I had a general maths question. I am trying to find the right topic to study for my problem. I would appreciate if someone could point me in the correct direction to study.
The general problem is: For every $x$ $\%$ change in variable $y$, there is an implied $z$ $\%$ change in variable $N$. E.g. for every $2$ $\%$ rise in unemployment, there is an implied $5$ $\%$ fall in the stock market.
I believe the correct topic to study for this type of problem is: Linear regression in statistics and $\beta$-Coefficients (in linear regression) to analyze sensitivity. Am I correct here?
If your data is empirical then the linear regression is what you need to use in order to answer your question. Normally in linear regression, we have predictors (inputs) $x_i$ and a criterion $y$ (output) [note that we can also have multiple outputs]. The linear regression can be stated as
$$y_n = w_0 + w_1x_{1,n}+...+w_px_{p,n} + \varepsilon_n,$$
in which $n$ designates the $n^{\text{th}}$ observation and $x_{i,n}$ the value of the $i^{\text{th}}$ predictor for the $n^{\text{th}}$ observation. As the linear model will always have some error associated with it we also need to add the error $\varepsilon_n$ of the $n^{\text{th}}$ observation. The parameters $w_j$ for $j=0,...,p$ are the weights (in statistics $\beta_j$).
Your statements can only be made if you have a statistically insignificant $w_0$ then you can interpret the $w_j$ as slopes. E.g. if we consider $w_1 = 0.5=\Delta y / \Delta x_1$. We must assume that all other predictors do not change. Then we can say if we increase $x_1$ by $\Delta x_1$ the output $y$ will increase by $\Delta y = w_1~\Delta x_1 =0.5\Delta x_1$.
If you have a statistically significant $w_0$ then you could redefine your output as $y_n-w_0$. Then you can do the same with this new output.