Source: Intermediate Microeconomics, Problem Set 2, Texas A&M University
Tags: budget constraint, budget line, utility maximisation, optimal bundle, price-consumption curve, individual demand curve, consumer choice, indifference curve, pizza and burritos example, ECON 323
A consumer's budget constraint shows every combination of two goods they can afford at given prices and income. The utility-maximising bundle sits where the highest attainable indifference curve just touches the budget line. When one good's price changes, the optimal bundle shifts, tracing out a price-consumption curve in goods-space and a demand curve in price-quantity space.
Budget constraint
The set of all bundles a consumer can afford given their income and the prices of goods. Expressed as P₁X₁ + P₂X₂ = M, where M is income.
Utility-maximising point (optimal bundle)
The bundle on the budget constraint that gives the highest utility. Graphically, it is the tangency point between the budget line and the highest reachable indifference curve.
Price-consumption curve (PCC)
The curve connecting all optimal bundles as the price of one good changes, holding income and the other good's price constant. Drawn in the same goods-space graph as the budget lines.
Individual demand curve
A curve showing the quantity demanded of a good at each price, derived from the price-consumption curve. Plotted with price on the vertical axis and quantity on the horizontal axis.
Indifference curve
A curve showing all bundles that give the consumer the same level of utility. Higher curves represent higher utility. Assumed convex to the origin under standard preferences.
The budget constraint equation takes the form:
P_Z · Z + P_B · B = M
where Z and B are the two goods, P_Z and P_B their prices, and M is income.
To graph it:
Find the intercept on each axis by setting the other good to zero
The horizontal intercept is M / P_Z
The vertical intercept is M / P_B
The slope of the budget line is −P_Z / P_B (the negative price ratio)
When the price of one good changes, the budget line pivots around the intercept of the good whose price stayed the same.
John has $300 per week. Burritos cost $15 each (on the vertical axis). Pizza prices change.
At P_Z = $5:
Horizontal intercept: 300 / 5 = 60 pizzas
Vertical intercept: 300 / 15 = 20 burritos
Optimal bundle: Z = 30, B = 10
At P_Z = $10:
Horizontal intercept: 300 / 10 = 30 pizzas
Vertical intercept: still 20 burritos
Optimal bundle: Z = 18, B = 8
At P_Z = $15:
Horizontal intercept: 300 / 15 = 20 pizzas
Vertical intercept: still 20 burritos
Optimal bundle: Z = 14, B = 6
Each price change pivots the budget line inward around the vertical intercept (burritos), because only the price of pizza is changing.
Plot all three optimal bundles on the same graph: (30, 10), (18, 8), (14, 6). Connect them with a smooth curve. This is the price-consumption curve for pizzas.
The PCC slopes downward here, meaning that as pizza becomes more expensive, John buys fewer of both goods. This tells you something about the nature of the goods and the strength of the income effect.
Take the price-quantity pairs from the optimal bundles and plot them on a separate graph with price on the vertical axis and quantity of pizzas on the horizontal axis:
(Z = 30, P_Z = $5)
(Z = 18, P_Z = $10)
(Z = 14, P_Z = $15)
Connect these points. The resulting downward-sloping curve is John's individual demand curve for pizzas. It obeys the law of demand: higher price, lower quantity demanded.
Budget constraint (general form):
P₁X₁ + P₂X₂ = M
Slope of the budget line:
Slope = −P₁ / P₂
(This equals the negative of the price ratio of the good on the horizontal axis to the good on the vertical axis.)
Intercepts:
Horizontal intercept = M / P₁
Vertical intercept = M / P₂
⚠️ When drawing budget lines for different prices, only the intercept of the good whose price changed moves. The other intercept stays fixed. A common mistake is shifting both intercepts.
⚠️ The price-consumption curve is drawn in the same two-good space as the budget lines and indifference curves, not in a separate price-quantity space. The demand curve is the one plotted in price-quantity space.
⚠️ Make sure the axes match the question. This problem specifies pizzas (Z) on the horizontal and burritos (B) on the vertical. Swapping them will produce incorrect slopes and intercepts.
⚠️ Verify each optimal bundle lies on its budget line. For example, at P_Z = $5: (30 × 5) + (10 × 15) = 150 + 150 = 300. Checks out.
Q: John has $300. Burritos cost $15 and pizzas cost $10. What is the equation of his budget line, and what are the intercepts?
A: The budget line is 10Z + 15B = 300. Horizontal intercept (set B = 0): Z = 30. Vertical intercept (set Z = 0): B = 20.
Q: If the price of pizzas rises from $5 to $10 while burritos stay at $15 and income stays at $300, which intercept changes and in which direction?
A: Only the horizontal intercept (pizzas) changes. It falls from 60 to 30. The vertical intercept (burritos) remains at 20. The budget line pivots inward around the burritos intercept.
Q: What is the difference between a price-consumption curve and an individual demand curve?
A: The price-consumption curve connects optimal bundles in goods-space (both goods on the axes) as one price changes. The individual demand curve plots the quantity of that one good against its own price. The demand curve is derived from the PCC by extracting the price-quantity pairs.
budget line, budget set, consumer equilibrium, tangency condition, optimal consumption bundle, price-consumption path, price-offer curve, demand schedule, law of demand, income and substitution effects, pivoting budget constraint, price ratio slope