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

Principle of Marginal Rate of Technical Substitution (MRTS)

The rate at which units of two inputs are substituted for one another to maintain the same level of output is known as the marginal rate of technical substitution. In other words, the MRTS of capital for labor shows the amount of labor that the producer must forgo in order to gain an additional unit of capital while maintaining the same level of production. The ratio of the change in labor units to the change in capital units is known as the MRTS. The slope of isoquant is reflected in the MRTS.

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:

• 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
• The ratio of the marginal productivities of two inputs is also reflected in the slope of the isoquant.
• The MRTS trend is declining. Isoquant has a negative slope because of the declining MRTS, and any points on this curve produce the same amount of output for the producer.

Cobb-Douglas Production function

Cobb-Dowlis production function is the name given to the function that two economists Cobb and Douglas created. With 3/4 of the labor force and 1/4 of the capital employed, this production function shows the overall impact on output. Theoretically, it represents a homogenous production function or constant returns to scale. In mathematics, it is expressed 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, the Cobb-Douglas Production function has a number of characteristics. Following is a brief discussion of a few of them:

Average product: The result of dividing the entire product by the total units of the input used is an average product. It speaks of the output for every unit of input. The average input productivities 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: The increase made to the overall product by using one more unit of the input is known as the marginal product. It is, in other words, the ratio of the change in the input units to the change in the total product. the production function's initial derivative with regard to an input (i.e. capital or labor). These are the inputs' marginal productivities:

• 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 switched out for one another to keep output at the same level. In other words, MRTS is the quantity of labor that the producer must forgo in order to gain an additional unit of capital while maintaining the same level of production. 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 speaks about the rate of change in output caused by an equal proportionate or percentage change in the input, which includes labor (L) and capital (K) (L). We refer to this as a shift in the scale of production when all inputs are altered in the same ratio. Returns to Scale is the study of how output changes as a result of production scale changes. 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.

• Better values indicate higher levels of production efficiency.
• A lower value indicates a lower level of production efficiency.

Reference

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