Functioning Markets

Economics & Law - NIT Northern Institute of Technology / TUHH, Hamburg · part of my Technology Management MBA · study notes for revision.

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So far we’ve drawn demand and supply as two lines that cross, and said the crossing point is “good”. But that hides the real question: why does trade make people better off in the first place? A demand curve tells us how much someone buys at a price; it doesn’t show why a swap between two people can leave both of them happier.

To see that, we need to model a person’s preferences more carefully - not just “how many units at what price”, but the whole map of what someone would trade for what. This chapter builds that map (indifference curves), adds the wallet (the budget line), then puts two people in one diagram (the Edgeworth box) to watch gains from trade appear out of nothing but a good starting mismatch. That sets up a clean idea: a well-functioning competitive market squeezes out every mutually beneficial trade - it is efficient.

Tip

The whole chapter in one line: draw what a person is willing to trade (indifference curves), draw what they can afford (budget line), and trade keeps happening until no swap can help one person without hurting the other. That end-state is what “the market works” really means.

1 · Preferences and the indifference curve

Picture a consumer choosing between just two goods - say pizza and burgers. We want to describe everything about their taste: is 10 pizza and 20 burgers better than 15 of each? Better than 5 and 30? Listing every pair is hopeless, so economists use a neat trick.

An indifference curve joins together all the bundles that give the consumer exactly the same satisfaction - the same utility (utility is just a made-up “happiness score” we attach to a bundle so we can rank bundles). The consumer is genuinely indifferent between every point on one curve: swap along it and they neither gain nor lose. That’s where the name comes from.

Bundle	Pizza	Burgers	Feeling
A	4	1	equally happy
B	3	2	equally happy
C	2	3	equally happy
D	1	4	equally happy

All four bundles sit on one indifference curve. Notice the pattern: give up a pizza and you must be handed more burgers to stay level. Now the key move - there isn’t just one curve. There are infinitely many, one for every possible level of satisfaction, stacked like contour lines on a map.

IC₃ - highest utilityfar from origin: more of both goods

▲ more preferred

IC₂ - medium utilitya better bundle than anything on IC₁

▲

IC₁ - lowest utilityclose to origin: less of both goods

An indifference map: burgers on the vertical axis, pizza on the horizontal. Each curve is a set of equally-good bundles. Because people prefer more to less, a curve sitting further from the origin is unambiguously better - every point on IC₃ beats every point on IC₁.

The single most useful fact: higher (further-out) curve = higher satisfaction. More of both goods is better, so the consumer always wants to climb onto the highest curve they can reach.

2 · The shape of a curve: three properties

Indifference curves aren’t drawn any old way. Three properties fall straight out of how sensible people rank things.

Downward-sloping the trade-off

If you take away some pizza, the only way to keep the consumer equally happy is to hand back more burgers. Less of one good must be paid for with more of the other - so the curve slopes down from left to right.

Never cross no contradiction

If two curves crossed at a point, that shared bundle would have to give two different utility levels at once - a logical contradiction. So curves can sit above or below one another, but they never intersect.

Convex to the origin bowed inward

The curve bends in toward the corner. This says people like balanced bundles more than extremes: 3 apples and 3 bananas beats 6 apples and 0 bananas, even though both are “six fruits”.

Why the inward bow specifically? Because a good you have little of feels precious, and a good you’re swimming in feels cheap. When you’re down to your last banana you’ll demand a lot of apples to part with it; when bananas are piled high you’ll let one go for almost nothing. That changing willingness to swap is exactly what bends the curve - and it has a name, which is the next section.

Note

This is the same idea as the law of demand, wearing different clothes. A convex curve says the consumer will “pay” (give up other goods) less and less for each additional unit of a good they already have plenty of. Diminishing willingness to substitute here mirrors the downward-sloping demand curve from Chapter 2 - the underlying human fact is diminishing marginal utility.

