Misunderstand "other things being equal"

   In the past summer, I have interviewed many high-school students wanting to study economics in university. As I mentioned in my earlier post, interviews is a useful source for me to know what students have learned in economics.
   In one case, a student (a pretty good one) gave us (I and another interviewer) a question: In economics, a usual assumption made is "ceteris paribus" or "other things being equal". However, in the real world, other things are always not equal, Then, how can we apply the economic theory for analyzing the real world? The theory may never be useful.
   I guess that not only this student but also many laymen may have the same feeling about this assumption used in economics. However, most doubts cast on this assumption is due to a misunderstanding. There is a need to clear up these doubts and remove the misunderstanding if we want to properly learn economics.
   At that time, I gave another answer to thus student. Upon reflection, there is an easier way and more constructive way to answer this student so that he can understand why his suspicion involves a misinterpretation of the "other things equal" assumption. Hence, let me answer his question (again) here.
   For example, when income increases, economic theory (demand and supply) will say that, other things being equal, the demand curve for a normal good will shift rightwards while the supply curve is unchanged. So, both the price and the quantity traded of the good will go up. However, other things are always not equal. But does it mean our analysis is not useful? Certainly not. If we know what "other things" change when income increases, then the economic analysis should simply take also this into account. For instance, if we also know that the cost of production decreases when income increases, then the theory tells us that supply curve will shift down when demand curve shift rightwards. The result: quantity traded certainly increases although price may go up or down. If we want to know whether the price will go up or down, then we have to know some more about the facts: the income increases by how much and the cost goes down by how much and what is the income elasticity of demand and cost elasticity of supply.
   Hence, you can see: The theory is useful. It can predict what happen even if "other things are not equal". But the more complex the event is about (only income increases or cost also changes), the more information (income and cost elasticities) we need for an analysis.
   So, "other things being equal" does not really prevent us from using the theory to analyze the real world. But then you may wonder: if so, why "other things being equal" is assumed? It seems that the assumption of "other things being equal" has not been satisfied and adopted in the case of "income increases while cost also goes down".
   The situation is: the assumption has really been used in the analysis above. When a demand curve shifts, say, from D1 to D2, other things are assumed to be equal. Only income increases and so the demand curve shifts to the right. When a supply curve shifts, say, from S1 to S2, other things are assumed to be equal. Only cost goes down and so the supply curve shifts down. Well, when both things (income increases and cost goes down) happen, wouldn't it be wrong to say "other things being equal"? No. You have to understand that a conducive approach to analyze anything is to analyze things step by step, instead of messing up everything together. To make step-by-step analysis possible, in each step, you had better consider only one thing to change (or you may mess too many things up). So, when you want to know what happens to the demand side, consider only demand factor (income). Other things (cost) are assumed equal. When you move on to the next step - the supply side, consider only supply factor (cost). Now the income has been considered to have increased to the new level already and no longer change further. Other things (income) are equal.
   From this example, you can see that "other things being equal" should not be misinterpreted as really assuming everything else unchanged in the real world. It is, however, necessary for analysis in steps, not messing things up.
   Well, you may still find this interpretation not convincing enough. You may say: when income increases, if other demand factors are also not constant, the demand curve may not shift to the right. Thus, we need to assume unrealistically that other demand factors are equal when predicting a rightward shift in demand curve. But this does not affect my interpretation above: "other things being equal" is assumed for any step-by-step analysis.
   In economics, we often express demand as an equation like this:
Quantity = 100 + 0.8(average personal income) - 1.2(price of the good) + 0.2(price of substitutes) - 0.15(price of complements) + 0.06(other factors)
   Demand equation can be estimated when economists got sufficient data about consumers' choices under different prices, income levels, etc. The above suspicion is like saying that we cannot be sure the demand curve shifts to the right when income increases by 1 unit as price of substitute may fall by more than 4 units at the same time (other things not equal), offsetting the effects from income.
   There are two responses to this suspicion: First, when we say the demand curve shifts rightwards, we have to make sure the income effect will not be offset by "other things" such as price reduction of substitutes. But it is still true that if "other things being equal", income will shift demand to the right. In fact, the parameter 0.8 for income effect says exactly that if "other things are equal", 1 unit increase in income increases quantity demanded by 0.8. It is meaningless to talk about 0.8 if "other things are not equal". If we do not have the parameter 0.8, we cannot even tell that when income increases by 1 unit while the price of substitute reduces by 4 units ("other things not equal") demand will be unchanged. Meanwhile, if we do not know what happen when "other things are equal" (0.8 parameter), we cannot know what happen when "other things are not equal" (both income and price of substitute change). There is nothing paradoxical involved. As mentioned, "other things being equal" is simply an assumption adopted to facilitate the step-by-step analysis. If we cannot divide things in steps and know in each step what would happen (know what would happen "when other things are equal"), we also cannot know what would happen when all steps are taken.
   Second, perhaps someone may concentrate on the last item in the demand equation above. There are always other things other than the factors identified as price of substitutes/complement, and so on, in a model (of demand in this case). In fact, when someone criticize the "other things being equal" assumption, I suspect that most of them have exactly this type of criticism in mind. They may think: there are so many factors in the real world and so how can we not miss some factors? But this problem is not unique to economics. For example, when scientists predict the weather of tomorrow, there must always be some factors other than those that have been identified in their weather model. That's one reason why scientific predictions may sometimes fail. Scientists can only try their best in identifying as many relevant factors as possible. There is nothing wrong and special about this. Of course, social scientists, economists in particular, may find their task of this identifying jobs particularly difficult as society/economy is a complex thing. As scientists, or economists, or even economics students, one should not therefore blame the theory, saying that "other things being  equal" prevents them from doing a good job. If we haven't tried to identify as many relevant factors as possible, it is our fault, not the theory.