Preferences & choice
Economics starts with a simple model of the consumer: she can rank bundles of goods, and she picks the best one she can afford.
Rational preferences
A bundle is a quantity of each good, e.g. \((a,b)\). We write \(X\succeq Y\) ("\(X\) is at least as good as \(Y\)"), \(X\succ Y\) (strictly better) and \(X\sim Y\) (indifferent). Preferences are rational if they are
- complete: for any two bundles, \(X\succeq Y\) or \(Y\succeq X\) (or both). Indifference is allowed; "can't compare" is not.
- transitive: if \(X\succeq Y\) and \(Y\succeq Z\), then \(X\succeq Z\).
The default is a cycle: \(X\succ Y\succ Z\succ X\). Nobody can be a utility maximizer with preferences like these, because there is no best bundle. Try making them transitive again.
More is better, and convexity
More is better (monotonicity): a bundle with more of both goods, or more of one and no less of the other, is strictly better.
Convexity: the average of two equally good bundles is at least as good as either of them. People like variety rather than extremes. Equivalently, \(|\mathrm{MRS}_{ab}|\) is decreasing in \(a\): the more \(a\) you already have, the less \(b\) you'd give up for another unit of \(a\), so the indifference curve flattens as \(a\) grows.
Convexity says the mix is at least as good, not strictly better. Perfect substitutes (\(u=a+b\)) are the borderline case: the mix is exactly as good, and the indifference curve is a straight line.
Choice and revealed preference
We can't read minds, but we can observe choices. If a consumer picks \(X\) when \(Y\) was also affordable, she reveals \(X\succeq Y\). If \(Y\) was strictly inside her budget (she could have afforded a bit more of both goods), then with more-is-better she reveals \(X\succ Y\).
Her choices are consistent with rational preferences only if these revealed rankings never contradict each other. A contradiction: \(X\succ Y\) from one observation and \(Y\succeq X\) from another.
If prices change and she still can afford her old bundle, her new choice must be revealed-preferred to it, or be the old bundle itself. This lets you narrow down where the new optimum can be without knowing her utility function (see the questions below).
Choice functions
A choice function \(C(A)\) says which option a consumer picks from a set \(A\) of available options. For a rational consumer (complete and transitive preferences) the choice from a big set is her favorite available option, so you can predict it from smaller observed choices.
Worked example
A rational consumer chooses \(C(a,b)=b\), \(C(b,c)=c\) and \(C(b,d)=b\). What is \(C(a,b,c,d)\)?
Show solution
Read each choice as a preference: \(b\succ a\), \(c\succ b\), \(b\succ d\). By transitivity \(c\succ b\succ a\) and \(c\succ b\succ d\), so \(c\) beats everything: \(C(a,b,c,d)=c\).
What could she choose after prices change?
Suppose a consumer chose \(A\) at old prices, and prices change (same preferences, same income). Her new choice is not completely free: if \(A\) is still affordable, she must pick something at least as good as \(A\). Any bundle that she could have afforded before with money left over was strictly worse than \(A\) (more is better), so she can't pick it now. If \(A\) is no longer affordable, there is no contradiction possible, and any bundle on the new budget line could be her choice.