Often at Oxbridge interviews, they will ask candidates to sketch graphs. Sometimes they will be contextualised (spoiler: the next Q later this week will be!) but sometimes they will just be testing your ability to sketch a mathematical function in the abstract.

They are looking to see how methodical, intuitive and clear your approach to explaining the sketch is.

Try this for starters:

Sketch the curve:

y = x³ / (x² − 1)

for all real values of x for which the function is defined.

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 below — but challenge yourself first! (Answer comes in the next edition! Subscribe to receive it directly to your inbox!)

0.7% → 1.3%

This week’s marginal number is 1.3%.

That is the revised estimate of the UK’s average annual productivity growth between 2009 and 2019.

The old estimate was 0.7%.

At first glance, that sounds like good news: perhaps Britain’s productivity performance after the financial crisis was not quite as dire as we thought.

But here is the catch. The revision does not mean the economy suddenly produced more output. It mainly comes from the denominator. If measured hours worked are lower than previously thought, then output per hour worked looks higher.

In other words, productivity can rise not because the numerator has gone up, but because the denominator has gone down.

That is a lovely economics lesson hiding inside a statistic. Before asking whether productivity has improved, always ask: improved relative to what?

The textbook claim is this:

If a firm faces a straight-line downward-sloping demand curve, then:

Let’s prove it.

But first, one important qualification:

This result is true for a linear demand curve. In other words, it is true when the AR curve is a straight line. If demand is curved rather than straight, MR will still usually lie below AR, but it will not necessarily be exactly twice as steep.

So the textbook diagram is not just assuming downward-sloping demand. It is assuming straight-line downward-sloping demand.

Suppose the firm faces the following demand curve:

P = a − bQ

where:

Since average revenue is price, we can write:

AR = P

So:

AR = a − bQ

This is the firm’s average revenue curve.

It is also the demand curve.

Total revenue is price multiplied by quantity:

TR = P × Q

Since:

P = a − bQ

we substitute this into the total revenue expression:

TR = (a − bQ)Q

Now multiply out the brackets:

TR = aQ − bQ²

So total revenue is:

TR = aQ − bQ²

This is important because marginal revenue comes from total revenue.

Marginal revenue is the extra revenue from selling one more unit.

Mathematically, marginal revenue is the rate of change of total revenue with respect to quantity.

So we differentiate total revenue with respect to Q.

Starting with:

TR = aQ − bQ²

Differentiate term by term:

The derivative of aQ is:

a

The derivative of bQ² is:

2bQ

So:

MR = a − 2bQ

That is the marginal revenue curve.

We now have two equations:

AR = a − bQ

MR = a − 2bQ

Look carefully at the two equations.

They have the same vertical intercept:

a

When Q = 0:

AR = a − b(0) = a

and:

MR = a − 2b(0) = a

So AR and MR start from the same point on the vertical axis.

But their slopes are different.

The slope of AR is:

−b

The slope of MR is:

−2b

So MR is twice as steep as AR.

That proves the first part of the textbook claim.

Now let’s find where each curve hits the horizontal axis.

A curve hits the horizontal axis when its vertical value is zero.

So first, find where AR = 0.

We know:

AR = a − bQ

Set AR equal to zero:

0 = a − bQ

Rearrange:

bQ = a

So:

Q = a / b

Therefore, AR hits the horizontal axis at:

Q = a / b

Now do the same for MR.

We know:

MR = a − 2bQ

Set MR equal to zero:

0 = a − 2bQ

Rearrange:

2bQ = a

So:

Q = a / 2b

Therefore, MR hits the horizontal axis at:

Q = a / 2b

Now compare the two horizontal intercepts:

AR hits the horizontal axis at:

Q = a / b

MR hits the horizontal axis at:

Q = a / 2b

But:

a / 2b = half of a / b

So MR hits the horizontal axis at exactly half the quantity where AR hits the horizontal axis.

That proves the second part of the textbook claim.

If demand is linear, the firm’s AR curve can be written as:

AR = a − bQ

The firm’s MR curve is:

MR = a − 2bQ

So the two curves have the same vertical intercept, but MR has twice the negative slope.

That is why, in the standard monopoly diagram, MR lies below AR and reaches the horizontal axis halfway along.

Now for the intuition.

Why does MR fall faster than AR?

When a firm with market power sells one more unit, there are two effects.

First, it gains revenue from selling the extra unit.

But second, because it faces a downward-sloping demand curve, it must lower the price to sell that extra unit.

And that lower price does not just apply to the extra unit. It also applies to the units it was already selling.

That is the key point.

The marginal revenue from selling one more unit is not simply the price of that unit. It is:

the revenue from the extra unit minus the revenue lost from lowering the price on previous units

This is why MR lies below AR.

AR tells us the price per unit.

MR tells us the extra revenue from increasing output.

For a firm facing a downward-sloping demand curve, those are not the same thing.

Suppose demand is:

P = 100 − 2Q

Since AR equals price, we have:

AR = 100 − 2Q

So the AR curve has:

Total revenue is:

TR = P × Q

So:

TR = (100 − 2Q)Q

Multiply out:

TR = 100Q − 2Q²

Marginal revenue is:

MR = 100 − 4Q

So the MR curve has:

The MR slope is twice as steep as the AR slope.

AR hits the horizontal axis when:

0 = 100 − 2Q

So:

Q = 50

MR hits the horizontal axis when:

0 = 100 − 4Q

So:

Q = 25

MR hits the horizontal axis at half the AR quantity.

That is exactly what the textbook diagram shows.

The textbook diagram works because it assumes a straight-line demand curve.

If:

AR = a − bQ

then:

MR = a − 2bQ

So:

The diagram is not magic. It is just algebra.

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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