## What is the Isocost Curve?

An isocost curve in the production theory that is a crucial part of economics depicts different combinations of factor inputs at a given market cost. In other words, this shows all the combinations of production factors that cost the same for the firm to employ. Usually, this can be depicted as a downward sloping straight line when the firms act as the price-takers in the factor markets. The slope of this mainly depicts the relative prices of different production factors.

A firm in the market operates for earning a higher income but this can become possible only when there is a reduction in production cost of firms to produce a given amount of product. Thus, a profit-maximization condition will intend to maximize the earning by minimizing the cost of the factors of the production required to produce a specified level of production. Furthermore, the optimal production point for a firm can be determined at the tangency between the isocost and the isoquant.

## Isoquant curve

The concept of the isoquant is related to the study of microeconomics wherein this curve shows the graph of charts for all the inputs required to produce specified units of output. Generally, this graph is used as a metric to estimate the impact of all the inputs on the products that can be obtained.

### Equilibrium

Parallel to the consumer choice theory concept in economics (the budget constraint), the production theory also depicts the isocost line. Like the indifference curve that is convex to origin, there is an isoquant wherein the graphical intersection of isocost line and isoquant helps to minimize the production cost within the production constraint.

### Isocost Constraints

Assuming that there are only two-factor inputs- Labor and Capital, the isocost constraint can be written as-

$r\ast K+w\ast L=C$

Here, the term r depicts the rental rate of capital, K depicts the capital, w depicts the wage rate, L depicts the labor, and C depicts the total cost.

Also, keeping the units of Labor used on the horizontal axis and the units of Capital used on the vertical axis we can derive the slope of the isocost as –w/r or slope = - (wages/rent).

Now, it is important to note that the isocost constraint is combined with an isoquant map to measure the optimal production point for any given unit of output. More precisely, the tangency point of any isoquant with an isocost line gives the least or lowest cost combination of inputs required to produce the given units of output related to that isoquant.

Equivalently, this tangency point will give the maximum output level that can be produced for a given total cost. Also, note that the line joining the tangency points of isoquants and isocosts (with constant input price) is known as the expansion path. The beneath graph shows the above-explained concept appropriately-

In the above image, Q1 depicts Isoquant 1, Q2 depicts Isoquant 2, and Q3 depicts Isoquant 3. So there is an isoquant map. The blank inverse sloping line is the isocost line and the intersection of this isocost line with the isoquant depicts the tangency points that are the same as the equilibrium points. Furthermore, the joining of the points P, Q, and R shows the expansion path.

### Marginal Rate of Technical Substitution

Economics gives a more detailed study regarding the understanding of the concept, “the marginal rate of technical substitution”. In most cases, it is written as MRTS. We know that the slope of an isoquant at any given point is the slope of a tangent line at that point. This slope of the isoquant is known as the marginal rate of technological substitution that tells a firm how much more capital is required to replace one unit of labor to maintain the same output.

In simple terms, this depicts the rate by which a firm can substitute one-factor input in terms of other input, keeping the result the same. The basic formula of this is-

MRTS = ΔK/ΔL= MPL/MPK

In the above, Delta K depicts the change in the capital uses, Delta L depicts the change in the labor used, MPL depicts the marginal productivity of the labor, and MPK depicts the marginal productivity of capital.

From the above, it seems that the ratio of the change in capital and labor is the same as the fraction of the marginal productivity of labor and capital. Now consider the beneath image and determine the MRTS at each unit of labor used.

Clearly from above, the marginal rate of technological substitution is-

Between a and b = 5, between b and c = 3, between c and d = 2, between d and e = 2.

The above numerical denotes that as the firm employs more and more units of labor, the sacrifice of capital units to raise the labor units will decline. So, this means the marginal rate of technical substitution for a production process is diminishing like MRS in the consumer theory. This further means that its standard shape is convex to the origin.

