Question

Calculate input demand and the cost-minimizing combination of inputs and use isoquant analysis to illustrate optimal input substitution.

Short answer

Cost minimization refers to the lowest possible outcome of input bundles that reduces the cost incurred by the producer due to the problem of scarcity in real life.

Step-by-step solution

Isoquant 

It defines the combination of capital and labor inputs that yields the producer's same output level. They are convex as both capital and labor inputs are not perfectly substitutable.

Marginal rate of technical substitution (MRTS):

It is the rate at which both the inputs capital and labor can substitute for each other. MRTS is the ratio between marginal products of both inputs.

$\mathrm{MRTS}=\frac{{\mathrm{MP}}_{\mathrm{L}}}{{\mathrm{MP}}_{\mathrm{K}}}$

Cost Minimization and Inputs Mix

Let us take two bundles on the same isoquant line to understand this. The bundles are A and B, respectively. They both produce the same level of units. However, they are in two different iso-cost lines. B is more costly than A. Thus, the producer will choose B upon A. Thus, at the cost-minimization input mix, the slope of the iso-cost $-\frac{w}{r}$is equal to the slope of isoquant

$MRTS=\left(\frac{w}{r}\right)$

The diagram shows that the cost of capital is higher in B than in A. So, the producer will substitute capital with labor and produce the same amount of output. Thus, A is considered to be a cost minimization input bundle.

Optimal Input Substitution

Suppose the price of an input changes; this will eventually change the cost-minimizing input bundle. Let us assume that the wage rate has increased, thus making labor inputs to be expensive. Let the initial iso-cost line be MN. So, the producer will produce bundle A initially at a low cost, as in the diagram below.

However, with an increase in the labor wage, the slope of the iso-cost line becomes MP. Thus, with the increased wage rate, the producer will substitute labor and choose capital-intensive production. Thus, point B becomes the new cost-minimizing input bundle.