# Marginal Rate of Technical Substitution

*Sundaram holds a Master's degree in Economics and has worked in marketing, search engine indexing, banking, and research analysis.*

## Meaning

The marginal rate of technical substitution (MRTS) is the rate at which one input can be substituted for another input without changing the level of output. In other words, the marginal rate of technical substitution of Labor (L) for Capital (K) is the slope of an isoquant multiplied by -1.

Since the slope of an isoquant is moving down, the isoquant is given by –ΔK/ΔL.

MRTS = –ΔK/ΔL = Slope of the isoquant.

## Table 1

Combinations | Labor (L) | Capital (K) | MRTS (L for K) | Output |
---|---|---|---|---|

A | 5 | 9 | -- | 100 |

B | 10 | 6 | 3:5 | 100 |

C | 15 | 4 | 2:5 | 100 |

D | 20 | 3 | 1:5 | 100 |

In the above table, all the four-factor combinations A, B, C and D produce the same level of 100 units of output. They are all iso-product combinations. As we move from combination A to combination B, it is clear that 3 units of capital can be replaced by 5 units of labor. Hence, MRTS_{LK} is 3:5. In the third combination, 2 units of capital are substituted by 5 more units of labor. Therefore, MRTS_{LK} is 2:5.

In figure 1,

MRTS_{LK} at point B = AE/EB

MRTS_{LK} at point C = BF/FC

MRTS_{LK} at point D = CG/GD

## Isoquants and Returns to Scale

Let us now examine the responses in output when all inputs are varied in equal proportions.

Returns to scale refer to output responses to an equi-proportionate, change in all inputs. Suppose labor and capital are doubled, and then if output doubles, we have constant returns to scale. If output is less than double, we have decreasing returns to scale, and if output is more than double, we have increasing returns to scale.

Depending on whether the proportionate change in output equals, exceeds or falls short of the proportionate change in both inputs, a production function is classified as showing constant, increasing or decreasing returns to scale.

For computing the returns to scale in a production function, we calculate the function co-efficient represented by the symbol ‘Ɛ’. The ratio of the proportionate change in output to a proportionate change in all inputs is called the function co-efficient Ɛ. That is Ɛ = (Δq/q)/(Δλ/λ) where the proportionate change in output and all inputs are shown by Δq/q and Δλ/λ. Then the returns to scale is classified as follows:

Ɛ < 1 = Increasing returns to scale

Ɛ = 1 = Constant returns to scale

Ɛ > 1 = Decreasing returns to scale

**Increasing returns to scale**

When output increases by a proportion that exceeds the proportion by which inputs increase, increasing returns to scale prevail.

The line OP is the scale line because a movement along this line shows only a change in the scale of production. The proportion of labor to capital along this line remains the same because it has the same sloe throughout. The operation of increasing returns to scale is shown by the gradual decrease in the distance between the isoquant. For example OA > AB > BC.

**Causes of increasing returns to scale**

Several technical and/or managerial factors contribute to the operation of increasing returns to scale.

*1. Increasing specialization of labor*

Increasing returns to scale can be the result of increase in the productivity of inputs caused by increased specialization and division of labor as the scale of operations increase.

*2. Indivisibilities*

In general, indivisibility implies that equipment is available only in minimum sizes or in definite ranges of size. Specialized machines are generally far more productive than less specialized machines. In large-scale operations the possibility of using specialized machines is higher, so productivity will also be higher.

*3. Geometric necessity*

For some production processes, it is a matter of geometric necessity. A larger scale of operation makes it more efficient. For example, to double the grazing area, a farmer need not have to double the length of fencing. Similarly, doubling the cylindrical equipment (like pipes and smoke stacks) and spherical equipment (like storage tanks) requires less than twice the quantity of metal.

**Decreasing returns to scale**

Decreasing returns to scale prevail when the distance between consecutive isoquants increases. For example, OA < AB < BC.

Decreasing returns arise when diseconomies are greater than economies. Difficulties in coordinating the operations of many factories and communication problems with employees may contribute to decreasing returns to scale. More than proportionate increases in managerial inputs may be required to expand output when an organization becomes very large. (see figure 3)

**Constant returns to scale**

Constant returns to scale prevail when output also increases by the same proportion in which input increases. In the case of constant returns to scale, the distance between successive isoquants remains constant. For example OA = AB = BC (see figure 4)

Constant returns arise when economies exactly balance with diseconomies. As economies of scale are exhausted, a phase of constant returns to scale may set in operation.

## Comments

**dossa** on January 17, 2018:

i like the explanation of the marginal rate of technical substution