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unempl[f82,jmc]		Mathematical models of unemployment

The object is to give the simplest models in which unemployment occurs.
What is the minimum number of people required for the different
kinds of unemployment phenomena?

1. Model 1.  A makes bread from his wheat.  A pound of bread requires
15 minutes of labor.
It takes 5 pounds of wheat
and an hour of labor to make a pound of bread.  A himself is the
consumer of the bread.  Along comes B with a special proces
developed by himself at great expense.  He can make a pound of bread
out of 3 pounds of wheat and a half hour of labor.  He can then
take 4 pounds of wheat from A, return a pound of bread, and have
a pound jof wheat left over for his labor.  Now A discovers a process
whereby he can make a pound of bread with 2 pounds of wheat and
10 minutes of labor.

	Now B is absolutely unemployed.  No matter
how little he is willing to take for his labor, he can't afford
to buy the wheat from A and make bread to be sold back to A,
because it takes him a pound more wheat to make a pound of bread
than it takes A, and the extra pound of wheat costs A more labor
than it takes him to make the bread.  B's only chance of continuing
in the bread business with this process is getting some third party
to subsidize him or to force A to continue trading on the old terms.

	This model is too extreme.  B can be unemployed under weaker
conditions.  Moreover, it only models the unemployment of B.  We
want a model in which  A  also will suffer.  This may require more
than just 3 commodities (wheat, bread and labor) and/or more than
just two people (A and B).  It may also require the introduction
of money and processes that require time.

	However, before we abandon this model, we will abstract it,
replacing the above numbers by variables and then give inequalities
that show how the phenomena depend on the values of the variables.

	We introduce variables as follows:

	wla - the number of units of labor required by A
to produce a unit of wheat.
	bla - the number of units of labor required by A to produce
a unit of bread from sufficient wheat.
	bwa - the number of units of wheat required by A to produce
a unit of bread.
	bwb - the number of units of wheat required by B to produce
a unit of bread.
	blb - the number of units of labor required by B to produce
a unit of bread from sufficient wheat.
	pwb - the number of units of wheat B charges for a unit of bread.

	The criterion for B's absolute unemployment is

(1)	(bwb - bwa) * wla > bla,

i.e. under these conditions it costs A less labor to make the bread
himself than to make the extra wheat required to employ B even if B
gets nothing for his labor.  A is motivated to deal with B provided

(2)	(pwb -pwa) * wla < bla.

In this case B gets (pwb - bwb)/blb units of bread for each unit
of labor.  A gets  1/(pwb * wla)  units of bread for each unit of
his labor.  If B has the option of going into  A's line of work,
i.e. growing the wheat himself and using A's breadmaking process,
then we must have

(3)	(pwb - pwa)/blb ≥ 1(pwb * wla)

to justify a separate business.  This is only possible in view of (2)
provided

(4)	

	Notice that B's unemployment is technological, i.e. it is
the availability to A of a new process for making bread that puts
B out of work.