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We define marginal product of labor, MPL as the derivative of f with respect to the L - which is, as (approximately) how much Y will increase when L increases by one unit. We also describe the marginal product of capital, MPK as the derivative of f with respect to K. Note that MPL and MPK will rely on both L and K (MPL and MPK are functions, not variables).
MPL = df/dL, MPK= df/dK
When we view Y as a function of L holding K fix, Y will be increasing in L however at a decreasing pace (Because of the fact that MPL is positive though decreasing in L).
We state labour productivity as Y/L that is, as GDP per hour worked. Labour productivity tells us how much we can produce by employing one hour of labour and it relies on the amount of capital and the technology.
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