Econ 384 Intermediate Microeconomics II

1. Econ 384Intermediate Microeconomics II Instructor: Lorne Priemaza Lorne.priemaza@ualberta.ca

2. A. Intertemporal Choice A.1 Compounding A.2 Present Value A.3 Present Value Decisions A.4 Lifecycle Model

3. A.1 Compounding If you invest an amount P for a return r, After one year: • You will make interest on the amount P • Total amount in the bank = P(1+r) = P + Pr After another year: • You will make interest on the initial amount P • You will make interest on last year’s interest Pr • Total amount in the bank = P(1+r)2 This is COMPOUND INTEREST. Over time you make interest on the interest; the interest compounds.

4. A.1 Compounding Investment: $100 Interest rate: 2% Derived Formula: S = P (1+r)t S = value after t years P = principle amount r = interest rate t = years

5. A.1 Compounding Choice Given two revenues or costs, choose the one with the greatest value after time t: A: $100 now B:$115 in two years, r=6% (find value after 2 years) S = P (1+r)t SA =$100 (1.06)2 = $112.36 SB =$115 Choose option B

6. A.1 Compounding Loss Choice This calculation also works with losses, or a combination of gains or loses: A: -$100 now B: -$120 in two years, r=6% (find value after 2 years) S = P (1+r)t SA =-$100 (1.06)2 = -$112.36 SB =-$120 Choose option A. (You could borrow $100 now for one debt, then owe LESS in 2 years than waiting)

7. A.2 Present Value What is the present value of a given sum of money in the future? By rearranging the Compound formula, we have: PV = present value S = future sum r = interest rate t = years

8. A.2 Present Value Gain Example What is the present value of earning $5,000 in 5 years if r=8%? Earning $5,000 in five years is the same as earning $3,403 now. PV can also be calculated for future losses:

9. A.2 Present Value Loss Example You and your spouse just got pregnant, and will need to pay for university in 20 years. If university will cost $30,000 in real terms in 20 years, how much should you invest now? (long term GIC’s pay 5%) PV = S/[(1+r)t] = -$30,000/[(1.05)20] = -$11,307

10. A.2 Present Value of a Stream of Gains or Loses If an investment today yields future returns of St, where t is the year of the return, then the present value becomes: If St is the same every year, a special ANNUITY formula can be used:

11. A.2 Annuity Formula PV = A[1-(1/{1+r})t] / [1- (1/{1+r})] PV = A[1-xt] / [1-x] x=1/{1+r} A = value of annual payment r = annual interest rate n = number of annual payments Note: if specified that the first payment is delayed until the end of the first year, the formula becomes PV = A[1-xt] / r x=1/{1+r}

12. A.2 Annuity Comparison Consider a payment of $100 per year for 5 years, (7% interest) PV= 100+100/1.07 + 100/1.072 + 100/1.073 + 100/1.074 = 100 + 93.5 + 87.3 + 81.6 + 76.3 = $438.7 Or PV = A[1-(1/{1+r})t] / [1- (1/{1+r})] PV = A[1-xt] / [1-x] x=1/{1+r} PV = 100[1-(1/1.07)5]/[1-1/1.07] = $438.72

13. A.3 Present Value Decisions When costs and benefits occur over time, decisions must be made by calculating the present value of each decision -If an individual or firm is considering optionX with costs and benefits Ctx and Btx in year t, present value is calculated: Where r is the interest rate or opportunity cost of funds.

14. A.3 PV Decisions Example • A firm can: • Invest $5,000 today for a $8,000 payout in year 4. • Invest $1000 a year for four years, with a $2,500 payout in year 2 and 4 • If r=4%,

15. A.3 PV Decisions Example 2) Invest $1000 a year for four years, with a $2,500 payout in year 2 and 4 If r=4%, Option 1 is best.

16. A.4 Lifecycle Model • Alternately, often an individual needs to decide WHEN to consume over a lifetime • To examine this, one can sue a LIFECYCLE MODEL*: *Note: There are alternate terms for the Lifecycle Model and the curves and calculations seen in this section

17. A.4 Lifecycle Budget Constraint Assume 2 time periods (1=young and 2=old), each with income and consumption (c1, c2, i1, i2) and interest rate r for borrowing or lending between ages If you only consumed when old, c2=i2+(1+r)i1 If you only consumed when young: c1=i1+i2 /(1+r)

18. Lifecycle Budget Constraint The slope of this constraint is (1+r). Often point E is referred to as the endowment point. i2+(1+r)i1 Old Consumption i2 E O i1 i1+i2 /(1+r) Young Consumption

19. A.4 Lifecycle Budget Constraint Assuming a constant r, the lifecycle budget constraint is: Note that if there is no borrowing or lending, consumption is at E where c1=i1, therefore:

20. A.4 Lifetime Utility • In the lifecycle model, an individual’s lifetime utility is a function of the consumption in each time period: U=f(c1,c2) • If the consumer assumptions of consumer theory hold across time (completeness, transitivity, non-satiation) , this produces well-behaved intertemporal indifference curves:

21. A.4 Intertemporal Indifference Curves • Each INDIFFERENCE CURVE plots all the goods combinations that yield the same utility; that a person is indifferent between • These indifference curves have similar properties to typical consumer indifference curves (completeness, transitivity, negative slope, thin curves)

22. Intertemporal Indifference Curves c2 • Consider the utility function U=(c1c2)1/2. • Each indifference curve below shows all the baskets of a given utility level. Consumers are indifferent between intertemporal baskets along the same curve. • • 2 • • U=2 1 U=√2 0 c1 1 2 4

23. Marginal Rate of Intertemporal Substitution (MRIS) • Utility is constant along the intertemporal indifference curve • An individual is willing to SUBSTITUTE one period’s consumption for another, yet keep lifetime utility even • ie) In the above example, if someone starts with consumption of 2 each time period, they’d be willing to give up 1 consumption in the future to gain 3 consumption now • Obviously this is unlikely to be possible

24. A.4 MRIS • The marginal rate of substitution (MRIS) is the gain (loss) in future consumption needed to offset the loss (gain) in current consumption • The MRS is equal to the SLOPE of the indifference curve (slope of the tangent to the indifference curve)

25. A.4 MRIS Example

26. A.4 Maximizing the Lifecycle Model • Maximize lifetime utility (which depends on c1 and c2) by choosing c1 and c2 …. • Subject to the intertemporal budget constraint • In the simple case, people spend everything, so the constraint is an equality • This occurs where the MRIS is equal to the slope of the intertemporal indifference curve:

27. Maximizing Intertemporal Utility c2 Point A: affordable, doesn’t maximize utility Point B: unaffordable Point C: affordable (with income left over) but doesn’t maximize utility Point D: affordable, maximizes utility IBL D • B • • C IIC2 • A IIC1 c1 0

28. A.4 Maximization Example

29. A.4 Maximization Example 2

30. A.4 Maximization Conclusion Lifetime utility is maximized at 817,316 when $797,619 is consumed when young and $837,500 is consumed when old. *Always include a conclusion

31. Maximizing IntertemporalUtility c2 Utility is always maximized at the tangent to the indifference curve • U=817,316 c1 0

32. A. Conclusion • Streams of intertemporal costs and benefits can be compared by comparing present values • To examine consumption timing, one can use the LIFECYCLE MODEL: • An intertemporal budget line has a slope of (1+r) • The slope of the intertemporal indifference curve is the Marginal Rate of Intertemporal Substitution (MRIS) • Equating these allows us to Maximize