Time and the Discount Rate
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![Page 1: Time and the Discount Rate](https://reader030.fdocuments.net/reader030/viewer/2022032708/56812a46550346895d8d81de/html5/thumbnails/1.jpg)
Time and the Discount Rate
• Project flows of costs and benefits are not even over time.
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Flow of Costs and Benefits
Option1
Year Cost Benefit Net
1 100 0 -100
2 10 0 -10
3 10 150 140
Sum 120 150 30
Option 2
Year Cost Benefit Net
1 40 50 10
2 40 50 10
3 40 50 10
Sum 120 150 30
Consider 2 Investment Options
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Flow of Costs and BenefitsNet Benefits by Year
-150
-100
-50
0
50
100
150
200
1 2 3
Year
Ne
t B
en
efi
ts Option 1 Option 2
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Flow of Costs and Benefits
• These two options have very different flows of costs and benefits over time.
• Is there any difference between these two options?
• Which is preferred?
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Flow of Costs and Benefits
• In order to address this question, need to understand how the future is valued relative to the present:– Intertemporal time preferences– The interest rate
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Time Preference of Consumers
• Consider consumers have a stock of wealth today, and must choose to consume between today and tomorrow.
• Intertemporal indifference map– Time indifference– Intertemporal indifference curve
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Intertemporal Consumption Choice
Consumption T0
Con
sum
ptio
n T
1
C0
S0
S1
I0 (Intertemporal Indifference Curve)
C1
x
x
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Intertemporal Consumption Choice
Consumption T0
Con
sum
ptio
n T
1A Family of Indifference Curves
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Time Preference of Consumers
• Expect consumers to have “positive time preference” :– prefer consumption today rather than in the future
• Why?– There is a chance that will not be able to consume in
the future • (in the long run we are all dead)
– Expectation that income will be higher in the future (economy will grow)
• (Goods and services will be more abundant in the future as a result of economic growth)
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Intertemporal Consumption Choice
Consumption T0
Con
sum
ptio
n T
1
“PPF” (Slope=-1)
C0
S0 = C0
Ce
Se < Ce
450
Zero time preference
Positive time preference
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The price of current consumption
• Income or wealth that is consumed today is not available to be saved for consumption in the future.
• What is the price of consuming today?
• Interest rate: amount that savings today can be increased in the future, through investment.
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Interest Rate
• The price (or opportunity cost) of consuming a dollar of income or wealth today rather than saving for consumption in the future
• Expressed as a percentage increase:– (K1/K0)-1} * 100
K0 money invested in time 0
K1 money obtained from investment in time 1
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Interest Rate
• Example:
• Invest $100 in time 0
• Return = $112 in time 1
• Interest rate=(112/100-1) * 100
= 12%
• A unit-free measure, expressed as a ratio
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Interest Rate
Consumption T0
Con
sum
ptio
n T
1
100
100
i = 0%
112
150
i = 12%
i = 50%
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Time Preference of Consumers
• From consumers’ time preferences, derive the supply of savings.
Determine amount of income and wealth that households will save for the future at different interest rates.
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Response to change in interest rate
Consumption T0
Con
sum
ptio
n T
1
S0
S1
S2
i0
i1
i2
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Response to change in interest rate
i0
Interest Rate
Savings ($)S0
x
i1
S1
i2
S2
x
x
Ssavings
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Investment opportunities
• Investment opportunities generate the demand for savings (capital)– Intertemporal production possibility frontier
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Intertemporal production possibility frontier
Production0
Pro
duct
ion 1
P0P1
I1
I0
PPF
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Demand for Investment
• Rank potential investments according to the rate of return, from highest to lowest.
• This will give the derived demand schedule for savings
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Demand for Investment
Production0
Pro
duct
ion 1
PPF
i0
i1
i2
I0
I1
i2
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Response to change in Interest Rate
Interest Rate
Investment Demand ($)
i0
I0
x
i1
I1
i2
I2
x
x
Demand for Invesment
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Capital Market
i Supplyof
Savings
Demandfor
Investment
Quantity ($)
ieq
Qeq
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Interest rate
• In market equilibrium
• interest rate (r) = MRS = MRT
• In reality: transactions costs associated with matching up suppliers and demanders of savings:– Financial intermediation (banks)– Spread between savings and borrowing rate– Risk premia