Clifton Economics · ECON10005 · Week 1

Foundations: sets, intervals and absolute values

Notation you will use all year

Before any calculus, you need the language for describing which numbers are allowed. Almost every question this year quietly depends on it — a price cannot be negative, a quantity cannot exceed capacity, and a logarithm refuses anything at or below zero.

A set is just a collection of objects. Each object in it is an element. We write x ∈ A to say x is an element of A, and B ⊂ A to say every element of B also sits inside A.

ℕ — natural numbers

The counting numbers 1, 2, 3, … Nothing negative, nothing fractional.

ℤ — integers

Zero together with every whole number, positive or negative: 0, ±1, ±2, …

ℝ — real numbers

Every point on the number line, including fractions and irrationals such as π.

ℝ₊ and ℝ₊₊

The non-negative reals (zero allowed) and the strictly positive reals (zero excluded).

Figure 1 — the three sets of numbers are nested, not separate

An interval names a stretch of the real line. The only thing to get right is whether the endpoints are included:

Figure 2 — the four interval types

Finally the absolute value strips off a minus sign — it is the distance from zero, and so never negative:

|x| = x if x ≥ 0, |x| = −x if x < 0

Insight. The definition of |x| is a piecewise one: two rules stitched together at a single point. That stitch is exactly why the absolute value turns up again at the end of this week as the standard example of a function with no derivative.

Functions, domain and range

Core concept

A function connects an input to an output, with one iron rule: each input gives exactly one output. Write it y = f(x), where x is the independent variable you choose, y is the dependent variable that results, and f is the rule connecting them.

Watch the difference between a variable and a parameter. In y = αx^β, the letters α and β are parameters — fixed numbers that set the shape of the curve. Only x varies.

Domain

Every input value the function is allowed to take.

Range

Every output value that actually results.

The domain is rarely "all real numbers" in economics. Take a pottery at Zennor selling glazed planters, facing demand Q = D(P) = 84 − 12P. Price cannot be negative, so P ≥ 0. Quantity cannot be negative either, so 84 − 12P ≥ 0, giving P ≤ 7. The economics, not the algebra, sets the boundaries.

Domain: P ∈ [0, 7] → Range: Q ∈ [0, 84]

Figure 3 — domain along the input axis, range along the output axis

How do you check something really is a function? Use the vertical line test: if any vertical line crosses the graph more than once, one input is producing two outputs, and the rule is not a function.

Figure 4 — the vertical line test

One-to-one mappings and inverse functions

Core concept

Every function sends each input to one output. A one-to-one function does the reverse as well: each output traces back to exactly one input. Test it with the horizontal line test — the mirror image of the vertical one.

Figure 5 — the horizontal line test

Why this matters. A function has an inverse if and only if it is one-to-one. If two inputs shared an output, running the machine backwards would face an impossible choice between them.

To find the inverse, swap the roles: write the equation, then solve for the other variable. For the Zennor stall:

Q = 84 − 12P ⟹ 12P = 84 − Q ⟹ P = D⁻¹(Q) = 7 − Q/12

This is the inverse demand function. The original answers "how many units at price P?"; the inverse answers "what price will clear Q units?". Same relationship, read from the other end.

Graphically, the inverse is the original curve reflected in the line y = x. Every point (a, b) on f becomes the point (b, a) on f⁻¹.

Figure 6 — reflection in the 45° line

Some functions are one-to-one only after you restrict the domain. f(x) = x² fails the horizontal line test on all of ℝ, because 3 and −3 both map to 9. Restrict it to x ≥ 0 and it passes, with inverse f⁻¹(y) = √y.

Differentiation and rates of change

Core technique

The derivative f ′(a) measures how steep the graph is at the single point x = a. Precisely, it is the slope of the tangent to the curve y = f(x) at the point (a, f(a)).

