COST MINIMIZATION OF A CORN INDUSTRY

A SET THEORY APPLICATION

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?

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)

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.

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

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.

Graphically,

z= (z1,z2)

z’ (z1’,z2’)

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

isoquant

isocost

Optimal choice