Econ 384 Intermediate Microeconomics II Instructor: Lorne Priemaza

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

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.

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

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 S A =$100 (1.06) 2 = $ S B =$115 Choose option B

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 S A =-$100 (1.06) 2 = -$ S B =-$120 Choose option A. (You could borrow $100 now for one debt, then owe LESS in 2 years than waiting)

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

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:

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

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

PV = A[1-(1/{1+r}) t ] / [1- (1/{1+r})] PV = A[1-x t ] / [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-x t ] / r x=1/{1+r} A.2 Annuity Formula

Consider a payment of $100 per year for 5 years, (7% interest) PV= / / / / = = $438.7 Or PV = A[1-(1/{1+r}) t ] / [1- (1/{1+r})] PV = A[1-x t ] / [1-x] x=1/{1+r} PV = 100[1-(1/1.07) 5 ]/[1-1/1.07] = $ A.2 Annuity Comparison

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 C t x and B t x in year t, present value is calculated: Where r is the interest rate or opportunity cost of funds.

A.3 PV Decisions Example A firm can: 1)Invest $5,000 today for a $8,000 payout in year 4. 2)Invest $1000 a year for four years, with a $2,500 payout in year 2 and 4 If r=4%,

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.

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

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

18 Lifecycle Budget Constraint Young Consumption Old Consumption O i 1 +i 2 /(1+r) The slope of this constraint is (1+r). Often point E is referred to as the endowment point. i1i1 i2i2 i 2 +(1+r)i 1 E

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 c 1 =i 1, 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(c 1,c 2 ) 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 U=√2 U=2 c1c1 c2c2 Consider the utility function U=(c 1 c 2 ) 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

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 c 1 and c 2 ) by choosing c 1 and c 2 …. 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 c2c2 c1c1 IIC 2 IBL 0 C B IIC 1 A D 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

28 A.4 Maximization Example

29 A.4 Maximization Example 2

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 c2c2 c1c1 0 U=817,316

32 A. Conclusion 1)Streams of intertemporal costs and benefits can be compared by comparing present values 2)To examine consumption timing, one can use the LIFECYCLE MODEL: a)An intertemporal budget line has a slope of (1+r) b)The slope of the intertemporal indifference curve is the Marginal Rate of Intertemporal Substitution (MRIS) c)Equating these allows us to Maximize