Answer to Question #9742 in Economics of Enterprise for monika

Question #9742

From this week's 7.1 lecture, I learned that when average product (AP) reaches its maximum when it equals to marginal product (MP). MP is the slope of the Total Product (TP) vs. Labor (L) (assume we are considering short-run with capital fixed). AP is the linear slope of the line from origin to the current point of the total product curve.
From my understanding this is a strong conclusion. So I can formulate a differential equation problem to see when AP reaches maximum from the relationship of the independent variable L and the dependent variable TP:
At the maximum point of AP:
MP = AP
=> dTP/dL = TP/L
=> dTP/TP = dL/L
=> integrating both sides we can have: ln(TP) - ln(TP0) = ln(L) - ln(L0), where TP0 and L0 are the initial total product and labor. TP and L are their current values.
=> ln(TP) = ln(L) - ln(TP0/L0), TP0/L0 is the initial AP noted as AP0
=> TP = e-AP0 * L
=> TP/L = e-AP0, again AP = TP/L, which is the current AP - maximum AP
=> APmax = e-AP0
So if the total product curve is continuous and derivable, we can claim that the average product reaches its maximum when its value is the natural exponential to the power of the negative beginning AP level. Each business always require some labors to do the work, though startups might have less. So basically we can get this AP0 easily. Now here comes the question:
Does this mean that we are able to predict the maximum AP according to the beginning configuration in a short-run? (Assume capital is fixed)

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