There are a number of ways to solve this system. For example. you can solve the

middle equation forS in terms of C. substitute this relation into the first and third

equations in (I), and then easily solve the resulting system of two equations in

two unknowns to compute

C = S,YS6. S = 4,702, and F = 35.737,

rounded to the nearest dollar. The next chapter is devoted to the solution of such

systems of linear equations. For the moment, note that the firm™s after-tax and

after-contribution p&its are $53_hOS.

We can use this linear model to compute (Exercise 6.1) that the firm would

have had after-tax profits of $57,000. if it had not made the Red Cross donation.

So. the $5.956 donation really only cost it $3.395 (= $57.000 $53.605). Later,

we will develop a formula for C. S, and T in terms of unspecified before-tax profits

P and even. in Chapter 26. in terms of the tax t&s and contribution percentages.

INTRODLKTION TO ALGEUKA 161

110 LINEAR

Example 2: Linear Models of Production

are perhaps the simplest production models to de-

Linear models of production

scribe. Here we will describe the simplest of the linear models. We will suppose

that our economy has n + I goods. Each of goods 1 through II is produced by one

production process. There is also one commodity, labor (good 0), which is not

produced by any process and which each process uses in production. A production

process is simply a list of amounts of goods: so much of good 1. so much of good

2, and so on. Thcsc quantities are the amounts of input needed to produce one

unit of the process™s output. For example, the making of one car rcquircs so much

steel, so much plastic, so much labor, so much electricity, and so forth. In fact,

some production processes, such as those for steel or automobiles, use some of

their own output to aid in subsequent production.

The simplicity of the linear production model is due to two facts. First, in these

models, the amounts of inputs needed to produce two automobiles are exactly twice

those required for the production of one automobile. Three cars require 3 times as

much of the inputs, and so on. In the jargon of microcconomics, each production

constant returns to scale. The production of 2. 3, or k cars

prw˜s exhibits

k times the amounts of inputs required for the production of I

requires 2, 3, or

car. Second. in these models thcrc is only one way to produce a car. There is no

way to substitute electricity for labor in the production of cars. Output cannot hc

increased hy using more of any one factor alone: more of all the factors is needed,

and always in the ˜ilmc proportions. .This simplifies the analysis of production

problems, hecauss the optimal input mix for the production of. bay. 1000 cars.

does not have to hc computed. It is simply 1000 times the optimal input mix

required for the production of I car.

Before undertaking an ahstmct analysis. we will work nut an cxamplc to

illustrate the key fcaturcs of the model. Consider the ccon˜nny of en org;mic

farm which products twu eoods: corn and fertilircr. Corn is produced using corn

(to plant) and fertilizer. F&tilircr is made from old corn stalks (and pcrhaph hy

feeding the u,rn tu cows, who then product useful end products). Suppose that

the production of I ton of corn requires as inputs 0. I ton of corn and 0.X ton of

fertilixr. The production US 1 ton of fertilizer rcquircs no fertilizer and 0.5 ton of

c,,m.

WC can describe each of the ˜MO production processes hy pairs of number\

(u, /I). whcrc i! represents the corn input ;md /J represents the fcrtilizcr input.

The corn producti<m process is dcscrihed hy the pair of number\ (0. I, 0.X). The

fcrtilizcr production process is dcscrihed by the pair of numhcrs (0.5, 0).

The most important question to zxsk of this model is: Whet can hc produced for

consumptiwi! Corn i:, used both in the production of corn and in the production

of fcrtilizcr. Fertilizer is used in the production of corn. Is thcrc any way o1

running both pn˜˜sscs so iis to leaw some corn and some fertiliar for individual

consumption™? If so. what combinations of corn end fcrtilircr for comumption are

fcasihlc?

EXAMPLES OF LINEAR MODELS 111

16.21

Answers to these questions can be found by examining a particular system of

linear equations. Suppose the two production processes are run so as to produce

XC tons of c”rn and X˜ tons of fertilizer. The amount of corn actually used in the

production of corn is 0. lxc- the amount of c”rn needed per ton of c”rn output

times the number of t”ns to be produced. Similarly, the amount of corn used in

the production of fertilizer is 0.5˜˜. The amount of corn left “ver for consumption

will be the total amount produced minus the amounts used for production of corn

and fertilizer: xc: - 0. Ixc - 0.5x,7, or 0.9˜˜ ˜ 0.5˜˜ tons. The amount of fertilizer

needed in production is 0.8˜˜ tons. Thus the amount left “ver for consumption is

xr 0.8xr t o n s .

Suppose we want our farm to produce for consumption 4 tons of c”rn and 2

tons of fertilizer. How much total production of corn and fertilizer will be required?

Put another way, how much c”rn and fertilizer will the farm have to produce in

order t” have 4 tons of c”rn and 2 tons of fertilizer left “ver for consumers? We

can answer this question by solving the pair of linear equations

0.9x, OSXF = 4 ,

-O.&y + xp =2

This system is easily solved. Solve the second equation for xr in terms of xc:

nfi = o.sxc + 2. (2)

Substitute this expression for XF into the first equation:

0.4X( O.S(O.Sx, + 2) = 4

and solve for xc:

0.5xc = 5, 20 X(™ = 10,

Finally, substitute xc. = 10 back into (2) to compute

u,: = 0.8 10 + 2 = 10.

In the general case. the production process for good j can be described by a set

of input-output coefficients {alri, a ,,, , u”, }, where a, denotes the input of good

i necdcd ro output one unit “f good j. Keep in mind that the first subscript stands

for the input good and the second stands for the output good. The production of

.rj units of good j requires a<,j,r; units of good 0, aljxj units of good 1, and so on.

Total output of g”od i must be allocated between production activities and

consumption. Denote by c; the consumer demand for good i. This demand is given

exogenously. which is t” say that it is not solved for in the model. Let c,, be

the coIIsumer™s supply of labor. Since good 0 (labor) is supplied by consumers

rather than demanded by consumers, cg will bc a negative number. An n-tuple

(Q, c,, , c˜) is said to be an admissible rr-tuple of consumes demands if c,, is

negative, while all the other G™S arc nonnegetive. WC want each process to produce

an output that is sufficient to meet both consumer demand and the input require-

ments of the )I industries. For our simple linear economy, this is the law of supply

and demand: output produced must be used in production or in consumption. Let

xi denote the amount of output produced hy process j. If process j produces xi

units of output, it will need U;˜IX, units of good i. Adding these terms up over all

the industries gives the demand for good i: ailxl + ai2xL + + a+˜,, + ci, The

law of supply and demand then requires

It is convenient to rearrange this equation to say that consumer demand musr equal

gross output less the amount of the good needed as an input for the production

processes. For good I, this says

-q,*, ” u ,/,I I,, = (˜,I,

This lads to the following system of II + I equation\ in II unknowns, which

summarizes the cquilihrium output levels for the cn(irr windustry economy:

(3)

- U,,J.X1 ˜. ” + (I (I ,,,, )I,, = (˜,,

-“,,I.˜1

˜,,,˜i˜

-%,*, m,,x,, = CII