Note on Principle of Marginal Rate of Technical Substitution and Cobb-Douglas Production Function

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Principle of marginal rate of technical substitution (MRTS)

The marginal rate of technical substitution is the rate in which units of two inputs are substituted for each other to maintain the same level of output. In other words, MRTS of capital for labour represents the amount of labour, which the producer has to sacrifice for the gain of one additional unit of capital so that his level of output remains the same. In short, MRTS is the ratio of the change in units of labour with the change in units of capital. The MRTS reflects the slope of isoquant.

Since, iso-quant reflects the production function with two variable inputs, which is written as,
Q=f(K,L)……………….(i)
Let us now suppose that the producer substitutes capital for labour such that his total output remains the same. When he sacrifices some units of labour, a stock of labour decreases by ΔL and he loses a part of his total output which is expressed as, -ΔL. MP(l)……………(ii).

On the other hand, the stock of capital increases by ΔK as a result of the substitution of capital for labour and he gains in total output which is expressed as +ΔK. MP(k)…………(iii)

By rearranging the equations (ii) and (iii) simultaneously, we get,

-ΔL.MP(l) = ΔK.MP(k)

Therefore, -ΔL/ΔK = MPk / MPl……………………..(iv)

Based on equation (iv), we conclude that:

1. The expression -ΔL/ΔK reflects the slope of isoquant ( for MRTSkl) when capital is substituted for labour. But when labour is substituted for capital, ΔK/ΔL gives MRTSlk

Symbolically,
MRTSlk = -ΔK/ΔL

MRTSkl = -ΔL/ΔK

2. The slope of isoquant also reflects the ratio between marginal productivities of two inputs.

3. The trend of MRTS is diminishing. Due to diminishing MRTS, isoquant has a negative slope and any points lying on such curve yield the same level of output to the producer.

 

Marginal rate of substitution
                         Marginal rate of substitution

In the above figure, we can see that all the combinations of labour and capital which are A, B, C, D and E are plotted on a graph. According to the figure, subsequent units of capital substituting labour go on decreasing with equal change (or increase) in units of labour, MRTSlk goes on decreasing. Various points are joined in order to form an isoquant. All these combinations produce the same level of output. This makes the producer indifferent. It implies that MRTS is diminishing. Due to the operation of the law of diminishing MRTS, isoquant is convex to the origin (or slopes downwards from left to right as rectangular hyperbola).

 

Cobb-Douglas Production function

The production function developed by two economists Cobb and Douglas is popularly known as Cobb- Douglas production function. This production function reveals the total effect on the output with the employment of 3/4th proportion of labour and 1/4th proportion of capital. Theoretically, it reflects constant returns to scale or homogeneous production function. Mathematically, it is written as:

Q = A Lα Kβ

where,

Q = the quantity of output or product

L = the quantity of labor employed

K = the quantity of capital used

A = a positive constant

α and β = constants between 0 and 1

 

Properties of the Cobb-Douglas Production function

Basically, there are several properties of the Cobb-Douglas Production function. Some of them are briefly discussed below:

Average product: An average product is the outcome of the total product divided by the total units of the input employed. It refers to the output per unit of the input. The average productivities of inputs are:

  • Average productivity of labor (APL) = Q / L = (A Lα Kβ ) / L
  • Average productivity of capital (APK) = Q / K = (A Lα Kβ ) / K

Marginal Product: Marginal product is the additional made to the total product by employing one more unit of the input. In other words, it is the ratio of the change in the total product with the change in the units of the input. First derivative of the production function with respect to an input (i.e. capital or labor). The marginal productivities of inputs are:

  • Marginal productivity of labor (MPL) = ∆Q / ∆L = ∂Q / ∂L = ∂ (A Lα Kβ) / ∂L =α (APL)
  • Marginal productivity of capital (MPK) = ∆Q / ∆K = ∂Q / ∂K =∂ (A Lα Kβ) / ∂K =β (APK)

Marginal rate of substitution: It is the rate at which units are substituted for each other to maintain the same level of output. In other words, MRTS represents the amount of labour which the producer has to sacrifice for the gain of one additional unit of capital so that the level output remains the same.

Mathematically, MRTSLK = (∂Q / ∂K) / (∂Q / ∂L) = α (Q / L) / β (Q / K) =αK / βL

Output Elasticity: It is the proportionate or percentage change in output in respond to a change in levels of capital or labor. (∂Q / Q) / (∂ L/ L) = (∂Q / ∂L) / (Q / L). If output of elasticity is less than 1, the production function is inelastic and vice versa.

Returns to scale: It describes the rate of change in output due to the same proportionate or percentage change in input i.e. capital (K) and Labor (L). When all the inputs are changed in same proportion, we call this change in scale of production. The study of the change in output as a result of a change in scale of production is known as Returns to Scale. The returns to scale can be measured by taking the sum of α and β. Let, α + β = V.

  • If V > 1, there is an operation of increasing returns to scale.
  • If V = 1, there is an operation of constant returns to scale.
  • If V < 1, there is an operation of decreasing returns to scale.

Efficiency of production: The efficiency of production can be measured by the coefficient β.

  • If it is higher, there is a higher degree of efficiency of production.
  • If it is lower, there is a lower degree of efficiency of production.

 

Reference

Koutosoyianis, A (1979), Modern Microeconomics, London Macmillan

  • Marginal rate of technical substitution is the rate in which units of two inputs are substituted for each other to maintain the same level of output.
  •  In other words, MRTS of capital for labour represents the amount of labour, which the producer has to sacrifice for the gain of one additional unit of capital so that his level of output remains the same. In short,
  • MRTS is the ratio of the change in units of labour with the change in units of capital.
  • The MRTS reflects the slope of isoquant.
  • The production function developed by two economists Cobb and Douglas is popularly known as Cobb - Douglas production function.
  •  This production function reveals the total effect on the output with the employment of 3/4th proportion of labor and 1/4th proportion of capital.
  • Theoretically, it reflects constant returns to scale or homogeneous production function.
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