The Mathematics of Planning: A Simple Example

The People's Democratic Republic of Erewhon is a poor labor-abundant country with two sectors, agriculture (X1) and heavy industry (X2). Currently, they are producing 10 million dinars worth of agricultural products and 2 million dinars worth of industrial goods. However, not all of this can be used as a final good, but instead is an input to production. To produce 10 million dinar's worth of X1 requires 1 million dinar's worth of X1 to be channeled back in, along with 500 thousand dinar's worth of X2. To produce 2 million dinar's worth of X2 requires 400 thousand dinar's worth of X1 and 600 thousand dinar's worth of X2. So, 8.6 million in X1 and 0.9 million in X2 is available for final use.

After the revolution, the new people's government wants you to set up a plan that will replace the market and allow central control of resource allocation, production, and distribution. How do you do it?

First, calculate your input coefficients. Call this aij, where i is the input and j is the output. Since producing one unit of X1 requires 0.10 units of X1 and 0.05 units of X2, a11 = 0.10 and a21 = 0.05. Similarly, a12 = 0.20 and a22 = 0.30.

Next, set up the input-output relationships:

X1 >=  a11 X1 + a12 X2 + Y1
or, X1 >= 0.10 X1 + 0.20 X2 + Y1

X2 >=  a21 X1 + a22 X2 + Y2
or, X2 >= 0.05 X1 + 0.30 X2 + Y2

That is, you must produce at least enough X1 to supply production of X1, production of X2, and final demand Y1, and you must produce at least enough X2 to supply production of X1, production of X2, and final demand Y2. In linear algebra, this can be written as X = A X + Y, or X = (I-A)-1 Y. You may graph these equations in X1 and X2 space as upward-sloping inequalities, and together they form a cone of feasible gross production that depends on how much is to be left over for final use:
 
X2 <= [(1-a11)/a12]X1 - Y1/a12
or, X2 <= 4.5 X1 - 5 Y1
X2 >= [a21/(1-a22)]X1 + Y2/(1-a22)
or, X2 >= X1/14 + Y2/0.7
Unfortunately, there is no perpetual motion machine, and outputs require inputs of scarce resources, called factors of production.  Suppose that production requires two resources, capital (K) and labor (L). To produce one dinar worth of X1 requires four units of L and one unit of K, while producing one dinar worth of X2 requires two units of L and five units of K. Suppose that there are 50 million labor units available, though there is currently a 12% unemployment rate (that is, 40 million labor units are in agriculture, 4 million are in industry, and 6 million are unemployed), and 20 million capital units are available (of which 10 million are in agriculture and 10 million are in industry). Measuring in millions, these resource constraints can be written as:
 
L >= aL1X1 + aL2X2
or, 50 >= 4 X1 + 2 X2
or, X2 <= 25 - 2 X1
K >= aK1X1 + aK2X2
or, 20 >= 1 X1 + 5 X2
or, X2 <= 4 - (1/5) X1

These resource constraints are downward-sloping lines, and together these two inequalities define a production possibilities set. However, feasible output also depends on material inputs, so combining all four constraints yields a feasible set consisting of only one small area.  If there is no slack, the area converges to a point.

In this example, the capital constraint is binding while the labor constraint is not. Producing more X2 is not possible without either expanding capital stock K or severely reducing final demand for food Y1 . You have a bottleneck, for you need to produce more heavy industry (capital) goods in order to produce more heavy industry (capital) goods.  On the other hand, decreasing unemployment without wasting labor requires that you produce more labor-intensive agricultural goods and less capital-intensive industrial goods, but this won't solve your capital constraint in the long run.

Note that while this example used the dinar value of input-output relationships, prices have played no role other than an accounting function. It is presumed in this model that there is no substitution between capital and labor, and so prices cannot affect resource allocation. Prices do affect distribution of income among workers, however, and they affect the distribution of income between capital and labor.