Substitute Prices and Demand Shifts

Why the Price of Another Good Matters

A buyer choosing what to drink might compare tea with coffee. If coffee becomes more expensive, some buyers may switch toward tea. At the same price of tea, those buyers are now willing and able to purchase more tea than before. The tea demand curve shifts; the change is not a movement along the original tea demand curve.

The central question is whether the goods serve as alternatives for buyers. If they do, a change in the price of one can affect demand for the other. The effect is evaluated while holding the other determinants of demand fixed, including the price of the good whose demand is being studied.

Definition: Two goods are substitutes when buyers can use one as an alternative to the other and an increase in the price of one tends to increase demand for the other, other things equal. For goods $X$ and $Y$, a rise in $P_Y$ increases demand for $X$; a fall in $P_Y$ decreases demand for $X$. The goods need not be identical, and how readily buyers switch can vary.

Reading a Substitute Effect in a Demand Equation

A convenient way to represent the relationship is a linear demand equation for good $X$ that includes the price of good $Y$:

$$ Q_X = a - bP_X + cP_Y $$

Here $Q_X$ is quantity demanded of $X$, $P_X$ is its own price, and $P_Y$ is the price of the other good. The constants $a$, $b$, and $c$ describe the relationship over the range being modeled. For the usual downward-sloping own-price relationship, $b>0$. In this linear model, $c>0$ represents a substitute-price effect: a higher $P_Y$ raises demand for $X$. The equation holds only while its other included factors are held fixed and the linear form is a reasonable approximation; it is not a universal description of every market or price range.

Key result: In $Q_X=a-bP_X+cP_Y$, holding $P_X$ and the other determinants fixed, a change in $P_Y$ changes quantity demanded of $X$ by $c$ times that price change. If $c>0$, a rise in $P_Y$ shifts the demand curve for $X$ to the right, and a fall shifts it to the left.

The sign of $c$ captures the direction of the relationship, not its importance in every possible setting. Its size measures how much the model’s quantity demanded changes for a one-unit change in $P_Y$, with the other variables fixed. For example, if $c=2$, a one-dollar rise in $P_Y$ raises $Q_X$ by two units in this equation.

Why the Demand Curve Shifts

The demand curve for $X$ relates its own price $P_X$ to quantity demanded $Q_X$, holding other influences constant. To see why a change in $P_Y$ shifts that curve, compare quantities at the same $P_X$. For an initial price $P_Y$ and a new price $P_Y+\Delta P_Y$, subtract the initial quantity from the new quantity:

\begin{aligned} \Delta Q_X &= \left(a-bP_X+c(P_Y+\Delta P_Y)\right) - \left(a-bP_X+cP_Y\right)\\ &=c\Delta P_Y \end{aligned}

The terms $a$ and $-bP_X$ cancel because they are unchanged. Thus, when $c>0$, a rise in $P_Y$ produces a higher $Q_X$ at every fixed $P_X$ in the model. That is a rightward shift in demand for $X$. A fall in $P_Y$ produces a lower $Q_X$ at every fixed $P_X$, a leftward shift.

The same result can be expressed in inverse-demand form by solving for $P_X$:

P_X=\frac{a+cP_Y-Q_X}{b}

For a fixed quantity $Q_X$, an increase in $P_Y$ raises the price buyers are willing to pay for $X$ in this model when $c>0$. This is another way to see the rightward shift. The conclusion comes from comparing the demand relationships at fixed values of the other variables, not from assuming that the market’s actual price of $X$ stays unchanged after all market adjustments.

Worked Example: A Higher Coffee Price and Tea Demand

A hypothetical local market’s weekly demand for tea is $Q_T=80-4P_T+2P_C$, where prices are dollars per package and quantity is packages per week. Coffee rises in price from $4 to $7. Calculate the change in tea demand at tea prices of $5 and $8, and identify the shift.

Method. Hold the tea price fixed at each value and substitute the old and new coffee prices into the tea demand equation. This compares demand at the same own price, which is the right comparison for identifying a shift.

