With \(g\) a function, \(e\) , a dependent variable, and \(\xi \) , the variable that spans \(e\) , the solutions to choice problems in either finance or economics can depend on the existence of some fixed point, \({e}_{t}^{*}=g\left({\xi }_{t}^{*}\right)={\xi }_{t}^{*}\) , whose ‘neighborhood’ is, simultaneously populated by some alternate non-fixed points, \({\ddot{e}}_{t}\) say, satisfying, \({\ddot{e}}_{t}=g\left({\ddot{\xi }}_{t}\right)\ne {\ddot{\xi }}_{t}\) . Hitherto, the rationality (economic cum financial) conditions that govern the two sets of possibilities had yet to be deciphered. Let \(e\) denote effort; \(\xi \) , an agent’s ‘effort capacity’; and \(U\) , utility. This study’s formal theory infers the rationality conditions that govern either \({e}_{t}^{*}=g\left({\xi }_{t}^{*}\right)={\xi }_{t}^{*}\) or \({\ddot{e}}_{t}=g\left({\ddot{\xi }}_{t}\right)\ne {\ddot{\xi }}_{t}\) as follows. Whereas \({e}_{t}^{*}={\xi }_{t}^{*}\) in the certainty equilibrium states, \({\Xi }^{*}\) , in the alternate states, \(\widehat{\Xi }\) , which are parameterized by choice under uncertainty, rational agents are restricted to be parameterized by, \({\ddot{e}}_{t}=g\left({\ddot{\xi }}_{t}\right)<{e}_{t}^{*}\) (abbreviated, \({\ddot{e}}_{t}<{e}_{t}^{*}\) ), the exclusion, as such, of \({\ddot{e}}_{t}>{e}_{t}^{*}\) . The existence of points, \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{e} _{t} \) satisfying each of, \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{e} _{t}<{\ddot{e}}_{t}<{e}_{t}^{*}\) and \(U\left({\ddot{e}}_{t}|\widehat{\Xi }\right)\ge U\left({e}_{t}^{*}|\widehat{\Xi }\right)>U\left(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{e} _{t}|\widehat{\Xi }\right)\) establishes the rationality and non-triviality of a search for the qualifying, \({\ddot{e}}_{t}.\) Importantly, if \(\xi \) is to increase over time, \(e\) is segmented into ‘capacity building (innovation) effort’, \(\widetilde{e}\) , and ‘output effort’, \(\widehat{e}\) . Further, either \({\widetilde{e}}_{t}\ngtr {\widetilde{e}}_{t-1}\) or \({\widehat{e}}_{t}\ngtr {\widehat{e}}_{t-1}\) is a necessary and sufficient condition for, \({\xi }_{t}\ngtr {\xi }_{t-1}\) . Applying the enumerated necessary and sufficiency conditions, with \(\Pi \) denoting agents’ welfare (the quality of life), \(\Upsilon\) , the distribution of wages, and ‘ ~ ’, ‘agents’ indifference’, a first-best progression to welfare is supported by, [ \(\widetilde{e}\sim \widehat{e}\) ]; [ \(\Upsilon(\widetilde{e})\sim \Upsilon(\widehat{e})\) ]; \(\Pi (\widetilde{e})\equiv \Pi (\widehat{e})\) ; and \(U(\widetilde{e})\equiv U(\widehat{e})\) ; that is, is bounded by a conferring of an equal importance on the activities of innovation and the activities of production; equivalently is more likely to be achieved if all agents are incentivized to ‘Learn Whilst Doing’. Applying the inference, either a neglect of, or an emphasis on manufacturing, respectively, non-manufacturing industries, is sub-optimal.
