Comparative-Statics Workshop With Multiple Market Shocks

Why Compare Equilibria?

A market shock changes the conditions under which buyers or sellers make decisions. Comparative statics compares the market equilibrium before the shock with the equilibrium after it, holding the model’s other features fixed. The central task is not merely to say that a curve shifts, but to determine how the equilibrium price and quantity respond—especially when demand and supply shift at the same time.

This tutorial uses a linear market model to organize that reasoning. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. The formulas describe changes between two equilibria; they do not describe the path or timing by which the market adjusts.

The Linear Market Model

Definition: Let market demand and supply be \(Q_D=a-bP\) and \(Q_S=c+dP\), where \(P\) is price, \(Q_D\) is quantity demanded, and \(Q_S\) is quantity supplied. Assume \(b>0\) and \(d>0\). The equilibrium is an interior point where \(Q_D=Q_S\). A change in \(a\) shifts demand while holding its slope fixed; a change in \(c\) shifts supply while holding its slope fixed. Write these intercept changes as \(\Delta a\) and \(\Delta c\), respectively.

With price on the vertical axis and quantity on the horizontal axis, a positive \(\Delta a\) shifts demand to the right: buyers demand more at every price. A positive \(\Delta c\) shifts supply to the right: sellers offer more at every price. The parameters \(b\) and \(d\) describe the slopes and are assumed not to change in this workshop.

These are shifts of curves, not movements along unchanged curves. A price change by itself changes quantity demanded and quantity supplied along their respective curves; a change in \(a\) or \(c\) changes the curve itself. Comparative statics compares the equilibrium resulting from the new curves with the original equilibrium.

Deriving the Comparative-Statics Formulas

Let the original equilibrium be \((P,Q)\), and let the new equilibrium be \((P+\Delta P,Q+\Delta Q)\). The derivation follows directly from the two equilibrium conditions. It also shows why the effects of the two shocks receive different weights when we calculate the quantity change.

Write the new equilibrium conditions.
After the shifts, demand is \(a+\Delta a-bP_{\text{new}}\) and supply is \(c+\Delta c+dP_{\text{new}}\). At the new equilibrium, both equal \(Q_{\text{new}}\).

Subtract the original equilibrium condition.
The original condition is \(a-bP=c+dP\). Subtract it from the new condition \(a+\Delta a-b(P+\Delta P)=c+\Delta c+d(P+\Delta P)\).

Solve for the price change.
Collecting terms gives \(\Delta a-\Delta c=(b+d)\Delta P\). Since \(b+d>0\), division gives the equilibrium price change.

\Delta P=\frac{\Delta a-\Delta c}{b+d}

For quantity, substitute this price change into either curve. Demand gives \(\Delta Q=\Delta a-b\Delta P\), while supply gives \(\Delta Q=\Delta c+d\Delta P\). They must agree because the new point is an equilibrium. Substitution yields:

\Delta Q =\Delta a-b\left(\frac{\Delta a-\Delta c}{b+d}\right) =\frac{d\Delta a+b\Delta c}{b+d}

The demand shock is weighted by \(d\) in the quantity formula, and the supply shock is weighted by \(b\). These weights arise from the slopes in the two equations; they are not interchangeable. The formulas assume the same positive slopes before and after the shocks and an interior equilibrium. They should not be applied without modification if, for example, a market has a binding quantity limit or the curves change slope.

Theorem: In this fixed-slope linear model, \(\Delta P=(\Delta a-\Delta c)/(b+d)\) and \(\Delta Q=(d\Delta a+b\Delta c)/(b+d)\). Because \(b+d>0\), the sign of each change is determined by its numerator.

Reading the Directions Before Calculating

The equations give a useful sign check. If demand alone increases, then \(\Delta a>0\) and \(\Delta c=0\), so both \(\Delta P\) and \(\Delta Q\) are positive. If supply alone increases, then \(\Delta c>0\) and \(\Delta a=0\), so price falls and quantity rises. These familiar one-curve results are special cases of the two formulas.

