Scarcity and the margin

Economics is not the study of money, and a course that begins with money will be lost by the third lesson.

Lionel Robbins gave the definition that still works, in his Essay on the Nature and Significance of Economic Science of 1932: economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses. Every word of that is load-bearing. There are ends, plural, so ranking is required. The means are scarce, so not all ends can be met. And the means have alternative uses, so committing them to one end withdraws them from another. Strip out the money and the markets and what is left is a maximisation subject to a constraint, which is a problem in calculus. This course is that problem, solved over and over with different objects in it.

Scarcity is a constraint, not a shortage

A shortage is a temporary failure to supply something: no eggs on the shelf this morning. Scarcity is permanent and applies to things that are abundant. There is a great deal of drinking water on Earth, and water is scarce in the economic sense, because using a litre to cool a data centre is using a litre that cannot irrigate a field. Scarcity means only that the resource has more claims on it than it can meet at once.

So the honest form of any economic question is not "is this worth doing" but "is this worth doing with resources that could do something else". Time is the cleanest example because nobody can accumulate it: a day is 24 hours for a billionaire and for a bankrupt, and every hour spent is an hour withheld from every other use. Diamonds are expensive and time is priceless, but only one of them is genuinely fixed in supply.

The subject splits into two kinds of claim, and mixing them is the most common way to argue past someone. A positive claim says what is: a tax of this size will cut consumption by that much. It can be wrong, and evidence settles it. A normative claim says what ought to be: that cut is worth its cost. It cannot be settled by evidence alone, because it rests on how the losses to some are weighed against the gains to others. Most of this course is positive. The welfare lessons later on are where the boundary has to be watched hardest, and the honest position there is that the model can say what is efficient and cannot, by itself, say what is good.

Opportunity cost

If the means have alternative uses, then the cost of using them one way is the best alternative given up. That is the opportunity cost, and it is the only cost that belongs in a decision. Money paid out matters only when it stands for something forgone; money not paid out matters just as much when something was forgone anyway.

Two consequences follow immediately, and both catch people. First, a cost you would bear whichever way you choose is not a cost of the choice. Second, a resource you already own is not free to use, because it could have been sold or rented to someone else.

Example. A student is deciding whether to take a three-year degree in England, where the tuition fee cap for a home undergraduate was £9,250 a year from 2017 to 2024. The alternative is a job paying £20,000 a year. Rent and food come to £9,000 a year and would be paid either way. What is the opportunity cost of the degree?

The fees are £9,250 × 3 = £27,750, and they are paid only in the degree branch, so they count. The forgone wages are £20,000 × 3 = £60,000, and they count too, even though no cheque is written for them, because they are the best alternative use of the same three years. Rent and food do not count at all: £27,000 over three years appears in both branches and cancels. So the opportunity cost is £27,750 + £60,000 = £87,750, of which more than two thirds is money nobody ever sees on an invoice. This is why the fee debate and the cost of a degree are different subjects.

Now you. The same student instead considers a two-year apprenticeship with no fees, paying £12,000 a year, against the same £20,000 job. What is the opportunity cost of the apprenticeship?

Answer

There are no fees, and rent and food cancel again. The only sacrifice is the wage gap: £20,000 minus £12,000 is £8,000 a year for two years, so £16,000. The apprenticeship costs about a fifth of what the degree costs, and neither figure says anything yet about what either is worth.

Economists are not reliably better at this than anyone else. Paul Ferraro and Laura Taylor put a four-option opportunity cost question to professional economists at the 2005 American Economic Association meetings, about the value of a free concert ticket, and 21.6 per cent chose the right answer. Random guessing would have scored 25. The concept is easy to state and genuinely hard to apply, which is why it is worth practising rather than merely reading.

