Budget Constraints With a Free Good

Introduction

A standard budget constraint relates the quantities a consumer can afford to the prices of goods and available income. If one good has a price of zero, its quantity contributes nothing to expenditure. That changes the shape of the budget set: increasing the quantity of the free good does not require reducing the quantity of the priced good.

The central task is to translate prices and income into the affordability condition without dividing by zero or treating a free quantity as spending. The result describes which bundles are affordable under the model; by itself, it does not say which affordable bundle a consumer will choose.

Model and assumptions: Consider two goods, $x$ and $y$, with quantities $x\geq 0$ and $y\geq 0$. The price of $x$ is $p_x>0$, the price of $y$ is zero, and available income is $m\geq 0$. Quantities are divisible, the consumer may spend less than all available income, and there are no taxes, fixed fees, or quantity limits. A bundle is affordable when its total expenditure is no greater than income.

Deriving the Budget Set

For these assumptions, total expenditure is the price of each good multiplied by its quantity, summed across goods. Since $y$ costs zero per unit, its expenditure term is zero regardless of how large $y$ is.

p_xx+0y\leq m \quad\Longleftrightarrow\quad x\leq \frac{m}{p_x},\qquad y\geq 0.

The budget set is therefore every nonnegative quantity of the free good paired with a priced-good quantity between zero and $m/p_x$. Its boundary of full expenditure is $x=m/p_x$, for every $y\geq 0$. In a graph with $x$ on the horizontal axis and $y$ on the vertical axis, this boundary is a vertical ray, not a downward-sloping line. The affordable region includes all points between the vertical axis and that ray, extending upward without bound.

Key result: With one priced good and one unlimited free good, the maximum affordable quantity of the priced good is $m/p_x$. The free good does not reduce that maximum. The budget set is $0\leq x\leq m/p_x$, $y\geq 0$.

This result also explains why a familiar intercept procedure needs care. When both prices are positive, setting one quantity to zero can give an intercept on the other axis. Here, setting $x=0$ leaves $y$ unrestricted by the budget equation, so there is no finite $y$-axis intercept. Dividing income by the free good’s price would require division by zero and is not a valid way to find an intercept.

Why the Free Quantity Does Not Reduce Spending

The result follows directly from how expenditure is calculated. For any two bundles with the same $x$ but different nonnegative quantities $y_1$ and $y_2$, their expenditures are identical:

$$ p_xx+0y_1=p_xx=p_xx+0y_2. $$

Thus changing only the free-good quantity cannot use up income. Conversely, if the consumer increases $x$, expenditure rises by $p_x$ for each additional unit, whether $y$ is small, large, or zero. Solving the affordability inequality for $x$ gives the maximum $m/p_x$; the value of $y$ does not enter the calculation.

Write total expenditure.
Multiply each quantity by its price and add the terms, including the zero-price term.

Apply the income limit.
Require total expenditure to be no greater than $m$, rather than assuming all income must be spent.

Solve for the priced quantity.
Because $p_x>0$, divide by $p_x$ to obtain $x\leq m/p_x$. Keep the nonnegativity condition $y\geq 0$.

Represent the set.
Include every bundle satisfying both conditions. If the free good is unlimited, the set extends indefinitely in the $y$ direction.

Worked Examples

Worked Example: Testing Bundles With an Unlimited Free Good

A household has $84 available for two goods. Good $x$ costs $7 per unit, and good $y$ is free and unlimited. Determine the maximum affordable amount of $x$, then classify the bundles $(9,100)$, $(12,240)$, and $(13,0)$.

Method. Write the expenditure inequality first. The model assumes nonnegative quantities, no quantity cap on $y$, and that spending may be less than $84.

The budget condition is $7x+0y\leq 84$, so $7x\leq 84$. Dividing both sides by 7 gives $x\leq 12$. Therefore the maximum affordable quantity of $x$ is 12 units.

For $(9,100)$, expenditure is $7(9)+0(100)=63+0=63$ dollars. Since $63\leq 84$, the bundle is affordable, with $84-63=21$ dollars unspent.

For $(12,240)$, expenditure is $7(12)+0(240)=84+0=84$ dollars. The bundle is affordable and lies on the full-expenditure boundary.

For $(13,0)$, expenditure is $7(13)+0(0)=91$ dollars. Since $91>84$, the bundle is not affordable.

Check. Only the quantity of the priced good determines expenditure. The free-good quantities do not change any of the three results. A common mistake is to treat 240 units of $y$ as if they used some of the household’s $84; the price term $0(240)$ is exactly zero.

Worked Example: Describing the Budget Set and Its Boundary

A student has $96 to spend on a priced snack $x$ costing $8 per unit and a free resource $y$ with no availability limit. State the budget set, describe its full-expenditure boundary, and decide whether $(0,500)$ is affordable.

Method. Use the expenditure inequality and keep the nonnegativity conditions separate from the spending condition.

The total expenditure is $8x+0y$. Affordability requires $8x+0y\leq 96$, which reduces to $8x\leq 96$, or $x\leq 12$. Including nonnegative quantities, the budget set is $0\leq x\leq 12$ and $y\geq 0$.

