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The Consumer's Budget

Formulate the budget constraint, calculate intercepts and slope, and diagram shifts vs. pivots.

8 micro-lessons

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What Can You Afford?

Hook introducing the income constraint.

The Problem of Choice

Imagine you have a fixed budget of Rs 20 to spend on two items: bananas and mangoes.

If both goods cost Rs 5 each, you obviously can't buy 5 bananas and 5 mangoes (that would cost Rs 50!). You face a limit. This fundamental boundary between what you can and cannot afford is what economists call the consumer's budget constraint.

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Available Bundles

Finding affordable integer bundles given prices and income.

Listing Affordable Bundles

Problem. Consider a consumer who has Rs 20 ( $M = 20$ ). Suppose both goods are priced at Rs 5 ( $p_1 = 5$ , $p_2 = 5$ ) and are available only in integral (whole) units. What bundles can the consumer afford?

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Equation of the Budget Line

Formal equation, intercepts, and slope derivation.

Equation of the Budget Line

p_{1} x_{1} + p_{2} x_{2} = M

By rearranging the terms to solve for $x_{2}$ , we can write the budget line in slope-intercept form:

x_{2} = \frac{M}{p _{2}} - \frac{p _{1}}{p _{2}} x_{1}

Horizontal Intercept

\frac{M}{p _{1}}

Max good 1 if all income is spent on it

Vertical Intercept

\frac{M}{p _{2}}

Max good 2 if all income is spent on it

Slope

- \frac{p _{1}}{p _{2}}

Rate of market substitution

Deriving the Slope

If a consumer moves between two points on the budget line, the total spending remains $M$ :

1. $p_{1} x_{1} + p_{2} x_{2} = M$
2. $p_{1} (x_{1} + Δ x_{1}) + p_{2} (x_{2} + Δ x_{2}) = M$
3. Subtracting (1) from (2): $p_{1} Δ x_{1} + p_{2} Δ x_{2} = 0$
4. Rearranging terms: $\frac{Δ x _{2}}{Δ x _{1}} = - \frac{p _{1}}{p _{2}}$

Finding Intercepts

Calculating intercepts and slope for a specific budget constraint.

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To find the intercepts and slope of a budget line, we first identify the income and prices. In this scenario, the consumer's income is $M = 20$ , the price of good 1 is $p_1 = 4$ , and the price of good 2 is $p_2 = 5$ . The budget constraint is written algebraically as $4x_1 + 5x_2 = 20$ . To find the maximum amount of good 1 the consumer can buy, we set the quantity of good 2 to zero ( $x_2 = 0$ ), which gives us a horizontal intercept of units. Similarly, spending the entire Rs 20 income purely on good 2 yields a vertical intercept of units. Finally, the slope of the budget line measures the rate at which the consumer can substitute one good for the other in the market. Using the price ratio formula $-p_1/p_2$ , the slope of this specific budget line is .

Changes in Income and Prices

Explanation of how M and p1 changes affect the geometric line.

What happens to the budget set if your income ( $M$ ) changes, but market prices stay exactly the same?

The equation becomes $p_1x_1 + p_2x_2 = M'$ .

Rewritten for the graph: $x_2 = \frac{M'}{p_2} - \frac{p_1}{p_2}x_1$ .

Notice that the slope ( $-p_1/p_2$ ) remains unchanged. However, the intercepts $\frac{M'}{p_1}$ and $\frac{M'}{p_2}$ change. This creates a parallel shift of the budget line. If income increases, it shifts outward. If income decreases, it shifts inward.

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Double Everything

Testing understanding of proportional changes.

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Suppose a consumer's budget set is originally bounded by the line $p_1x_1 + p_2x_2 = M$ . Due to a sudden period of inflation, the consumer's income exactly doubles, and the prices of both goods exactly double as well. What happens to the budget line?

← previousOrdinal Utility And Indifference Curves next →Optimal Choice Of The Consumer

Available Bundles

Finding affordable integer bundles given prices and income.

Listing Affordable Bundles

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Finding Intercepts

Calculating intercepts and slope for a specific budget constraint.

0 of 3 blanks filled

Changes in Income and Prices

Explanation of how M and p1 changes affect the geometric line.

What happens to the budget set if your income ( $M$ ) changes, but market prices stay exactly the same?

The equation becomes $p_1x_1 + p_2x_2 = M'$ .

Rewritten for the graph: $x_2 = \frac{M'}{p_2} - \frac{p_1}{p_2}x_1$ .

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Double Everything

Testing understanding of proportional changes.

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