Scale

Abstract

One possible measure of efficiency for an organization is the number of administrators per operative or the total number of employees per operative which is the previous number plus one. The first measure was defined as a_R. For a constant span of control we have

$$\matrix{{{\rm{a}}_{\rm{R}} } \hfill & { = {{1 + {\rm{s}}_{\rm{R}} {{{\rm{s}}^{\rm{R}} - 1} \over {{\rm{s - 1}}}} - {\rm{s}}_{\rm{R}} {\rm{s}}^{{\rm{R - 1}}} } \over {{\rm{s}}_{\rm{R}} {\rm{s}}^{{\rm{R - 1}}} }}} \hfill \cr {} \hfill & { = {{1 + {{{\rm{s}}_{\rm{R}} } \over {{\rm{s - 1}}}} \cdot \left( {{\rm{s}}^{{\rm{R - 1}}} - 1} \right)} \over {{\rm{s}}_{\rm{R}} {\rm{s}}^{{\rm{R - 1}}} }}} \hfill \cr {} \hfill & { = {1 \over {{\rm{s - 1}}}} + {\rm{s}}^{1 - {\rm{R}}} \cdot \left[ {{1 \over {{\rm{s}}_{\rm{R}} }} - {1 \over {{\rm{s - 1}}}}} \right].} \hfill \cr }$$

Thus

$${\rm{a}}_{\rm{R}} \left\{ {\matrix{ { \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle=}\vphantom{_x}}$}} {1 \over {{\rm{s - 1}}}}\,{\rm{if}}\,{\rm{s}}_{\rm{R}} < {\rm{s}}} \hfill \cr { = {1 \over {{\rm{s - 1}}}}\,{\rm{if}}\,{\rm{s}}_{\rm{R}} = {\rm{s - 1}}{\rm{.}}} \hfill \cr } } \right.$$

(1)

Proposition (1) contains two statements: The ratio of administrators to operatives takes on its minimum value when the president’s span of control assumes its admissible maximum. More importantly, the minimum ratio ${1 \over {{\rm{s - 1}}}}$ depends only on the span of control and is independent of the scale R of the organization. A constant span of control is consistent with constant returns to scale in administration provided the president’s span of control is large. Here we have defined returns to scale in terms of the ratio of administrators to operatives.