Introduction: In economics the property of the slope of a tangent line to a level curve is used in different settings. In consumer behavior this property is known as the marginal rate of substitution and it is denoted by $MRS(x,y)$, while in producer behavior it is known as the marginal rate of technical substitution and is denoted by $MRTS(L,K)$.
Definition: For a utility function $U(x,y)$ of a consumer the quotient $\dfrac{U_x'(x,y)}{U_y'(x,y)}$ at the point $(x,y)$ is called the marginal rate of substitution and is denoted by $MRS(x,y)$:
\[
MRS(x,y)=\frac{U_x'(x,y)}{U_y'(x,y)}.
\]
For a production function $F(L,K)$ of a producer the quotient $\dfrac{F_L'(L,K)}{F_K'(L,K)}$ is called the marginal rate of technical substitution and is denoted by $MRTS(L,K)$:
\[
MRTS(L,K)=\frac{F_L'(L,K)}{F_K'(L,K)}
\]
or, in terms of the marginal physical product of the input factors,
\[
MRTS(L,K)=\frac{MPP_L(L,K)}{MPP_K(L,K)}.
\]
Definition: For a utility function $U(x,y)$ of a consumer the quotient $\dfrac{U_x'(x,y)}{U_y'(x,y)}$ at the point $(x,y)$ is called the marginal rate of substitution and is denoted by $MRS(x,y)$:
\[
MRS(x,y)=\frac{U_x'(x,y)}{U_y'(x,y)}.
\]
For a production function $F(L,K)$ of a producer the quotient $\dfrac{F_L'(L,K)}{F_K'(L,K)}$ is called the marginal rate of technical substitution and is denoted by $MRTS(L,K)$:
\[
MRTS(L,K)=\frac{F_L'(L,K)}{F_K'(L,K)}
\]
or, in terms of the marginal physical product of the input factors,
\[
MRTS(L,K)=\frac{MPP_L(L,K)}{MPP_K(L,K)}.
\]