What is a budget constraint and what does it represent?

A budget constraint shows all combinations of two goods a consumer can afford given their income and the prices of both goods. The budget line (P_x times X + P_y times Y = M) represents the frontier where all income is spent. The budget set includes all affordable bundles on or below this line (P_x times X + P_y times Y is less than or equal to M).

What does the slope of the budget line tell you?

The slope of the budget line equals negative P_x divided by P_y, representing the opportunity cost of Good X in terms of Good Y. A slope of negative 0.5 means the consumer must give up 0.5 units of Good Y to obtain one additional unit of Good X. It reflects the market rate of exchange between the two goods.

How does a price change shift the budget line?

A price change causes the budget line to pivot. If the price of Good X increases, the budget line pivots inward around the Y-intercept because you can still buy the same amount of Y but less X. The slope becomes steeper. If Good X price decreases, it pivots outward and the slope becomes flatter.

How do income and substitution effects relate to budget constraints?

When a price changes, the budget line pivots. Economists decompose the total effect on consumption into substitution and income effects, but this requires knowing consumer preferences via indifference curves. This calculator shows how the budget constraint itself changes, which is the first step in that analysis. The full decomposition requires additional preference information.

Budget Constraint Calculator | Income & Price Effects

Enter Values

Income (M)

Good X Name

Good Y Name

Price of Good X (P_x)

Price of Good Y (P_y)

Comparative Statics (Optional)

New Income

New Price of Good X

New Price of Good Y

Budget Constraint Formulas

P_x × X + P_y × Y = M
Y = (M / P_y) − (P_x / P_y) × X

X-intercept: M / P_x
Y-intercept: M / P_y
Slope: −P_x / P_y

Model Assumptions

Two-good economy (simplification of reality)
Linear budget constraint (constant prices, no quantity discounts)
The budget line represents full expenditure of income; bundles below the line involve spending less than total income
Prices are given (consumer is a price taker)
Non-negativity: quantities must be non-negative (X ≥ 0, Y ≥ 0)
No taxes, subsidies, or transaction costs

For educational purposes only. This model simplifies real-world consumer choice to two goods with fixed prices.

Calculator by Ryan O'Connell, CFA

Budget Constraint Results

Budget Line Equation

5X + 10Y = 100

Y = 10 − 0.5X

Key Values

X-Intercept 20.00 units of Good X

Y-Intercept 10.00 units of Good Y

Slope −0.5000

Interpretation

Opportunity Cost Each Good X costs 0.50 Good Y

Equal-Spending Bundle (10.00, 5.00)

Equal-spending bundle: illustrative point where spending is split equally between both goods.

Budget Line Diagram

Original Budget Line

Formula Breakdown

P_x × X + P_y × Y = M

Understanding Budget Constraints

What Is a Budget Constraint?

A budget constraint represents all combinations of two goods that a consumer can afford given their income and the prices of both goods. The budget line (P_x × X + P_y × Y = M) is the frontier where all income is spent. The budget set includes all affordable bundles on or below this line.

Key Formulas (Mankiw Ch. 21)

Budget Line: P_x × X + P_y × Y = M
Slope Form: Y = (M / P_y) − (P_x / P_y) × X
Slope: −P_x / P_y (opportunity cost of X in terms of Y)
Source: Mankiw, Principles of Microeconomics, Ch. 21

How Changes Shift the Budget Line

Income changes cause a parallel shift: an increase shifts the line outward (more affordable bundles), while a decrease shifts it inward. The slope remains unchanged because relative prices have not changed.

Price changes cause the budget line to pivot. If P_x rises, the line pivots inward around the Y-intercept (you can still afford the same amount of Good Y). If P_y rises, it pivots around the X-intercept.

Verification Example (Mankiw Ch. 21): M = $100, P_x = $5, P_y = $10.
X-intercept = 20, Y-intercept = 10, Slope = −0.5.
If P_x rises to $10: new X-intercept = 10, Y-intercept unchanged = 10, new slope = −1.

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Frequently Asked Questions

A budget constraint shows all combinations of two goods a consumer can afford given their income and the prices of both goods. The budget line (P_x × X + P_y × Y = M) represents the frontier where all income is spent. The budget set includes all affordable bundles on or below this line (P_x × X + P_y × Y ≤ M). Points above the budget line are unaffordable.

The slope of the budget line equals −P_x/P_y and represents the opportunity cost of Good X in terms of Good Y. A slope of −0.5 means the consumer must give up 0.5 units of Good Y to obtain one additional unit of Good X. It reflects the market’s rate of exchange between the two goods, determined solely by their prices.

An income increase shifts the budget line outward in a parallel fashion — both intercepts increase proportionally while the slope stays the same, since relative prices haven’t changed. An income decrease shifts it inward. This is because more (or less) of both goods becomes affordable without changing the trade-off rate between them.

A price change causes the budget line to pivot. If the price of Good X increases, the budget line pivots inward around the Y-intercept (you can still buy the same amount of Good Y, but less of Good X). The slope becomes steeper. If the price of Good Y increases, the line pivots inward around the X-intercept and the slope becomes flatter. Price decreases cause the opposite pivots.

A budget constraint shows what a consumer can afford (determined by income and prices), while an indifference curve shows what a consumer prefers (combinations giving equal satisfaction). The optimal consumption bundle is typically where the highest indifference curve is tangent to the budget constraint, though corner solutions are possible when the consumer maximizes utility by spending all income on one good.

When a price changes, the budget line pivots. Economists decompose the total effect on consumption into substitution and income effects, but this requires knowing consumer preferences (indifference curves). This calculator shows how the budget constraint itself changes — the first step in that analysis. The substitution effect reflects the change in relative prices, while the income effect reflects the change in purchasing power. The full decomposition requires additional preference information beyond what the budget constraint alone provides.

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Disclaimer

This calculator is for educational purposes only. It uses a simplified two-good model with linear budget constraints and constant prices. Real-world consumer choice involves many goods, non-linear pricing, taxes, and transaction costs not captured here. This tool should not be used for business or policy decisions.

Budget Constraint Calculator