Consumer Choice: Utility Maximization

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*

0.00

Optimal Y*

0.00

Max utility U*

0.00

MRS at optimum

0.00

Your bundle (drag the yellow dot on the budget line)

x=0.00, y=0.00 → U=0.00

Utility lost vs optimum

0.00 (0.0%)

Budget line: p_xx + p_yy = I

Indifference curve family

Highlighted curve at U*

Optimal bundle (tangency point)

Your draggable bundle

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

Indifference-curve family — faint blue curves each hold utility constant at a different level; the highlighted curve sits at exactly U*, the highest utility reachable on the budget line.
Orange budget line — every bundle the consumer can just afford. Anything above the line costs more than income I.
Yellow marker — the optimal bundle (x*, y*), the single point where the highlighted indifference curve is tangent to the budget line.
Red draggable dot — drag it anywhere along the orange budget line to pick a different affordable bundle; its own (fainter) indifference curve shows how much utility that bundle actually delivers, always at or below U*.
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 and the whole curve family with it.

Drag the red dot along the budget line

Consumer Choice: Utility Maximization (2D)