Marginal rate of technical substitution

The marginal rate of technical substitution of L for K (MRTS_LK) is defined as the additional amount of capital needed to replace a unit of labor, holding output constant. Mathematically, the MRTS_LK can also be expressed as:

Let's examine this expression carefully. The vertical bar at the end of this expression is a mathematical term that indicates that the expression to the left is evaluated given the condition stated at the bottom right of this line. This condition requires that Q = Q-bar (that's how this is pronounced). What this means, more intuitively, is that the expression to the left is evaluated only for points on a given isoquant (one corresponding to an output level of Q-bar). The term in parentheses is nothing more than the negative of the slope of a line connecting two points on the isoquant. Taking the limit of this expression as the change in L tends to zero, however, results in this being the negative of the slope of a tangent line to the isoquant.

The diagram below illustrates how the negative of the slope of the tangent line serves as a measure of the MRTS.