Download C* (1+r)

Chapter 16
Capital and Time
© 2006 Thomson Learning/South-Western
Time Periods and the Flow of
Economic Transactions

Ways transactions can occur across periods.



Individual Savings--The Supply of Loans.


2
Durable goods that last more than one period.
An individual can borrow or lend.
Savings frees up resources that can be used to
produce investment goods.
Savings provide funds for firms to finance investment
goods.
Two-Period Model of Saving

Suppose there are only two time periods.





3
C0 is consumption this year.
C1 is consumption in the following year.
Only consumption yields utility which can be
purchased with current income, Y.
Income saved earns interest (at a real interest
rate of r) before it is used to buy C1.
The consumers goal is to maximize utility.
A Graphical Analysis



The indifference curves in Figure 16-1
show the utility obtainable from various
combinations of C0 and C1.
When C0 = Y, no income is saved for the
second period.
When C0 = 0, C1 = (1 + r)Y.

4
The person can consume all income in the
second period plus what is earned in interest.
FIGURE 16-1: The Savings Decision
C
1
(1+r) Y
U3
U2
U1
Y
5
C0
A Graphical Analysis


Between these two endpoints, the budget
constraint is the black straight line.
Utility is maximized at C*0, C*1 where the
MRS equals (1 + r).

6
Utility is maximized where the rate the
individual is willing to trade C0 for
C1 equals the rate he or she is able to trade
these in the market through savings.
FIGURE 16-1: The Savings Decision
C
1
(1+r) Y
C*
1
U3
U2
U1
C*
0
7
Y
C0
Substitution and Income Effects
of a Change in r


A change in r changes the “price” of future
versus current consumption.
The substitution effects of an increase in r are
shown in Figure 16-2.



8
The move along U2 to S.
The higher opportunity cost of C0 rises and the
person substitutes C1 for C0.
The person saves more do to the increase in r.
FIGURE 16-2: Effect of an Increase
in r on Savings Is Ambiguous
C1
(1+r’) Y
(1+r) Y
S
C*
1
U2
C*0
9
Y
C0
Substitution and Income Effects
of a Change in r

The income effect is S to C0**, C1**.


The net effect of increased r on C0 (and on
savings) is ambiguous.


10
If consumption in both periods is a normal
good, both should increase.
Savings increase if the substitution effect
dominates (as shown in Figure 16-2), but
decrease if the income effect dominates.
Savings probably increase with higher r.
FIGURE 16-2: Effect of an Increase
in r on Savings Is Ambiguous
C1
(1+r’) Y
(1+r) Y
C**
1
S
U3
C*
1
U2
C** C*
0 0
11
Y
C0
Rental Rates and Interest Rates

If depreciation (d) and borrowing (r = interest)
costs are proportional to the market price of the
equipment being rented (P) we have the following
expression for the per-period rental rate, v.
Rentalrate  v  Depreciation  Borrowing Costs
 dP  rP  (d  r ) P.
12
Rental Rates and Interest Rates

This equation shows why there is an
inverse relationship between the demand
for equipment and the interest rate.


13
When the interest rate is high, rental rates will
be high and firms will try to substitute toward
cheaper inputs while low interest rates induce
firms to rent more equipment.
This will change the demand for loans, with
low rates encouraging greater borrowing.
Ownership of Capital Equipment

Firms that own equipment are really in two
businesses.


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14
They produce goods.
They lease capital equipment to themselves.
The implicit rates they pay for leasing
capital equipment are the same as for a
firm that rents such equipment.
Determination of the Real Interest Rate


Figure 16-3 shows the supply of loans assumed
to be an upward sloping function of the interest
rate, r.
The demand for loans is negatively related to
the interest rate.

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15
Higher rates increase the equipment rental rate.
Q*, r* is the equilibrium, with the rate that links
economic time periods together.
FIGURE 16-3: The Real Interest Rate
Is Determined in the Market for Loans
Real interest
rate
S
r*
D
Q*
16
Quantity of loans
per period
Changes in the Real Interest Rate

Factors that increases firms’ demand for
capital equipment will increase the demand
for loans. These include:
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17
Technical progress that makes equipment more
productive.
Declines in the equipment market prices.
Optimistic views of the demand for products.
The increased demand causes an increase in
the real interest rate.
Changes in the Real Interest Rate

Factors that affect savings by individuals
will shift the supply curve of loans. These
include