In this model, it is quite applicable that both labor and capital are considered as the perfect substitute for each other because MRTS(L, K) is constant at all the points on the isoquants. However, it can be stated as an unrealistic case because capital and labor are not the same production factors and hence they can’t be accounted as a perfect substitute.

## Isoprofit Curve

This curve shows the combination of two or more variables that helps generate the same level of gain for an organization. For instance, when there is only a single organization then the isoprofit can be determined for alternative combinations. However, when there are two organizations then the isoprofit can be determined from the combinations of the output level of both organizations that causes a fixed level of benefit for the organization.

It can be estimated by taking a positive difference between the revenue earned from sale proceeds and the total production cost.

### Isoprofit Equation

Assuming the total cost function as C (Q) and revenue as the product of price and quantity, the equation of profit can be written as:

$\pi =PQ-C(Q)$

It is to note that the isoprofit curve belongs to the family of the QP-plane, each of which is consistent with the given profit. So, an isoprofit equation can be written as:

$k=PQ-C(Q)$

Here, k is the constant profit.

It is to note that there exist different isoprofit curves for each value of k. Graphically the P is written on the vertical axis so re-writing the above in terms of P as a function of Q will help to understand the relationship between each variable appropriately. That is

$P=\frac{C(Q)+K}{Q}$

Clearly, from above, there is a direct relationship between variables P and K for any given of Q (Quantity). This means in a family of isoprofit lines, a higher is preferred over a lower.

By breaking the above equation we can write-

$P=\frac{C(Q)}{Q}+\frac{K}{Q}$

The above equation is equal to

$P=AC+\frac{K}{Q}$

Assume K is equal to 0. So, P is equal to AC which depicts there is a zero economic profit and any point that lies below it depicts a loss.

Now, consider the case where K > 0. Therefore, the slope of the above function can be estimated by taking a derivate of the above function. That is

$\frac{dP}{dQ}=\frac{-K}{{Q}^{2}}<0$, and from the double derivation $\frac{{d}^{2}P}{d{Q}^{2}}=\frac{2K}{{Q}^{3}}>0$

This is a convex function.

To understand the concept more precisely give the numerical value to each given term. That is

P = $5, Q = 200 units, and C(Q) = $700. Thus,

k = PQ - C(Q)

k = $5 * 200 - $700

k = $300

Similarly, the slope of the above question is equal to $\frac{dP}{dQ}=-\frac{300}{4000}<0$.

Also,

$\frac{{d}^{2}P}{d{Q}^{2}}=\frac{2(\$300)}{8000000}>0$

**Optimal Production**

The marginal cost is the cost incur to produce an additional unit of product in the market. It is mainly determined by-

MC = TC (n) - TC (n-1)

The marginal revenue is the benefit received from selling an additional unit of product in the market. It is mainly determined by-

MR = TR (n) - TR (n-1)

Although, maximizing the profit is the aim of each organization operating in the market yet the optimal production point can be estimated at a quantity where the MR is the same as MC. This is also known as equilibrium.

Different factors affect an organization’s decisions regarding price and quantity such as market power, the elasticity of the product, and production cost. So, only satisfying the profit-maximization condition is not only enough.

**Future aspects of using isocost and isoprofit curves**

Nowadays, a greater emphasis is given on the reduction of the production cost so that earnings will upsurge. The major reason behind this reduction in the production cost is the continuous up-gradation technology and its adaptation. In economics, the study of this concept plays a central role to determine the operational scale in the market. Further, this topic has a vast scope for future studies as it makes the books learning practical learning for an individual. Different scholars and economists also do extensive studies on these topics as a part of their higher studies. The other concept related to this topic is demand, utility, income, etc.

## Context and Application

This topic is significant in the professional exams for both undergraduate and graduate courses like

- Bachelor in Science in Mathematics.
- Bachelor in science in Non-medical.
- B.tech in Mechanical Engineering.
- Masters in Science in Mathematics.

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