Consumer Utility Maximization (3D)

Consumer Choice

Income (budget) 100

Price of X 4

Price of Y 5

Preference weight α 0.5

Optimal X*

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Optimal Y*

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Max utility U*

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How it works

A consumer's satisfaction from two goods X and Y is modelled with a Cobb-Douglas utility function, plotted here as a surface. The consumer wants the highest point on that surface reachable within a fixed budget — the budget constraint slices the goods plane along a straight line, and the highest surface point directly above that line is the utility-maximizing bundle. Geometrically it is exactly where an indifference curve (a level contour of the surface, shown on the base plane) is tangent to the budget line.

U(X,Y) = X^α · Y^(1-α)
budget: Px·X + Py·Y = I
max U(X,Y) s.t. budget
 → X* = α·I / Px
 → Y* = (1-α)·I / Py
 → U* = X*^α · Y*^(1-α)

Income — total budget I available to spend on the two goods; raising it pushes the budget line (and the optimum) outward.
Price of X / Y — rotates the budget line; a more expensive good pulls the optimal bundle away from it.
Preference weight α — how much of the budget the consumer optimally allocates to X versus Y; higher α tilts the optimum toward more X.

Unlike the flat 2D supply-and-demand chart, this is a genuinely 3D optimization: the surface height is utility itself, and the optimum is a tangency point in three dimensions, not an intersection of two lines.

Utility surface Budget line Indifference curve Optimal bundle

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Consumer Utility Maximization in 3D