What are the most used approximations in economics? As I progress in my self-study, I'm tired of having to search for several different approximations techniques used in economic textbooks. Sometimes the authors don't even tell the reader they are using an approximation, they just assume the reader is already familiarised with the approximation.

Because of this, and to try to help me, and my fellow students of economics, I ask the following. What are the most used/useful approximations (and techniques) used in economics?

My wish is that anyone whether studying alone or not, can come to this question, as they'll be sure that it will have at least some approximation related to what they want. Any help would be appreciated.

Here's an example:

I've read today that «$\frac{\Delta m_{t+1}}{m_t}=\frac{\Delta M_{t+1}}{M_t}-\frac{\Delta P_{t+1}}{P_t}$.»

Well this is not an equality but an approximation, for if $z=\frac{x}{y}$, then $\frac{\Delta z}{z}\approx \frac{\Delta x}{x}-\frac{\Delta y}{y}$
There are four categories of approximations used in economics.  Unfortunately, none of them are economic approximations, but rather mathematical or statistical approximations.  As such, they do not appear in the economic literature, so much so as in the mathematical literature.  You are not seeing in the economic literature because students should be exposed to the content by mathematicians.

The difficulty comes from the type of math used in economics.  There are two ways you could look at calculus.  One is that it is the study of change.  The other is that it is the study of approximations.  All of calculus is a set of approximations, though if you will mentally agree that $\lim_{n\to\infty}\frac{1}{n}=0$ then you can get "exact" answers.  Imagine, however, that the smallest unit is a penny, then you cannot go to zero and now you have an approximation if a penny is the smallest possible size.  The question economists have to answer within the profession is whether or not the difference has a large enough impact to matter.

It is rare in economics for anything to be anything except an approximation.  There are some categories though to these approximations.

The first category can be thought of as either equilibrium approximations or local approximations.  Mathematicians would use the phrase "local approximations."  The main one of these is the Taylor expansion, though there are others.  In classes, I have to be very careful when I teach undergraduates because they often substitute their own mental approximations for the actual ones in work.  While it is wonderful that they are trying to work problems through from first principles, its usually the ones who didn't study the answer in the book and who are panicking.

The second one can be thought of as functional approximations.  The actual functions in the real world are discrete and usually created by the separate functions of many different firms.  While there are certain mathematical properties created by the topology of the problem.  We know that certain types of problems are forced to have a certain general form, but the specific form will be unique to the real world problem.  As an example, a taxi company needs one driver per cab.  If you add cabs but no drivers then no revenue gain is created.  Likewise, if you add cab drivers but no cabs, you get not revenue gains.  Indeed, in both cases you can take losses from paying the cabies and the costs for the cabs such as insurance.  So the production function, which is a Leontief production function is assumed to be present almost all the time, ignoring cabbies who quit, or wrecked cabs resulting in a temporary mismatch between workers and cabbies.

There is also the issue that utility functions are ordinal and any function that preserves preference ordering is equally valid.  This is not an approximation so much as an arbitrary choice for convenience.

The third category could be thought of as model approximations.  For example, using game theory, is an arbitrary choice as a way to model something.  The use of a general equilibrium method versus a model that is sub-game perfect will create different approximations in the end.  Of course the goal is for empiricism to judge between the modeling methods, but for your purposes they should both be thought of as approximations.

Finally, all of statistics is a form of approximation.  The danger is to believe statistical models and all economic models are statistical models.  They can be excellent or poor.  Statistical significance is not enough, it needs economic significance, that is "what is the effect size."  Likewise, how sensitive is the statistical model to the data and to reality.  You can easily create a statistical model that is not at all robust.  That will make your approximations very fragile.

There isn't a list.  The most common are Taylor expansions, finding the slope, $\hat{\beta}$ in models as an approximation of the derivative, and production functions.  The best way to tell you are dealing with an approximation will always be seeing $\Delta$ in a problem.  If you see $\Delta$ then you are dealing with an approximation of $\mathrm{d}$.  Although $\mathrm{d}x$ is really an approximation of $\Delta{x}$, economics always solving inverse problems so in economic practice $\Delta{x}$ is an approximation of $\mathrm{d}x$.