A SET THEORY APPLICATIONCOST MINIMIZATION OF A CORN INDUSTRY
FIRM’S PROBLEMGenerally, we can break firm’s problem into three:1. Which combinations of inputs produce a givenlevel of output?2. Given input prices, what is the cheapest way toattain a certain output?3. Given output prices, how much output shouldfirm 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 aproduction 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 theleast 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. Thecost 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 twoinput bundles z= (z1,z2) and z’= (z1’,z2’), z1 ≥ z1’ and z2 ≥ z2’. This impliesthat f (z1,z2) ≥ f (z1’,z2’) or in words, “more is better.”2. Continuity The preference relation is continuous because the neighboringpoints 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 twopoints in the isoquants (the counterpart of indifference curves in theproduction set), the line drawn is within the preferred set. To simplyput it, averages are preferred than extremes.
We can get the optimal solution here by thetangency condition. *Isoquant- bundles of input that yield thesame output *Isocost- the set of inputs with the same costor amount of money isocost Optimal choice isoquant