Solving and Graphing Linear Demand Equations

Representing Demand With a Linear Equation

A demand schedule lists quantities buyers plan to purchase at different prices. A linear demand equation is a compact way to represent such a relationship when, for the range being studied and other relevant conditions held constant, quantity changes at a constant rate as price changes. It lets us calculate quantities at prices not explicitly listed in a schedule and construct a graph from a few points.

In this tutorial, $P$ denotes the price per unit and $Q_d$ denotes the quantity demanded per period. The quantity unit and time period must be stated: for example, units per week or hundreds of tickets per month. A common linear form is:

Q_d=a-bP,\qquad a>0,\quad b>0.

Definition: In $Q_d=a-bP$, $a$ is the quantity predicted when price is zero, and $b$ is the amount by which predicted quantity decreases for each one-unit increase in price. The equation describes demand with other influences held constant. The units of $b$ are quantity units per period per unit of price.

The minus sign expresses the downward relationship between price and quantity demanded in this linear model: increasing $P$ by one unit reduces the equation’s predicted $Q_d$ by $b$ units. This is consistent with the law of demand, under its ceteris paribus condition. The equation should not be interpreted as saying that price is the only influence on demand; rather, it describes a specified relationship while other relevant conditions are held fixed.

Calculate Quantities Before Drawing the Line

To calculate quantity at a stated price, substitute that price for $P$, then perform the arithmetic in the equation. Keep the units attached to the result. For a useful graph, calculate several points, including the intercepts when possible. Each point on the graph is written as $(Q_d,P)$: quantity is the horizontal coordinate and price is the vertical coordinate.

The intercepts are found by setting one variable equal to zero. At $P=0$, the predicted quantity is $a$, giving the quantity-axis intercept $(a,0)$. At $Q_d=0$, solving the equation gives $P=a/b$, so the price-axis intercept is $(0,a/b)$. These are graph coordinates in the order $(Q_d,P)$, not in the order in which the variables appear in the equation.

P=0 \implies Q_d=a \qquad\text{and}\qquad Q_d=0 \implies P=\frac{a}{b}.

The price intercept is sometimes called the choke price: it is the price at which this linear equation predicts quantity demanded to be zero. This name refers to the intercept of the specified model, not a universal price applying under every set of market conditions.

Worked Example: Calculating Points and Finding Both Intercepts

A hypothetical weekly demand equation for a specialty food item is $Q_d=180-6P$, where $Q_d$ is packages per week and $P$ is dollars per package. Calculate quantity demanded at prices of $10 and $20, find both intercepts, and list points to plot.

Method. Substitute each price into the equation, then set $P=0$ and $Q_d=0$ separately to find the intercepts.

At $P=10$:

$$ Q_d=180-6(10)=180-60=120 $$

The predicted quantity is 120 packages per week, giving the graph point $(120,10)$.

At $P=20$:

$$ Q_d=180-6(20)=180-120=60 $$

The predicted quantity is 60 packages per week, giving the graph point $(60,20)$.

Quantity-axis intercept: Set price equal to zero.

$$ Q_d=180-6(0)=180. $$

The quantity-axis intercept is $(180,0)$, meaning 180 packages per week at a price of zero dollars per package.

Price-axis intercept: Set quantity equal to zero and solve for the price that makes the equation zero.

0=180-6P,\qquad 6P=180,\qquad P=30.

The price-axis intercept is $(0,30)$. The points to plot are $(0,30)$, $(60,20)$, $(120,10)$, and $(180,0)$. Plot quantity in packages per week horizontally and price in dollars per package vertically, then draw a straight line through the points. Check: at each listed price, substitution reproduces its plotted quantity. A common mistake avoided here is writing the intercept coordinates as $(P,Q_d)$ when the graph axes require $(Q_d,P)$.