With simultaneous shifts, do not infer both directions from the phrase “both curves shift.” If demand and supply both increase, quantity rises, but price can rise, fall, or remain unchanged depending on the relative intercept changes. If demand increases while supply decreases, price rises, but quantity can move either way. In each case, calculate the sign of each numerator separately.

Worked Example: Demand Falls While Supply Increases

A market initially has demand \(Q_D=100-2P\) and supply \(Q_S=20+3P\). A change in buyer preferences reduces the demand intercept by 10, while improved production conditions increase the supply intercept by 4. Find the new equilibrium and the changes in price and quantity.

Method. Use the fixed-slope comparative-statics formulas, then verify the answer by solving both equilibria. The intercept changes are \(\Delta a=-10\) and \(\Delta c=4\); the slopes are \(b=2\) and \(d=3\).

First check the original equilibrium:

100-2P=20+3P,\qquad 80=5P,\qquad P=16

At \(P=16\), quantity is \(100-2(16)=68\), and supply also gives \(20+3(16)=68\). Thus the original equilibrium is \((16,68)\).

The price change is:

\Delta P=\frac{-10-4}{2+3}=\frac{-14}{5}=-2.8

The quantity change is:

\Delta Q=\frac{3(-10)+2(4)}{2+3} =\frac{-30+8}{5} =\frac{-22}{5}=-4.4

So the new price is \(16-2.8=13.2\), and the new quantity is \(68-4.4=63.6\). Verify using the shifted curves: demand is \(90-2P\), and supply is \(24+3P\). At \(P=13.2\), demand is \(90-26.4=63.6\), and supply is \(24+39.6=63.6\). The results agree. The method avoids treating the two shocks as if they had identical effects on quantity.

Worked Example: Both Curves Shift Right

Suppose demand is \(Q_D=60-2P\) and supply is \(Q_S=12+4P\). An increase in buyer interest raises the demand intercept by 8, and a production improvement raises the supply intercept by 12. Find the changes and the new equilibrium.

Method. Calculate price and quantity changes separately. Since both curves shift right, quantity should rise; the direction of price is not determined by that fact alone.

Here \(b=2\), \(d=4\), \(\Delta a=8\), and \(\Delta c=12\). The original equilibrium is found from:

60-2P=12+4P,\qquad 48=6P,\qquad P=8

The original quantity is \(60-2(8)=44\), which matches supply: \(12+4(8)=44\). Now calculate the changes:

\Delta P=\frac{8-12}{2+4}=-\frac{2}{3}

\Delta Q=\frac{4(8)+2(12)}{2+4} =\frac{32+24}{6} =\frac{28}{3}

Price falls by \(2/3\), while quantity rises by \(28/3\). The new equilibrium is \(P=8-2/3=22/3\) and \(Q=44+28/3=160/3\). As a direct check, shifted demand is \(68-2P\) and shifted supply is \(24+4P\). At \(P=22/3\), demand is \(68-44/3=160/3\); supply is \(24+88/3=160/3\). The equality confirms the calculation. Here the supply-intercept increase is large enough, relative to the slopes, to lower price despite the increase in demand.

Worked Example: Price Rises While Quantity Falls

A market has demand \(Q_D=100-2P\) and supply \(Q_S=10+3P\). A new use for the product raises the demand intercept by 4, while a costly input shortage lowers the supply intercept by 10. Determine which equilibrium changes are certain and calculate them.

Method. Apply the two formulas with \(\Delta a=4\), \(\Delta c=-10\), \(b=2\), and \(d=3\). A demand increase and supply decrease both put upward pressure on price, but their quantity effects oppose one another.

