Clifton Economics · Week 1 — Introduction to Economics & Budget Constraints

What Is Economics About?

Core idea

Economics starts from one fact: resources are scarce, wants are unlimited. Every choice forces a trade-off — to have more of one thing you must give up something else. This is the engine of all economic reasoning.

Microeconomics — how individuals and firms decide, and how they interact in markets.
Macroeconomics — economy-wide outcomes: inflation, unemployment, growth. But macro emerges from micro decisions, so the two are linked.

Economists use models — simplified representations of reality. A model of a sugar tax needn't model a person's entire life. Simplification is a feature, not a bug.

Positive vs normative: Positive economics makes testable factual claims ("a tax reduces consumption"). Normative economics makes value judgements ("we should introduce a tax"). Science deals in positives; policy involves both.

Opportunity Cost

Definition

Opportunity cost is the value of the best alternative forgone when making a choice. It always includes both explicit costs (money paid) and implicit costs (earnings you gave up).

A night out example: Explicit costs — £10 entry + £20 drinks + £10 taxi = £40. Implicit costs — could have worked 3 hrs at £15/hr = £45. Total opportunity cost = £85 — far more than the cash spent.

The Budget Constraint

Model

A consumer's consumption bundle (x₁, x₂) describes how much of each good they choose. With income m and prices (p₁, p₂), the budget set is all affordable bundles:

p₁x₁ + p₂x₂ ≤ m (budget set) | p₁x₁ + p₂x₂ = m (budget line)

Rearranging: x₂ = m/p₂ − (p₁/p₂)x₁ — a straight line with y-intercept m/p₂, x-intercept m/p₁, and slope −p₁/p₂.

Quick drawing trick: find how much of good 2 you could buy spending everything on it (m/p₂), then how much of good 1 (m/p₁). Plot those two intercepts and connect them.

Interactive

Drag the sliders to change income and the two prices, and watch the budget line move.

Income m (£) £50

Price of good 1 p₁ (£) £5

Price of good 2 p₂ (£) £2.5

Drag income up → the line shifts parallel outward (same slope, both intercepts move). Drag a price up → the line pivots inward on that axis only (one intercept fixed, slope changes). These two effects are fundamentally different.

Why the Slope Equals −p₁/p₂

Derivation

Rather than memorising the slope, derive it. Suppose a consumer is on the budget line and decides to consume Δx₁ more of good 1. They must change their consumption of good 2 by Δx₂ to remain on the line. Writing the budget constraint before and after:

p₁x₁ + p₂x₂ = m and p₁(x₁+Δx₁) + p₂(x₂+Δx₂) = m

Subtracting gives p₁Δx₁ + p₂Δx₂ = 0

therefore Δx₂/Δx₁ = −p₁/p₂

This is the slope — and it is the opportunity cost of good 1 measured in units of good 2. Consuming one extra unit of good 1 requires sacrificing exactly p₁/p₂ units of good 2. The slope tells you the rate at which the market will swap good 1 for good 2.

What Moves the Budget Line?

Comparative statics

Step through each scenario to see exactly what changes — and why.

Base: m=£50, p₁=£5, p₂=£2.50 — intercepts: 10 (y-axis), 20 (x-axis)

The ad valorem tax key insight: taxing all goods proportionally leaves relative prices p₁/p₂ unchanged — slope is the same — making it equivalent to a pure income reduction (parallel inward shift).

The Composite Good & the Numeraire

Extension

The composite good: good 2 can represent "everything else the consumer might buy," measured in pounds. Setting p₂ = 1 (one pound costs one pound), the budget constraint becomes p₁x₁ + x₂ ≤ m. This isn't a special case — all standard results carry over unchanged.

The numeraire: only relative prices and real income determine the budget set — not the absolute price level. We can always divide through by one price to reduce the problem. The good whose price we set to 1 is called the numeraire. For example, dividing the budget line by p₂ gives (p₁/p₂)x₁ + x₂ = m/p₂ — same budget set, one fewer free variable.

Key insight: if every price and income scale up by the same factor, the budget line is completely unchanged. Only relative prices matter — a doubling of all prices and income leaves consumer choices unaffected.

