Introduction
A demand shift changes the quantity buyers want at each price. With supply held fixed, the market-clearing price and quantity generally both change. Linear equations make it possible to calculate those changes directly and explain why the effect is divided between price and quantity.
This analysis focuses on a change in the demand intercept: the demand curve’s slope stays the same, but its quantity demanded at every given price changes by the same amount. The result is a comparative-statics calculation: compare the original equilibrium with the new equilibrium after the demand change, without tracing the adjustment process between them. The method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
The equations describe the relevant range of prices and quantities. In particular, the algebraic equilibrium must be economically meaningful for the question at hand: a negative price or quantity may indicate that the linear equations should not be used outside their relevant range. The analysis also holds the supply curve and the demand slope fixed; changing either one would be a different comparative-statics question.
Equilibrium Before and After the Demand Change
At the initial equilibrium, buyers’ desired quantity equals sellers’ offered quantity. Equating the two equations and solving for price gives the initial equilibrium price. Substituting that price into either curve gives equilibrium quantity.
Substituting the equilibrium price into supply gives the corresponding quantity. The same result follows from substituting into demand.
After the demand intercept changes to \(a+\Delta a\), the new equilibrium is found by replacing \(a\) with \(a+\Delta a\). The difference between the new and original equilibrium values gives the effects of the change.
Why the Change Is Divided Between Price and Quantity
The result follows by comparing the two equilibrium equations directly. Before the shift, equilibrium satisfies \(a-bP^*=c+sP^*\). After the intercept change, it satisfies \(a+\Delta a-bP_{\text{new}}^*=c+sP_{\text{new}}^*\). Subtracting the initial equation from the new one isolates the changes in price and quantity.
The quantity change follows from the supply equation, which has not shifted: \(\Delta Q^*=s\Delta P^*\). Substituting the price change gives \(\Delta Q^*=\frac{s\Delta a}{b+s}\). This also explains the economic interpretation. At any fixed price, the demand shift changes desired quantity by \(\Delta a\). The resulting price change reduces the quantity demanded by \(b\Delta P^*\), while increasing quantity supplied by \(s\Delta P^*\). At the new equilibrium, the extra demand is exactly accounted for by those two effects:
Thus, the demand-intercept change is not generally equal to the change in equilibrium quantity. Some of the shift is absorbed by the price increase, which reduces quantity demanded along the unchanged demand curve. The slopes determine how much goes to each effect.
Worked Examples
Worked Example: A Higher Demand Intercept in a Local Market
A hypothetical market has quantity measured in units per week and price measured in dollars per unit. Its equations are \(Q_s=20+4P\) and \(Q_d=140-6P\). A change in buyers’ preferences raises the demand intercept from 140 to 170, while the supply equation and demand slope stay fixed. Find the initial and new equilibria, then calculate the changes. Method: solve for each equilibrium and check the change formula.
Initial equilibrium. Set supply equal to demand: \(20+4P=140-6P\). Adding \(6P\) and subtracting 20 gives \(10P=120\), so \(P^*=12\). Supply gives \(Q^*=20+4(12)=20+48=68\). Demand gives \(140-6(12)=140-72=68\), so the initial equilibrium checks.
New equilibrium. The shifted demand equation is \(Q_d=170-6P\), not \(170-6P\) with a changed slope or supply. Set the new demand equal to supply: \(20+4P=170-6P\). Hence \(10P=150\), so \(P_{\text{new}}^*=15\). The new quantity is \(Q_{\text{new}}^*=20+4(15)=80\). Demand also gives \(170-6(15)=170-90=80\).
Changes and check. The intercept change is \(\Delta a=170-140=30\), with \(b=6\) and \(s=4\). Therefore \(\Delta P^*=30/(6+4)=3\), and \(\Delta Q^*=4(30)/(6+4)=12\). The calculated changes agree with \(15-12=3\) and \(80-68=12\). The common mistake this method avoids is using the entire 30-unit demand shift as the equilibrium quantity change: here, 18 units of the shift are offset by the rise in price reducing quantity demanded, and 12 units become additional equilibrium quantity.
Worked Example: A Lower Demand Intercept
For a hypothetical produce market, quantity is measured in thousands of crates per week and price in dollars per crate. Supply is \(Q_s=12+3P\), and demand initially is \(Q_d=132-5P\). A change in market conditions lowers the demand intercept by 20, leaving the demand slope and supply equation unchanged. Calculate both equilibria and interpret the changes. Method: treat the decrease as a negative \(\Delta a\), then verify each equilibrium in both equations.
Initial equilibrium. Equating the equations gives \(12+3P=132-5P\). Thus \(8P=120\), so \(P^*=15\). The quantity is \(Q^*=12+3(15)=12+45=57\) thousand crates. Demand confirms \(132-5(15)=132-75=57\).
New equilibrium. The new intercept is \(132-20=112\), so demand becomes \(Q_d=112-5P\). Equating it with supply gives \(12+3P=112-5P\), or \(8P=100\). Therefore \(P_{\text{new}}^*=12.5\). Supply gives \(Q_{\text{new}}^*=12+3(12.5)=12+37.5=49.5\) thousand crates. Demand gives \(112-5(12.5)=112-62.5=49.5\).