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18
Government-provided pension plans that
reduce individuals’ current savings which
increases the real interest rate.
Reductions in taxes on savings increase the
supply of loans and decrease the real interest
rate.
1961~2006年三種利率變動趨勢
利率
16
14
12
10
8
6
4
2
0
50
55
60
65
70
75
80
85
90
95
民國
中央銀行利率(期底)--重貼現率　銀行業牌告利率(期底)*--一年期存款
十年期中央政府--公債次級市場利率**
*係指台灣銀行、合作金庫銀行、第一銀行、華南銀行及彰化銀行五大銀行平均利率。
**係指距到期日接近十年之政府公債殖利率；資料來源係根據櫃檯買賣中心資料再加權平均。
19
資料來源：中央銀行
央行利率、銀行業利率與政府公債
利率之比較
民國
中央銀行利率(期底)--重貼現率
銀行業牌告利率(期底)*--一年期存款
十年期中央政府--公債次級市場利率**
50
14.400
--
--
55
11.520
--
--
60
9.250
--
--
65
9.500
10.750
--
70
11.750
13.000
--
75
4.500
5.000
--
80
6.250
80262
--
85
5.000
6.020
6.04
90
2.125
2.410
4.03
91
1.625
1.860
3.46
92
1.375
1.400
2.16
93
1.750
1.520
2.66
94
2.250
1.990
2.05
95
2.750
2.200
1.98
*係指台灣銀行、合作金庫銀行、第一銀行、華南銀行及彰化銀行五大銀行平均利率。
20
**係指距到期日接近十年之政府公債殖利率；資料來源係根據櫃檯買賣中心資料再加權平均。
資料來源：中央銀行
Present Discounted Value


Transactions that take place at different
times cannot be compared directly
because of the interest that is received or
paid.
A promise to pay a dollar today is not the
same as a promise to pay a dollar in one
year.

21
A dollar today is more valuable because it can
be invested at interest for the year.
Single-Period Discounting

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In a two period model, a dollar invested
today will grow by a factor of (1 + r) next
year.
The present value of a dollar that will not
be received until next year is 1/(1 + r)
dollars.

22
The present value of $1 a year from now, with
r = 0.05 is $0.95 [$0.95 = $1/(1.05)].
Present Value


The present value is discounting the
value of future transactions back to the
present day to take account of the effect of
potential interest payments.
Table 16-1 demonstrates the discount
factor for various interest rates.

23
The first row demonstrates that the higher the
interest rate, the smaller the discount factor.
Table 16-1: Present Discounted Value of $1
for Various Time Periods and Interest Rates
Interest Rate
Years until Payment
Is Received
1
2
3
5
10
25
50
100
24
1 Percent
$.99010
.98030
.97059
.95147
.90531
.78003
.60790
.36969
3 Percent
$.97087
.94260
.91516
.86281
.74405
.47755
.22810
.05203
5 Percent
$.95238
.90703
.86386
.78351
.61391
.29531
.08720
.00760
10 Percent
$.90909
.82645
.75131
.62093
.38555
.09230
.00852
.00007
Multiperiod Discounting

The present value of $1 that is not to be
paid until n years in the future is given by:
$1
Present Value of $1in n years 
.
n
(1  r )

Table 16-1 shows various interest rates
and different values of n.

25
16.2
For example, with r = 0.10 and n = 10, the
present value of $1 is $0.39.
Present Value and Economic
Motives

The goal of the firm making decisions over
time is changed to “maximize the present
value of all future profits.”
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26
This yields nearly the same results as we
have shown for one period profit
maximization.
This is sometimes stated as the firm makes
decisions to “maximize the present value of
the firm.”
Present Value and Economic
Decisions


27
For individuals, present value enters the
utility maximization decision through the
budget constraint.
In some cases, individuals may “discount”
the future in that they would prefer to
consume in the present relative to the
future.
Pricing of Exhaustible Resources

Scarcity costs are the opportunity costs of
future production foregone because current
production depletes exhaustible resources.

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28
These are in addition to the usual production
costs.
In Figure 16-4, the usual production
marginal costs are reflected in the supply
curve, S.
FIGURE 16-4: Scarcity Costs
Associated with Exhaustible Resources
Price
S
P*
D
0
29
Q*
Quantity
per week
Pricing of Exhaustible Resources



30
Scarcity costs shift the marginal cost curve
up to S’.
Because of scarcity costs, current output
falls from Q* to Q’, and the market price
increases from P* to P’.
The charges effectively encourage
“conservation” of the exhaustible resource.
FIGURE 16-4: Scarcity Costs
Associated with Exhaustible Resources
S’
Price
P’
S
P*
D
0
31
Q’
Q*
Quantity
per week
The Size of Scarcity Costs

The actual value depends upon the future
resource price.

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32
For example, suppose the firm believes that copper
will sell for $1 per pound in 10 years.
Selling one pound today will mean $1 foregone in
the future since copper supply is fixed.
If r = 5 percent, the present value equals $0.61.
If production marginal costs = $0.35 per pound,
scarcity costs = $0.26 per pound ($0.61-$0.35).
Time Pattern of Resource Prices

In the absence of change in real
production costs or firms’ expectations
about future prices, the relative price of
resources should be expected to rise over
time at the real rate of interest.

33
In the previous example, with r = 5 percent,
copper prices would increase by 5 percent
per year to equal $1 in 10 years.
Time Pattern of Resource Prices



34
If resource prices rose more slowly that the
real rate of interest, firms would invest
elsewhere decreasing supply and increasing
the resource price.
If resource prices rose faster than the real
rate of interest, firms would increase supply
and decrease its price.
Equilibrium could only occur if the price
increase equaled the real rate of interest.