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Intermediate Microeconomics Unit 1: Budget Constraints

The budget constraint describes every bundle a consumer can afford; it is half of every consumer choice problem, so you must be able to draw it, shift it and read its slope instantly.

Covers: Budget set and budget line, Slope as the relative price, Income and price changes, Taxes, subsidies and rationing, Kinked budget lines and in-kind transfers, Endowments and intertemporal budgets

Study guide

A consumer with income m who buys two goods at prices p₁ and p₂ can afford any bundle (x, y) with p₁x + p₂y ≤ m. That region is the budget set, and its upper boundary, p₁x + p₂y = m, is the budget line. Putting x on the horizontal axis, the line hits the axes at m/p₁ and m/p₂, and solving for y gives y = m/p₂ − (p₁/p₂)x. The slope −p₁/p₂ is the market's rate of exchange: to get one more unit of x, the consumer must give up p₁/p₂ units of y. That ratio is the opportunity cost of x. Economists often treat y as a composite good, 'all other spending,' with a price of $1, so units of y are simply dollars.

Changes in income and prices move the line in predictable ways. A higher income raises both intercepts and shifts the line outward in parallel, because the slope depends only on prices. A change in p₁ moves only the horizontal intercept, so the line pivots around the vertical intercept: a lower p₁ makes it flatter and larger. Multiplying every price and income by the same factor leaves the line exactly where it was, which is why only relative prices and real income matter. This also lets us pick a numeraire, a good whose price is set to 1, and measure everything else relative to it.

Government policies show up as changes in the budget line. A quantity tax of t per unit raises the effective price of x to p₁ + t, and an ad valorem tax of rate τ raises it to (1 + τ)p₁; both make the line steeper. A per-unit subsidy lowers the effective price and flattens the line. A lump-sum tax or transfer changes income only, shifting the line in parallel. Rationing caps the quantity of a good, which cuts off part of the budget set with a vertical boundary. In-kind transfers such as food vouchers add a flat segment: the household can buy more food, but it cannot turn vouchers into other goods.

Many real budget lines are kinked. A quantity discount after the tenth unit makes the line flatter past the kink; a tax on units above a threshold makes it steeper past the kink. A two-part price, such as a membership fee plus a lower per-use price, gives a second line that starts lower but is flatter, and the consumer's true budget set is the outer envelope of the options. When a consumer owns goods instead of cash, her income is the market value of her endowment, p₁ω₁ + p₂ω₂, and price changes pivot the line around the endowment point. The same logic gives the intertemporal budget line, where consuming a dollar today costs 1 + r dollars of future consumption.

Key ideas

• The budget set is every bundle with p₁x + p₂y ≤ m; the budget line is its boundary.
• The slope −p₁/p₂ is the opportunity cost of x measured in units of y.
• Income changes shift the line in parallel; a price change pivots it around the other good's intercept.
• Scaling all prices and income by the same factor does not change the budget set.
• Quantity and value taxes raise the effective price; lump-sum taxes act like an income cut.
• Rationing, quantity discounts, thresholds and in-kind transfers create kinks or cut-off segments.
• With an endowment, the budget line always passes through the endowment point.

Formulas and rules

Common mistakes

• Inverting the slope as −p₂/p₁ instead of −p₁/p₂ with x on the horizontal axis.
• Saying a price change shifts the line in parallel instead of pivoting it.
• Treating an in-kind transfer as identical to cash even when the voucher limit binds.
• Applying a threshold tax or discount to every unit instead of only the units past the kink.
• Adding future income to current income without discounting it by 1 + r.

Before the test, make sure you can

• Write the budget line and compute both intercepts from income and prices.
• Explain the slope as an opportunity cost and compute it from prices.
• Show how income changes, price changes and balanced inflation move the line.
• Draw budget lines with quantity taxes, ad valorem taxes, subsidies and lump-sum taxes.
• Construct kinked budget lines for discounts, thresholds, rationing and membership fees.
• Compare in-kind transfers with equal-value cash grants.
• Build budget lines from endowments and from income in two periods with interest.

Words to know

Budget set

All bundles a consumer can afford: p₁x + p₂y ≤ m with x, y ≥ 0.

Budget line

The bundles that cost exactly the consumer's income, p₁x + p₂y = m.

Relative price

The ratio p₁/p₂, the number of units of y that trade for one unit of x in the market.

Opportunity cost (budget)

The amount of the other good given up to buy one more unit of a good, equal to p₁/p₂ for good x.

Composite good

A stand-in for 'all other goods,' usually priced at $1 so its units are dollars.

Numeraire

A good whose price is set to 1 so other prices are expressed relative to it.

Quantity tax

A tax of a fixed amount t on each unit, raising the effective price to p + t.

Ad valorem tax

A tax set as a percentage of the price, raising the effective price to (1 + τ)p.

Lump-sum tax

A fixed tax that does not depend on what is bought; it shifts the budget line inward in parallel.

Per-unit subsidy

A payment of s per unit purchased that lowers the effective price to p − s.

Rationing

A legal limit on the quantity of a good, which truncates the budget set.

Kinked budget line

A budget line whose slope changes at some quantity because the price changes there.

In-kind transfer

A benefit that must be spent on a specific good, such as food vouchers.

Endowment

The bundle of goods a consumer owns before trading; its market value is her income.

Intertemporal budget constraint

The present value of consumption equals the present value of income: c₁ + c₂/(1 + r) = m₁ + m₂/(1 + r).

Two-part tariff (budget view)

A fixed fee plus a per-unit price, which lowers the intercept but flattens the line.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.