Taxes, Subsidies and Rationing

Policy

Three distinct policy tools each affect the budget line differently:

Quantity (specific) tax

Fixed amount t per unit of good 1. Effective price rises to p₁ + t. Line pivots inward around y-intercept — x-intercept falls, slope steepens.

Ad valorem (value) tax

Percentage τ on price, so effective price becomes (1+τ)p₁. On one good: pivots. On all goods equally: parallel inward shift — same as an income cut.

Lump-sum tax

Fixed amount taken regardless of what is consumed. Reduces m directly → parallel inward shift. Slope unchanged. Less distorting than quantity taxes.

Quantity subsidy

Government pays s per unit consumed, lowering effective price to p₁ − s. Mirror of a quantity tax: line pivots outward, expanding the budget set.

Rationing places a hard cap x̄₁ on how much of good 1 can be consumed. The affordable region is the normal triangle with the section beyond x̄₁ removed — producing a kinked budget line that turns vertical at x̄₁.

Real-world example: A means-tested food voucher scheme worth £400/month that can only be spent on groceries creates a kinked budget line — the consumer can spend up to £400 on groceries freely, but once that is exhausted, each extra pound of groceries must come at the expense of other goods.

Exam tip: a quantity tax on one good tilts (pivots) the budget line. A lump-sum tax or an equal ad valorem tax on all goods shifts it in parallel. Always identify which type of tax applies before drawing.

Your score

0 / 7

Write your answer, then reveal the model answer and mark yourself honestly.

Foundations

Q1 · Opportunity cost

A student has a free Sunday afternoon (6 hours). They can either revise economics or pick up a shift at their part-time job paying £10/hour. They also spend £3 on a revision guide.

(a) Identify the explicit and implicit costs of spending the afternoon revising.

(b) Calculate the total opportunity cost of revising.

(d) Under what condition would revising be the economically rational choice?

Model Answer

(a) Explicit cost: £3 (the revision guide). Implicit cost: the forgone wages from not working — 6 hrs × £10 = £60.
(b) Total opportunity cost = £3 + £60 = £63.
(c) The friend counts only explicit monetary costs, ignoring the implicit cost of time. Since time is scarce and could be used productively elsewhere, it always carries an opportunity cost — "free" activities are never truly free in an economic sense.
(d) Revising is rational when its expected benefit — higher exam scores, a better degree classification, greater future earnings or utility — exceeds £63. Economics compares the full opportunity cost with the full expected benefit.

Q2 · Constructing the budget constraint

A consumer has a weekly budget of £80. They spend it on two goods: cinema tickets at £8 each and takeaway meals at £4 each.

(a) Write down the budget constraint, defining your variables clearly.

(b) Calculate both intercepts and state what each represents.

(d) Check whether the bundle (5 cinema tickets, 10 meals) lies inside, on, or outside the budget set.

Model Answer

(a) Let c = cinema tickets, f = takeaway meals. Budget constraint: 8c + 4f = 80.
(b) Max cinema (f=0): c = 80/8 = 10; max meals (c=0): f = 80/4 = 20. Each intercept is the amount affordable if the whole budget is spent on that good.
(c) Slope = −p_c/p_f = −8/4 = −2. Each extra cinema ticket costs 2 takeaway meals — the opportunity cost of a ticket in terms of meals.
(d) Spending: 8(5) + 4(10) = 40 + 40 = £80 = m. The bundle costs exactly the full budget, so it lies precisely on the budget line — affordable, using all income. ✓

Shifts, pivots and policy

Q3 · Income and price changes

Start from the budget in Q2 (m=£80, p_c=£8, p_f=£4). For each scenario, describe precisely what happens to the budget line — which intercept(s) change, which stay fixed, how the slope changes, and whether the consumer is better or worse off.

(a) A pay rise increases the weekly budget to £120.

(b) A streaming service makes cinema tickets cheaper at £5 each.

(d) The government introduces a 25% ad valorem tax on all leisure spending.

