Preferences and utility

The marginal rule from the previous lesson needs a benefit function, and for a person choosing between goods there is no obvious unit to measure benefit in.

Jeremy Bentham thought there was. His felicific calculus of 1789 treated pleasure as a quantity with an intensity and a duration, addable across people, and for a century economists wrote as though a unit of satisfaction were a physical measurement waiting for a better instrument. It is not. Nobody has ever measured a util, and no experiment could, because there is nothing there to measure. What can be observed is choice: offered two bundles of goods, a person takes one. Everything in this lesson is built out of that observation and nothing else, which turns out to be enough.

The ranking is the primitive

Start with a consumption bundle, a list of quantities of each good. With two goods it is a point $(x, y)$ in the positive quadrant, and the two-good case is not a simplification so much as a drawing convention: let $y$ stand for "money spent on everything else" and any problem becomes two-dimensional.

The primitive object is a relation on bundles. Write $A ⪰ B$ for "the consumer finds $A$ at least as good as $B$ ". From it, two others are defined rather than assumed: $A$ is strictly preferred to $B$ when $A ⪰ B$ holds and $B ⪰ A$ does not, and $A$ is indifferent to $B$ when both hold. Notice what has not been said. There is no number attached to $A$ , no claim about how much better it is, and no comparison with anyone else's ranking. A ranking is all the data there is.

Four assumptions, and what each buys

An arbitrary relation is useless, so preferences are restricted. Four assumptions do the work, and it is worth knowing which result each one is paying for, because they are not equally innocent.

Completeness. For any two bundles, at least one of $A ⪰ B$ and $B ⪰ A$ holds. The consumer is never unable to say. This rules out "I have no idea", which is a real state of mind, and it is the assumption that makes the whole quadrant rankable rather than partly rankable.

Transitivity. If $A ⪰ B$ and $B ⪰ C$ then $A ⪰ C$ . Without it there is no best element to find: a consumer preferring $A$ to $B$ , $B$ to $C$ and $C$ to $A$ can be walked around the cycle for a small fee at each step and returned to where they started, poorer, forever. Transitivity is what makes maximisation meaningful at all.

Monotonicity, or non-satiation. More of a good is better. This is what makes them goods, and it fixes the direction of every diagram: north-east is up.

Continuity. If a sequence of bundles all at least as good as $B$ converges to some bundle $A$ , then $A$ is at least as good as $B$ too. Preferences have no sudden jumps. This is the technical assumption, the one with no intuitive content, and it is the one that earns the representation theorem.

Gerard Debreu proved in 1954 that complete, transitive, continuous preferences over a connected set of bundles can be represented by a continuous utility function: a function $u$ with $u (A) \geq u (B)$ exactly when $A ⪰ B$ . That theorem is the licence for everything that follows. It says a numerical function exists, so calculus applies, and it says nothing whatever about the function being unique or meaningful.

Indifference curves

Fix a value $\overline{u}$ and look at the set of bundles satisfying $u (x, y) = \overline{u}$ . That level set is an indifference curve, and the family of them is a contour map of the preference ranking. Francis Edgeworth drew the first ones in Mathematical Psychics in 1881, and Vilfredo Pareto in 1906 made the decisive point that the contours carry all the information and the labels on them do not.

Three properties follow from the assumptions rather than from drawing habit. Indifference curves slope downward, because monotonicity says that giving up some $x$ must be compensated with more $y$ to stay level. Curves further from the origin are better, for the same reason. And two indifference curves cannot cross: if they did, the crossing point would be indifferent to a bundle on each curve, and transitivity would force those two bundles to be indifferent to each other, contradicting the fact that one lies north-east of the other.

Utility represents, it does not measure

Here is the point that separates modern consumer theory from Bentham. If $u$ represents a preference ranking and $f$ is any strictly increasing function, then $f (u)$ represents exactly the same ranking, because applying an increasing function to both sides of an inequality leaves the inequality alone. Utility is ordinal: only the order of the numbers means anything. Their size, their differences, and their ratios do not.

