Chapter two elaborates choice, inherent in the preferences of chapter 1 of MWG. Specifically, it is about consumer choice in a *market economy*. This is an ambiguous term, but here it means a setting where consumers trade at known prices.

## Commodities

While everyone uses the word commodity, Mas-Colell defines it as a good available for purchase or trade. All available goods make the *commodity vector* or *bundle.*

Each good incorporates time and location. Good x_{i }can be an apple now in the consumer's apartment. Consumers must approximate "now". It could be a few seconds or a year, depending on the situation. The proper approximation will depend on the specific choice. Also, there is nothing limiting each x_{i} to only physically identical goods. While Mas-Colell is not explicit, these are subjective goods. If Budweiser and Miller are interchangeable (my Wisconsin friends just gasped) to the consumer, they are the same commodity.

## Consumption Sets and Budgets

Given a commodity space, only some specific goods are physically and institutionally available to the consumer. For example, the consumer cannot consumer more than 24 hours of leisure in a day. For now, there are only positive bundles. The consumer chooses from this *consumption set*. His choice does not depend on goods that are not available. The writers assume the consumption set is convex. (Note: Most *normal* consumption sets are convex so I will not elaborate.)

The real choice the consumer faces is also limited by his budget or wealth. He can only consume what he can afford. No matter many Lamborghinis are available, he only has $200. The sum of all prices, which are assumed to be given, multiplied by the quantities consumed must be less than the budget. To add jargon, every affordable set of commodities is called a *Walrasian* or *competitive budget set*.

For a budget to make sense, all commodities must trade at known prices or exchange rates, every commodity must have a price. Otherwise, it does not make sense to talk about a budget. This is similar to the completeness axiom for rational preferences.

## Demand Functions

For each budget and set of prices, the consumer prefers a certain consumption bundle. Actually, there can be multiple preferred bundles, but we will ignore those cases for now. The set of bundles a consumer will pick at each price makes up a *demand function*. To simplify the analysis, Mas-Colell assumes the budget is fully spent. If he has money remaining, he will spend it on something. In order to do the math that economists love, Mas-Colell also assumes continuity and differentiability.

To better understand demand functions, it is natural to ask how they change with changes in the prices or the budget, the two parameters. Since we assume differentiability, calculating the change is easy if we assume we know the demand function. To find how consumption of goods change with a change in wealth, called the *wealth effect*, simply compute the derivative of the demand function for each good with respect to wealth. If the consumer is richer, he will consume more of certain goods. He could consume less, but that is not the case with normal goods.

The effect of a price change of one good on another can be similarly calculated, by taking the derivative with respect to the changing price. Economists call this the *price effect*. If the price of one good increase, the demand will normally change.

Note only relative price changes matter. If all prices increase in the same proportion, the proportional consumption will stay the same. It will be lower, since the budget is still restricting consumption.

## Weak Axiom of Revealed Preference

Much of the second chapter applies the weak axiom of revealed preference to budget sets. I will not go through the math, but try to explain the logic. Remember from chapter 1 that the weak axiom says that if I always choose x from a bundle of x and y, y can never be picked from new bundle x,y,z. This makes sense. Why would another added option, z, make me change my preference to y?

Imagine two optimal- on the demand function- bundles for a certain budget, B and B*_{,} with corresponding prices, p and p*. If the consumer can afford the bundle B at the prices of p*, the consumer still chose B*.

This means consumer prefers B* over B when both are available. If he could afford B* at p prices, then he would choose B*, but he chooses B at p. This consumer will choose B over B* only when B* is not affordable._{
}

This relates to the original weak axiom, since it shows that introduction of new options from a different budget will not make the consumer change his preference for B* over B. (For a formal definition, check out page 29.)

From this, Mas-Colell shows that demand and prices move in opposite directions. If the price of a good rises, consumers purchase less. Obvious to most economists, it might impress some mathematicians.

## Analysis

While choice is more interesting than preference, chapter two is still an introduction. Readers learn a few interesting results, such as substitution effects. But after much struggling with the mathematics, I am not closer to understanding the economy or individuals. After reading one chapter in Rothbard or Stigler, I feel like I am learning economics. It does not feel that way with MWG. I need patience. It is a a marathon, not a sprint. These concepts and strategies will hopefully build a solid foundation for the rest of the book. If I wanted to learn the basics, I would reread *Economics in One Lesson*.

This chapter is limiting, because of its focus on a market economy. Whether economics is the study of action, choice, or allocation, the subject does include more than trading of commodities. I would prefer to see "trading" that includes internal tradeoffs at "prices" of the opportunity cost. I cannot adjust Mas-Colell´s model to do this, but maybe someone can.

This is not the place to have the debate on whether preferences, utility functions, or consumption bundles are differentiable. However, it does seem important. Almost all the analysis of so far relies heavily on this assumption and it is not defended at all. Again, I will have to wait to see more how useful the assumption is. At this point, Rothbard's analysis seems much more realistic and informative. Again, it is early and I had strong priors for Rothbard. My Bayesian updating will necessarily take time.