The production possibility frontier

Put the idea on a diagram. Take an economy with one resource, 100 hours of labour a day, and two goods, fish and coconuts. Suppose catching $F$ fish takes $F^{2} / 100$ hours and gathering $C$ coconuts takes $C^{2} / 25$ hours. Both are convex in output, which is the assumption that the easy fish are caught first and each additional one is harder. Using all the labour,

\frac{F^{2}}{100} + \frac{C^{2}}{25} = 100

or, cleared of fractions, $F^{2} + 4 C^{2} = 10000$ . That curve is the production possibility frontier: every combination on it uses the whole resource, everything inside it is feasible but wasteful, and everything outside it is impossible with today's technology. Setting $C = 0$ gives 100 fish, and setting $F = 0$ gives 50 coconuts.

Three readings of the picture matter. A point strictly inside the curve means unemployment or misallocation, and moving to the frontier gives more of both goods with no sacrifice, which is why the phrase "no such thing as a free lunch" is a statement about points on the frontier only. A point outside is reachable only by acquiring more resources or better technology, which is what growth is. And on the frontier itself, more of one good is available only at the price of less of the other, which is opportunity cost drawn as a slope.

The slope is a price

Differentiate the frontier implicitly, which is what implicit differentiation is for: $2 F d F + 8 C d C = 0$ , so

\frac{d C}{d F} = - \frac{F}{4 C}

The magnitude of that slope is the marginal rate of transformation: how many coconuts one more fish costs at the margin. It is not a fixed number. At $F = 60$ the constraint gives $4 C^{2} = 10000 - 3600 = 6400$ , so $C = 40$ , and the rate is $60 / 160 = 0.375$ coconuts per fish. Push production to $F = 80$ and $C$ falls to 30, and the rate rises to $80 / 120 = 0.67$ coconuts per fish. The more fish already being caught, the more each extra fish costs, which is exactly the curvature the assumption about hard fish put in.

Example. On the same frontier $F^{2} + 4 C^{2} = 10000$ , the economy is at $F = 60$ . It wants ten more fish. Estimate the coconut cost using the marginal rate, then compute it exactly, and say why the two differ.

The marginal rate at $F = 60$ is 0.375, so ten fish look like a sacrifice of 3.75 coconuts. Exactly: at $F = 70$ , $4 C^{2} = 10000 - 4900 = 5100$ , so $C = 35.71$ , a fall of 4.29 coconuts. The linear estimate understates the cost because the rate is rising over the interval, and it is evaluated only at the start. Marginal reasoning is a first derivative, so it is exact for a small step and an approximation for a large one, a caveat worth carrying through the whole course.

Now you. On the same frontier, find the marginal rate of transformation at $F = 80$ , and the exact coconut cost of moving from $F = 80$ to $F = 90$ .

Answer

At $F = 80$ , $C = 30$ and the rate is $80 / (4 \times 30) = 0.67$ coconuts per fish. At $F = 90$ , $4 C^{2} = 10000 - 8100 = 1900$ , so $C = 21.79$ . The exact cost of the ten fish is $30 - 21.79 = 8.21$ coconuts, against a linear estimate of 6.7. The gap is much wider than before, because the frontier is far steeper here.

The same slope reasoning is what settles who should produce what. If a second island gives up only 0.2 coconuts per fish where this one gives up 0.375, the second island has the lower opportunity cost in fish, and both gain from specialising and trading at any rate between the two. That is comparative advantage, and it depends on the ratio of the slopes, never on who is better at things in absolute terms.

Thinking at the margin

Now the rule the rest of the course runs on. Suppose an activity produces benefit $B (x)$ and costs $C (x)$ , both differentiable. The net gain is $B (x) - C (x)$ , and at an interior maximum its derivative vanishes:

B' (x) = C' (x)

Do the thing until the marginal benefit of one more unit equals the marginal cost of it. This is not a new economic principle. It is the first-order condition from calculus, with the two derivatives given names, and it is worth saying so plainly, because it means the economics is entirely in the choice of $B$ and $C$ and never in the mathematics.