3 · The marginal rate of substitution (MRS)

The marginal rate of substitution (MRS) is the star of this chapter. Definition in one breath:

The MRS is how much of one good a consumer is willing to give up to get one more unit of the other good, while staying equally happy (i.e. staying on the same indifference curve).

Geometrically, the MRS is just the slope of the indifference curve at a point (taken as a positive number). Because MRS compares the value of the two goods at the margin, it also equals the ratio of their marginal utilities:

MRSMRS = MUₓ / MUₖ = −(ΔY / ΔX)

MUₓ marginal utility of good X - the extra happiness from one more X

MUₖ marginal utility of good Y - the extra happiness from one more Y

ΔY / ΔX burgers given up per extra pizza - the curve’s slope

A worked feel for it. Say a consumer starts with lots of donuts and little coffee - 5 donuts, 2 coffees. Coffee is scarce for them, so it’s precious: they’d happily give up 3 donuts to get 1 more coffee. Here MRS equals 3 - one coffee is worth three donuts to this person, right now.

Now the crucial twist. As they keep trading donuts for coffee and move along the curve, coffee stops being scarce and donuts become the rare thing. Their willingness to hand over donuts collapses:

Coffee gained	Donuts they’ll sacrifice	MRS
1st extra coffee	3 donuts	3
2nd extra coffee	2 donuts	2
3rd extra coffee	1 donut	1

This falling MRS is the convex shape - the slope gets gentler as you slide down the curve. Diminishing MRS and “bowed toward the origin” are two descriptions of the same picture. Hold on to the MRS: it’s the quantity that has to match up for a person to be optimally set, and later, for a trade between two people to be exhausted.

4 · What you can afford: the budget line

Preferences tell us what a consumer wants. They don’t care about money. The budget line brings in the hard constraint: a limited income and given prices.

A budget line shows every bundle the consumer can just afford if they spend their whole income. With income I, price Pₓ for good X and Pₖ for good Y, the affordable-and-fully-spent bundles satisfy:

Budget linePₓ·X + Pₖ·Y = I

Slopeslope = −(Pₓ / Pₖ)

Meaning the price ratio - how many burgers the market makes you give up to buy one more pizza

Concretely: a student has 100 euros, pizza costs 10, a burger costs 5. Spend it all and the affordable frontier runs like this:

Pizza	Burgers	Spend
10	0	100 €
8	4	100 €
5	10	100 €
0	20	100 €

The slope is minus 10 over 5, that is minus 2: to buy one more pizza you must give up two burgers. That number is set by the market, not by taste. Where a bundle sits relative to the line matters:

On the line - affordable, whole income usedInside the line - affordable, but money left unspentOutside the line - unaffordable, out of reach

And the line moves. If income rises, the whole line shifts outward, parallel to itself - you can buy more of both goods, the trade-off unchanged. If one price changes, the line rotates around the unchanged intercept - cheaper pizza swings the pizza-end outward, so the price ratio itself changes.

5 · The consumer’s optimum: MRS = price ratio

Now put the wants and the wallet on the same diagram. The consumer wants the highest indifference curve possible, but is trapped on or below the budget line. Push out to a curve too high and it never touches the line - unaffordable. Settle for a curve that cuts through the line in two places and you’ve left happiness on the table - you could slide to a higher curve still within budget.

The best you can do is the curve that just kisses the budget line at a single point - it is tangent to it. Call it point E.

Willingness to tradeMRS = slope of indifference curve

must equal at the optimum

Market trade-offprice ratio = slope of budget line

↓

Optimal bundle Ehighest reachable curve, tangent to budget

At the tangency point E the two slopes are identical, so the consumer’s personal exchange rate between the goods exactly matches the market’s. Nowhere else can they climb higher without spending money they don’t have.

