# Profit Maximization Through the Technique of Isoquant and Isocost Line

## Introduction

A rational firm seeks maximization of its profit. Maximization of profit implies minimization of cost. The cost is minimum, when the input combination is optimal. Therefore, choosing the right input combination leads to cost minimization and hence ensures maximum profits. In the theory of consumer behavior, we analyze the equilibrium of a consumer with the help of the indifference curve analysis. Similarly, the producer’s equilibrium, which represents the least cost combination of inputs, can be examined with the help of isoquants. It should be remembered that the isoquants in the theory of production are, in fact, the counterpart of the indifference curves in the theory of consumption.

## Assumptions

The principle of least-cost combination rests on the following assumptions:

- Capital and labor are the two factors involved in production.
- All the units of both the factors are homogeneous
- The prices of the factor units are given
- The total money outlay is also given
- There is perfect competition in the factor-market.

In order to analyze producer’s equilibrium, the firm under consideration should know its isoquant map an its isocost line.

**Isoquant Map**

Isoquants indicate various possibilities of combining two inputs. For each level of output, there will be a different isoquant. When a set of isoquants are depicted on a graph it is called an isoquant map.

**Isocost Line**

The concept of isocost line is not a new one. It is the counterpart of the budget line in the theory of consumption. The isocost line is the producers’ resource line.

In the words of Prof. Barthwal, “it is a locus of all combinations of two (or more) inputs which the producer can buy using his fixed outlay at fixed input prices.”

We shall now draw the isocost line on the basis of an imaginary example.

Let us assume that a firm has a sum of $500 to spend on two factors, labor and capital. Further, let us assume that a unit of labor costs $10 and a unit of capital (machine) costs $100. With the total outlay of $500, the firm could hire 50 units of labor and no capital, or it could hire 5 units of capital and no labor; or some combination of labor and capital in between. OM in the diagram represents 50 workers and ON represent 5 machines.

If we connect the two points N and M, we get the isocost line. Thus, an isocost line gives all combinations of labor and capital at equal cost. The isocost line will shift when the prices of factors change, the outlay remaining the same. Likewise, the isocost line will shift to the right if the outlay of the firm increases. Hiring more of both inputs will cost more. When the total outlay is $600, the isocost line is N_{1}M_{1}. N_{2}M_{2} is the isocost line when the outlay is $700.

Thus, the isocost line depends upon two things:

1. The prices of the factors of production

2. The total outlay which the firm wants to make on the two factors of production.

The slope of an isocost line is P_{L}/P_{K}, which is the ratio of the price of labor to the price of capital, when labor is shown on the X-axis and capital is shown on the Y-axis.

**The Optimum Combination of Inputs**

Let us consider the geometry of the producer’s equilibrium.

Now the problem confronting the firm is to reach the highest possible isoquant with its given isocost line. In other words, it is the problem of getting the highest amount of output from the given outlay. Towards this end, the equal product map has been super imposed on the isocost line NM.

In figure 2, NM is the firm’s isocost line. Isoquants IQ_{1}, IQ_{2} and IQ_{3} represent different levels of output. Equilibrium is attained at the point where the isoquant is tangent to the isocost line. The isocost line NM sets the upper boundary for the purchase of the inputs when outlay and input prices are given.

Outlay is not sufficient to move to IQ_{3}. Likewise, the segments of isoquants falling below the isocost line indicate under-utilization of his outlay fully. Rationality on the part of the producer requires full utilization of resources for optimization of output.

Points A and B also satisfy the tangency condition and they lie within the reach of the producer. However, at these points the firm remains at a lower isoquant IQ_{1}, which yields a lesser level of output than that on IQ_{2}. Thus, E is the point of equilibrium from where there is no tendency on the part of the producer to move away. The firm will get its maximum output when it employs OL_{0} units of labor and OK_{0} units of capital. The equilibrium position of the firm can also be explained in terms of the equality between MRTS and the factor price ratio. The slope of the isoquant is the marginal rate of technical substitution (MRTS) and the slope of the isocost line indicates the factor price ratio. It follows that while in equilibrium,

MRTS_{LK }= P_{L}/P_{K}

Thus, marginal rate of technical substitution can also be written as the ratio of the marginal product of labor to that of the marginal product of capital.

MP_{L}/MP_{K} = P_{L}/P_{K} or MP_{L}/P_{L} = MP_{K}/P_{K}