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.^{[9]} The precise mathematical formula is MP_{L} = (dAP_{L}/dL)**q** + AP_{L}. Graphically the MP_{L} 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 [9], 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 AP_{L}, average product of labor):

MP_{L} = lim _{L→L0} (TP(L) – TP(L_{0})) / (L- L_{0}) =

= lim _{L}_{→L0} ( (TP(L)/L – TP(L_{0})/L) / L(1 – L_{0}/L) ) =

= lim _{L}_{→L0} ( ((TP(L)/L – TP(L_{0})/ L_{0}) + TP(L_{0})/ L_{0} – TP(L_{0})/L) / L(1- L_{0}/L) ) =

= lim _{L}_{→L0} ( ((TP(L)/L – TP(L_{0})/ L_{0}) / L(1- L_{0}/L) + (TP(L_{0})/ L_{0} – TP(L_{0})/L) / L(1- L_{0}/L) ) =

= lim _{L}_{→L0} ( (((TP(L)/L – TP(L_{0})/ L_{0})/(L- L_{0})) L ) + lim _{L}_{→L0} ( TP(L_{0}) (L – L_{0})/LL_{0}/(L- L_{0})/L ) =

= (dAP_{L}/dL) L|_{L=L0} + AP_{L}(L_{0})

Therefore it looks like *MP*_{L} = (dAP_{L}/dL) **L** + AP_{L}, 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 L_{0}; it is, however, of good import to economics.

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