Product Rule Fun

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I feel a little guilty for Zach remembering the product rule in class yesterday, then deciding to “simplify” the problem by taking natural logs, and thus not using his contribution to show the final result. So, because you all know how I worry that you’ll lose sleep over such things, Here are the steps we would take to get the result without taking logs. Enjoy, and have a great weekend.

(1) w_m = p_m MPL_m.
(2) dw_m = p_m dMPL_m+MPL_m dp_m .
Dividing by w_m we have:
(3) (dw_m/w_m) = (p_mdMPL_m/w_m) + (MPL_m dp_m/w_m).
Recalling from equation (1) that w_m = p_m MPL_m, we have:
(4) (dw_m/w_m) = (p_m dMPL_m)/(p_m MPL_m) + (MPL_m dp_m)/(p_m MPL_m).
Cancelling like terms in the numerators and denominators on the right-hand side, we have the final result (which is identical to the version in which we started off by taking logs):
(5) (dw_m/w_m) = (dMPL_m/MPL_m) + (dp_m/p_m).
If you’re bored this weekend, try this out on the example we did using log differences to show that the Cobb-Douglass production function has the property of constant returns to scale.

If you’ve read this far, you’ll probably enjoy knowing that while at VMI, since I wore a uniform, I also had to maintain haircut and shave standards that were also roughly in line with military uniform codes. Have a good weekend.