Assume there are only two time periods. Also, assume that prevailing deposit and lending rates are 10% • Now assume that in Pakistan, the lending rate is 13% while deposit rate is 7%. How would the budget constraint change? Reflect the change in Japanese consumer budget constraint if the lending and deposit rates are 8% and 20% respectively. • Is the Pakistani consumer worse off or better off as a result of interest rate change. What about the Japanese consumer? Use both the graphs and the equation while explaining your answer.