Let me recap. In the first part, I laid out a model of the economy. In the second part, I derived marginal productivity conditions and noted equations for equilibrium prices. In the third part, I outlined how to construct the factor price frontier, and asserted that that analysis is equivalent to marginal productivity analysis when they both are applicable.
Marginal productivity is not a theory of income distribution. I consider it an analysis, like the construction of the factor price frontier, of the choice of technique.
Consider two nested models of production. In the first, the coefficients of production are fixed. Equilibrium conditions in the production model would consist of the system of equations arising from the condition that no pure economic profit exists. These equilibrium conditions and the choice of the numeraire determine, with one degree of freedom, the wage, the interest rate, and prices. No marginal productivity conditions exist, for interior solutions, in this model.
In the more general model, one allows coefficient of production to vary, instead of taking them as given parameters. Marginal productivity equilibrium conditions arise for each coefficient of production. The model still has a single degree of freedom.
If one took the wage as given from outside, the model shows how firms choose processes in which the present value of the marginal product of labor is equal to the wage. The wage is not determined by the value of the marginal product of labor. Rather, it is more appropriate to say the value of the marginal product of labor is determined by the wage.
Samuelson, however, when calculating the marginal product of labor, noted that the interest rate could be taken as given instead. The factor price frontier shows possible distributions of income, given the possible coefficients of production. But marginal productivity cannot determine a specific location on the frontier. By taking an important distributive variable, the interest rate, as given, an economist, in theory, can determine the location on that frontier and the value of the marginal product of labor. (Notice that in case of reswitching, one cannot even map from the chosen technique, out of the set of all possible techniques, to the wage and the value of the marginal product of labor.)
Interestingly enough, Walras understood that marginal productivity is not a theory of income distribution:
"[The theory of marginal productivity] introduces into the problem of production a system of equations...in which the number of equations is equal to the number of coefficients of production and in which these coefficients are represented as unknowns...[The theory] makes possible a definitive criticism and refutation of the English theory of rent, by showing that the consideration of marginal productivity is relevant to the determination of coefficients of production, but is not relevant to the determination of the price of services." (Leon Walras, Elements of Pure Economics, Appendix III, (Translated by William Jaffe), 1954.)
Occasionally, you may see somebody saying that changes in technology or relative physical productivity are what drives changes in income distribution, at least in theory:
"To a good neoclassical economist, the statement that the relative price of a factor of production--like the labor of the elite top 1% of America's wage and salary distribution--has risen is the same thing as the statement that the relative productivity of that factor of production has risen." -- Brad DeLongDoes DeLong get neoclassical economics correct?
For now, that's all.