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(14

= ,вЂ˜q<>J + p-˜l.z + + p вЂ˜вЂњ).

Since (a + + aвЂќ)( I U) = a aвЂќ.вЂ˜, as one can easily check,

Substituting a = c-вЂ˜˜ and II = N from (12) yields a present value for the annuity

of

To calculate the present value of an annuity which pays A dollars a year forever,

we let N - = in (14):

PV= fвЂ™1вЂ™

A (15)

It is sometimes convenient to calculate the present value of an annuity using

annual compounding instead of continuous compounding. In this case, equation

(12) becomes

A

pv = A

1+i- +вЂњвЂ˜+(l+r)N

Apply equation (13) with u = 1 /(l + r) and n = N:

To calculate the present value of an annuity which pays A dollars a year forever at

interest rate I compounded annually, we let N - e= in (16):

PV = A. (17)

I

The intuition for (17) is straightforward; in order to generate a perpetual flow ofA

dollars a year from a savings account which pays interest annually at rate r. one

must deposit A/r dollars into the account initially.

Optimal Holding Time

Supposc that you own some real estate the market value of which will be V(t)

dollars I years from now. If the interest rate remains constant at I over this period,

the corresponding time stream of present values is V(t)emrвЂ™. Economic theory

suggests that the optimal time rU to sell this property is at the maximum value of

this time stream of present value. The first order conditions for this maximization

prohlcm are

(V(r)e ˜вЂњ)вЂ™ = VвЂ™(r)r-вЂќ rV(t)P = 0,

W)

˜ r at I = the optimal selling time lo. (18)

V(l)

Condition (18) is a natural condition for the optimal holding time. The left-hand

side of (I 8) gives the rate of change of V divided by the amount of V ˜ a quantity

called the percent rate of change or simply the growth rate. The right-hand side

EXPONENTS AND LOGARITHMS 151

100

gives the interest rate, which is the percent rate of change of money in the bank.

As long as the value of the real estate is growing more rapidly than money in the

bank, one should hold on to the real estate. As soon as money in the bank has a

higher growth rate, one would do better by selling the property and banking the

proceeds at interest rate r. The point at which this switch takes place is given by

(18), where the percent rates of change are equal.

This principle of optimal holding time holds in a variety of circumstances, for

example, when a wine dealer is trying to decide when to sell a case of wine that is

appreciating in value or when a forestry company is trying to decide how long to

let the trees grow before cutting them down for sale.

Example 5.8 You own real estate the market value of which f years from now is

given by the function V(t) = 10,OOOefi. Assuming that the interest rate for the

foreseeable future will remain at 6 percent, the optimal selling time is given by

вЂ˜imaximizing the present value

F(f) = 10,000e˜˜e-˜вЂњ˜вЂ™ = 10,000e˜˜-˜Q˜вЂ˜.

˜

The first order condition for this maximization problem is

0 = P(t) = 10,00O˜вЂ˜li˜.вЂњ˜вЂ™

which holds if and only if

1

- = .06 o r la = (A)2 = 6 9 . 4 4

24%

Since FвЂ™(t) is positive for 0 < f < fll and negative for I > r,,, f,, = 69.44 is

indeed the &ha/ max of the present value and is the optimal selling time of the

I real estate.

Logarithmic Derivative

Since the logarithmic operator turns exponentiation into multiplication, multi-

plication into addition, and division into subtraction, it can often simplify the

computation of the derivative of a complex function, because, by Lemma 5.2,

uвЂ™(x) = (In u(x))вЂ™ U(X).

and therefore (19

If In u(x) is easier to work with than u(x) itself, one can compute uвЂ™ mox easily

using (19) than by computing it directly.

[5.61 APPLICATIONS 101

Example 5.9 LetвЂ™s use this idea to compute the derivative of

JF2

JJ=-. (20)

x2 + 1

The natural log of this function is

,вЂќ g=J = ; In (2 - 1) - In (2 + I).

( x2 + 1 1 (21)

It is much simpler to compute the derivative of (21) than it is to compute the

derivative of the quotient (20):

Now, вЂњse (1вЂ™)) to compute yвЂ™:

E вЂ˜= -3x3 + 5x ,E

( 1 2(x2 - I)(,9 + 1) x3 + 1

x.1 + 1

-3xJ +5x

= qx2 - ,y,yx2 + 1)вЂќ

Ewmple 5. IO A favorite calculus problem, which can only be solved by this

mohod 15 the computation of the derivative of g(x) = xx. Since

I .вЂ˜.

(Inxr)вЂ™ = (xlnx)вЂ™ = Inx + I,

the derivative of xx is (1n.r + I) PI by (19).

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