Isocost Line

Isocost Line Equation

Now, let's dive deeper into the isocost line equation to further comprehend the concept.

We know that an isocost line shows all the possible combinations of labor and capital for a certain amount of cost. Now, to find an isocost line equation, let us assume that \(C\) is the total cost of producing a particular amount of output, and \(r \times K\) and \(w \times L\) denote the rental cost of capital and total wage of labor, respectively.

The equation of the isocost line is:

\(C=r \times K + w \times L\)

\(\hbox{Where:}\)

\(K\) - the amount of capital employed

\(r\) - the rental rate of capital

\(L\) - the amount of labor employed

\(w\) - the wage rate.

The Slope of the Isocost Line

Now, let us have a look at how we can determine the slope of the isocost line by using the isocost line equation.

The equation of an isocost line can be rearranged as the equation of a straight line. The equation of a straight line is written as follows:

\(y = ax+b\)

Now, let's rearrange the equation of the isocost line in the form of a straight-line equation.

\(C=r \times K + w \times L\)

\(r \times K = C-w \times L\)

\(K = C/r - (w/r)L\)

Fig. 1 - The Slope of the Isocost Line

Figure 1 depicts the isocost line I₁, along with all the possible combinations of inputs for the total estimated cost. A firm can either use a combination of (L₁ ; K₁) or (L₂ ; K₂) or (L₃ ; K₃), depending on how they want to operate. If a firm decides to make its production labor-intensive, then it might choose the input combination at point C with L₁ of labor and K₁ amount of capital. Similarly, if a firm decides to make its production capital-intensive, then it will choose the combination at point A. It all depends on how the firm wants to operate and what helps them maximize its profit.

Shifts of the Isocost Line

Let us have a look at what causes the shifts of the isocost line.

Fig. 2 - Shifts of the Isocost Line

The major reason behind the shifts of the isocost line is the change in the number of inputs used by a firm. Sometimes a firm might decide to expand its operations and increase its production, and sometimes it might look to reduce its production. Depending on the organizational strategy, the amount of inputs they use for production also varies.

In Figure 2 above, the initial isocost line is at I₁. Now, if a firm decides to expand its operations, it will require an increased amount of inputs (labor and capital). In this case, the isocost line shifts to the right from I₁ to I₃ as the increased amount of inputs means increased total cost. Likewise, if a firm decides to contract its operations, the number of inputs they require will decrease. In this case, the isocost line will shift from I₁ to I₂.

Isocost Line and Isoquant

Let's have a look at the isocost line and isoquant together.

An isocost line helps us determine the combination of inputs for a defined amount of cost. An isoquant curve is a little bit different. An isoquant curve helps us determine the combination of inputs to produce the same amount of output.

Fig. 3 - Isocost Line and Isoquant

Figure 3 is a graph representing the isocost line (I₁) and an isoquant curve (Q₁). For a firm to produce maximum output at the minimum cost, the firm should aim to produce at the point where isoquant is tangent to isocost. At the tangent point, the slope of both isocost and isoquant is the same.

Isocost Line Properties

Let us now examine some of the properties of the isocost line to see how firms can produce given output at the lowest possible cost and how firms react when input prices change.

• The properties of the isocost line are as follows:1. Helps the firm to produce given output at minimum cost.2. Helps the firm to adjust between two inputs when the price of one input changes.

Isocost Line Properties: Minimum Cost Output

Let us first have a look at how the firms can produce the required output at a minimum cost.

Fig. 4 - Minimum Cost Output

Figure 4 depicts various isocost lines and an isoquant curve. Every firm strives to create its output at the lowest possible cost. Figure 4 depicts a firm attempting to create output Q₁. How can a firm produce at the lowest possible overall cost?

Assume the firm wishes to invest an I₁ amount of money. However, any combination of labor and capital on I₁ will not be sufficient to generate Q₁. At the cost levels I₂ and I₃, output Q₁ is possible. At I₃, the output can be obtained by combining K₃ units of capital and L₃ units of labor or K₁ units of capital and L₁ units of labor. However, the same output can be generated at the lowest cost at isocost I₂ by using K₂ units of capital and L₂ units of labor.

• The isoquant curve is tangent to the isocost line at point A in Figure 4, indicating that this is the point where a firm can create the most output at the lowest cost.

Isocost Line Properties: Substituting Input When Input Price Changes

Now, let's look at how a firm substitutes inputs when relative input prices change.

Fig. 5 - Isocost Line Properties

Let us Assume that the price of one of the inputs, say, labor, has increased. When the price increases, the slope of an isocost line increases in magnitude, which makes the isocost line rotate clockwise. Hence, it becomes steeper. How will a firm manage such a situation? Well, the firm will substitute the amount of labor it uses with capital.

In Figure 4 above, the firm was initially producing a Q₁ level of output using a combination of L₁ units of labor and K₁ units of capital. Because labor costs have risen, the firm has decided to utilize a new combination of K₂ units of capital and L₂ units of labor to achieve the same output. Hence, the new cost minimization point for the firm is at point B.