Consumer Choice: Utility Maximization Simulator

Prices & Income

Price of Good X (p_x) $2.00

Price of Good Y (p_y) $1.00

Income (I) $100

Preference weight on X (α) 0.50

Consumer's Optimum

Optimal X*

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Optimal Y*

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Max utility U*

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MRS at optimum

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Utility surface U(x,y) = x^αy^1-α

Budget line: p_xx + p_yy = I

Indifference curve at U*

Optimal bundle (tangency point)

How it works

The consumer chooses quantities of two goods X and Y to maximise a Cobb-Douglas utility function subject to a budget constraint:

maximize  U(x,y) = x^α · y^(1-α)
subject to  p_x·x + p_y·y = I

Solving with a Lagrange multiplier gives closed-form demand functions — the consumer always spends a fixed share α of income on good X and (1-α) on good Y:

x* = α·I / p_x
y* = (1-α)·I / p_y
U* = (x*)^α · (y*)^(1-α)

At the optimum the indifference curve is tangent to the budget line: the marginal rate of substitution (the slope of the indifference curve) equals the price ratio (the slope of the budget line).

MRS = (α/(1-α)) · (y/x)  =  p_x / p_y   at the optimum

Utility surface — height at each (x,y) point on the floor grid is U(x,y). The peak of the whole surface would need infinite income; the budget constraint caps how far along it the consumer can actually move.
Orange budget curve — traces the actual utility achieved while sliding along the budget line p_xx + p_yy = I. Its highest point is the consumer's optimum — everywhere else on the budget line yields strictly less utility.
Blue indifference curve — every bundle on it delivers exactly U*; it touches the budget line at exactly one point, the optimum, and nowhere crosses above it.
Yellow marker — the optimal bundle (x*, y*, U*), i.e. the point of tangency.
Raising α shifts spending toward X; raising a price shrinks how much of that good the fixed income can buy, re-tilting the whole optimum.

Optimal bundle Budget curve Indifference curve

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