1 / 32

Econ 384 Intermediate Microeconomics II

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

aoki
Download Presentation

Econ 384 Intermediate Microeconomics II

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related