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.