Calculus of Utility Maximization

Step 1: Setup & The Budget Constraint

The consumer's problem: Choose quantities of good X and good Y to maximize utility, subject to a budget constraint.

Budget Constraint:

P_x × Q_x + P_y × Q_y = Income

With current values:

$2.00 × Q_x + $3.00 × Q_y = $60

X-intercept (spend all on X): Q_x = 30.0 | Y-intercept (spend all on Y): Q_y = 20.0

Key Concept — The Budget Line: The red line on the graph represents all affordable bundles. The slope equals −P_x/P_y = −0.67. This is the rate at which the market trades X for Y.

Indifference Curve & Budget Line

Step 2: The Cobb-Douglas Utility Function

We work with a specific utility function that has clean mathematical properties:

U = Q_x^α × Q_y^(1−α)

With current α = 0.60:

U = Q_x^0.60 × Q_y^0.40

What does α tell us?

α = 0.60 means the consumer has a relative preference for good X
The consumer will spend 60% of income on X
The consumer will spend 40% of income on Y
This spending split is independent of prices — only α matters!

Try it: Change α with the slider. Notice how the spending percentages in the Optimal Solution box update immediately — and the graph shows the optimal bundle shift.

Why Cobb-Douglas?

Generates smooth, convex indifference curves (the teal curves on the graph)
Has diminishing marginal utility in each good
Produces clean, closed-form solutions (no messy algebra)
Widely used in economics and finance

Step 3: Computing Marginal Utilities

Marginal utility measures the extra utility from consuming one more unit. We find these by taking partial derivatives of the utility function.

Marginal Utility of X:

MU_x = ∂U/∂Q_x = α × Q_x^(α−1) × Q_y^(1−α)

With α = 0.60:

MU_x = 0.60 × Q_x^−0.40 × Q_y^0.40

Note: MU_x decreases as Q_x rises (since the exponent on Q_x is negative) — this is diminishing marginal utility.

Marginal Utility of Y:

MU_y = ∂U/∂Q_y = (1−α) × Q_x^α × Q_y^(−α)

MU_y = 0.40 × Q_x^0.60 × Q_y^−0.60

Marginal Rate of Substitution (MRS):

MRS = MU_x / MU_y = (α / (1−α)) × (Q_y / Q_x)

With current α: MRS = 1.50 × (Q_y / Q_x)

The MRS is the slope of the indifference curve: how many units of Y the consumer would give up for one more X.

Marginal Utilities at the Optimal Point

Calculated with current parameter values
Quantity	Value	Interpretation
Q_x*	18.0	Optimal units of X
Q_y*	8.0	Optimal units of Y
MU_x at optimum	0.70	Extra utility from 1 more X
MU_y at optimum	1.05	Extra utility from 1 more Y
MRS at optimum	0.67	Should equal P_x/P_y

Step 4: The Optimization Condition

The golden rule of consumer optimization: At the optimal bundle, the last dollar spent on each good delivers the same marginal utility.

MU_x / P_x = MU_y / P_y

Equivalently — MRS equals the price ratio:

MRS = P_x / P_y

The slope of the indifference curve equals the slope of the budget line. This is the tangency condition you see on the graph.

Substituting our MRS expression:

(α / (1−α)) × (Q_y / Q_x) = P_x / P_y

After rearranging:

α / (1−α) = (P_x × Q_x) / (P_y × Q_y)

With current values: α/(1−α) = 1.50

This means: Spending on X / Spending on Y = 1.50

Verification with current values: Spending on X = $36.00 | Spending on Y = $24.00
Ratio = 1.50 ✓ (matches α/(1−α))

Step 5: Solving for Optimal Quantities

We have two equations and two unknowns (Q_x and Q_y):

Equation 1 — Optimization condition (from Step 4):

P_x × Q_x = [α / (1−α)] × P_y × Q_y

Equation 2 — Budget constraint:

P_x × Q_x + P_y × Q_y = I

The Demand Functions:

Q_x* = (α × I) / P_x

Q_y* = ((1−α) × I) / P_y

With current values (α = 0.60, I = $60):

Q_x* = (0.60 × $60) / $2.00 = 18.0

Q_y* = (0.40 × $60) / $3.00 = 8.0

Key Insight: The consumer spends exactly α = 60% of income on X and (1−α) = 40% on Y, regardless of prices. This is a unique property of Cobb-Douglas preferences.

Step 6: The Optimal Solution & Insights

Bringing it all together: the full optimal solution with current parameters.

Optimal Consumer Solution
Variable	Value	How computed
Q_x*	18.0	α × I / P_x
Q_y*	8.0	(1−α) × I / P_y
Spending on X	$36.00	α × I
Spending on Y	$24.00	(1−α) × I
Total Utility (U*)	12.60	Q_x^α × Q_y^(1−α)
MRS at optimum	0.67	Must equal P_x/P_y
MU_x/P_x	0.35	Both equal (optimality check)
MU_y/P_y	0.35	Both equal (optimality check)

Five Key Takeaways from Cobb-Douglas Optimization:

Consumer spends 60% on X and 40% on Y — set by α alone
Doubling P_x cuts Q_x in half, but spending on X is unchanged
Doubling income doubles both quantities (unit income elasticity)
At the optimum: MRS = P_x/P_y (indifference curve tangent to budget line)
At the optimum: MU_x/P_x = MU_y/P_y (equal marginal utility per dollar)

Experiment: Adjust the sliders on the left and watch the table and graph update in real time. Try doubling P_x — does Q_x* halve while spending on X stays constant?