Minimum Average Cost
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The average cost of producing output q is of the form
$${\rm{\bar C}}\left( {\rm{q}} \right) = {{\rm{F}} \over {\rm{q}}} + {\rm{c}} \cdot {\rm{q}}^{{\rm{\eta - 1}}}.$$
(1)
Since
$${\rm{\eta = }}\left\{ {\matrix{ {{1 \over {\left( {1 - {\rm{\beta }}} \right)}}} \hfill & {{\rm{in}}\,{\rm{the}}\,{\rm{short}}\,{\rm{run}}} \hfill \cr {{1 \over {\left( {1 - {\rm{\beta }}^{\rm{R}} } \right)}}} \hfill & {{\rm{in}}\,{\rm{the}}\,{\rm{medium}}\,{\rm{run}}} \hfill \cr } } \right.$$
and η > 1 in any case, the average cost curve C(q) must be U-shaped.