Budget Constraint and Consumer Optimum

The consumer wants to maximize utility but is limited by income. The budget constraint is the mathematical expression of this limitation. In combination with preferences, it determines the optimal choice.

Budget Constraint

For two goods X and Y: Px × X + Py × Y = M Where M is income (or budget), Px, Py are prices, X, Y are quantities.

The budget line is a graphical representation. All combinations of X and Y available given the budget.

Slope of the budget line: −Px/Py — the relative price of goods. Shows how much Y must be given up for one unit of X in the market.

Intersection points with axes: On the X axis: M/Px (all budget spent on X) On the Y axis: M/Py (all budget spent on Y)

Shifts of the Budget Line

Change in income: Increase in M → parallel shift outward Decrease in M → parallel shift inward Slope does not change (relative prices are the same)

Change in the price of one good: Increase in Px → the budget line pivots inward along the X axis The point on the Y axis does not change, and the point on the X axis moves closer to the origin Slope changes

Proportional change in all prices: Equivalent to income change in the opposite direction Doubling all prices = halving income

Indifference Curves

Indifference curve — the set of combinations of goods that bring the same utility. The consumer is indifferent to which point on the curve is chosen.

Properties of indifference curves: Downward slope: to maintain utility when X decreases, Y must increase Do not intersect: logical requirement (transitivity of preferences) Convex to the origin: reflects the diminishing marginal rate of substitution The further from the origin — the higher the utility: more of both goods is better

Marginal Rate of Substitution (MRS)

Marginal Rate of Substitution — the amount of Y the consumer is willing to sacrifice for one unit of X while maintaining utility. MRS = −ΔY/ΔX = −dY/dX (along the indifference curve) MRS = MUx / MUy

Decreasing MRS: the more X and the less Y, the less Y is ready to be given up for another unit of X. This explains the convexity of the curve.

Intuition: if you have a lot of pizza and little beverage, you value beverages highly. If you have a lot of beverages and little pizza — vice versa.

Consumer Optimum

The consumer chooses the point on the budget line that reaches the highest indifference curve.

Graphically: the tangent point of the budget line and the indifference curve.

The optimum condition: MRS = Px / Py Or equivalently: MUx / Px = MUy / Py

Intuition: MRS — subjective willingness to exchange Y for X Px/Py — objective market opportunity for exchange At optimum, they are equal

If MRS > Px/Py — it is advantageous for the consumer to buy more X (subjectively values X above market price)

Deriving the Demand Curve

Method: vary the price Px, find the optimum at each price, construct the relationship X*(Px).

When Px decreases: The budget line pivots outward along the X axis The new optimum is on a higher indifference curve Usually X* increases (law of demand)

The “price-consumption” curve: connects optima at different prices. Projection onto the X axis at different prices gives the demand curve.

Income Effect and Substitution Effect

A price change affects the choice through two channels:

Substitution effect: The good becomes relatively cheaper → buy more Pure change in relative prices at constant utility Always negative (in the direction of the cheaper good)

Income effect: Price reduction = increase in real income For normal goods: income rise → greater consumption For inferior goods: income rise → lower consumption

For normal goods: both effects in the same direction → price reduction increases demand (law of demand). For inferior goods: effects in different directions. If the income effect is stronger (Giffen good) — violation of the law of demand is possible. Practically almost never encountered.

Practical Significance

For policy: analysis of the impact of taxes and subsidies on welfare through changes in the budget constraint.

For marketing: understanding the trade-offs consumers make helps to position products.

For the investor: how changes in prices (inflation, tariffs) affect consumer choice and demand for companies’ products.