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Suppose we have a production system where there is a known production function $f(x_1,x_2,\dots,x_n)$ that gives the amount of the commodity produced as a function of the amounts $x_i$ of the inputs, $i=1,2,\dots,n$. the unit price of the produced commodity is $q$ and the unit prices of the input are $p_1, p_2, \dots, p_n$. The producer wishing to maximize profit must solve the problem :

maximize $qf(x_1,x_2,\dots,x_n)-p_1x_1- p_2x_2-\dots p_nx_n$ Now if I apply the first order necessary condition to this function then the equation we get can be interpreted as "at the solution the marginal value due to a small increase in the $i$-th input must be equal to the price $p_i$". I am not understanding the intuitive meaning behind this.

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I presume they meant to write "at the solution the marginal increase in produced value due to a small increase in the i-th input must be equal to the price $p_i$".

Let $\phi(x) = qf(x_1,x_2,\dots,x_n)-p_1x_1- p_2x_2-\dots p_nx_n$. At a solution, you will have $\frac{\partial \phi(x)}{\partial x} = 0$, or, in component form, $\frac{\partial \phi(x)}{\partial x_k} = 0$.

Since $\frac{\partial \phi(x)}{\partial x_k} = q\frac{\partial f(x)}{\partial x_k} -p_k = 0$, you have $q\frac{\partial f(x)}{\partial x_k} = p_k$.

For small $\delta$, $q\frac{\partial f(x)}{\partial x_k} \delta$ is the marginal increase in produced value (due to a $\delta$ change in the $k$th input), and $p_k \delta$ is the increase in input cost.

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