<p>\begin{equation}
MRTS_{x_{i};x_{j}}=\frac{\partial f(X)}{\partial x_{i}}/\frac{\partial f(X)}{\partial x_{j}} \qquad \forall i\neq j
\end{equation}</p>

CC BY-SA 3.0
2015-07-06T11:50:19.203Z
2015-07-06T11:50:19.203Z
Héctor Garrido
<p>Suppose I have a vector <em><strong>x</strong></em> = ($x_1,x_2,x_3,x_4,x_5$) of inputs. How it is the MRTS($x_1,x_2,x_3,x_4,x_5$) ?</p>

<p>And which is the relation with the marginal products of inputs 1-5 ?</p>

CC BY-SA 3.0
2015-06-30T23:31:18.510Z
Could you please elaborate slightly on what you have tried?
Jamzy
CC BY-SA 3.0
2015-07-06T12:41:44.673Z
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
<microeconomics>
Marginal rate of technical substitution for multi-dimension inputs?
sparkle
# Marginal rate of technical substitution for multi-dimension inputs?

(Asked by: username_0 on 2015-06-30)

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 ?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Could you please elaborate slightly on what you have tried?  
> **username_2**: 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).  

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## 1 answers

### Answer by username_3 on 2015-07-06. Score: 1 | Accepted answer

\begin{equation}
MRTS\_{x\_{i};x\_{j}}=\frac{\partial f(X)}{\partial x\_{i}}/\frac{\partial f(X)}{\partial x\_{j}} \qquad \forall i\neq j
\end{equation}


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