Consider a simple production function with only one factor of production L, where L is Labor and Q is the total output.

Q=18L\(^{2}\) -L\(^{3}\)

If L =0, Q= 0. If L=1, Q=17. If L=2, Q=64. If L=3, Q= 135 and so forth....

The **Marginal Product of Labor, MPL**, is defined as the change in output that results from employing an added unit of labor. If that is the case, we can easily calculate the MPL given the production function Q.

When L goes from 0 to 1, Q went from 0 to 17. When L goes from 1 to 2, Q went up by 64 as shown previously. Therefore, the MPL when Labor=2 is (64-17/1-0)= 47. You can also do the same when you add the third labor, or when L=3. The corresponding MPL is (135-64)/(3-2)= 71.

We can also compute MPL as the first derivative of the Production Function Q where Q is Q=18L\(^{2}\) -L\(^{3}\) Therefore, dQ/dL= 36L - 3L\(^{2}\)

So, when L = 1, plug this into dQ/dL and we get 33. When L=2, dQ/dL= 36(2)-3(2)\(^{2}\)= 60.

See associated production function table here.

Question: Why do the results differ ( 17 against 33 and 47 against 60 and so forth) when we calculated MPL manually versus using the first derive function of Q?

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