What is the point of all the models in an economics degree? I am a student of Economics doing my masters degree and I have to say I am a bit tired of all the models.

On the one side, there is these models where I can see how they are derived, which always seems to be a mixture of using the Lagrangean with some - often rather unconvincing - assumptions. But I kind of fail to see the deeper meaning. For example how do New Keynesian (or RBC) models help me understand the real world? 


How would they be used in practice - is it reasonable to assume they could forecast inflation, growth or interest? It seems like these things are inherently impossible to predict. 
Do they help me understand how they work? The understanding I have about them comes more from non-mandatory applied courses and my personal readings about past financial crisis etc. 
But probably most importantly, do I believe anything that comes out of them which I did not assume already before? Certainly not. 


Or the Solow model, how is the fact that some countries seem to have followed a path that vaguely resembles that assumed by the Solow model a justification for its validity? Drawing a simplified picture of a human doesn’t help me understand why they became sick.
There is extreme cases like Barro's model of Ricardian Equivalence which seems like either the world's most boring circus trick or an attempt to fool policy-makers without basic maths skills. 

On the other side there is a few cases where I would say my understanding improved with models in a way it would not have without, e.g. the notion of externalities, Pigouvian taxation or comparative advantage. But I have to say - given the countless pages of Lagrangeans I have set up, hours I have spend maximising profit functions and arguments I had in my head trying to find ways to justify some micro axioms - this is not a lot.

Don't get me wrong, I am fascinated by the study of the Economy and everything it relates to, I just don't see how a lot of what we learn is actually a helpful tool. I am a good student and neither looking for an easy way out, nor do I want to rant about studying. I am just looking for some motivation where this all leads. Can somebody explain to me what the deeper meaning of economic modelling is and why we give it this much attention? 

And yes I know all models are wrong, but how are they useful?
Models are more than just math. Model is a simplification of a reality that allows you to study the underlying mechanisms. Models do not need to be mathematical. Many people actually create models without even realizing it. For example, if someone says "minimum wage will not lead to unemployment" that person is actually having a model even if they never present any mathematics. Such person definitely does not take into consideration full reality - every choice of every single individual, movement of every atom and so on. 

Hence, whether you use mathematics or not you are using some model anytime you are describing any mechanism, such as x causes y, or there is such and such relationship between x and y because of this or that. Thinking otherwise is just self-deception as no human mind can take into account full reality when examining any issue. You have to simplify whether you do it with math or without. However, I get that at the heart of your question is why economists use mathematical models, but I wanted to highlight that there is no way of escaping models. 

Mathematical modelling has advantages over plain text.  

Dani Rodrick put it best when he said:


  We need the math to make sure that we think straight--to ensure that our conclusions follow from our premises and that we haven't left loose ends hanging in our argument. In other words, we use math not because we are smart, but because we are not smart enough.
  
  We are just smart enough to recognize that we are not smart enough. And this recognition, I tell our students, will set them apart from a lot of people out there with very strong opinions about what to do about poverty and underdevelopment. 


Hence the reason why economists use mathematical models is that they are tremendously helpful in disciplining your thinking and by forcing you to say exactly what you mean. Written text is imprecise, words can have double meaning, people can easily cover up unsound logic by good use of literary devices or appealing to your emotions all of which can cloud your judgement.  

This makes text  harder to actually analyze analytically. For example, even in moral philosophy - the least mathematical field I can think of, to fully analyze argument made in text and determine whether it makes sense you have to often resort into translating it from plain English into symbolic logic - which is form of math. Mathematics itself is just a language based on logic, that's also why in mathematics we often talk of equations as sentences or statements. 

When you reduce model to set of mathematical expression you are forcing yourself to make your own thinking bare. You cannot easily conceal faulty logic anymore. You can argue that math can be used to 'fool policy-makers' as you put it but you cant easily fool other scholars with math while with text you can easily fool both policy makers and scholars.  My argument here is not that the math is bullet proof, as argued by Romer some scholars can try to mislead with using "mathiness", but lying with math is much much more harder than lying with text. 

Models actually do help predict what will happen. First, do not confuse prediction with forecasting. For example New Keynesian model predicts that if there will be sudden shock to demand wage rigidity will lead to higher than natural level of unemployment - that is a prediction but its different from forecasting as the prediction is based on the ceteris paribus conditions while the word is constantly changing. 

Predictions that model make can be useful in itself for several reasons. Predictions actually allow us to test whether one theory is superior than another. In fact the only way of testing a theory is to test its prediction against empirical observations. If someone has elaborate theory written in text that says that minimum wage does not affect employment and other person will make equally compelling case in text against that how will you decide which person is right?

One way how to do that is to turn the first person text into a mathematical model. The person says that minimum wage does not affect unemployment? Okay that means he postulates the following relationship:

$$ U(w_{min}) = a + bw_{min}$$ where $b=0$

Now converting the thinking of that person into mathematical model actually allows us to test it - we can based on the model above construct regression of unemployment on minimum wage:

$$U = \beta_0 + \beta_1 w_{min} +\epsilon $$ 

and test whether the first person was correct by testing the null hypothesis of $\beta_1=0$ against alternative hypothesis $\beta_1 \neq 0$.

Predictions also allow us to do counterfactual analysis. We can ask ourselves what would happen if in the presence of wage rigidities government increases spending during the recession. New Keynesian models provide you with framework to think analytically about such questions and fully explore all their logical conclusions. 

Does this mean that math is the only way of studying economics? No. Of course, every individual is different. Some people are visual learners, some people are auditory learners.  Some students respond better to narration than to math. Moreover, as you yourself admitted math helped you to understand some problems so it worked for you well at least a couple of times. While you might have learned better most concepts by reading about them  other student might have completely opposite experience. 

Moreover, depending on your career choices you might end up not using most of the math in real life. However, a master degree especially if its master of science and not just master of arts is often a stepping stone (at least in the Europe) towards PhD or more academic career in policy institutes or government institutions where you will need the math if not directly then at least indirectly to understand new advances presented in academic journals. For example, even routine psychologists must have some statistical understanding if they want to keep up to date with new techniques to help their patients as in order to see if some technique is better than other some testing has to be done. Otherwise you will be just at mercy of trusting what authors say in conclusions, or some intermediary like academicians who like to blog for wide audience etc.

Furthermore, you should not interpret anything I said above as saying math is the only way of doing economics or that narrative analysis is useless. Sometimes narrative analysis can be more nuanced, and with some problems we might not even yet discovered ways how to model them mathematically in satisfactory way. However, models and especially mathematical models are incredibly important tool in any economist's toolkit.