by Alin Voicu
The “Marginal product of labor” entry in Wikipedia (en.wikipedia.org/wiki/Marginal_product_of_labor) contains an error. Under the “Definitions” paragraph, it says:
The marginal product of labor is the change in output per unit change in labor. In discrete terms the marginal product of labor is ∆Q/∆L. In marginal terms the MPL is the first derivative of the total product curve or short run production function or ∂Q/∂L. The precise mathematical formula is MPL = (dAPL/dL)q + APL. Graphically the MPL is the slope of the total product curve/short run production function.
The “precise mathematical formula” is precisely wrong. It is not clear whether the formula comes from the work referred to in footnote , namely, Perloff, J: Microeconomics Theory & Applications with Calculus, page 173. Pearson 2008. In any case, it should be (with TP denoting Total Product, a.k.a., q, and APL, average product of labor):
MPL = lim L→L0 (TP(L) – TP(L0)) / (L- L0) =
= lim L→L0 ( (TP(L)/L – TP(L0)/L) / L(1 – L0/L) ) =
= lim L→L0 ( ((TP(L)/L – TP(L0)/ L0) + TP(L0)/ L0 – TP(L0)/L) / L(1- L0/L) ) =
= lim L→L0 ( ((TP(L)/L – TP(L0)/ L0) / L(1- L0/L) + (TP(L0)/ L0 – TP(L0)/L) / L(1- L0/L) ) =
= lim L→L0 ( (((TP(L)/L – TP(L0)/ L0)/(L- L0)) L ) + lim L→L0 ( TP(L0) (L – L0)/LL0/(L- L0)/L ) =
= (dAPL/dL) L|L=L0 + APL(L0)
Therefore it looks like MPL = (dAPL/dL) L + APL, i.e., L is the factor multiplying the derivative of the average product in the first term, not q.
The relationship is mathematically unremarkable and it is true for any “well-behaved” function differentiable in L0; it is, however, of good import to economics.