Figure 7 — the derivative is a slope at a point, not over a stretch

Point a40

Four notations mean exactly the same thing, and you will meet all four:

y ′ f ′(x) dy/dx d f(x)/dx

Assumed knowledge. The rules of differentiation — power, product, quotient and especially the chain rule — are treated as A-level material and are not re-taught. If they feel shaky, fix that now: every remaining week of this unit stands on them.

The sign of the derivative tells you the direction of travel, and whether flat stretches are permitted separates strict from weak:

Figure 8 — strictly and weakly monotonic functions

In economics the derivative gets a name: marginal. For a firm producing x units in a period —

Marginal cost

C ′(x), the extra cost of one more unit.

Marginal revenue

R ′(x), the extra revenue from one more unit.

Profit

π(x) = R(x) − C(x).

Marginal profit

π ′(x) = R ′(x) − C ′(x).

Worked example. Trelights Forge has cost C(q) = 2q² + 4q + 30 for q wrought-iron brackets.

C ′(q) = 4q + 4 → C(5) = 100 and C ′(5) = 24

So the 100 pounds already spent on five brackets tells you nothing about the 24 pounds the sixth will cost. Total and marginal are different questions.

Relative rate of change and elasticity

Core technique

A derivative is an absolute rate of change: pounds per tub, litres per hour. Often the more useful question is proportional — how big is the change relative to the level you started from? That is the relative rate of change:

relative rate of change of f at b = f ′(b) / f(b)

For Trelights at q = 5, this is 24 / 100 = 0.24. Producing one more bracket raises total cost by 24% of its current level.

Apply the same idea to demand and you get the single most used number in microeconomics. Price elasticity of demand scales the slope by price and quantity, which strips out the units entirely:

E_D = [D ′(p) / D(p)] · p = (dQ/dP) · (P/Q)

Read it as: a 1% rise in price changes quantity demanded by E_D per cent. Supply works identically, with E_S = [S ′(p) / S(p)] · p.

Two standard results worth recognising on sight:

D(p) = Mp^−c

Elasticity is exactly −c, the same at every price. This is why such functions are called constant-elasticity demand.

D(p) = K

A flat, unresponsive demand: D ′(p) = 0, so E_D = 0. Price changes do nothing — think of table salt.

The classic trap. Elasticity is not the slope. On a straight-line demand curve the slope is the same everywhere, yet elasticity runs from nearly zero at low prices to unboundedly large at high ones. The one place they seem to agree is the midpoint, where E_D = −1.

Figure 9 — elasticity along a constant-slope demand curve

Higher-order derivatives, concavity and convexity

Core technique

Differentiate f and you get f ′. If f ′ can itself be differentiated, the result is the second derivative f ″(x) — written also as y ″, d²y/dx² or d²f(x)/dx². It measures how the slope is changing.

That is precisely what curvature means, and it gives a test you will use in every optimisation week from here on:

Concave

f ″(x) ≤ 0 throughout the interval. The slope is falling; any chord lies below the graph. Strictly concave if f ″(x) < 0.

Convex

f ″(x) ≥ 0 throughout the interval. The slope is rising; any chord lies above the graph. Strictly convex if f ″(x) > 0.

Figure 10 — the chord test

A function need not be one or the other everywhere. Take f(x) = x³ − 9x, so f ′(x) = 3x² − 9 and f ″(x) = 6x. The second derivative changes sign at x = 0: concave to the left, convex to the right. That switching point is an inflection point.

Figure 11 — curvature flips at the inflection point

Note the asymmetry. A linear function has f ″(x) = 0 everywhere, so it satisfies both weak conditions at once — it counts as weakly concave and weakly convex. It is neither strictly.

Limits, continuity and L'Hôpital's Rule

Core technique

A limit asks what a function approaches near a point — deliberately ignoring what happens at the point. That is the whole trick: lim_x→a f(x) can exist perfectly well even where f(a) is undefined.

