# Marginal rate of technical substitution for multi-dimension inputs? [closed]

Suppose I have a vector x = ($x_1,x_2,x_3,x_4,x_5$) of inputs. How it is the MRTS($x_1,x_2,x_3,x_4,x_5$) ?

And which is the relation with the marginal products of inputs 1-5 ?

• Could you please elaborate slightly on what you have tried? – Jamzy Jun 30 '15 at 23:31
• MRTS is defined as holding output fixed while reducing input $i$ while increasing output $j$ so it is always a pairwise description of manipulating two inputs. But for all production functions this is not always possible. Can we assume that the production function is a smooth, differentiable function of degree 2 in all arguments? Can we assume that it is monotonically increasing in all arguments? Are you just looking for the definition of the MRTS in a multi-input setting? It is the same as in the two input case, the MRTS(i,j) = MarginalProduct(i)/MarginalProduct(j). – BKay Jul 6 '15 at 12:41

$$MRTS_{x_{i};x_{j}}=\frac{\partial f(X)}{\partial x_{i}}/\frac{\partial f(X)}{\partial x_{j}} \qquad \forall i\neq j$$