Given a function
$$C(x(t),y(t))=x*y$$
and discrete data for variables $x(t),y(t)$ at the points $t_0,t_1$, what is the contribution (total or percentage) of the change in variable $x$ to the change $\Delta C$
$$\Delta C = C(x(t_0),y(t_0))-C(x(t_1),y(t_1))$$
in the function $C$?
Background: Going through a paper recently I got stuck on the approach that the authors were using. I had not come across this before, so maybe there is an elegant way to explain this.
In a 2018 paper on solar photovoltaics (P.9 Main Body, P.1 in Supplementary Material), the authors have a cost function $C$ which describes the cost associated with manufacturing one unit. It depends on manufacturing variables $x,y$, which change over time (e.g. price of silicon, price of chemicals, etc.)
$$ C(x(t),y(t)) $$
They want to determine the contribution of a single variable $x$ to the total change of the cost function between two points in time $\Delta C (t_0, t_1)$. Variables are known only at discrete points in time ($t_0,t_1$).
They start by writing out the differential of the cost function $C$ as
$$ dC (x(t), y(t)) = \frac{ \partial C }{ \partial x } \frac{ \text{d} x }{ \text{d} t} \text{d} t + \frac{ \partial C }{ \partial y } \frac{ \text{d} y }{ \text{d} t} \text{d} t $$
where the contribution of the change in variable x over time $t_0 < t < t_1$ is then
$$ \Delta C_x = \int_{t=t_0}^{t_1} \frac{ \partial C }{ \partial x } \frac{ \text{d} x }{ \text{d} t} \text{d} t $$
Here they say
If it were possible to observe the (...) variables x in continuous time, (...) [this equation] would provide all that is needed to compute the contribution of each variable x.
Using logarithmic differentiation, they go on to rewrite the expression as
$$ \Delta C_x = \int_{t=t_0}^{t_1} C(t) \frac{ \partial \ln C }{ \partial x } \frac{ \text{d} x }{ \text{d} t} \text{d} t $$
and then for $C(t)$ assume a constant $C(t) \approx \tilde{C} $ which is ultimately chosen to be $\tilde{C} = \frac{ \Delta \tilde{C} }{ \Delta \ln \tilde{C} }$, such that $\Delta C_x + \Delta C_y = \Delta C$.
Questions:
Even if the time dependence of variables was known (eg. daily data on the price of silicon, etc.), then integrating would not yield what the authors are actually looking for.
They are interested in the contribution of single variables to the total change in $C$ (eg. what percentage of total manufacturing cost reductions are due to decrease in silicon price). But integrating using
$$\Delta C_x = \int_{t=t_0}^{t_1} \frac{ \partial C }{ \partial x } \frac{ \text{d} x }{ \text{d} t} \text{d} t $$
is dependent on the path of curves $x(t),y(t)$. This would yield different results for different time dependency of variables. A variable $x(t)$ would yield a different $\Delta C_x$ than a variable $x'(t)$, which is not what the authors seek to describe.
First of all I agree with you completely, what the authors are proposing does not make a lot of sense. That said, it is a proposition easier to criticise than to improve. The question has a mathematical aspect and a more applied one.
Mathematical aspect
You are (rightly) questioning the path dependence of the proposed solution. But I do not think there is a way to solve this in a path independent fashion. The reason for this negative statement is very simple: A product is not a sum!
What you are looking for is a way to split the $C$-difference $x_1 y_1 - x_0y_0$ in path independent fashion. In other words you are looking for real functions $\Delta_x$ and $\Delta_y$ such that:
This is impossible which you can see by calculating the partial differential with respect to $x_1$ and $x_0$ of the equation for the total difference: $$ \frac{\partial \Delta_x}{\partial x_1}=y_1\text{ and }\frac{\partial \Delta_x}{\partial x_0}=y_0.$$
The left hand side of each equation depends only $x_1$ and $x_0$ while the right hand side depends on $y_1$ respectively $y_0$, which is impossible unless everything is constant. So there is no way to split the product on the right hand side of the total difference equation into additive terms. In hindsight this is not surprising: A product is just not a sum!
Domain specific aspects
Since this leaves you with inevitable path dependence, what can you do about it? You need to add more constraints to make the split non-arbitrary. These constraints can not be derived from mathematical principles but only from aspects specific to your domain of application. Three different general ways to approach this are:
I have to admit, all solutions seem to be quite involved and will likely require subtle technical arguments and reasoning.
A further heads-up: The final (cost) allocation on the additive decomposition should not be done arbitrarily. You should always use the Shapley or Aumann-Shapley values. (see wikipedia entry )
A last way out
If a problem in applied science does not have a solution, it is possible you are looking at the wrong problem. In your case $x$ and $y$ may simply not be the right variables from an economic perspective for this kind of allocation. Maybe the truly relevant variable is $z=xy$. Obviously, this variable nicely describes the change in cost. Examples where this is encountered in practice are cases such as $x$ is a price of something in a foreign currency and $y$ is the exchange rate. Then the proper solution is to define the variable of interest $z$ as the price in own currency.