Technical Rate of Substitution (TRS)

Technical rate of substitution between two factors denotes the rate of substitution between two factors so as to keep the level of output constant.

Derivation
1. Rewrite production function as an implicit function $$f(x_{i}, x_{j}(x_{i})) \equiv y \,$$ 2. Differentiate $$\partial f(x_{i}, x_{j}(x_{i}))/\partial x_{i} + \partial f(x_{i}, x_{j}(x_{i}))/\partial x_{j} \cdot \partial x_{j}(x_{i})/ \partial x_{i} =0 $$ 3. Re-arrange $$\partial x_{j}(x_{i})/ \partial x_{i} = - \frac{\partial f(x_{i}, x_{j}(x_{i}))/\partial x_{i}}{\partial f(x_{i}, x_{j}(x_{i}))/\partial x_{j} }$$