Why the Graph Is a Straight Line

The equation is linear because its coefficients are constant and the variable $P$ appears only to the first power. More directly, compare the equation’s predictions at two prices. If the price changes by an amount $\Delta P$, the quantity changes by:

\begin{aligned} Q_d(P+\Delta P)-Q_d(P) &=\bigl(a-b(P+\Delta P)\bigr)-(a-bP)\\ &=-b\Delta P. \end{aligned}

The change in quantity is proportional to the change in price, with the same constant $b$ at every point. That constant rate of change is why the plotted points lie on a straight line. Because $b>0$, an increase in price gives a negative change in quantity. This justifies drawing the line downward as we move from left to right when quantity is on the horizontal axis and price is on the vertical axis.

The intercepts provide convenient endpoints for drawing the line over the nonnegative-price portion of the model. Other computed points help check that substitution and plotting are consistent. A graph should have labelled axes and units, and its scale should make the intercepts and calculated points readable.

Worked Example: Building a Graph From a Different Demand Equation

A hypothetical online service has weekly demand $Q_d=240-8P$, where $Q_d$ is subscriptions per week and $P$ is the weekly fee in dollars. Calculate demand at fees of $12.50 and $25, find the intercepts, and describe a suitable graph.

Method. Substitute each fee, then use the zero-price and zero-quantity conditions to identify the axis intercepts. Keep the fee in dollars and quantity in subscriptions per week throughout.

At $P=12.50$:

$$ Q_d=240-8(12.50)=240-100=140. $$

This gives the point $(140,12.50)$.

At $P=25$:

$$ Q_d=240-8(25)=240-200=40. $$

This gives the point $(40,25)$.

Quantity-axis intercept: At a zero fee, $Q_d=240-8(0)=240$, so the intercept is $(240,0)$.

Price-axis intercept: Set $Q_d=0$.

0=240-8P,\qquad 8P=240,\qquad P=30.

The price-axis intercept is $(0,30)$. Plot $(0,30)$, $(40,25)$, $(140,12.50)$, and $(240,0)$, using subscriptions per week on the horizontal axis and dollars per week on the vertical axis. Draw a straight line through them. Check: the calculated quantities are nonnegative at both stated fees, and each point satisfies the original equation. A common mistake avoided here is treating 12.50 as a quantity or changing the fee units partway through the calculation.

Interpreting the Nonnegative-Price Range

A demand equation is an algebraic model, and its predictions must be checked against the meaning of the variables. For $a>0$ and $b>0$, restrict attention to nonnegative prices. The equation predicts nonnegative quantities when $0\le P\le a/b$. For $P>a/b$, it predicts a negative quantity, which is not a feasible quantity demanded. In an economic application, that result signals that the linear equation should not be used as a literal quantity prediction at that price; do not interpret a negative quantity as actual purchases.

Negative prices are outside the usual price domain for this basic demand graph. Algebraically, however, if $P<0$, then $Q_d=a-bP>a>0$. Thus it would be incorrect to say that the equation predicts negative quantities at all prices outside the interval from zero to $a/b$. The prediction becomes negative above the price intercept, not below zero.

Worked Example: Checking the Equation’s Feasible Range

A hypothetical market for a digital add-on is represented by $Q_d=210-7P$, where $Q_d$ is licenses per month and $P$ is the monthly price in dollars. Determine the intercepts, calculate the prediction at $P=32$ and $P=-2$, and identify which predictions correspond to nonnegative quantities at nonnegative prices.

Method. Find the price intercept first. Then evaluate each specified price directly and distinguish the model’s algebraic output from a feasible quantity.

At $P=0$, $Q_d=210$, so the quantity-axis intercept is $(210,0)$. To find the price-axis intercept:

0=210-7P,\qquad 7P=210,\qquad P=30.

The price-axis intercept is $(0,30)$.

At $P=32$:

$$ Q_d=210-7(32)=210-224=-14. $$

The equation predicts $-14$ licenses per month. Since $32>30$, this is beyond the price intercept and is not a feasible quantity demanded.

