In economics, it is common to say that people maximize their happiness. What does that mean? How can we develop a model for maximizing happiness?

While chapter 2 focuses on choice rules as the basis for a theory of the consumer, chapter 3 returns to preferences. From these preferences, MWG develops a theory of consumer demand. Much of this chapter reiterates the first chapter. Part of this post will look familiar.

## Axioms and Assumptions

MWG assumes rational preferences. Rational preferences must be *complete* and *transitive*. Chapter 1 explained these. They are central to microeconomics.

In addition to the axioms, which can never be violated in this analysis, MWG adds more assumptions. They may not always hold in reality or be used in every model, but simply the model. This is always the trade-off.

*Desirability*-

Consumers always want more. They never have "enough". While these seems silly, at some point I have had enough ice cream and become sick, MWG extends the time-frame. 2 gallons are too much for today, but not for a year. This is different from Rothbard, who looks at specific action at a specific time.

Also, MWG is only looking at *goods*. Everything consumed is desirable. *Bads*, such as pollution, become goods simply by reframing the problem. Consumers might not want more pollution, but they can want more clean air. Here the good is clean air. This change does not affect the fundamental nature of the problem.

*Convexity-*

Throughout this chapter, MWG assumes that preferences are convex. Convexity is a mathematical term, but has an economic interpretation. As the consumer receives more of a good, say apples, he will want each additional unit less than the previous, relative to other goods. This makes sense in most cases. This formulates "a basic inclination of economics agents for diversification." This is the *diminishing marginal rate of substitution. *

It is a little disappointing that assumptions so central to the analysis of demand are just that, assumptions. Yes,they are realistic in most cases, but not always. The student of methodology in me is saddened. Nevertheless, the analysis trots on.

## Preference and Utility

We use these assumptions to construct a *utility function*. With a utility function, MWG can turn economics into calculus. However, even the assumptions made so far are not enough. One more in needed.

MWG assumes that preferences are *continuous.* If preferences are continuous, then it is possible to create a continuous utility function. While continuity means something specific in mathematics, for the economists it has two important parts.

**First**, every good can be broken into smaller and smaller goods. Consumers can have 2 beers or they can increase their consumption to 2.00000000001 beers. This may seem silly, but it has a more realistic explanation here at page 110. Instead of thinking of consumption at one particular time, consumption can be turned into a rate of consumption. Another partial beer at one moment might seem odd, but one extra beer consumed over a year is starting to approximate a continuous change in beer consumption.

**Second**, and less controversial, each small change in a consumption bundle has a small change in utility. The consumer's "utility" does not all of a sudden jump when he has a bit more of a good.

As a textbook, MWG makes these assumption clear, but does not defend them or acknowledge their problems . I will not harp on these too much more. From now on, we will just assume that preferences are continuous and are represented by a continuous and differentiable utility function.

## Maximization

Assuming a utility function with the above characteristics and rational actors want to maximize their utility, consumer behavior is turns into a simple system of equations.

The standard is the *Utility Maximization Problem (UMP). *Consumers will maximize their utility function based on some constraint. The common constraint is wealth. A consumer only consumes what he can afford, i.e. quantity of all goods multiplied by its price is less than his wealth.

For each set of prices and wealth, at least one bundle is **the** best. Most the time this will only be one bundle, like the top ranked bundle in Rothbard. The function that assigns the best bundle for any set of prices and wealth is the *Walrasian demand function.* Given any prices and wealth, the rational maximizing consumer will choose a specific bundle, **x**(p,w), to consume.

The solution to this problem is simple when the utility function and constraint are "normally behaved."

First, the consumer will spend all of his money. This comes from our assumption that more is better.

Second, the consumer will choose that proportion of goods relative to their prices. The rate that the consumer will trade goods and stay just as happy, his *marginal rate of substitution*, is equal to the price ratio. If it were not so, the consumer could trade one good for another at their market prices and get more utility. This would not be a maximum.

**Minimization**

This same problem can be examined in a multiple ways. The most common alternative way to view this is as an *expenditure minimization problem (EMP*). What is the least amount a consumer can spend to reach a certain level of utility? The math is slightly different, but the concept is the same. The UMP computes the utility of a maximizing agent given wealth and the EMP computes the minimum wealth need to reach a utility level. The constraint and the max/min function are reversed in the two problems.

MWG uses these properties to show some of the results already shown in chapter 2, namely, the law of demand. Prices and demand more in opposite directions. Demand curves slope down.

## Analysis

The chapter of consumer demand is extensive. I decided to break it up for convenience.

The dual approach of MWG is confusing, but helpful. Some results repeated, once using choice rules and the other using preference relations. It feels redundant, but the math is different.

MWG spends most of the chapter proving things that do not seem important and assuming things that seem important. While it is nice to prove that continuous preferences lead to continuous utility functions, that is not the problem. The questionable step is the continuous preferences. The mathematician in me is left somewhat satisfied, but the economist is deeply skeptical.

2 1/2 chapters in and I have learned a lot of math formalism, but little clear economics. It is unlike any book I have ever read. That is a good thing.