First, today's new question. Also, don't forget to scroll down for the worked solution to #002.

The textbook claim: If you've studied a monopolist or any firm facing a downward-sloping demand curve, you'll have seen the standard diagram: an average revenue curve (AR), which is the demand curve itself, and a marginal revenue curve (MR) sitting below it, twice as steep and hitting the horizontal axis at exactly half the quantity.

It's drawn that way in every A-level textbook. But why is it true? The textbook usually states it as a fact and moves on. Your job today is to prove it.

P.S. roll up your sleeves and explore the question… once you've grappled with it for a while, if you are still stuck… graduated hints at the bottom of this page — but challenge yourself first!

21% → 61%

Average US firm markup above marginal cost — comparing 1980 to recent estimates.

So what? The 61% is an average. The authors find the rise is driven almost entirely by a small group of high-markup firms at the top of the distribution. Which firms might those be — and what does that tell you about the modern economy?

Source: De Loecker, Eeckhout & Unger, Quarterly Journal of Economics (2020).

Last issue I gave you a consumption function — C = 40 + 0.9Y − 0.002Y², defined for 0 ≤ Y ≤ 200 — and asked you to work through what it tells us about how consumption, the marginal propensity to consume, and the multiplier all change as national income changes. Here's how I'd work through it.

This is a question about reading the shape of the function, not just the numbers in it.

Look at the function term by term:

The combined effect: at low Y, the linear +0.9Y term dominates, and consumption rises briskly with income. As Y grows, the −0.002Y² term grows faster than linearly, and starts to drag the rate of increase down. Consumption still rises, but each extra pound of income adds less to consumption than the pound before it.

The economic intuition is straightforward: rich households consume a smaller share of additional income than poor households do. Give £1,000 to a household earning £15,000 and most of it is spent. Give £1,000 to a household earning £150,000 and a much bigger share is saved. The −0.002Y² term is the mathematical way of building that empirical fact into a model.

This is the textbook idea that the marginal propensity to consume falls as income rises — and it's exactly what Q2 will make precise.

The marginal propensity to consume (MPC) is the rate at which consumption changes with respect to income — exactly the same ∆-to-d move we made in #001.

Differentiating C = 40 + 0.9Y − 0.002Y²:

So:

A quick sense check. At Y = 0, MPC = 0.9 — households spend 90p of every extra pound at low incomes. At Y = 100, MPC = 0.9 − 0.4 = 0.5 — they now spend only 50p of every extra pound. At Y = 200, MPC = 0.9 − 0.8 = 0.1 — they spend just 10p of every extra pound. The pattern matches the intuition from Q1 exactly: MPC starts high and falls as Y rises.

The multiplier formula given is:

So we need to solve:

For the multiplier to be greater than 2, we need 1 − MPC < ½, which gives:

Substituting our expression for MPC:

So the multiplier is greater than 2 for all values of Y below £100 billion. Above that level of national income, the MPC has fallen below 0.5 and the multiplier drops below 2.

A useful sense check at the boundary: at Y = 0, MPC = 0.9, so k = 1 / 0.1 = 10. At Y = 50, MPC = 0.7, so k = 1 / 0.3 ≈ 3.33. At Y = 100, MPC = 0.5, so k = 1 / 0.5 = 2. At Y = 200, MPC = 0.1, so k = 1 / 0.9 ≈ 1.11. The multiplier shrinks rapidly as Y rises — which matters more than you might expect, as Q6 will explore.

This is the boundary case from Q3. We need k = 2, which from the working above means MPC = 0.5. Setting:

So the multiplier equals exactly 2 at Y = £100 billion. Below this, the multiplier is bigger; above it, smaller.

[Diagram goes here: a graph with Y on the horizontal axis (0 to 200) and C on the vertical axis. The consumption curve starts at C = 40 (the autonomous consumption intercept), rises steeply at first, gradually flattens, and reaches its maximum at Y = 225 — but since we're only looking at 0 ≤ Y ≤ 200, we see the rising-but-flattening portion only. Mark Y = 100 on the horizontal axis as the "multiplier = 2" point. To the left of Y = 100, label the curve "Multiplier > 2 (steep section: high MPC)". To the right of Y = 100, label it "Multiplier < 2 (flattening section: low MPC)".]

Key features to include in the sketch:

The shape itself is the answer to the economic intuition question. Where the curve is steep, the MPC is high, and so the multiplier is high. Where the curve is flat, MPC is low, multiplier is low. The sketch is just a visual translation of what we've already proved algebraically.

This is the question that ties everything together — and it's the one an Oxbridge interviewer would press hardest on.

The mechanical answer comes straight from the maths: at low Y, MPC is high, so the multiplier k = 1 / (1 − MPC) is high. A £1bn injection of government spending creates a much larger total increase in Y when MPC is 0.9 than when MPC is 0.1. Compare: at Y = 0, the multiplier is 10, so a £1bn fiscal injection generates £10bn of additional national income. At Y = 200, the multiplier is about 1.11, so the same £1bn generates only £1.11bn.

But the economic intuition behind that result is what matters. Why is MPC higher at low Y?

When national income is low, the average household is poorer. Extra income gets spent — on food, on essentials, on goods that were previously unaffordable. There's no buffer of savings to absorb it. Money received quickly becomes money spent, which becomes someone else's income, which is mostly spent again. The multiplier process runs at full strength because each round of spending recirculates a large fraction.

When national income is high, the average household is richer and already meeting most of its needs. Extra income gets saved, paid down on debt, or invested abroad — all of which are leakages out of the circular flow. Each successive round of the multiplier loses more to leakage, and the process dies out faster.

This is the macroeconomic version of the same logic we've used throughout the question: the marginal effect of an extra pound depends on where in the income distribution that pound lands. And it's the empirical foundation for one of the most contested arguments in macroeconomic policy — that fiscal stimulus during recessions (when income is low and unemployment high) packs a much bigger punch than the same stimulus during booms.

This question is a stepping-stone from the textbook treatment of the multiplier (where MPC is a fixed number you're told, and k = 1/(1−MPC) is a single value) to the real economics — where MPC varies with income, time, household type, and economic conditions, and the multiplier varies along with it.

Every serious empirical estimate of the fiscal multiplier in the last 20 years has wrestled with this same idea: the multiplier isn't a constant of nature. It's bigger when the economy has slack (think 2009), smaller when it doesn't (think 2022). It's bigger when the policy targets low-income households (whose MPC is high) than high-income ones (whose MPC is low). It's bigger when monetary policy is at the zero lower bound, because central banks can't offset fiscal stimulus by raising rates. The IMF, the OBR, and the Bank of England all build models where the multiplier is allowed to vary — and the variation matters enormously for whether a given fiscal stimulus will work.

That's the move you're learning to make: from a textbook formula with a single number, to a model with a function. It's the same move we made for elasticity in #001. It will be the same move in every issue of The Margin from here.

Marginal gains… Answer for today’s question is out in two days… Come back then or subscribe to get it direct to your inbox!

P.S.

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