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who firs1 studicd (his type of system in the I93lls and latct won a Nobel Prize in

economics for his work. It is wid lo he open hccause the demand co,. , (;, is

exogenously given. while the supply of goods is endogenously determined. that

is. is determined by the equations under study. In this system of equations. the

ai,â€˜s and the c;â€˜s are given and we must solve for the xiâ€™s, the gross outputs of the

industries.

There BE a number of algebraic questions associated with these equations

whose answers are important for obtaining the economic insights the inter-industry

model has to offer. For example, what sets of input-output coefficients yield

a nonnegative solution of system (3) for some admissible n-tuple of consumer

demandsâ€™? What set of output n-tuples will achieve a specified admissible n-tuple

of consumer demands? What set of admissible n˜uples of consumer demands can

be obtained from some given set of input-output coefficients?

We have seen how this model sets up in terms of a system of linear equations.

But many insights into the workings of the Leontief model can best be understood

by studying the geometry of the model. We will study linear systems from the

geometric point of view in Chapter 27.

Example 3: Markov Models of Employment

Aggregate unemployment rates do not tell the whole story of unemployment. In

order to target appropriate incomes policies it is necessary to see exactly who

is unemployed. For example. is most unemployment due to a few people who

are unemployed for long periods of time, or is it due to many people, each of

whom is only briefly unemployedl? Questions like these can be answered by data

about the duration of unemployment and the transition between employment and

unemployment. Markov models are the probability models commonly used in

these studies.

If an individual is not employed in a given week. in the next week he or she

may either find a job or remain unemployed. With some chance. say probability 11,

the individual w,ill find a,job; and therefore with pmhability I - /7 that individual

will remain unemployed. Similarly. if an individual is employed in a given week.

wz let 4 be the probability that he or she will remain employed and therefore I - â€˜1

the probability of becoming unemployed. The probabilities [I. q. 1 ˜ p, and 1 - y

are called transition probabilities. In order to keep this model simple, we will

assume that the chances of finding a job are independent of how many weeks the

job seeker has been unemployed and that the chances of leaving a ,job are also

independent of the number of weeks worked. Then the random process of leaving

jobs and tinding new ones is said to be a Markov process. The two possibilities.

employed and unemployed. are the states of the process.

The wuxition probabilities can lead to a description of the pattern of unem-

ployment over time. For example. suppose that there are .Y males of working age

who are currently employed. and? w˜hc are currently unemployed. How will these

numbers change next weekâ€™? Of the .I˜ males currently employed. on average (1.˜

will remain employed and (I y).,˜ will become unemployed. Of the y males

currently unemployed. on average pr mill become employed while (I - p)!â€˜ will

remain unemployed. Summing up. the average number employed next week will

be I+!˜ + ,I?. and the average number unemployed will be (I q).l- + (1 - p)?. If

changes in the size of the labor force are ignored, the week-by-week dynamics of

average unemployment are described by the linear equations

x,+1 =w +PYr

(4)

Y,+1 = (1 d& + (1 ˜ ply,,

where x, and y, are the average numbers of employed and unemployed, respec-

tively, in week t. This system of equations is an example of a linear system of

difference equations.

Macroeconomist Robert Hall estimated the transition probabilities for various

segments of the U.S. population in 1966. For white males the corresponding

system (4) of equations is

I-,+, = .998x, + .136y,

(5)

y,+l = .002x, + .864y,.

For black males, the system is

x,+˜ = .996x, + .lOZy,

(6)

y,,, =.004x, + .898y,.

In the above three systems of equations, note that for any pair of numbers x,

and Y,,

4+1 + Y,+, ˜ & + Yi.

In particular, if we start out with data in perccnteges, so that x0 and yr, sum to 1.

then x, and y! will sum to 1 for all f. To see this, just add the two equations in (4).

Furthermore, it is easy to see that if xi and yi are nonnegative numhcrs, then .x,-,

andy,+, will be also. Thus. if the initial data we plug into the equation at time 0 is

a distribution of the population, the data at each time f will also be a distribution.

There are two questions that are typically asked of Markw proccsscs. First;

will x, and y, ever he constant over time? That is, is there a distribution of the

population between the two states that will replicate itself in the dynamics of

equation (4)? In other words, is there a nonnegative (x, y) pair with

x = qx f py

Y=(l-9)x+(1-PlY (7)

I =x+y.

Such a pair, if it exists, is called a stationary distribution, or a steady state of

(4). Once such a distribution occurs. it will continue to recur for all time (unless

p OT q changes).

16.21 EXAMPLESOF IUNEARMODELS 115

The second question is contingent on the existence of a stationary distribution.

Will the system, starting from any initial distribution of states, converge to a steady

state distribution? If so, the system is said to be globally stable. Both of these

questions can be answcrcd using techniques of linear algebra.

The first two equations of equation system (7) can be rewritten as

II = (y - 1)x + py

(8)

I) = (1 - y)x - ,I):

However, there is really only one distinct equation in (8), since the second equation

is just the negative of the first equation and therefore can hc discarded. Combining

the first equation in system (8) with the remaining equation in (7), conclude

WC

that candidates for steady stutcs will be solutions tcl the system of equations

(q-l)x+pyâ€™y=o

(9

r+y= I.

(We also hwc the nonnegativity constraint, but this will not he a problem.) To

wlvc system (â€˜2); multiply the second equation through by -p and add the result

to the tirst equation. The resulting equation will contain no yâ€™s and can easily hc

solved for .Y. Then, use either of the cqustions in (9) to solve for the corresponding

y. The resulting solution is

Applying this fcvmula to Hallâ€™s data gives 8 steady state unemployment rate uf

I .4 percent for w˜hitc m&x and 3.77 percent for black males. The ability question

asks: when is thcrc a tcndcncy to move toward thcsc ratesâ€™? This analysis is harder

than anything clsc we have done so f;˜r. but it still involycs linear techniques.

Note that we haw seen two ditfcrent linear systems in the Markov mwlcl:

cystcm (4) which dcxrihcs the dynamics ofthc population distribution. and system

(â€˜1) which dcscribcs the long-run steady sratc equilibrium.

Example 4: IS-LM Analysis

116 ,NTRODIJCTION 161

T O LINEAR ALGEBRA

by consumers (consumption) plus the spending I by firms (investment) plus the

spending G by government:

Y=C+ItG

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