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Given a profit-maximizing firm with production function $f(x_1,x_2)$, I understand that we can formulate a firm's cost function $C(q)$ by using the contingent demand functions $x_1^c$ and $x_2^c$. We can then use this cost function to see whether a firm has decreasing costs, namely if $C(tq)<tC(q)$. If the second derivative of the cost function, C''(q), is less than zero, does that also imply that this firm faces decreasing costs and thus increasing returns to scale?

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Second derivative of cost function is actually the first derivative of marginal cost function. i.e. $$ \frac{\partial^2C(q)}{\partial q^2} =\frac{\partial}{\partial q}\frac{\partial C(q)}{\partial q}=\frac{\partial}{\partial q}MC(q) $$ Now if $\frac{\partial^2C(q)}{\partial q^2}<0$, this means that marginal cost is decreasing in output. If marginal cost is decreasing then that implies that firm's average cost is decreasing and hence it exhibits increasing returns to scale.

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