At $P_T=5$ and the original coffee price $P_C=4$:

$$ Q_T=80-4(5)+2(4)=80-20+8=68 $$

At $P_T=5$ and the new coffee price $P_C=7$:

$$ Q_T=80-4(5)+2(7)=80-20+14=74 $$

At this tea price, demand increases by $74-68=6$ packages per week. Check at $P_T=8$: the original quantity is $80-4(8)+2(4)=80-32+8=56$; the new quantity is $80-32+14=62$. Demand again increases by 6 packages per week.

The coffee-price coefficient is $2>0$, so tea and coffee act as substitutes in this model. The rise in coffee’s price shifts tea demand right by 6 packages per week at every tea price represented by the equation. Common mistake avoided: changing $P_T$ would move along tea demand. Here $P_T$ is fixed and $P_C$ changes, so the result is a shift.

From Individual Demand to Market Demand

Market demand adds the quantities demanded by all buyers at each given price. If several buyer groups have demand equations that include the price of the same substitute, the market effect can be found by calculating each group’s response and then adding the quantities. When each group’s substitute-price coefficient is positive, the increase in the market quantity at a fixed own price is the sum of those group responses.

Worked Example: Adding Substitute Effects Across Buyer Groups

Two groups of buyers have weekly demands for tea given by $Q_1=30-2P_T+P_C$ and $Q_2=20-P_T+0.5P_C$, with prices in dollars per package. Coffee rises from $4 to $8. Find the market tea demand before and after the change at a tea price of $6, then check the change using the combined equation.

Method. Calculate each group’s quantity at the fixed tea price, add the group quantities for each coffee price, and then verify by adding the equations. This avoids treating one group’s change as the entire market response.

Initially, $P_T=6$ and $P_C=4$. Group 1 demands $30-2(6)+4=30-12+4=22$ packages. Group 2 demands $20-6+0.5(4)=20-6+2=16$ packages. Initial market demand is $22+16=38$ packages per week.

After coffee rises to $P_C=8$, group 1 demands $30-12+8=26$ packages. Group 2 demands $20-6+0.5(8)=20-6+4=18$ packages. New market demand is $26+18=44$ packages per week, an increase of $44-38=6$.

For a check, add the equations:

$$ Q_M=(30-2P_T+P_C)+(20-P_T+0.5P_C)=50-3P_T+1.5P_C $$

At $P_T=6$, the original market quantity is $50-3(6)+1.5(4)=50-18+6=38$. The new quantity is $50-18+1.5(8)=50-18+12=44$. The calculation agrees. Common mistake avoided: adding the equations means adding quantities at a common price; it does not mean adding the groups’ prices or their demand coefficients without regard to the equation.

Inferring a Substitute Relationship from Changes

Sometimes the equation is not provided, but two observations show how demand changes when the other good’s price changes. If the price of $X$ and the remaining demand conditions are unchanged, the direction of the change in $Q_X$ as $P_Y$ changes indicates the sign of the cross-price relationship. In a linear equation, the change can also be used to solve for its coefficient.

Worked Example: Recovering a Cross-Price Coefficient

A hypothetical demand model for coffee is $Q_C=a-2P_C+cP_T$, where prices are dollars per bag and quantities are bags per week. At $P_C=12$, tea priced at $3 is associated with coffee demand of 41 bags. When tea rises to $7, coffee demand is 53 bags. Find $c$, determine $a$, and calculate coffee demand at $P_C=15$ and $P_T=7$.

Method. First subtract the two observations. Because $P_C$ is unchanged, the change in quantity isolates the effect of the change in $P_T$:

$$ 53-41=c(7-3) $$

Thus, $12=4c$, so $c=3$. Use the first observation to find $a$:

$$ 41=a-2(12)+3(3)=a-24+9=a-15 $$

Therefore, $a=56$. At $P_C=15$ and $P_T=7$, predicted coffee demand is:

$$ Q_C=56-2(15)+3(7)=56-30+21=47 $$

The positive coefficient $c=3$ indicates that tea and coffee are substitutes in this model: tea’s price increase raises coffee demand, all else equal. Common mistake avoided: dividing the full quantity $53$ by the tea-price change would not isolate the cross-price effect. The relevant change is $53-41=12$, because the own price is held fixed.