Example. Take $u (x, y) = \sqrt{x y}$ , and compare the bundles $A = (4, 9)$ and $B = (8, 4)$ . Now repeat with $v = x y$ and with $w = \ln x + \ln y$ , and check that the ranking survives.

Under $u$ : $u (A) = \sqrt{36} = 6$ and $u (B) = \sqrt{32} = 5.66$ , so $A$ wins. Under $v$ : $v (A) = 36$ and $v (B) = 32$ , so $A$ wins. Under $w$ : $w (A) = \ln 36 = 3.584$ and $w (B) = \ln 32 = 3.466$ , so $A$ wins. All three are increasing transformations of one another, since $v = u^{2}$ and $w = \ln v$ , so no ranking could have differed. But note what did change: under $u$ the gap is 0.34, under $v$ it is 4, and under $w$ it is 0.118. Any statement of the form " $A$ is six per cent better" is a statement about the arbitrary labelling and not about the consumer.

Now you. Using the same three functions, rank $A = (4, 9)$ against $C = (6, 6)$ , and say what the answer means.

Answer

$u (A) = 6$ and $u (C) = \sqrt{36} = 6$ ; $v (A) = v (C) = 36$ ; $w (A) = w (C) = 3.584$ . The consumer is indifferent between them, on every representation, which is what indifference has to look like: the bundles sit on the same contour, and no relabelling can pull them apart.

The practical consequence is a rule for later lessons. Any result that changes when utility is relabelled is an artefact and must be thrown away. Interpersonal comparisons are the biggest casualty: since each person's utility number is arbitrary up to an increasing transformation, adding two people's utilities together is arithmetic on units that do not exist. That does not make the comparison meaningless as a moral matter, but it does mean the model cannot supply it, and the welfare lessons will have to work with something weaker.

The marginal rate of substitution

Something must survive relabelling, or the theory would say nothing. What survives is the slope of the indifference curve.

Marginal utility is a partial derivative, ${MU}_{x} = \partial u / \partial x$ , the rate at which utility rises with $x$ holding $y$ fixed. On its own it is as arbitrary as utility itself: replace $u$ by $u^{2}$ and every marginal utility doubles at $u = 1$ and changes by a different factor elsewhere. Take the ratio, though, and the arbitrariness cancels.

Move along an indifference curve. Total utility does not change, so ${MU}_{x} d x + {MU}_{y} d y = 0$ , giving

MRS = - \frac{d y}{d x} = \frac{{MU}_{x}}{{MU}_{y}}

The marginal rate of substitution is how many units of $y$ the consumer will give up for one more unit of $x$ and remain exactly as well off. It is a rate of exchange, measured in $y$ per $x$ , and it is a fact about the person that a competing offer can be tested against. Under a relabelling $f (u)$ , the chain rule multiplies both marginal utilities by the same $f' (u)$ , which cancels in the ratio. The MRS is the observable content of the utility function.

Example. A consumer has $u (x, y) = x^{0.4} y^{0.6}$ and currently holds $(20, 30)$ . Find the MRS, and check it against a finite trade.

Differentiating, ${MU}_{x} = 0.4 x^{- 0.6} y^{0.6}$ and ${MU}_{y} = 0.6 x^{0.4} y^{- 0.4}$ . The ratio simplifies to $MRS = (0.4 / 0.6) (y / x) = (2 / 3) (30 / 20) = 1$ . So one more unit of $x$ is worth exactly one unit of $y$ here. Checking: $u (20, 30) = 25.508$ and $u (21, 29) = 25.487$ . The trade very nearly holds the consumer level, and the small shortfall is real, because the MRS is a derivative and applies exactly only to an infinitesimal step. Over a full unit the curve has already bent.

Now you. A consumer has $u (x, y) = x^{0.25} y^{0.75}$ and holds $(10, 60)$ . Find the MRS and say in words what it means.

