The lethal mistake of your analysis of optimal employment/output is that it is partial. This mistake is widespread and ancient, so let us call this the Marshall Fallacy.
Marshallians are methodologically committed to microfoundations, that is, to the perspective of the individual firm owner. This is the wrong perspective for dealing with the employment of the economy as a whole. So the first methodological thing to do is to let the micro-boss go and to switch to the perspective of the macro-boss. In other words, the economy consist of one firm, you are the boss and you are supposed to determine employment and output, such that the firm can exist in future periods.
Accordingly, your parameters are defined with this axiom set:
(0) The objectively given and most elementary systemic configuration of the economy consists of the household and the business sector which in turn consists initially of one giant fully integrated firm.
(i) Yw=WL wage income Yw is equal to wage rate W times working hours L,
(ii) O=RL output O is equal to productivity R times working hours L,
(iii) C=PX consumption expenditure C is equal to price P times quantity bought/sold X.
These premises define the pure consumption economy, they are certain, true, and primary, and therefore satisfy all methodological requirements. The graphical representation is given on Wikimedia.
At any given level of employment L, the wage income Yw that is generated in the consolidated business sector follows by multiplication with the wage rate W. On the real side, output O follows by multiplication with the productivity R. Finally, the price P follows as the dependent variable under the conditions of budget balancing, i.e. C=Yw and market clearing, i.e. X=O. Note that the ray in the southeastern quadrant is NOT a linear production function; the ray tracks ANY underlying production function. Note also that the wage rate W is an AVERAGE if the individual wage rates are different among the employees, which is normally the case.
Under the conditions of market clearing and budget balancing in each period the price is given by P=W/R (1), i.e. the market clearing price is always equal to unit wage costs. This is the MOST ELEMENTARY form of the price theorem also called LAW OF SUPPLY AND DEMAND.
If the wage rate W is lowered, the market clearing price P falls. If the number of working hours L is increased the price remains constant, provided productivity R does not change. If productivity decreases the price P rises. If productivity increases the price falls. In any case, labor gets the whole product, the real wage W/P is invariably equal to the productivity R according to (1), and profit for the business sector as a whole is zero. All changes in the system are reflected by the market clearing price. The elementary market economy is indefinitely reproducible under the condition of no external/physical limitations like space, raw materials etcetera, which ALL have to be introduced later in the course of an ever more detailed analysis.
This has been the first step. With the second step the conditions of market clearing and budget balancing have to be lifted. This GENERALIZATION produces the phenomena of inventory changes (O-X greater than 0 or less than 0) and of saving/dissaving (Sm≡Yw-C greater than 0 or less than 0) and of monetary profit/loss (Qm≡C-Yw greater than 0 or less than 0).
It always holds Qm+Sm=0 or Qm=-Sm, in other words, the business sector’s surplus (deficit) equals the household sector’s deficit (surplus). Profit is the counterpart of dissaving and loss is the counterpart of saving. This is the most elementary form of the PROFIT LAW. Profit for the economy as a WHOLE has NOTHING to do with productivity, the wage rate, the working hours, exploitation, competition, or the smartness of the macro-boss. Overall profit/loss is determined by the change of the household sector’s debt.
Given the axioms (0) to (iii) one has to proceed top-down by successive DIFFERENTIATION until one arrives at the INDIVIDUAL agent. Differentiation is the opposite of bottom-up or aggregation.
The first thing to notice is that the macro-boss can realize any level of employment L under the condition of market-clearing and budget balancing. The macro-boss is indifferent with regard to the employment level because monetary profit Qm is zero at all employment levels. The real wage is given with W/P=R (1) and to simplify matters here it is assumed that the productivity R is equal on all levels of employment.
Now, total employment is given as the sum over the number n of workers L= L1+ ... +Li+ ... +Ln. The individual labor time can be formally split into the norm time U, e.g. 8 hours per day times working days per period, and an individual factor l1, l2 etc. such that Li=Uli. A value of li=1 means that the i-th worker works full-time, li=0.5 means half-time, and li greater1 means overtime. This gives the relationship between total labor input L, the number of workers n and individual labor input li as shown on Wikimedia.
The macro-boss is indifferent between all employment levels L and (practical organizational problems put aside for the moment) as a matter of principle also indifferent between the possible combinations of number of workers n and individual labor time Li. So, to begin with, there is NO optimization problem only an organizational problem.
Conclusion: In the pure consumption economy with market clearing and budget balancing the real wage is determined by the actual production conditions. Given the productivity (= real wage), each worker chooses his individual input factor li which is equal or less than 1. The norm time U is defined by law according to average health standards. The macro-boss is indifferent between all levels of employment L and all combinations of n, U, li. As a result there is NO conflict between labor and the macro-boss with regard to the realization of full employment.
There are concepts which are both economic and measurable, e.g. wage rate, labor time, output, price, consumption expenditures, profit etc. It seems to be a rather straightforward idea for economists to show how their foundational concepts are logically interrelated. This is the very precondition of coherent talk about the economy. It is a fact, though, that economists have not managed to define their basic concepts consistently. In marked contrast, every physics student knows how the basic concepts of their subject matter, e.g. mass, force, energy, velocity, acceleration, fit logically together and relate to reality.
As a result, we have after 200+ years of brain-dead blather that neither Walrasians, Keynesians, Marxians, Austrians nor Pluralists have any idea what profit is and how the price- and profit mechanism works. Not only this, the know-nothings of economics pester the world with economic policy proposals.
You, for example, talk about immiseration and that ‘the farm is run for the benefit of farmers’. What do you think that this is? Economics?
It is pretty obvious that whatever this is, it is NOT economics. Hence the title of this blog is misleading. What is taking place here is pollution of the scientific atmosphere and severe damage of the intellectual/spiritual environment. The time has come to stop this.