A consumer with income M chooses quantities x, y of two goods to maximize a Cobb–Douglas utility function subject to a budget constraint:
Utility: U(x,y) = x^α · y^(1-α)
Budget: Px·x + Py·y = M
Optimum: x* = α·M / Px
y* = (1-α)·M / Py
The optimum occurs where an indifference curve is tangent to the budget line — the marginal rate of substitution (how many units of Y the consumer would trade for one more unit of X) exactly equals the price ratio:
MRS = MUx / MUy = (α·y) / ((1-α)·x) = Px / Py
The green surface plots U(x,y) over the (x, y) plane. The bright curve traces utility along the budget line itself — every feasible bundle the consumer can afford — and its single peak is the optimal bundle (x*, y*). The grey ring on the floor is the indifference curve U = U* projected down; the straight floor line is the budget line Px·x + Py·y = M. Drag to orbit, scroll to zoom.
- Income M — shifts the budget line outward; the optimum scales up along the same ray from the origin (Cobb–Douglas income expansion path).
- Prices Px, Py — rotate the budget line and change relative affordability, shifting the optimal mix toward the cheaper good.
- α — how strongly the consumer prefers X; higher α tilts spending, and the optimal bundle, toward more X.
Real-world relevance: this is the textbook model behind demand curves, income and substitution effects, and how price changes reallocate household spending.