Econ 384 Intermediate Microeconomics II

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Econ 384Intermediate Microeconomics II
Instructor Lorne Priemaza Lorne.priemaza_at_ualbert
a.ca
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A. Intertemporal Choice

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

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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(1r) P Pr
• After another year
• You will make interest on the initial amount P
• You will make interest on last years interest Pr
• Total amount in the bank P(1r)2
• This is COMPOUND INTEREST. Over time you make
  interest on the interest the interest compounds.

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A.1 Compounding

• Investment 100
• Interest rate 2

Year Calc. Amount
1 100 100.00
2 1001.02 102.00
3 1001.022 104.04
4 1001.023 106.12
5 1001.024 108.24
Derived Formula S P (1r)t S value after
t years P principle amount r interest rate t
years
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A.1 Compounding Choice

• Given two revenues or costs, choose the one with
  the greatest value after time t
• A 100 now B115 in two years, r6
• (find value after 2 years)
• S P (1r)t
• SA 100 (1.06)2 112.36
• SB 115
• Choose option B

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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, r6
• (find value after 2 years)
• S P (1r)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)

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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
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A.2 Present Value Gain Example
What is the present value of earning 5,000 in 5
years if r8?
Earning 5,000 in five years is the same as
earning 3,403 now. PV can also be calculated
for future losses
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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
GICs pay 5) PV S/(1r)t
-30,000/(1.05)20 -11,307
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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
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A.2 Annuity Formula
PV A1-(1/1r)t / 1- (1/1r) PV
A1-xt / 1-x x1/1r 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
A1-xt / r x1/1r
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A.2 Annuity Comparison
Consider a payment of 100 per year for 5 years,
(7 interest) PV 100100/1.07 100/1.072
100/1.073 100/1.074 100 93.5 87.3
81.6 76.3 438.7 Or PV A1-(1/1r)t /
1- (1/1r) PV A1-xt / 1-x
x1/1r PV 1001-(1/1.07)5/1-1/1.07
438.72
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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.
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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 r4,

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A.3 PV Decisions Example
2) Invest 1000 a year for four years, with a
2,500 payout in year 2 and 4 If r4,
Option 1 is best.
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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

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A.4 Lifecycle Budget Constraint

• Assume 2 time periods (1young and 2old), 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,
• c2i2(1r)i1
• If you only consumed when young
• c1i1i2 /(1r)

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Lifecycle Budget Constraint
The slope of this constraint is (1r). Often
point E is referred to as the endowment point.
i2(1r)i1
Old Consumption
i2
E
O
i1
i1i2 /(1r)
Young Consumption
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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 c1i1, therefore
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A.4 Lifetime Utility

• In the lifecycle model, an individuals lifetime
  utility is a function of the consumption in each
  time period
• Uf(c1,c2)
• If the consumer assumptions of consumer theory
  hold across time (completeness, transitivity,
  non-satiation) , this produces well-behaved
  intertemporal indifference curves

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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)

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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


U2
1
Uv2
0
c1
1
2
4
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Marginal Rate of Intertemporal Substitution (MRIS)

• Utility is constant along the intertemporal
  indifference curve
• An individual is willing to SUBSTITUTE one
  periods consumption for another, yet keep
  lifetime utility even
• ie) In the above example, if someone starts with
  consumption of 2 each time period, theyd be
  willing to give up 1 consumption in the future to
  gain 3 consumption now
• Obviously this is unlikely to be possible

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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)

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A.4 MRIS Example
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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

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Maximizing Intertemporal Utility
c2
Point A affordable, doesnt maximize
utility Point B unaffordable Point C affordable
(with income left over) but doesnt maximize
utility Point D affordable, maximizes utility
IBL

D

B


C
IIC2

A
IIC1
c1
0
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A.4 Maximization Example
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A.4 Maximization Example 2
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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

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Maximizing IntertemporalUtility
c2
Utility is always maximized at the tangent to the
indifference curve


U817,316
c1
0
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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 (1r)
• The slope of the intertemporal indifference curve
  is the Marginal Rate of Intertemporal
  Substitution (MRIS)
• Equating these allows us to Maximize