Full expenditure occurs when $8x=96$, so $x=12$. Any bundle $(12,y)$ with $y\geq 0$ lies on the boundary. The boundary is a vertical ray at $x=12$, extending upward without a finite endpoint under the unlimited-availability assumption.

For $(0,500)$, expenditure is $8(0)+0(500)=0$ dollars. Since $0\leq 96$, it is affordable. It is inside the budget set, not on the full-expenditure boundary.

Check. At $x=0$, the budget constraint places no upper bound on $y$. A common mistake is to claim that the $y$-axis intercept is $96/0$; this expression is undefined. The correct statement is that there is no finite budget-imposed upper bound on $y$.

Worked Example: One Free Good Among Several Goods

A buyer has $60 for goods $a$, $b$, and $c$. Good $a$ costs $4 per unit, good $b$ costs $6 per unit, and good $c$ is free and unlimited. Determine whether $(6,5,80)$ and $(9,5,2)$ are affordable. Then find the maximum amount of $a$ if the buyer obtains 3 units of $b$.

Method. Add expenditure across all goods. The free good contributes a zero term; it does not remove the costs of the other goods.

The budget condition is $4a+6b+0c\leq 60$. For $(6,5,80)$, expenditure is $4(6)+6(5)+0(80)=24+30+0=54$ dollars. Since $54\leq 60$, the bundle is affordable, leaving $60-54=6$ dollars unspent.

For $(9,5,2)$, expenditure is $4(9)+6(5)+0(2)=36+30+0=66$ dollars. Since $66>60$, this bundle is not affordable.

If $b=3$, the cost of $b$ is $6(3)=18$ dollars. The remaining amount available for $a$ is $60-18=42$ dollars, so $a\leq 42/4=10.5$. Because quantities are divisible, the maximum is 10.5 units of $a$, for any nonnegative quantity of $c$.

Check. The first bundle costs $54, the second $66, and the maximum $a$ with $b=3$ costs $4(10.5)+6(3)=42+18=60$ dollars. The common mistake is to omit the priced good $b$ when finding what remains for $a$, or to let $c$ change the spending total.

Free Does Not Always Mean Unlimited

The vertical extension of the budget set depends on the assumption that the free good is available in any nonnegative quantity. A good can have a zero price but still be rationed or physically limited. If $y$ is free but at most $\bar y$ units are available, the budget condition remains $p_xx\leq m$, while the availability condition adds $0\leq y\leq \bar y$. The resulting set is bounded in the $y$ direction by availability, not by expenditure.

Similarly, a one-time access fee or a required purchase would need to be included separately; it is not represented by a zero per-unit price alone. The zero-price model says only that each additional unit of $y$ adds no expenditure. It does not establish that the good has no other costs or that unlimited quantities physically exist.

Common Mistakes and Why the Distinction Matters

Charging income for a free quantity. The expenditure term is $0y=0$, not $y$ dollars or a share of income.
Dividing by zero to find an intercept. The usual price-divided intercept formula requires a positive price. When a price is zero, describe the affordability condition directly instead.
Calling the boundary a downward-sloping line. With one priced good and one unlimited free good, full expenditure fixes $x=m/p_x$, regardless of $y$. The boundary is vertical.
Confusing the budget set with the boundary. The boundary includes bundles that spend exactly all income. The budget set also includes bundles that leave some income unspent.
Assuming a zero price means unlimited availability. A quantity cap is a separate constraint and must be stated separately.

Getting this distinction right matters whenever an affordability calculation includes a good with a zero price. It prevents a false trade-off between the free good and the priced good, and it gives the correct geometric representation of the set of affordable bundles. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, write the expenditure condition before deciding which bundles are affordable.

A buyer has $72. Good $x$ costs $9 per unit, while $y$ is free and unlimited. Find the maximum $x$, and classify $(7,250)$, $(8,500)$, and $(8.1,0)$. (Answer: maximum $x=8$; $(7,250)$ and $(8,500)$ are affordable, while $(8.1,0)$ is not.)
Income is $45, $x$ costs $5 per unit, and $y$ is free and unlimited. Give the budget set and full-expenditure boundary, and explain whether the set has a finite $y$-axis intercept. (Answer: $0\leq x\leq 9,\ y\geq 0$; boundary $x=9,\ y\geq 0$; there is no finite $y$-axis intercept.)
A consumer has $70 for goods $a$, $b$, and $c$, priced at $3, $7, and $0 per unit, respectively. If $b=4$, find the maximum affordable $a$ and state how $c$ affects it. (Answer: $a=(70-7(4))/3=14$ units; any nonnegative $c$ leaves the spending limit unchanged.)
A person has $108 for good $x$, priced at $12 per unit, and unlimited free good $y$. Calculate the spending and remaining income for $(6,70)$, then classify $(9,400)$ and $(10,0)$. (Answer: $(6,70)$ costs $72 and leaves $36; $(9,400)$ is affordable at exactly $108; $(10,0)$ is not affordable because it costs $120.)
A free good $y$ is available in quantities from 0 to 4 only. Income is $50 and $x$ costs $10 per unit. Write the budget set including availability, and classify $(4.5,4)$ and $(5,4)$. (Answer: $0\leq x\leq 5,\ 0\leq y\leq 4$; $(4.5,4)$ is affordable at $45, and $(5,4)$ is affordable at exactly $50.)

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