I disagree with the first sentence. Since my disagreement hinges on the difference between partial and total derivatives I hope it is broadly interesting.
When Milton Friedman titled one of his books Free To Chose his underlying model was that happyness was a function both of the number of choices and the quality of the choices:
). His theory is that q is a dependent variable:
). When choices, c, are few, then producers offer consumers poor choices, on a take-it or leave-it basis. When choices are many, producers compete and consumers are offered good choices.frac{dq}{dc} is positive and large.frac{partialh}{partialq} is positive and large. What of frac{partialh}{partialc}? Presumably it is negative, all that comparison shopping is a chore, but in this analysis it is seen as small. Choice is good,meaning
I see the consumerist position, that choice is good, meaning frac{partialh}{partialc}>0, as a crude vulgarisation of the argument above.
Trying to apply this to a 30 year old American contemplating polyamory, my assumption is that he has experience of how the inner dynamics of the modern American monogamous romance play out. Unhappy experience. Now he is wondering about the dynamics implicit in polyamory. He wants to know whether changing the rules produces a better game, and he knows that he cannot find out via the simple equation: more choice = better. He must consider how the players respond to the changed incentives produced by the new rules.
I disagree with the first sentence. Since my disagreement hinges on the difference between partial and total derivatives I hope it is broadly interesting.
If q is a function of c, then h becomes a function of one independent variable, and your use of partials here doesn’t make sense, because you can’t hold c constant while changing q or vice versa.
You are making me feel old. My notation was orthodox in 1958. Indeed, in A Course Of Pure Mathematics, Tenth Edition, section 157, Hardy writes:
The distinction between the two functions is adequately shown by denoting the first by frac{df}{dx} and the second by frac{partialf}{partialx}, in which case the theorem takes the form frac{df}{dx}=frac{partialf}{partialx}frac{partialf}{partialy}frac{dy}{dx}; though this notation is also open to objection, in that it is a little misleading to denote the functions
\}) and
) whose forms as functions of x are quite different from one another, by the same letter f in frac{df}{dx} and frac{partialf}{partialx}.
I think your notation is still orthodox, or at least fairly common, nowadays. Wikipedia uses it on its total derivative page, for example, and it seems familiar to me.
If q is a function of c, then h becomes a function of one independent variable, and your use of partials here doesn’t make sense, because you can’t hold c constant while changing q or vice versa.
I thought that this was the kind of situation partial derivatives are there for. AlanCrowe’s just applied the multivariable chain rule, if I’m getting it right.
I disagree with the first sentence. Since my disagreement hinges on the difference between partial and total derivatives I hope it is broadly interesting.
When Milton Friedman titled one of his books Free To Chose his underlying model was that happyness was a function both of the number of choices and the quality of the choices:
I see the consumerist position, that choice is good, meaning frac{partialh}{partialc}>0, as a crude vulgarisation of the argument above.
Trying to apply this to a 30 year old American contemplating polyamory, my assumption is that he has experience of how the inner dynamics of the modern American monogamous romance play out. Unhappy experience. Now he is wondering about the dynamics implicit in polyamory. He wants to know whether changing the rules produces a better game, and he knows that he cannot find out via the simple equation: more choice = better. He must consider how the players respond to the changed incentives produced by the new rules.
If q is a function of c, then h becomes a function of one independent variable, and your use of partials here doesn’t make sense, because you can’t hold c constant while changing q or vice versa.
You are making me feel old. My notation was orthodox in 1958. Indeed, in A Course Of Pure Mathematics, Tenth Edition, section 157, Hardy writes:
The distinction between the two functions is adequately shown by denoting the first by frac{df}{dx} and the second by frac{partialf}{partialx}, in which case the theorem takes the form frac{df}{dx}=frac{partialf}{partialx} frac{partialf}{partialy}frac{dy}{dx}; though this notation is also open to objection, in that it is a little misleading to denote the functions
I think your notation is still orthodox, or at least fairly common, nowadays. Wikipedia uses it on its total derivative page, for example, and it seems familiar to me.
I thought that this was the kind of situation partial derivatives are there for. AlanCrowe’s just applied the multivariable chain rule, if I’m getting it right.
Thanks, you (and Alan) are right. Sorry, it’s been a while.