Initially, \(100-2P=10+3P\), so \(90=5P\), \(P=18\), and \(Q=100-2(18)=64\). The price change is:

\Delta P=\frac{4-(-10)}{2+3}=\frac{14}{5}=2.8

The quantity change is:

\Delta Q=\frac{3(4)+2(-10)}{2+3} =\frac{12-20}{5} =-\frac{8}{5}=-1.6

Thus price rises to \(18+2.8=20.8\), and quantity falls to \(64-1.6=62.4\). The shifted curves are \(Q_D=104-2P\) and \(Q_S=3P\). At \(P=20.8\), demand equals \(104-41.6=62.4\), and supply equals \(3(20.8)=62.4\). The check confirms both levels and changes. The quantity direction is not fixed by the direction of either shock alone; it follows from the weighted sum.

A Reliable Calculation Routine

For a multiple-shock problem, keep the original equilibrium, the shocks, and the new equilibrium distinct. A compact routine reduces sign errors and makes the answer interpretable in context.

Write the original demand and supply equations, and identify \(b\) and \(d\) as positive slope magnitudes.
State \(\Delta a\) and \(\Delta c\) with signs. A demand decrease is a negative \(\Delta a\); a supply decrease is a negative \(\Delta c\).
Calculate \(\Delta P\) and \(\Delta Q\) separately, retaining the denominator \(b+d\).
Report new levels by adding changes to the old equilibrium: \(P_{\text{new}}=P+\Delta P\) and \(Q_{\text{new}}=Q+\Delta Q\).
Substitute the new price into both shifted curves. Both must produce the same new quantity.

The final check is especially useful when the question asks for a direction as well as a number. If demand and supply at the proposed new price do not agree, the calculation or sign assignment needs correction.

Common Exam Trap: Reversing the Quantity Weights

A tempting but incorrect quantity calculation is to weight the demand shock by \(b\) and the supply shock by \(d\). For the first example, this error would give:

\frac{2(-10)+3(4)}{2+3} =\frac{-20+12}{5} =-\frac{8}{5}

That is not the quantity change. The correct expression weights \(\Delta a\) by \(d\) and \(\Delta c\) by \(b\):

\Delta Q=\frac{3(-10)+2(4)}{5} =\frac{-22}{5}

A quick way to catch this particular error is to substitute the calculated \(\Delta P=-14/5\) into the demand equation for the change: \(\Delta Q=\Delta a-b\Delta P=-10-2(-14/5)=-22/5\). The supply equation gives the same result: \(\Delta c+d\Delta P=4+3(-14/5)=-22/5\).

Key takeaway: For fixed-slope linear demand and supply, compute price change from the difference between the intercept shocks and quantity change from their slope-weighted sum. Inspect the two numerators separately: one direction may be certain while the other is ambiguous.

Check Your Understanding

For each problem, identify the intercept changes, calculate equilibrium changes, and use the stated information to check the direction of the result.

Demand is \(Q_D=80-2P\) and supply is \(Q_S=20+3P\). The demand intercept increases by 5, with supply unchanged. Find the changes in equilibrium price and quantity. (Answer: Price rises by 1; quantity rises by 3.)
Demand is \(Q_D=120-4P\) and supply is \(Q_S=10+2P\). The demand intercept falls by 6 and the supply intercept rises by 3. Find both changes and state whether quantity changes. (Answer: Price falls by 1.5; quantity change is 0.)
In a market with \(b=1\) and \(d=4\), demand and supply intercepts both increase: \(\Delta a=5\), \(\Delta c=10\). Find the price and quantity changes. (Answer: Price falls by 1; quantity rises by 6.)
In a market with \(b=3\) and \(d=1\), demand rises by 12 and supply falls by 6. Find the changes and explain which direction is ambiguous before calculation. (Answer: Price rises by 4.5; quantity falls by 1.5; quantity direction is ambiguous before calculation.)
In a market with \(b=2\) and \(d=5\), the demand intercept rises by 9. What supply-intercept change would leave equilibrium price unchanged, and what would then happen to quantity? (Answer: The supply intercept must rise by 9; quantity rises by 9.)

What You'll Learn