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What links can be made between the marginal productivities of the factors of production and the costs of production

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In a perfectly competitive markets for factors of production the marginal productivity of factors is equal to price of factor, which in turn decides the cost of production. For example, consider two factor production function $Y=F(K,L)$. At equilibrium, under assumption of perfect competition in market for both factors, $F_K = r$ and $F_L = w$, where $F_K, F_L$ are marginal products with of $K$ and $L$ respectively, and $r, w$ are respective prices of these factors.

Cost of production is $$C(Y)=rK+wL=F_K\cdot K+F_L\cdot L$$

This is the simplest link between marginal productivities and cost of production. In a partial equilibrium setting, with $r, w$ given, this can be further simplified to eliminate $K$ and $L$ (see equation (6) in this answer)

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Assume the following profit function: $\pi = pY - wL -rK$ and a two factor economy $Y(L,K)$. Optimal labour and capital inputs for the firm can be found by taking the respective derivatives and setting them to zero:

$\frac{\delta \pi}{\delta L} = p \frac{\delta Y}{\delta L} -w = 0 \iff \; \frac{\delta Y}{\delta L} = \frac{w}{p} $

$\frac{\delta \pi}{\delta K} = p \frac{\delta Y}{\delta K} -r = 0 \iff \; \frac{\delta Y}{\delta K} = \frac{r}{p} $

Now you can see that marginal productivities are equal marginal costs. The derivative with respect to the input factor tells you what the next unit of capital or labour will produce and the right hand side tells you what they will cost (real wage or real interest rate). If this condition does not hold firms could be better off by adjusting their production towards this condition. For example assume marginal product of labour is greater than the real wage $(\delta Y / \delta L > w / p)$. The firm should hire an additional worker since she will produce more than she costs. This logic holds true until the conditions are fulfilled.

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