Choice Under Certainty — The Standard Model

Consider a set $X$ of alternatives. Think primarily of $X$ as the set of possible baskets of consumption goods/services, $x=(x_1,x_2,\dots,x_n)$ in $X$ is a basket that contains $x_1$ units of good 1, $x_2$ units of good 2, ..., $x_n$ units of good $n$.

The standard model assumes that any individual is able to compare and rank the elements of $X$. Fix some individual $i$.

We call the ranking $\succ_i$ individual $i$'s preferences.

Given a choice set $X$ and individual $i$ with preferences $\succeq$ we shall assume:

• Ability to rank (aka Completeness): for any two baskets $\mathbf{a}$ and $\mathbf{b}$ then $\mathbf{a}\succeq \mathbf{b}$ or $\mathbf{b}\succeq \mathbf{a}$.
• Coherence (aka Transitivity): for any three baskets $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$: $\mathbf{a}\succeq \mathbf{b}$ and $\mathbf{b}\succeq \mathbf{c}$ together imply that $\mathbf{a}\succeq \mathbf{c}$.
• More is better (I) (aka Monotonicity): for any two baskets, $\mathbf{a}=(a_1,\ldots,a_n)$ and $\mathbf{b}=(b_1,\ldots,b_n)$, whenever $a_1\ge b_1,\ldots,a_n\ge b_n$ it must be that $\mathbf{a}\succeq \mathbf{b}$.
• More is better (II) (aka Strict Monotonicity): for any two baskets, $\mathbf{a}$ and $\mathbf{b}$, whenever $a_1\ge b_1,\ldots,a_n\ge b_n$ and $\mathbf{a}\neq \mathbf{b}$, it must be that $\mathbf{a}\succ \mathbf{b}$.
• Continuity: for any basket $\mathbf{a}$ and a sequence of baskets $\mathbf{b}_n$: if $\mathbf{a}\succ \mathbf{b}_n$ and $\mathbf{b}=\lim b_n$ then $\mathbf{a}\succeq \mathbf{b}$.

1. When $\mathbf{a}\succeq \mathbf{b}$ and $\mathbf{b}\succeq \mathbf{a}$, we say that the individual is indifferent between the baskets $\mathbf{a}$ and $\mathbf{b}$ and we write $\mathbf{a}\sim \mathbf{b}$.
2. It is (implicitly) assumed that an individual's preferences are exogenous and stable: they are given and remain the same (at least for the period of our analysis).
3. Also, it is (implicitly) assumed that preferences are invariant with respect to different representations. That is, if $(x,y)$ and $(x,y)$ are different representations of the same basket, then $(x,y)$ $\sim$ $(x,y)$.
4. But the standard model does not assume one is conscious of one's own preferences. If an individual chooses basket $\mathbf{a}$ over basket $\mathbf{b}$, we will say that he/she reveals to prefer the basket $\mathbf{a}$ over the basket $\mathbf{b}$, $\mathbf{a}\succeq \mathbf{b}$.

1. Strawman argument. The “traditional” model (XIX–early XX century economics) of a selfish, rational, economic man is NOT standard economic theory.
2. Fallacy of composition. “Second, economic man is selfish, but even in traditional economics there are cases of a parent having altruistic feelings toward a child. In this situation we consider the decision maker as maximizing with respect to themselves and their descendants as a single unit, preserving the postulation of selfish man.”
3. Definition by contrast. Rationality’s broad meanings are stated—(1) means–end coherence, and (2) the ends themselves withstand scrutiny—but “economic rationality” is defined by contrast against a standard never stated.

1. Completeness rules out incomplete preferences.
   (being unable to judge is not possible).
2. Invariance rules out framing effects.
3. Stability rules out reference dependent preferences.

• An utility function $u_i$ is a scale for some preference $\succeq_i$. An utility function is just a convenient way of converting the rankings given by the preference to some numerical scale.
• We say that the utility function $u_i$ represents the preference $\succeq_i$ if for any two baskets $\mathbf{a}$ and $\mathbf{b}$: $$\mathbf{a}\succeq_i \mathbf{b}\Leftrightarrow u_i(\mathbf{a})\ge u_i(\mathbf{b}).$$
• As there are many temperature scales (Fahrenheit, Celcius, Kelvin, etc.), there are also infinitely many possible different scales (utility functions) that represent the same preference.

1. For any basket in the diagonal $(x,x)$ its utility is equal to $u(x,x)=x$.
2. For any other basket $c$, find a basket in the diagonal $d$ such that $c\sim d$ and set $u(c)=u(d)$.

We want to test whether a consumer's choices are rational. So we consider a sequence of prices and choices $(\mathbf{p}^t, \mathbf{x}^t)$, $t=1,\ldots,T$ that the consumer made over time.

Notice that $\mathbf{p}^t$ is a list of prices prevailing at time $t$. For example, in the case of two goods, X and Y: $\mathbf{p}^t=(p^t_{x},p^t_{y})$ and $\mathbf{x}^t=(x^{t},y^{t})$ is the basket chosen by the consumer at period $t$.

To save notation, we write $\mathbf{p}^t\cdot \mathbf{x}^t$ for the value of the basket under the prices, so $\mathbf{p}^t\cdot \mathbf{x}^t=p^t_{x}x^{t}+p^t_{y}y^{t}$.

Consider a sequence of prices and choices $(\mathbf{p}^t, \mathbf{x}^t)$, $t=1,\ldots,T$.

Consider a sequence of prices and choices $(\mathbf{p}^t, \mathbf{x}^t)$, $t=1,\ldots,T$.

If $\mathbf{x} \succeq_{DR} \mathbf{y}$ or $\mathbf{x} \succeq_{IR} \mathbf{y}$ we say that $\mathbf{x}$ is revealed to be preferred to $\mathbf{y}$ and write $\mathbf{x} \succeq_R \mathbf{y}$.

Main results:

1. If GARP is satisfied for all the choices in the data then there is a preference that satisfies all the standard assumptions (ability to rank, coherence, more is better and continuity) and that also generates these choices.
2. If GARP is violated, there are no rational preferences that can rationalize the data.

In this other example, GARP is violated. Can you explain why?

Assume two goods (X and Y) for simplicity and imagine that instead of maximizing some utility, the consumer chooses her optimal basket according to one of the behavioral rules below — we are assuming our consumer follows just one of the rules. We cannot dismiss the consumer as being irrational because perhaps the behavioral rule can be generated by rational preferences.

1. Spend 30% of income on good X and 70% of income on good Y.
2. Spend 20% of income on the cheapest good and 80% on the more expensive good.
3. Spend 90% of income on the cheapest good and 10% on the more expensive good.