Overview
This lecture explains how consumers make optimal choices given their preferences, budget constraints, and the prices of goods, introducing key concepts like the budget line, optimal choice rules, and demand functions.
Budget Constraint and Affordable Bundles
- A bundle is affordable if its total cost does not exceed the consumer's income: (P_X X + P_Y Y \leq M).
- The budget constraint is the set of bundles where (P_X X + P_Y Y = M).
- The budget line graphs this constraint, with vertical intercept (M/P_Y) and horizontal intercept (M/P_X).
- The slope (-P_X/P_Y) shows the market's trade-off rate between goods X and Y.
- Increasing income shifts the budget line outward; decreasing income shifts it inward, with the slope unchanged.
- Changing (P_X) or (P_Y) rotates the budget line around the intercepts.
Optimal Choice and Demand
- The optimal bundle lies on the budget line and is the most preferred affordable bundle.
- If the indifference curve is steeper than the budget line ((MRS_{XY} > P_X/P_Y)), spend more on X and less on Y.
- If the indifference curve is flatter ((MRS_{XY} < P_X/P_Y)), spend more on Y and less on X.
- Optimal choice rule:
- (i) If only X is consumed, (MRS_{XY} > P_X/P_Y), it's a corner solution at ((M/P_X, 0)).
- (ii) If only Y is consumed, (MRS_{XY} < P_X/P_Y), it's a corner solution at ((0, M/P_Y)).
- (iii) Otherwise, solve the system: (P_X X + P_Y Y = M) and (MRS_{XY} = P_X/P_Y) for the interior solution.
Demand Functions and Law of Demand
- The individual demand function relates quantity demanded to its price, income, and other goods' prices.
- A price increase for X causes movement along the demand curve: as (P_X) rises, quantity demanded falls (law of demand).
- Income changes shift the demand curve for normal goods: more income increases demand, less decreases it.
- With Cobb-Douglas preferences ((\sigma=0)), the consumer spends fixed income shares on X and Y, independent of prices.
- For Cobb-Douglas, X and Y are independent: the demand for X does not shift if (P_Y) changes.
- If (\sigma>0), X and Y are substitutes: increasing (P_Y) increases demand for X.
- If goods are homogeneous ((\alpha = \beta), large (\sigma)), the consumer buys only the cheaper good unless prices are nearly the same.
Key Terms & Definitions
- Budget Constraint — All bundles where spending exactly equals income.
- Budget Line — The graph of the budget constraint; shows maximum affordable combinations.
- Marginal Rate of Substitution (MRS) — The trade-off rate between two goods on an indifference curve.
- Corner Solution — Optimal choice is at an axis; consumer buys only one good.
- Individual Demand Function — Links the price of a good to the quantity demanded by one consumer.
- Normal Good — Demand increases as income rises.
- Independent Goods — Demand for one is unaffected by the price of the other.
- Substitutes — Demand for one increases if the other’s price rises.
- Law of Demand — Higher price leads to lower quantity demanded, all else equal.
Action Items / Next Steps
- Review mathematical derivations of the budget constraint, optimal choice conditions, and demand functions.
- Practice drawing and interpreting budget lines and optimal bundles under different scenarios.