Distinguishing a Shift from a Movement

A demand curve plots the relationship between the price of a good and the quantity demanded of that good, with other determinants held fixed. A change in the good’s own price changes quantity demanded and is shown as a movement along that curve. A change in the price of a substitute changes demand at each own price and is shown as a shift of the curve.

For example, if coffee becomes more expensive, the demand curve for tea shifts right when tea and coffee are substitutes. If tea’s own price then rises, buyers move along the new tea demand curve and choose a lower quantity of tea, other things equal. Keeping these two comparisons separate prevents a common directional error.

Change being considered	Effect on demand for $X$, if $X$ and $Y$ are substitutes	Graph interpretation
Price of $Y$ rises	Demand for $X$ increases	Demand for $X$ shifts right
Price of $Y$ falls	Demand for $X$ decreases	Demand for $X$ shifts left
Price of $X$ rises	Quantity demanded of $X$ decreases, other things equal	Movement along demand for $X$

Common Exam Traps and Why the Distinction Matters

Reversing the direction. If the price of a substitute rises, buyers are more likely to switch toward the good under study, so its demand increases.
Calling every quantity change a movement. A change in the price of the good under study causes a movement along its demand curve. A change in the substitute’s price shifts its demand curve.
Ignoring the ceteris paribus condition. The simple sign prediction holds with the other determinants fixed. If several factors change, their effects must be separated before deciding the net result.
Assuming all pairs are substitutes in every setting. The relationship depends on whether buyers regard the goods as alternatives. The positive cross-price effect is a model assumption or an observed pattern, not a label that follows merely because two goods are different.

Recognizing substitute effects helps interpret market news and demand estimates. A business forecasting demand for one product may need to consider the prices of alternatives, not just its own price. The same reasoning applies to hypothetical insurance products when buyers can choose between products that serve a similar coverage need: a price increase for one may shift demand toward the other, although the size and even the presence of that response would depend on buyers’ choices.

For demand analysis aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus, keep the comparison explicit: name the good whose demand is being analyzed, identify the substitute whose price changes, hold the own price fixed when measuring the shift, and state the direction of the result.

Check Your Understanding

For each problem, identify what is held fixed and distinguish the demand shift from any change in quantity demanded.

A demand equation is $Q_X=50-2P_X+P_Y$. If $P_X=8$ and $P_Y$ rises from $3 to $6, calculate the initial and final quantity demanded and state the shift direction. (Answer: $37$ to $40$, an increase of $3$ units and a rightward shift.)
A demand equation is $Q_X=90-3P_X+2P_Y$. At $P_X=10$, $P_Y$ falls from $5 to $2. Calculate the change in $Q_X$ and state the shift direction. (Answer: $70$ to $64$, a decrease of $6$ units and a leftward shift.)
Two groups have demands $Q_1=24-P_X+0.5P_Y$ and $Q_2=16-2P_X+P_Y$. At $P_X=6$, calculate market demand when $P_Y$ is $4 and when it is $8. (Answer: $28$ units at $P_Y=4$ and $34$ units at $P_Y=8$, an increase of $6$.)
A linear demand equation is $Q_X=a-4P_X+cP_Y$. When $P_X=5$, quantity is $31$ at $P_Y=2$ and $43$ at $P_Y=6$. Find $c$ and $a$, then calculate demand at $P_X=7$, $P_Y=6$. (Answer: $c=3$, $a=45$, and $Q_X=35$; the positive coefficient indicates substitutes in this model.)
Demand is $Q_X=75-3P_X+1.5P_Y$. At quantity $Q_X=30$, find the inverse-demand price when $P_Y=2$ and when $P_Y=6$. Explain the shift. (Answer: $P_X=16$ when $P_Y=2$, and $P_X=18$ when $P_Y=6$; the higher substitute price shifts demand right and raises the inverse-demand price by $2 at that quantity.)

Change being considered	Effect on demand for \(X\), if \(X\) and \(Y\) are substitutes	Graph interpretation
Price of \(Y\) rises	Demand for \(X\) increases	Demand for \(X\) shifts right
Price of \(Y\) falls	Demand for \(X\) decreases	Demand for \(X\) shifts left
Price of \(X\) rises	Quantity demanded of \(X\) decreases, other things equal	Movement along demand for \(X\)

What You'll Learn