Utility Maximization Solver Calculator

Enter Utility Maximization Inputs

Utility model

Income or budget

Result decimals

Price of good X

Price of good Y

Alpha parameter

Beta parameter

Rho parameter

Parameter guide

Alpha and beta usually act as preference weights. Rho matters mainly for CES. For complements, alpha and beta define the fixed consumption recipe.

Example Data Table

Model	Income	Px	Py	Alpha	Beta	Rho	Optimal X*	Optimal Y*	Max Utility
Cobb-Douglas	100	10	5	0.6	0.4	0.5	6	8	6.7317
Perfect Substitutes	90	6	9	3	2	0.5	15	0	45
Perfect Complements	120	8	4	2	1	0.5	20	10	10
CES	100	10	5	0.5	0.5	0.5	3.3333	13.3333	7.5
Quasi-Linear	80	4	2	12	1	0.5	6	28	49.5011

Formula Used

1) Budget Constraint

Px × X + Py × Y = Income

2) Cobb-Douglas Utility

U(X,Y) = X^α Y^β

Optimal demands: X* = Income × α / [(α + β)Px], Y* = Income × β / [(α + β)Py].

3) Perfect Substitutes

U(X,Y) = αX + βY

Compare α/Px with β/Py. Spend all income on the good with higher utility per price. Equal ratios imply many optimal bundles.

4) Perfect Complements

U(X,Y) = min(X/α, Y/β)

Keep the fixed recipe X:Y = α:β. The optimal scale is Income / (Pxα + Pyβ).

5) CES Utility

U(X,Y) = (αX^ρ + βY^ρ)^1/ρ

The solver uses the CES first-order condition and the budget line. The implied ratio is X/Y = [(βPx)/(αPy)]^1/(ρ-1).

6) Quasi-Linear Utility

U(X,Y) = α ln(X) + βY

Interior tangency gives X* = αPy / (βPx). If that bundle is unaffordable, the optimum becomes a corner.

7) Marginal Rate of Substitution

MRS = MUx / MUy. At an interior optimum, the solver checks whether MRS ≈ Px / Py.

How to Use This Calculator

Select the utility model that matches your problem.
Enter the available income or total budget.
Provide the price of good X and good Y.
Enter alpha and beta as weights, recipe terms, or coefficients.
Enter rho only when using the CES model.
Choose the number of decimals for the output table.
Press Solve Utility Maximum to display the result above the form.
Use the export buttons to save the result as CSV or PDF.

Frequently Asked Questions

1) What does this calculator solve?

It finds the bundle of two goods that gives the highest utility under a budget constraint. It also reports utility, spending shares, marginal utilities, and the solution type.

2) When does a corner solution appear?

A corner solution appears when the best choice puts all spending on one good. This often happens with perfect substitutes or with quasi-linear preferences under tight income.

3) What do alpha and beta mean?

They describe preference strength or required bundle proportions. Their interpretation changes by model, so always match them to the selected utility function before solving.

4) What is rho in the CES model?

Rho controls substitution flexibility between goods. Lower values make goods more complementary, while values closer to one make them more substitutable.

5) Why is MRS important?

MRS shows how much of one good a consumer gives up for another while keeping utility unchanged. At an interior optimum, it usually matches the price ratio.

6) Can this page help with homework checks?

Yes. It is useful for checking computed bundles, verifying tangency logic, and comparing different preference structures using the same prices and income.

7) Why do perfect complements use a fixed ratio?

Perfect complements are consumed together in exact proportions. Extra units of only one good do not raise utility, so the optimal bundle stays on the recipe ray.

8) What do the export buttons save?

They save the calculated results table. CSV is useful for spreadsheet work, while PDF is useful for reports, lectures, printed notes, and record keeping.