Changes and check. Here \(\Delta a=-20\), \(b=5\), and \(s=3\). The formulas give \(\Delta P^*=-20/(5+3)=-2.5\) dollars and \(\Delta Q^*=3(-20)/(5+3)=-7.5\) thousand crates. These match \(12.5-15=-2.5\) and \(49.5-57=-7.5\). The common sign error is to enter the 20-unit decrease as \(+20\), which would incorrectly predict rising price and quantity.
Worked Example: A Demand Increase With Fractional Slopes
In a hypothetical equipment market, quantity is measured in hundreds of items and price in dollars per item. The equations are \(Q_s=5+1.5P\) and \(Q_d=95-3.5P\). A new use for the equipment increases the demand intercept by 12, with the slopes and supply equation unchanged. Find the new equilibrium and check it against the initial one. Method: use the intercept-change formulas, then substitute into both curves.
Initial equilibrium. Set \(5+1.5P=95-3.5P\). Combining the price terms gives \(5P=90\), so \(P^*=18\). Then \(Q^*=5+1.5(18)=5+27=32\) hundreds of items. Demand agrees: \(95-3.5(18)=95-63=32\).
Effect of the intercept increase. Here \(\Delta a=12\), \(b=3.5\), and \(s=1.5\). The total slope in the denominator is \(b+s=3.5+1.5=5\). Thus \(\Delta P^*=12/5=2.4\) dollars, and \(\Delta Q^*=1.5(12)/5=18/5=3.6\) hundreds of items.
New equilibrium and check. The new price is \(18+2.4=20.4\) dollars, and the new quantity is \(32+3.6=35.6\) hundreds of items. Supply gives \(5+1.5(20.4)=5+30.6=35.6\); demand, whose new intercept is \(95+12=107\), gives \(107-3.5(20.4)=107-71.4=35.6\). Both curves agree. The common mistake this calculation avoids is treating the quantity change as 12: only \(3.6\) hundreds of items appear as added equilibrium quantity, while the rest of the shift is offset by the price-induced reduction in quantity demanded.
Interpreting the Slopes and Avoiding Traps
The denominator \(b+s\) combines the absolute demand slope and the supply slope because a price change affects both sides of the market. For a given intercept change, a larger \(b+s\) means a smaller equilibrium price change. The quantity effect depends specifically on \(s\): it is the movement along the unchanged supply curve caused by the new price. These formulas describe changes in the linear model, not percentage changes or elasticities.
| Change in demand intercept | Equilibrium price effect | Equilibrium quantity effect |
|---|---|---|
| \(\Delta a>0\) | \(\Delta P^*>0\) | \(\Delta Q^*>0\) |
| \(\Delta a<0\) | \(\Delta P^*<0\) | \(\Delta Q^*<0\) |
| \(\Delta a=0\) | \(\Delta P^*=0\) | \(\Delta Q^*=0\) |
A reliable calculation keeps the shift distinct from a movement along a curve. The demand intercept change shifts demand; after that shift, the new equilibrium price changes quantity demanded along the new demand curve and quantity supplied along the fixed supply curve. In particular, do not set the change in equilibrium quantity equal to \(\Delta a\). Also do not use \(b\) alone in the price-change denominator: both the demand response \(b\Delta P^*\) and the supply response \(s\Delta P^*\) contribute to clearing the market.
Before accepting a result, check the signs and substitute the new price into both equations. For a positive intercept change with positive slopes, a negative price or quantity change signals an algebra or sign error. If the equations imply an equilibrium outside their economically meaningful range, the linear model may no longer be an appropriate description there.
Check Your Understanding
For each hypothetical market, assume the stated linear equations apply at the equilibria and that only the demand intercept changes. Calculate the requested effects and use the displayed answers to check your work.
- A market has \(Q_s=8+2P\) and \(Q_d=68-4P\). The demand intercept rises by 12. Find the initial equilibrium and the changes in equilibrium price and quantity. (Answer: Initial \(P^*=10\), \(Q^*=28\); \(\Delta P^*=2\), \(\Delta Q^*=4\).)
- Supply is \(Q_s=10+2P\), and demand changes from \(Q_d=100-3P\) to \(Q_d=120-3P\). Find the new equilibrium price and quantity, and the changes from the initial equilibrium. (Answer: New \(P^*=22\), \(Q^*=54\); \(\Delta P^*=4\), \(\Delta Q^*=8\).)
- Supply is \(Q_s=6+3P\), and demand initially is \(Q_d=96-5P\). The demand intercept falls by 16. Find the new equilibrium and state the price and quantity changes. (Answer: New \(P^*=9.25\), \(Q^*=33.75\); \(\Delta P^*=-2\), \(\Delta Q^*=-6\).)
- A market has demand slope \(b=4\) and supply slope \(s=2\). The demand intercept increases by 24. Calculate the equilibrium price and quantity changes, and state how much of the intercept change is offset by reduced quantity demanded as price rises. (Answer: \(\Delta P^*=4\), \(\Delta Q^*=8\); reduced quantity demanded is \(b\Delta P^*=16\), and \(16+8=24\).)
- With supply and demand slopes \(s=1.5\) and \(b=3\), respectively, the observed equilibrium price rises by 4 and quantity rises by 6. Assuming supply is fixed and the change is due only to a demand-intercept shift, infer that shift and check whether the quantity change is consistent. (Answer: \(\Delta a=(b+s)\Delta P^*=(3+1.5)(4)=18\); the implied \(\Delta Q^*=1.5(18)/4.5=6\), consistent with the observation.)