Answer

$MRS = (0.25 / 0.75) (y / x) = (1 / 3) (60 / 10) = 2$ . The consumer would give up two units of $y$ for one more unit of $x$ and be no worse off, so any seller offering $x$ at a price below two units of $y$ has a deal.

Convexity: averages beat extremes

One more assumption is standard, and unlike the first four it is an empirical claim rather than a coherence requirement. Preferences are convex when, given two bundles the consumer is indifferent between, any mixture of them is at least as good as either. Drawn, that is an indifference curve bowed towards the origin. Read as a rate, it says the MRS falls as $x$ rises: the more $x$ someone already has, the less $y$ they will surrender for another unit of it.

Example. With $u (x, y) = \sqrt{x} + \sqrt{y}$ , the bundles $(64, 4)$ and $(4, 64)$ both give $8 + 2 = 10$ . What does the consumer think of an even mixture of the two?

The mixture is $(34, 34)$ , giving $2 \sqrt{34} = 11.66$ , which is strictly better than 10. Averaging the two extremes gained the consumer real ground, and it is the concavity of the square root that did it: the units of $x$ moving from the bundle that had 64 of them were worth much less than the units arriving at the bundle that had 4.

Now you. Same utility function. Compare $(100, 4)$ and $(4, 100)$ with their even mixture.

Answer

Both extremes give $10 + 2 = 12$ . The mixture is $(52, 52)$ , worth $2 \sqrt{52} = 14.42$ , again strictly better. Convexity is what will make the consumer's optimum a smooth tangency rather than a jump to one axis, which is why the next lesson can use calculus at all.

Three shapes worth knowing

Not every preference is smoothly convex, and three special cases carry most of the exceptions.

Perfect substitutes have straight indifference curves, $u (x, y) = a x + b y$ , and a constant MRS of $a / b$ . A consumer who genuinely does not care whether a pound of sugar arrives in one bag or two has these preferences over bag sizes. The constant rate matters because it means the consumer will spend everything on whichever good is cheaper per unit of $a$ or $b$ , with no compromise bundle in sight.

Perfect complements are the opposite: $u (x, y) = \min (a x, b y)$ , with L-shaped indifference curves. Left and right shoes are the standard case, and cars and their engines the honest industrial one. There is no MRS at the corner, because the curve has no derivative there, and extra units of either good alone are worth exactly nothing.

Quasilinear preferences take the form $u (x, y) = g (x) + y$ , linear in one good. The MRS is $g' (x)$ , which depends on $x$ alone, so the indifference curves are vertical shifts of one another and the demand for $x$ does not move when income does. That is unrealistic for food and rent, and a reasonable approximation for a good that takes a tiny share of the budget, which is why quasilinear utility is the workhorse of the surplus calculations later in this course.

Where the axioms break

Every assumption here has been tested, and the results are mixed in an interesting way rather than a fatal one.

Transitivity fails reproducibly. Amos Tversky showed in 1969 that when options differ on two dimensions and one difference is small enough to ignore, subjects cycle: gamble $A$ beats $B$ , $B$ beats $C$ , and $C$ beats $A$ , with the cycles appearing consistently rather than randomly. Completeness fails whenever the goods are hard to compare, and asking someone to rank a career against a relationship gets a genuine refusal rather than a slow answer.

Continuity fails in a way worth being precise about, because it looks like the safest axiom and is not. Lexicographic preferences, where the consumer always prefers more $x$ and only looks at $y$ to break exact ties in $x$ , are complete, transitive and monotonic. Debreu showed that no utility function whatsoever can represent them, not merely no continuous one. There are too many indifference sets to label with real numbers. So the representation theorem is doing real work, and continuity is what buys it.

None of this stops the framework being useful, and the next several lessons will use it hard. What the failures do is fix its status. The model is not a description of how people deliberate; it is a way of turning observed choices into a function you can differentiate. The final lesson comes back to the evidence and asks how much of the theory survives it.

That function is now available, and it still chooses nothing. Monotonicity says more is better, so the ranking on its own points straight out to infinity. What stops it is that bundles have to be paid for.