Note: This chapter was long so the summary has two parts. The first part is here.
Out of the utility maximization problem (UMP) and the expenditure minimization problem (EMP), MWG develops four different functions that help understand consumer demand. Two are from the UMP and two from the EMP.
- Walrasian Demand Function x(p,w)- I have money and goods cost a certain price, what bundle of goods will I buy?
- Indirect Utility Function v(p,w)- With some money and prices, how happy can I become?
- Expenditure Function e(p,u)- I want to be*this* happy. How much do I need to spend?
- Hicksian (or compensated) Demand Function h(p,u)- If my wealth could be compensated to reach a level of utility, what goods would I buy?
These distinctions are subtle, but each explains certain characteristics of demand. Also, economists can measure some of these, at least in theory. While we never be able to measure utility and therefore cannot estimate the Hicksian. However, in theory we can estimate the Walrasian. Therefore, through the Walrasian can be used to test the Hicksian.
Each of these represents a different economic concept. These concepts and connections between them make up a bulk of the third chapter.
Wealth and Substitution Effects
Much of economics deals with price changes. If the price of a good goes up, people buy less of it. This is clear. However, this change in behavior has two different theoretical parts.
Consider gasoline. When the price goes up, people consume less because some people switch to other options such as natural gas or wood. This is call the substitution effect. However, people also consume less because they are relatively poorer. I have less real wealth, so I consume less. This is the wealth effect. In reality it is impossible to tell what part of the change in consumption is due to each, but it helps to break them apart theoretically.
Most of the time when economists talk about price changes they ignore the wealth effect. This is a good estimate if the good is a small part of someone's budget. A change in the price of brown sugar does not cut my real wealth in any meaningful sense.
In the Hicksian demand function, the wealth effect does not exist since wealth is "automatically" compensated to cancel it out. Because the wealth effect is not a problem, the Hicksian focus on the substitution effect. A substitute for a good is another good for which consumption increases when the first good's price increases. The consumer switches between the goods. Because the preferences are assumed to be convex, every good has at least one substitute. There is always something to switch between.
MWG goes through a long mathematical proof showing that preferences can be inferred from people's actions. Therefore, we can know people's real preferences and judge whether people are better or worse off. This is how economists can do welfare analysis.
If the price of a good goes up, consumers of that good are worse off. They will be unable to consume as much. This is obvious. The tricky part is when MWG tries to quantify how much better off a person is.
Basically, how much would the person be willing to pay to avoid or make the change? If he would pay $20, we can say that he is $20 better or worse off. With this, welfare analysis across people is the summing of all of these quantities. To be even more fancy, it is an integration of all of these quantities.
This part of chapter three was much more in depth mathematically. Some very simple economic concepts were explained in great detail in mathematics. It is an exercise in deriving economics without needing to know economics. A pure mathematical theorist could get to the same results, but not use the same terminology.
Yet, it is helpful to be so explicit with some fuzzy subject. The distinction between wealth and substitution effects can be blurry in the real world and in our minds, but when put into a specific function, they are clear. They are clear if you understand the math.
I still cannot follow how we got from utilities are ordinal to they are cardinal and we can attribute a dollar amount to them. I believe I follow the math derivation, but not the methodological process. It will forever be in my mind until someone explains it.