Relation between marginal cost and output elasticities Does anyone know of any results that show the link between marginal cost and the output elasticities analytically? I am looking at production and cost theory books but can't find any results that posit a direct dependence.
I figured it out:
The first-order condition of the cost minimization problem for, say, material inputs $m_{it}$ gives:
$ \lambda \frac{\partial F}{\partial M} = P_M $
Where F is the production function, $P_M$ the material input prices. Multiply by $\frac{M}{F}$ and rearrange,
$ \lambda = \frac{P_M M}{\beta_M F}  $,
where $\beta_M$ is the output elasticity with respect to material inputs. Similar results obtain for other inputs. Summing them all gives the relation to the short-run returns to scale.