Post on 29-Mar-2015
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Time and the Discount Rate
• Project flows of costs and benefits are not even over time.
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
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
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?
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
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
Intertemporal Consumption Choice
Consumption T0
Con
sum
ptio
n T
1
C0
S0
S1
I0 (Intertemporal Indifference Curve)
C1
x
x
Intertemporal Consumption Choice
Consumption T0
Con
sum
ptio
n T
1A Family of Indifference Curves
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)
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
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.
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
Interest Rate
• Example:
• Invest $100 in time 0
• Return = $112 in time 1
• Interest rate=(112/100-1) * 100
= 12%
• A unit-free measure, expressed as a ratio
Interest Rate
Consumption T0
Con
sum
ptio
n T
1
100
100
i = 0%
112
150
i = 12%
i = 50%
Time Preference of Consumers
• From consumers’ time preferences, derive the supply of savings.
Determine amount of income and wealth that households will save for the future at different interest rates.
Response to change in interest rate
Consumption T0
Con
sum
ptio
n T
1
S0
S1
S2
i0
i1
i2
Response to change in interest rate
i0
Interest Rate
Savings ($)S0
x
i1
S1
i2
S2
x
x
Ssavings
Investment opportunities
• Investment opportunities generate the demand for savings (capital)– Intertemporal production possibility frontier
Intertemporal production possibility frontier
Production0
Pro
duct
ion 1
P0P1
I1
I0
PPF
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
Demand for Investment
Production0
Pro
duct
ion 1
PPF
i0
i1
i2
I0
I1
i2
Response to change in Interest Rate
Interest Rate
Investment Demand ($)
i0
I0
x
i1
I1
i2
I2
x
x
Demand for Invesment
Capital Market
i Supplyof
Savings
Demandfor
Investment
Quantity ($)
ieq
Qeq
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