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The partial means we hold everything else constant, but I am unsure of what we mean by everything else (does it mean all other arguments -- in which case we would hold $w+y$ constant -- or does it mean all other endogenous variables are held constant -- in which case $w+y$ changes but $w$ doesn't. or maybe it means something else)?

Edit 2: (additional question) If $\frac{\partial f}{\partial y}$ refers to the partial derivative holding the variable $y$ constant then does $\frac{\partial f}{\partial y} = f_1 + f_2$ , where $f_i$ denotes the partial derivative w.r.t to the $i$'th argument?

  • My logic behind this being that changing $y$ by an small amount changes both the first and second arguments by a small amount.

Edit: i realize that an answer is probably just "This is bad notation, avoid it". I've come across this kind of notation sometimes and it confuses me though. Maybe the answer is that context needs to be used to tell if they mean differentiation w.r.t an argument or w.r.t a variable.

user106860
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1 Answers1

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When you take the partial of a function of several variables, the variables might be independent of each other, or might be related in some other way.

When all the arguments of a function are independent variables, then the partial of the function with respect to one of them is just the derivative of the function with regard to that variable, holding all the others constant. But this only applies when all the variables are independent.

On the other hand, when the arguments have a relationship between them, as in your case here, one applies the chain rule. That is, for a function $f(w+y,y),$ we have that $$f_y=f_{w+y}(w+y)'+f_y(y)'=f_{w+y}+f_y,$$ where the primes denote differentiation with respect to $y.$

Allawonder
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  • but isn't $f_{w+y}(w+y)' + f_y(y)'$ the total derivative w.r.t to y (and not the partial derivative)? – user106860 Sep 10 '19 at 16:07
  • @user106860 As indicated, this is the partial of $f$ with respect to $y.$ The function has the two independent variables $w,y,$ and we've taken the partial with respect to the second. As for total derivative, I don't know what you mean. I know about total differentials, I know about gradients (or derivatives, as some call them), but I don't know what's meant by a total derivative. Care to elucidate? – Allawonder Sep 11 '19 at 06:25
  • https://en.wikipedia.org/wiki/Total_derivative – user106860 Sep 13 '19 at 15:18
  • @user106860 Oh, I see. Another way to understand the total differential. No, I have not calculated a total derivative at all since I've not even considered the derivatives with respect to the other independent variable, namely $w.$ That's what I was talking about -- considering whether your variables are interrelated or independent is important. – Allawonder Sep 13 '19 at 15:47