Amount by which the quantity of one input can be reduced when one extra unit of another input is used, so that output remains constant.
$$ MRTS = -\frac{\Delta K}{\Delta L} = -\frac{MP_1}{MP_2} $$
$$ \min w_1x_1 + w_2x_2 \hspace{0.5cm} \text{ s.t. } y \le f(x_1,x_2) $$
where,
$w_1$ is cost of input 1 and $x_1$is amount of input 1.
$$ x_1^* = x_1^(w_1,w_2,y) \\ x_2^ = x_2^*(w_1,w_2,y) $$
Graph showing all possible combinations of labor and capital that can be purchased for a given total cost.
$$ y = \alpha x_1 + \beta x_2 $$
Minimised Cost:
$$ \min (\frac{w_1}{\alpha}, \frac{w_2}{\beta}) . y
$$
$$ x_1^* = \begin{cases} \frac{w_1}{\alpha}.y && \frac{w_1}{\alpha} < \frac{w_2}{\beta} \\ = 0 && \frac{w_1}{\alpha} > \frac{w_2}{\beta} \\ = y - \beta x_2^* && \frac{w_1}{\alpha} = \frac{w_2}{\beta} \end{cases} $$
Cost Function: