Cost minimization of a corn industry. A Set Theory application.

1. A SET THEORY APPLICATION
COST MINIMIZATION OF A CORN INDUSTRY

2. FIRM’S PROBLEM
Generally, we can break firm’s problem into three:
1. Which combinations of inputs produce a given
level of output?
2. Given input prices, what is the cheapest way to
attain a certain output?
3. Given output prices, how much output should
firm produce?

3.  The second one is a problem of cost
minimization. This can be solved using set theory.
 In modelling a firm as a production function, we
first assume the following:
1. The firm produces a single output.
2. The firm has N possible inputs (z1,....zN)
3. Inputs are translated into an output by a
production function q= f (z1,z2)

4. To illustrate this model, we consider a farmer’s
technology. Here, the single output is corn. It
has 2 inputs: labor and capital (i.e. machinery)
denoted as z1 and z2, respectively.

5. COST MINIMIZATION PROBLEM
 Choose a production bundle z in the production set Z that yields the
least cost of producing certain output q.
min (z1,z2) subject to f (z1,z2)≥ q
 This problem yields the firm’s input demands denoted by:
z* (r1,r2,q)
where r is the input price (r1= wage for labor and r2 = rent for capital)
 The money used by the firm to attain its target output is its cost. The
cost function therefore is:
c (r1,r2, q)= min (z1,z2) subject to f (z1,z2)≥ q

6. 1. Monotonicity
The production function is monotone because for any two
input bundles z= (z1,z2) and z’= (z1’,z2’), z1 ≥ z1’ and z2 ≥ z2’. This implies
that f (z1,z2) ≥ f (z1’,z2’) or in words, “more is better.”
2. Continuity
The preference relation is continuous because the neighboring
points of z and z’ follows the same order, and that is z ≥ z’
3. Convexity
The preference relation is convex because if we take any two
points in the isoquants (the counterpart of indifference curves in the
production set), the line drawn is within the preferred set. To simply
put it, averages are preferred than extremes.

7.  Graphically,
z= (z1,z2)
z’ (z1’,z2’)

8.  We can get the optimal solution here by the
tangency condition.
 *Isoquant- bundles of input that yield the
same output
 *Isocost- the set of inputs with the same cost
or amount of money
isocost
Optimal choice
isoquant