Labor productivity can be a useful calculation; however, it is helpful to consider its formulation.  Most think of Average Product (APL) as one formulation; it is an output/input ratio.  It is often represented as output per hour worked when we calculate the output of a group of workers.  

One drawback is taking the capital stock (and other variables) as given, which means the impact of capital equipment is not separated out in APL.

The Marginal Product of Labor (MPL) is another useful calculation.  It is the derivative of output with respect to the labor input, holding the other variables constant.  This rate of change is useful in some applications; however, it also has limitations.  It would be challenging to calculate the MPL  in your example of one head.  

If we could calculate the MPl of the executive, it might increase if the output of the organization if the workers are energized by the executive to produce more.  It is worth noting that the workers are the ones doing the work and if they are faster at doing the same work than before because of this executive, then MPL increases.  It is a rather large leap to suggest that the executive alone did all of this.  After all, the workers who actually performed the work (faster) are partially responsible for the outcome in this example.

Let's go back to your question in the title. Perhaps it is better to suggest that productivity may be one element in evaluating effectiveness.  

Salary is another issue; typically depends on several factors.  This might include some measure of productivity and/or the demand an supply of labor, among others  

I hope this helps and remember, this is just one example.

Rob

A great question - maybe we can conceptualise productivity as the production of wellbeing. Then we need to ask the question of wellbeing of who or what?

This is an intimately moral question.

https://www.ons.gov.uk/peoplepopulationandcommunity/wellbeing/articles/measuringnationalwellbeing/sept2016

Dear colleagues. Thank you for Your answers. Apparently the use of a single parameter is not adequate assessment of effectiveness.

And it is important not to forget about the paradoxes in statistics, mathematics, Economics, such as Productivity paradox, and others...:)

https://en.m.wikipedia.org/wiki/Productivity_paradox

I agree Mikhail.

The OECD produced a guide 16 years ago - it is interesting now to see how they think about productivity (see link), and some of the limitations that Mikhail reminds us of.

I also like this other article, which is a much more humanistic approach:

A humanistic historian viewing quantitative social research sees much in it that is admirable and indispensable as an aid to the understanding of society and men. As a guide to the significance of problems, however, it is misleading when it exudes the assumption that only problems susceptible to quantitative solutions are important. As a means of explaining human or social behavior, it is powerful but profoundly incomplete. As a source of a theory of human nature and of the universe, it is but a new formulation of an ancient romantic myth.

https://www.oecd.org/std/productivity-stats/2352458.pdf

http://web.uconn.edu/weakliem/3841/schlesinger.pdf

In term of innovation, the answer is NO!

Can be effective as increasing the gross domestic product is the sum of two terms: productivity growth and increase the total amount of working time. See

http://ru.pressfrom.com/finance/finance-news/-59078-kak-tramp-dobetsya-trehprocentnogo-rosta-ekonomiki-ssha/