At that tangency the slopes are equal, which gives the single most important condition in consumer theory:

Consumer optimumMRS = Pₓ / Pₖ

Left side the rate at which the consumer is willing to swap goods (taste)

Right side the rate at which the market lets them swap goods (prices)

The intuition is lovely. Suppose in the market one pizza trades for two burgers, but you personally value a pizza at three burgers. Then buying a pizza (paying two burgers’ worth) hands you three burgers’ worth of happiness - a bargain, so you keep buying pizza. Doing so slides your MRS down until it hits 2, and the bargain vanishes. You stop adjusting exactly when your MRS equals the price ratio. If your personal valuation is below the market’s, you’d run the swap the other way. Only when the two are equal is there no move left worth making.

6 · Two people, one box: the Edgeworth box

Everything so far is one person. The magic starts with two. The Edgeworth box is a diagram that squeezes two consumers and a fixed total of two goods into a single rectangle - and lets us watch them trade.

The course’s example is Robinson, stranded on an island, who one day (a Friday) discovers he isn’t alone. Their situation is a pure-exchange economy: two people, two goods, fixed quantities, no production - just whatever nature handed out, to be swapped.

Robinson west coast

• Humid coast - bananas thrive, coconuts don’t
• Endowment: 8 bananas, 2 coconuts
• Drowning in bananas, starved of coconuts

Friday east coast

• Dry coast - coconuts thrive, bananas don’t
• Endowment: 2 bananas, 8 coconuts
• Drowning in coconuts, starved of bananas

The island holds 10 bananas and 10 coconuts in total, and always will - nobody grows more. Those two totals set the size of the box: 10 wide, 10 tall. Here’s the clever construction:

Bottom-left corner = Robinson’s originmeasure his goods rightward & upward

the SAME box, read two ways

Top-right corner = Friday’s originmeasure his goods leftward & downward, upside-down

↓

Any single point = one full allocationit splits all 10 bananas & all 10 coconuts between them

The trick: Friday’s diagram is flipped 180° and pinned to the top-right corner. Because the box is exactly 10×10, whatever Robinson doesn’t hold, Friday must - so one dot fixes both people’s bundles at once. The starting dot, the endowment, is (8 bananas, 2 coconuts) for Robinson, which automatically leaves (2, 8) for Friday.

So a point is an allocation. Move the dot and you’re re-dividing the same fixed pie between the two islanders - no goods created or destroyed, just shifted. Each person carries their own indifference map into the box. Robinson’s utility rises as his dot moves toward the top-right (more of both, for him); Friday’s rises toward the bottom-left (more of both, for him, since his diagram is upside-down). Their two families of curves are laid over each other in the one box.

7 · Gains from trade and the contract curve

Start at the endowment: Robinson with 8 bananas and 2 coconuts, Friday with 2 bananas and 8 coconuts. Both are lopsided and both would love a more balanced diet. Can they help each other? Draw the indifference curve each person is on at that starting point. Between the two curves there’s a little almond-shaped region - the lens.

Tip

The lens is the deal zone. Every allocation inside the lens sits on a higher curve for both Robinson and Friday than the endowment does. Any point in there makes both better off at once - a genuine win-win. Economists call such a move a Pareto improvement: at least one person gains and nobody loses. So Robinson trades away some surplus bananas, Friday trades away some surplus coconuts, and they step into the lens together.

But once inside, a new, smaller lens usually opens up between their new curves - so they trade again. When does it stop? Trading can only end when there’s no lens left at all - when the two indifference curves no longer overlap but merely touch, tangent to each other.

No trade left to makeMRS(Robinson) = MRS(Friday)

Why it’s the end both value the two goods at the same exchange rate, so no swap can help one without hurting the other

Think about why equal MRS kills all further trade. A swap only helps both people when they disagree about relative value - when Robinson prizes coconuts more than Friday does, there’s room for Robinson to buy coconuts cheaply from Friday and both feel richer. The instant their MRS values line up, that disagreement is gone, and with it every remaining bargain.