Model Answer

(a) Parallel outward shift. New intercepts: max cinema = 15, max meals = 30. Slope unchanged at −2. Unambiguously better off — all original bundles remain affordable and new ones are added.
(b) Pivot outward about the meals intercept. Max cinema rises to 80/5 = 16; max meals fixed at 20. Slope flattens to −1.25. At least as well off — no original bundle is lost.
(c) Pivot inward about the cinema intercept. Max meals falls to 80/6 ≈ 13.3; max cinema fixed at 10. Slope steepens to −1.33. Worse off — meal-heavy bundles are no longer feasible.
(d) Effective prices: p_c = 1.25×8 = £10, p_f = 1.25×4 = £5. Slope = −10/5 = −2 (unchanged); intercepts fall to max cinema 8, max meals 16. Parallel inward shift — equivalent to cutting income to 80/1.25 = £64. Worse off; the tax acts like an income cut because relative prices are untouched.

Q4 · Algebraic budget line changes

A consumer faces budget line p₁x₁ + p₂x₂ = m. The price of good 1 triples, the price of good 2 doubles, and income rises by a factor of 6.

(a) Write the new budget line in terms of the original p₁, p₂ and m.

(b) Is the new line steeper or flatter? Show your reasoning.

Model Answer

(a) New prices p₁′ = 3p₁, p₂′ = 2p₂, m′ = 6m, so the line is 3p₁x₁ + 2p₂x₂ = 6m.
(b) New slope = −3p₁/2p₂ = −(3/2)(p₁/p₂). Since 3/2 > 1 it is steeper — good 1 has become relatively more expensive.
(c) Intercepts: x₁ = 6m/3p₁ = 2m/p₁ (doubled), x₂ = 6m/2p₂ = 3m/p₂ (tripled). Both grow, so the budget set expands; every affordable bundle stays affordable. The consumer cannot be worse off, whatever their preferences.

Q5 · Tax policy comparison

A government will collect the same total from a consumer under either: (A) a lump-sum tax of £20, or (B) a quantity tax of £2 per unit on good 1.

(a) Describe each policy's effect on the budget line.

(b) If the consumer originally bought 10 units of good 1, how much does policy B raise? Is it the same as A?

Model Answer

(a) Policy A (lump-sum): income falls by £20 → parallel inward shift, slope unchanged. Policy B (quantity tax): effective price of good 1 rises by £2 → pivots inward about the y-intercept, slope steepens.
(b) Revenue = £2 × 10 = £20 — the same as A here. But B changes relative prices, so the consumer may substitute away from good 1, pushing revenue below £20 if consumption falls.
(c) The lump-sum tax leaves relative prices unchanged, so the consumer still faces the original trade-off. The quantity tax distorts the price of good 1, forcing an inefficient substitution away from it — a deadweight loss. Same revenue, less distortion.

Numeraire & methodology

Q6 · Numeraire and relative prices

A consumer faces p₁ = £6, p₂ = £3 with income m = £90.

(a) Write the budget constraint and calculate both intercepts.

(b) Set good 2 as the numeraire (p₂ = 1). Rewrite the constraint in normalised form.

(d) If everything doubles (p₁=£12, p₂=£6, m=£180), what happens to the line, and why?

Model Answer

(a) 6x₁ + 3x₂ = 90. Max x₁ = 15, max x₂ = 30, slope = −2.
(b) Divide by p₂ = 3: 2x₁ + x₂ = 30 — normalised price of good 1 is 2, normalised income is 30.
(c) Identical intercepts (15 and 30) and slope (−2); normalising by p₂ changes the units, not the economics.
(d) Completely unchanged. 12x₁ + 6x₂ = 180 divides by 2 to 6x₁ + 3x₂ = 90. Only the ratio p₁/p₂ = 2 and real income m/p₂ = 30 matter. ✓

Q7 · Positive vs normative

Classify each statement as positive or normative, and briefly justify.

(a) "Raising the minimum wage to £12/hr will reduce low-skilled employment by about 3%."

(b) "The government should raise the minimum wage because workers deserve a fair income."

(d) "Economic growth should be prioritised over reducing inequality."

Model Answer

(a) Positive. A testable empirical claim about a policy's effect — confirmable or refutable with labour-market data.
(b) Normative. "Should" signals a value judgement about fairness that data alone cannot settle.
(c) Positive. A comparative empirical claim about cost-effectiveness, assessable with abatement-cost and emissions data.
(d) Normative. A value judgement about what matters more — output versus distribution. Two people with identical data could rationally disagree.

Self-check — fill in the figures

Type each value, then press Check answers. Cells turn green when correct; any you miss show the right figure beside them.