http://andrewgelman.com/2016/07/14/about-that-claim-that-police-are-less-likely-to-shoot-blacks-than-whites/

My thinking is that in all this discussion of police violence, different academic papers (and news headlines) charge different accusations of police bias. As the author said in response to you, these two papers by Ross and Fryer come with opposite But when I crunch the numbers in my own rudimentary (but accurate) way, I get the same results: Saying a “20% more likely to be shot or use of force” with all controls in place still means a difference of 1 time more or less out of 100 instances “on the ground.” I think I’m onto something that hasn’t been reported elsewhere, but I honestly don’t have the statistical aptitude to say with authority, “this is right.”

If interested in reading more and getting in contact (organic.design(at)gmail.com) let me know. I’m interested in vetting a couple ideas and publishing:

https://medium.com/@agent.orange.chicago/how-roland-fryers-controversial-study-on-racial-bias-by-police-actually-shows-negligible-bias-ea3a8b1fd293#.45pzqqvkv

You seem to have taken Keen at his word that there is a “traditional approach” which assumes “firms do nothing” in response to the actions of other firms and that approach is to be contrasted with a new approach invented by Steve Keen which is more realistic. That is simply not true, as you can easily verify for yourself.

Keen’s solution is a very confused exposition of a special case of a very traditional, in fact, archaic, approach (again, look up “conjectural variations”). And much like in other cases in which he claims to have invented something that is actually well-known, Keen proceeds to try to solve his new invention himself and doesn’t get the right answer.

With regard to your math: Of course I do not think and in no place did I imply that “a derivative implies a relationship where the variable in the denominator is independent and the variable in the numerator is dependent.” Taking derivatives with respect to Q is an ill-defined operation because “what happens to the profits at firm_i if it and/or any other firm changes its output” depends on the distribution if that change in output, not just its magnitude (it’s not a “sufficiently well-behaved variable”). Suppose profits_1 = [ q_1^2 – ln(q_2) ] and Q=(q_1 + q_2). What is d(profits_1)/dQ?

Differentiating with respect to Q does not “describe what happens to the profits at firm_i if it and/or any other firm changes its output.” If firm i changes its own output (and that’s what we should be modeling, as own-output is the firm’s choice variable) we want to evaluate d(profits_i)/dq_i. If we seek a Nash equilibrium, we set dq_j/dq_i = 0 forall j ~i when evaluating that expression, and then solve the resulting system of equations for an equilibrium. That is not the same as assuming that firms believe their competitors won’t make strategic responses (look up the concept of a “reaction function”). If we wanted to “describe what happens to the profits of firm_i if it and/or any other firm changes its output” we would write an expression for d(profits_i) letting the entire vector of outputs change. That is not the same as the derivative of firm i’s profits with respect to Q. The standard Cournot (1838, yes, that’s the year of publication) model, which is what this is despite Keen’s claims to have invented it, generates a position of the *system* which is an equilibrium in a formally-defined sense.

Finally, as I said above, none of this is new. You can find countless expositions of the conjectural variations model Keen claims to have invented in textbooks written 50 to 70 years ago, and many mathematically rigorous expositions of standard game theoretic representations of the modern model in any modern IO textbook (along with much more sophisticated dynamic and stochastic models, and with extensive empirical evidence). For example, consider this textbook’s exposition of these basic models, along with some pointers to some modern extensions and empirics:

http://www.econ.yale.edu/~steveb/Econ600/chapter2.pdf

Generally, I encourage you to go look this stuff up in credible sources rather than trying to learn economics from Steve Keen.

]]>Also, if you’re wondering why Keen takes the derivative of profits with respect to Q rather than q_i, it’s because it stands as a marker for the entire system. In other words, it describes what happens to the profits at firm_i if it and/or any other firm changes its output, whereas the expression usually used here dpi/dq_i (which should really be a partial derivative) only describes what happens to profits at firm_i if only its own output changes and other outputs is held constant, an unrealistic situation. A firm is doing well if it’s maximizing profits with respect to changes in its own output, but it’s doing even better if it’s maximizing profits with respect to changes in the outputs of all the firms. Notice that this doesn’t have to imply that firms are either colluding or reacting to each other, just that they’re allowed to change their outputs.

While the traditional approach is to examine what one business should do if all other businesses do nothing, Keen’s approach is to examine what one business should do if it has a reasonable expectation that other businesses are doing something. My instinct, without having run the math, is that for markets where outputs change infrequently, we would expect to see results closer to the traditional approach, while for markets with frequent output changes, it should be closer to Keen’s equation. In the former, the assumption that other businesses are leaving their outputs constant is reasonably good. As a non-economist, I couldn’t answer the question as to whether such things actually exist.

Also, your language in this post seems to imply that you think a derivative implies a relationship where the variable in the denominator is independent and the variable in the numerator is dependent, but that’s really not how derivatives work at all. And the part where you say that the derivative of dpi/dQ is undefined is very much incorrect: you can take derivatives with respect to any sufficiently well-behaved variable, either partial derivatives or total derivatives. The answer may be a function, including of variables besides just the two mentioned, or it may be zero if there is no relationship between the two. There certainly is a relationship between Q and pi_i, although it’s not obvious at first glance what it might be until Keen walks us through it.

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