At $P=-2$:

$$ Q_d=210-7(-2)=210+14=224. $$

This algebraic prediction is positive, as expected when $P<0$, but the negative price is outside the usual domain of this basic demand model. For nonnegative prices, the equation predicts nonnegative quantities from $P=0$ through $P=30$, inclusive. Check: the zero-quantity price is $210/7=30$, and the prediction at $P=32$ is negative because that price exceeds 30. A common mistake avoided here is claiming that every price outside the nonnegative feasible interval produces a negative quantity; the calculation at $P=-2$ shows otherwise.

A Reliable Graphing Procedure

A few orderly steps prevent common calculation, labelling, and interpretation errors. In particular, remember that this graph places quantity on the horizontal axis and price on the vertical axis, so a calculated pair must be written in the order $(Q_d,P)$.

Record the equation and units.
Identify the quantity unit, time period, and price unit before substituting values.

Calculate requested quantities.
Substitute each stated price for $P$, evaluate carefully, and label the resulting quantity.

Find both intercepts.
Use $P=0$ for the quantity-axis intercept and $Q_d=0$ for the price-axis intercept.

Plot and check.
Write points as $(Q_d,P)$, label both axes with units, and verify that plotted points satisfy the equation.

A useful final check is to confirm that the graph slopes downward from left to right and that the zero-quantity intercept is at a nonnegative price when $a>0$ and $b>0$. If the equation gives a negative quantity at a nonnegative price, compare that price with $a/b$; it lies above the price intercept.

Common Mistakes

Reversing the graph coordinates. With quantity on the horizontal axis and price on the vertical axis, plot $(Q_d,P)$, not $(P,Q_d)$.
Dropping units. A price measured in dollars per subscription cannot be substituted as though it were a quantity in subscriptions per week.
Finding the price intercept incorrectly. Set $Q_d=0$, then solve $0=a-bP$, which gives $P=a/b$. The quantity intercept is instead found by setting $P=0$.
Treating a negative prediction as a feasible quantity. For nonnegative prices above $a/b$, the linear equation predicts a negative number; that is an extrapolation beyond the feasible quantity range.
Misstating what happens at negative prices. For $P<0$, the equation gives $Q_d=a-bP>a$. Negative prices are outside the usual domain, but they do not produce negative predicted quantities in this model.

Key takeaway: For $Q_d=a-bP$ with $a,b>0$, calculate quantities by substitution, find the quantity intercept at $P=0$, and find the price intercept by setting $Q_d=0$. Plot $(Q_d,P)$, label the units, and interpret the equation only over a meaningful range.

Check Your Understanding

For each problem, calculate carefully and check that graph coordinates use quantity first and price second.

A hypothetical weekly demand equation is $Q_d=125-5P$, where quantity is units per week and price is dollars per unit. Find quantity at $P=9$ and both intercepts. (Answer: $Q_d=80$ units per week at $P=9$; intercepts $(125,0)$ and $(0,25)$.)
A market has $Q_d=360-12P$, with quantity in packages per month and price in dollars per package. Calculate quantity at $P=15$ and find the price-axis intercept. (Answer: $Q_d=180$ packages per month; price-axis intercept $(0,30)$.)
A demand equation is $Q_d=84-3P$, with quantity in units per day and price in dollars per unit. Find the largest nonnegative price for which the equation predicts a nonnegative quantity, and calculate quantity at $P=20$. (Answer: largest price $P=28$; at $P=20$, $Q_d=24$ units per day.)
A hypothetical ticket demand equation is $Q_d=500-20P$, where quantity is tickets per month and price is dollars per ticket. Calculate quantity at $P=12.50$, then find both intercepts and state the graph’s coordinate order. (Answer: $Q_d=250$ tickets per month; intercepts $(500,0)$ and $(0,25)$; plot coordinates as $(Q_d,P)$.)
A model is $Q_d=72-4P$, with quantity in units per week and price in dollars per unit. Calculate the predictions at $P=21$ and $P=-3$, and state which price is outside the usual nonnegative-price domain and which prediction is infeasible. (Answer: at $P=21$, $Q_d=-12$, an infeasible negative quantity; at $P=-3$, $Q_d=84$, a positive algebraic prediction, although the negative price is outside the usual domain.)

What You'll Learn