Now the payoff idea. There isn’t just one such tangency - there’s a whole string of them, one for each possible starting balance of power. Join up all the points where a Robinson-curve is tangent to a Friday-curve and you trace out the contract curve.

Contract curve = every allocation where MRS values are equalEvery point on it is Pareto-efficientNo further mutually beneficial trade exists there

Pareto-efficient means: you cannot make anyone better off without making someone else worse off - the mutual gains are fully used up. The contract curve is the complete set of those efficient allocations.

Caution

Efficient is not the same as fair. The contract curve is a whole line, running from “Robinson gets almost everything” to “Friday gets almost everything”. Every point on it is efficient, yet they differ wildly in fairness. Which one you land on depends on the starting endowment - who owned what before trading began. Efficiency tells you the pie has no waste left in it; it says nothing about how justly the pie is sliced. That gap between efficient and fair is where law and policy step in - and where the next chapter lives.

8 · Why this means “the market works”: the First Welfare Theorem

Here’s the quiet punchline that all of this was building toward. Suppose that instead of haggling face to face, Robinson and Friday simply respond to a market price for bananas in terms of coconuts. Each of them, acting purely selfishly, buys and sells until their own MRS equals that price ratio - exactly the optimum from Section 5.

But if both set their MRS equal to the same market price ratio, then their MRS values must be equal to each other - which is precisely the contract-curve condition. So the market, with no one intending it, lands the pair on a Pareto-efficient allocation.

Each trader is selfishsets own MRS = market price ratio

same price ratio for both

So both MRS values matchMRS(Robinson) = MRS(Friday)

↓

Outcome is on the contract curvePareto-efficient - no waste left

The First Welfare Theorem in miniature: a competitive market, under the right conditions, delivers an efficient allocation automatically. Self-interest plus a common price does the work that careful bargaining would.

This is the First Welfare Theorem: a competitive market, under the right conditions, produces a Pareto-efficient outcome. It’s the formal spine of Adam Smith’s “invisible hand” - the market quietly exhausts every gain from trade. That’s what we mean by a functioning market: not that it’s fair, but that it wastes nothing.

Notice the load-bearing phrase: “under the right conditions.” Everything above assumed tidy things - many price-taking traders, no cheating, no spillovers onto third parties, everyone fully informed, goods you can own and exclude others from. When those assumptions hold, the market is efficient. When they break - monopoly power, pollution, hidden information, public goods - the invisible hand fumbles, and the outcome drifts off the contract curve. Naming and diagnosing those breakdowns is exactly the job of the next chapter.

Revision summary

Tip

The one-minute recap

• An indifference curve joins bundles giving the same utility; the consumer is indifferent along it. Curves slope down, never cross, and bow inward (convex); a curve further from the origin is better.
• The marginal rate of substitution (MRS) is how much of one good you’ll give up for one more of the other while staying equally happy - the slope of the curve. It diminishes as you move along the curve (that’s the convex shape).
• The budget line, Px·X + Py·Y = I, shows what’s affordable; its slope is the price ratio Px / Py. Income shifts it out (parallel); a price change rotates it.
• The consumer’s optimum is where the highest reachable curve is tangent to the budget line: MRS = price ratio - personal valuation equals the market’s.
• The Edgeworth box stacks two people (origins at opposite corners) and a fixed total of two goods; one point is a full allocation. From the endowment, trade moves both into the lens (a Pareto improvement) and continues until curves are tangent - where MRS(A) = MRS(B) and no win-win remains.
• The set of all such tangencies is the contract curve - the Pareto-efficient allocations. Efficient is not the same as fair; which point you reach depends on the starting endowment.
• First Welfare Theorem (intuition): a competitive market, under the right conditions, puts both MRS values equal to one common price ratio - landing on the contract curve. A working market is efficient. When the conditions break, it isn’t.

Next: Market Failure, Information & Behaviour → - what happens when the conditions for a good market break down.