Proof mathematically that MRTS is rario of MP the two factors

Answer 1

The Marginal Rate of Technical Substitution (MRTS) refers to the rate at which one input (say, capital \(K\)) can be substituted for another input (say, labor \(L\)) while keeping the output constant. Mathematically, the MRTS between two factors is defined as:

\[ \text{MRTS}_{L,K} = -\frac{dK}{dL} \]

This expression represents the slope of an isoquant — a curve that represents all combinations of \(K\) and \(L\) that produce the same level of output.

Now, we need to show that the MRTS is equal to the ratio of the Marginal Products (MP) of the two factors. The Marginal Product of Labor (MPL) is the additional output produced by using one more unit of labor, holding capital constant. Similarly, the Marginal Product of Capital (MPK) is the additional output produced by using one more unit of capital, holding labor constant.

Formally, MPL and MPK are given by the partial derivatives of the production function \(Q = f(L, K)\):

\[ \text{MPL} = \frac{\partial Q}{\partial L} \]
\[ \text{MPK} = \frac{\partial Q}{\partial K} \]

### Derivation

To derive this relationship, we'll start with the condition of keeping output constant. Suppose the production function \(Q = f(L, K)\) remains constant at a level \(Q_0\):

\[ f(L, K) = Q_0 \]

Totally differentiating this equation with respect to \(L\) and \(K\) while keeping \(Q\) constant gives:

\[ \frac{\partial f}{\partial L} dL + \frac{\partial f}{\partial K} dK = 0 \]

This can be written using the marginal products as:

\[ \text{MPL} \cdot dL + \text{MPK} \cdot dK = 0 \]

Rearranging to isolate \(\frac{dK}{dL}\):

\[ \text{MPL} \cdot dL = -\text{MPK} \cdot dK \]
\[ \frac{dK}{dL} = -\frac{\text{MPL}}{\text{MPK}} \]

Thus, the MRTS can be expressed as:

\[ \text{MRTS}_{L,K} = -\frac{dK}{dL} = \frac{\text{MPL}}{\text{MPK}} \]

### Conclusion

We have shown mathematically that the Marginal Rate of Technical Substitution (MRTS) is indeed the ratio of the Marginal Products of the two factors:

\[ \text{MRTS}_{L,K} = \frac{\text{MPL}}{\text{MPK}} \]

This relationship is fundamental in understanding how changes in the employment of one input can be compensated by changes in the employment of another, without changing the level of output.