Exponential distribution

The density function for the exponential distribution is f(x) = 1e x=

and the distribution function is

F(x) = 1 e x= :

It is de…ned for x > 0. The exponential distribution is often used for the failure rate of equipment: the probability that a piece of equipment will fail

5 2.5 0 -2.5 -5 0.3 0.2 0.1 0 x y x y

Figure 11.2: Changing the mean of the normal density

5 2.5 0 -2.5 -5 0.3 0.2 0.1 0 x y x y

0 1 2 3 4 5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 x y

Figure 11.4: Density function for the lognormal distribution

by time x is F(x). Accordingly, f(x) is the probability of a failure at time x.

11.4.5

Lognormal distribution

The random variablex~has thelognormal distributionif the random vari- able y~= ln ~x has the normal distribution. The density function is

f(x) = _p1 2

h

e (lnx)2=2i 1 x and it is de…ned for x 0. It is shown in Figure 11.4.

11.4.6

Logistic distribution

F(x) = 1

1 +e x:

Its density function is

f(x) = e

x

5 2.5 0 -2.5 -5 0.25 0.2 0.15 0.1 0.05 x y x y

Figure 11.5: Density function for the logistic distribution

It is de…ned over the entire real line and gives the bell-shaped density function shown in Figure 11.5.

12

Integration

If you took a calculus class from a mathematician, you probably learned two things about integration: (1) integration is the opposite of di¤erentiation, and (2) integration …nds the area under a curve. Both of these are cor- rect. Unfortunately, in economics we rarely talk about the area under a curve. There are exceptions, of course. We sometimes think about pro…t as the area between the price line and the marginal cost curve, and we some- times compute consumer surplus as the area between the demand curve and the price line. But this is not the primary reason for using integration in economics.

Before we get into the interpretation, we should …rst deal with the me- chanics. As already stated, integration is the opposite of di¤erentiation. To make this explicit, suppose that the function F(x) has derivative f(x). The following two statements provide the fundamental relationship between derivatives and integrals:

Z b a

f(x)dx=F(b) F(a); (12.1)

and _Z

f(x)dx=F(x) +c; (12.2) where c is a constant. The integral in (12.1) is a de…nite integral, and its distinguishing feature is that the integral is taken over a …nite interval. The integral in (12.2) is an inde…nite integral, and it has no endpoints. The reason for the names is that the solution in (12.1) is unique, or de…nite, while the solution in (12.2) is not unique. This occurs because when we integrate the function f(x), all we know is the slope of the function F(x), and we do not know anything about its height. If we choose one function that has slope f(x), call it F (x), and we shift it upward by one unit, its slope is still f(x). The role of the constant c in (12.2), then, is to account for the indeterminacy of the height of the curve when we take an integral.

The two equations (12.1) and (12.2) are consistent with each other. To see why, notice that _Z

f(x)dx=

Z ₁ 1

f(x)dx;

so an inde…nite integral is really just an integral over the entire real line

( ₁;₁). Furthermore, Z b a f(x)dx = Z b 1 f(x)dx Z a 1 f(x)dx = [F(b) +c] [F(a) +c] = F(b) F(a):

Some integrals are straightforward, but others require some work. From our point of view the most important ones follow, and they can be checked by di¤erentiating the right-hand side.

Z xndx = x n+1 n+ 1 +cfor n6= 1 Z ₁ xdx = lnx+c Z erxdx = e rx r +c Z lnxdx = xlnx x+c

There are also two useful rules for more complicated integrals: Z af(x)dx = a Z f(x)dx Z [f(x) +g(x)]dx = Z f(x)dx+ Z g(x)dx:

The …rst of these says that a constant inside of the integral can be moved outside of the integral. The second one says that the integral of the sum of two functions is the sum of the two integrals. Together they say that integration is a linear operation.

12.1

Interpreting integrals

Repeat after me (okay, way after me, because I wrote this in April 2008):

Integrals are used for adding.

They can also be used to …nd the area under a curve, but in economics the primary use is for addition.

To see why, suppose we wanted to do something strange like measure the amount of water ‡owing in a particular river during a year. We have not …gured out how to measure the total volume, but we can measure the ‡ow at any instant in time using our Acme Hydro‡owometerTM. At time t the volume of water ‡owing past a particular point as measured by the Hydro‡owometerTM _{is h(t).}

Suppose that we break the year intonintervals of lengthT =neach, where T is the amount of time in a year. We measure the ‡ow once per time interval, and our measurement times are t1; :::; tn. We use the measured ‡ow at time

ti to calculate the total ‡ow for periodi according to the formula

h(ti)

T n

where h(ti) is our measure of the instantaneous ‡ow and T =n is the length

of time in the interval. The total estimated volume for the year is then V(n) = tn X t=t1 h(ti) T n: (12.3)

We can make our estimate of the volume more accurate by taking more measurements of the ‡ow. As we do thisn becomes larger andT =nbecomes smaller.

Now suppose that we could measure the ‡ow at every instant of time. Then T =n _! 0, and if we tried to do the summation in equation (12.3) we would add up a whole bunch of numbers, each of which is multiplied by zero. But the total volume is not zero, so this cannot be the right approach. It’s not. The right approach uses integrals. The idea behind an integral is adding an in…nite number of zeroes together to get something that is not zero. Our correct formula for the volume would be

V =

Z T

0

h(t)dt:

The expression dt takes the place of the expression T =n in the sum, and it is the length of each measurement interval, which is zero.

We can use this approach in a variety of economic applications. One major use is for taking expectations of continuous random variables, which is the topic of the next chapter. Before going on, though, there are two useful tricks involving integrals that deserve further attention.

12.2

Integration by parts

Integration by parts is a very handy trick that is often used in economics. It is also what separates us from the lower animals. The nice thing about integration by parts is that it is simple to reconstruct. Start with the product rule for derivatives:

d

dx[f(x)g(x)] =f

0_{(x)g(x) +f(x)g}0_(x):

Integrate both sides of this with respect to x and over the interval[a; b]:

Z b a d dx[f(x)g(x)]dx= Z b a f0(x)g(x)dx+ Z b a f(x)g0(x)dx: (12.4) Note that the left-hand term is just the integral (with respect to x) of a derivative (with respect to x), and combining those two operations leaves

the function unchanged: Z b a d dx[f(x)g(x)]dx = f(x)g(x)j b a = f(b)g(b) f(a)g(a): Plugging this into (12.4) yields

f(x)g(x)_jb_a = Z b a f0(x)g(x)dx+ Z b a f(x)g0(x)dx: Now rearrange this to get the rule for integration by parts:

Z b a f0(x)g(x)dx= f(x)g(x)_jb_a Z b a f(x)g0(x)dx: (12.5) When you took calculus, integration by parts was this mysterious thing. Now you know the secret –it’s just the product rule for derivatives.