Two conditions have to hold for the rule to be right, and both fail in real problems often enough to be worth stating. The stationary point must be a maximum rather than a minimum, so $B'' < C''$ there, which is the usual second derivative test and is the formal content of "diminishing returns". And the optimum must be interior. If marginal benefit is below marginal cost at every positive $x$ , the answer is $x = 0$ and the derivatives never meet: the right amount of a bad investment is none of it, and no first-order condition will tell you that. Corner solutions come back properly in the consumer's problem two lessons from now.

The rule also explains why totals mislead. Water is worth more to a human being than diamonds are, and yet diamonds trade for more. Adam Smith found this genuinely puzzling in 1776. The resolution is that no one ever chooses between all the water and all the diamonds. The choice on offer is one more litre against one more carat, and where water is abundant the marginal litre is worth almost nothing while the marginal carat is not. Price tracks marginal value, not total value, and almost every paradox about what things are worth dissolves into that distinction.

Equalising at the margin

When one fixed resource is split between several uses, the marginal rule becomes an equalisation. If it were true that the last pound spent in one place bought more than the last pound spent in another, moving a pound would raise the total, so no such gap can survive at the optimum.

Example. A firm has £26,000 to spend on advertising in two regions. Revenue in region A is $60 \sqrt{a}$ thousand pounds when $a$ thousand is spent there, and revenue in region B is $40 \sqrt{b}$ . How should the budget be split?

The marginal returns are $d R_{A} / d a = 30 / \sqrt{a}$ and $d R_{B} / d b = 20 / \sqrt{b}$ . Setting them equal gives $30 / \sqrt{a} = 20 / \sqrt{b}$ , so $\sqrt{a} = 1.5 \sqrt{b}$ and $a = 2.25 b$ . With $a + b = 26$ that gives $3.25 b = 26$ , so $b = 8$ and $a = 18$ . Both marginal returns are then $30 / \sqrt{18} = 20 / \sqrt{8} = 7.07$ thousand pounds per thousand spent. Total revenue is $60 \sqrt{18} + 40 \sqrt{8} = 367.7$ thousand, against 360.6 thousand for an even split of 13 and 13. The even split feels fair and costs £7,100.

Now you. The budget is £40,000, region A still returns $60 \sqrt{a}$ , and region B now returns only $30 \sqrt{b}$ . Find the split and the total revenue.

Answer

Equalising, $30 / \sqrt{a} = 15 / \sqrt{b}$ , so $\sqrt{a} = 2 \sqrt{b}$ and $a = 4 b$ . With $a + b = 40$ , $b = 8$ and $a = 32$ . Revenue is $60 \sqrt{32} + 30 \sqrt{8} = 339.4 + 84.9 = 424.3$ thousand pounds, against 402.5 thousand for an even split.

What is not a cost

The mirror image of opportunity cost is the sunk cost: money or effort already committed and unrecoverable whatever happens next. It appears in no branch of the decision, so it cancels exactly as the rent did, and the correct treatment is to ignore it entirely.

People do not. Hal Arkes and Catherine Blumer ran the clean demonstration in 1985, selling season tickets to the Ohio University theatre and randomly discounting some buyers to two thirds or roughly half the normal price without telling them the discount was coming. All three groups had identical tickets and identical plays ahead of them, and the only difference was money already spent. Full-price buyers attended significantly more plays in the first half of the season. The past payment changed behaviour that it could not rationally affect. The pattern has a name in policy too: the British and French governments continued funding Concorde long after the projected returns had collapsed, on the argument that too much had already been spent to stop.

This is the first place where the model and the people come apart, and it will not be the last. The claim that people ignore sunk costs is false as a description and useful as a prescription, and those are different claims about the same sentence. Keeping them separate is most of what it takes to use this subject honestly. The final lesson returns to the evidence in full.

What the rule still lacks is a measure of benefit. Every optimisation so far has assumed a function $B (x)$ handed over from outside: revenue in pounds, coconuts on a frontier. For a consumer choosing between goods there is no such natural unit, only the fact that they prefer some bundles to others. Turning a ranking into a differentiable function is the next lesson's problem, and it turns out to need surprisingly little.