A function is continuous at a when all three of these hold:

1. f(a) is defined 2. lim_x→a f(x) exists 3. lim_x→a f(x) = f(a)

Fail any one and f is discontinuous there. The most common failure is a jump, where the function approaches different values from either side. Those two values are the one-sided limits, and the limit exists only when they agree:

lim_x→a⁻ f(x) = lim_x→a⁺ f(x)

Figure 12 — a jump discontinuity

Limits also describe end behaviour — what happens as x runs off to ±∞:

Figure 13 — behaviour at the extremes

When both parts of a limit behave, the rules of limits let you break the problem up. If lim f(x) = A and lim g(x) = B, then sums, products and powers all pass straight through, and quotients do too provided B ≠ 0.

The interesting cases are the ones where they do not behave. Evaluate carelessly and you land on an indeterminate form — an expression that carries no information until you dig further:

0/0 ±∞/±∞ 0 × ∞ ∞ − ∞ 0^∞ 1^∞ ∞⁰

L'Hôpital's Rule rescues the first two. If f and g are differentiable and the quotient gives 0/0 or ±∞/±∞, then

lim_x→a f(x)/g(x) = lim_x→a f ′(x)/g ′(x)

differentiating top and bottom separately — this is not the quotient rule. Applied to a 0/0 case:

lim_x→2 (x² − 4)/(x³ − 8) = lim_x→2 2x / 3x² = 4/12 = 1/3

Two extensions. The rule can be applied repeatedly — lim_x→∞ x²/e^x needs two passes before it resolves to 0. And a 0 × ∞ form can be forced into a fraction first: x² ln x becomes ln x ÷ x⁻², which the rule then handles.

When the derivative fails to exist

Watch for this

Not every function has a derivative at every point. There are two ways it breaks down:

A kink

The graph has a sharp corner. The slope approaching from the left disagrees with the slope from the right, so no single tangent exists.

A discontinuity

The graph jumps. You cannot draw a tangent across a gap.

Figure 14 — continuous everywhere, differentiable everywhere except one point

The direction only runs one way. Differentiability implies continuity, but continuity does not imply differentiability — |x − 4| is the standing counterexample. So "the function is differentiable at a" is a sufficient condition for continuity there, never a necessary one.

Compound interest and geometric series

Core technique

Invest a principal P at rate r per period and reinvest the interest each time, and the balance grows by a factor of (1 + r) every period:

A = P(1 + r)^t

Put 500 pounds into a Polzeath Quarry bond at 20% a year and after three years it is worth 500 × 1.2³ = 864 pounds. Compare that with a flat 100 pounds a year, which reaches only 800 — the gap widens with every period, because compounding earns interest on interest.

Figure 15 — compound against simple growth

Rearranging that same formula answers the reverse question — how long, or what is a future sum worth today? Since t sits in the exponent, you take logarithms to release it:

t = [ln(A) − ln(P)] / ln(1 + r) and P = A · (1 + r)^−t

That second form is the present value of money received t periods in the future — the standard way to compare payments arriving at different dates.

Now string several such payments together. A geometric series multiplies each term by a constant k:

a + ak + ak² + … + akⁿ⁻¹ = a · (1 − kⁿ) / (1 − k) (k ≠ 1)

If the ratio is small enough that terms shrink away, the sum settles on a finite total even with infinitely many terms:

a + ak + ak² + … = a / (1 − k) provided |k| < 1

Figure 16 — partial sums converging

Count your terms. The single most common error here is an off-by-one. The formula above runs from ak⁰ to akⁿ⁻¹, which is n terms in total. A stream of payments in years 1 to 4 is four terms — check by writing the first and last out in full before reaching for the formula.

Introduction

Foundations: sets, intervals and absolute values

Functions, domain and range

One-to-one mappings and inverse functions

Differentiation and rates of change

Relative rate of change and elasticity

Higher-order derivatives, concavity and convexity

Limits, continuity and L'Hôpital's Rule

When the derivative fails to exist

Compound interest and geometric series