Why Trace Shocks One at a Time?
A market may experience several changes before a new equilibrium is observed. For example, a production improvement might shift supply, followed by a change in consumer preferences that shifts demand. Comparing only the initial and final equilibria gives the total change, but it does not show what happened at each stage. Tracing the sequence identifies the intermediate equilibrium and the effect associated with each shock.
The method is straightforward: hold all schedules fixed except for the one that changes, solve the new equilibrium, and record the change from the immediately preceding equilibrium. Repeat for each shock. The step effects must add to the total change from the initial equilibrium to the final one.
This method applies to changes in equilibrium price and quantity. A shift in a curve is an exogenous change in the schedule; it is not the same as a movement along an unchanged curve when price changes. The earlier material on distinguishing curve shifts from movements along curves makes this distinction explicit. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Solving and Recording Each Equilibrium
Suppose market demand and supply are written as quantity-form schedules: quantity demanded and quantity supplied are each expressed as functions of price \(P\). Equilibrium occurs when the quantity buyers plan to purchase equals the quantity sellers plan to sell. With linear schedules, use the same price in both equations, set the quantities equal, and solve. