Chapter 16Chapter 16

© 2006 Thomson Learning/South-Western

Capital and Time

2

Time Periods and the Flow of Economic Transactions

Ways transactions can occur across periods. Durable goods that last more than one period. An individual can borrow or lend.

Individual Savings--The Supply of Loans. Savings frees up resources that can be used to

produce investment goods. Savings provide funds for firms to finance investment

goods.

3

Two-Period Model of Saving

Suppose there are only two time periods. 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.

4

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. The person can consume all income in the

second period plus what is earned in interest.

5

C1

U2

U1

U3

(1+r) Y

C0Y

FIGURE 16-1: The Savings Decision

6

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). 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.

7

C1

U2

U1

U3

C*1

(1+r) Y

C0C*0

Y

FIGURE 16-1: The Savings Decision

8

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. 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.

9

C1

U2

(1+r’) Y

C0C*0 Y

FIGURE 16-2: Effect of an Increase in r on Savings Is Ambiguous

C*

(1+r) Y

1

S

10

Substitution and Income Effects of a Change in r

The income effect is S to C0**, C1

**. If consumption in both periods is a normal

good, both should increase.

The net effect of increased r on C0 (and on savings) is ambiguous. 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.

11

C1

U2

U3

C**1

(1+r’) Y

C0C*0 Y

FIGURE 16-2: Effect of an Increase in r on Savings Is Ambiguous

C*

(1+r) Y

C**0

1

S

12

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.

.)( PrdrPdP

v

Costs Borrowing onDepreciati Rentalrate

13

Rental Rates and Interest Rates

This equation shows why there is an inverse relationship between the demand for equipment and the interest rate. 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.

14

Ownership of Capital Equipment

Firms that own equipment are really in two businesses. 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.

15

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. Higher rates increase the equipment rental rate.

Q*, r* is the equilibrium, with the rate that links economic time periods together.

16

S

D

r*

Quantity of loansper periodQ*

Real interestrate

FIGURE 16-3: The Real Interest Rate Is Determined in the Market for Loans

17

Changes in the Real Interest Rate

Factors that increases firms’ demand for capital equipment will increase the demand for loans. These include: 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.

18

Changes in the Real Interest Rate

Factors that affect savings by individuals will shift the supply curve of loans. These include 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.

19

1961~2006年三種利率變動趨勢

*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。

資料來源：中央銀行

0

2

4

6

8

10

12

14

16

50 55 60 65 70 75 80 85 90 95民國

利率

( ) - - 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

20

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。**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

21

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. A dollar today is more valuable because it can

be invested at interest for the year.

22

Single-Period Discounting

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. The present value of $1 a year from now, with

r = 0.05 is $0.95 [$0.95 = $1/(1.05)].

23

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. The first row demonstrates that the higher the

interest rate, the smaller the discount factor.

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Table 16-1: Present Discounted Value of $1 for Various Time Periods and Interest Rates

Interest RateYears until Payment

Is Received 1 Percent 3 Percent 5 Percent 10 Percent 1 $.99010 $.97087 $.95238 $.90909 2 .98030 .94260 .90703 .82645 3 .97059 .91516 .86386 .75131 5 .95147 .86281 .78351 .62093 10 .90531 .74405 .61391 .38555 25 .78003 .47755 .29531 .09230 50 .60790 .22810 .08720 .00852100 .36969 .05203 .00760 .00007

25

Multiperiod Discounting

The present value of $1 that is not to be paid until n years in the future is given by:

Table 16-1 shows various interest rates and different values of n. For example, with r = 0.10 and n = 10, the

present value of $1 is $0.39.

2.16.)1(

1$nr

yearsn in $1 of ValuePresent

26

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.” 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.”

27

Present Value and Economic Decisions

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.

28

Pricing of Exhaustible Resources

Scarcity costs are the opportunity costs of future production foregone because current production depletes exhaustible resources. 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.

29

Price

D

S

P*

Quantityper week

0Q*

FIGURE 16-4: Scarcity Costs Associated with Exhaustible Resources

30

Pricing of Exhaustible Resources

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.

31

Price

D

S

S’

P’

P*

Quantityper week

0Q’ Q*

FIGURE 16-4: Scarcity Costs Associated with Exhaustible Resources

32

The Size of Scarcity Costs

The actual value depends upon the future resource price. 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).

33

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. In the previous example, with r = 5 percent,

copper prices would increase by 5 percent per year to equal $1 in 10 years.

34

Time Pattern of Resource Prices

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.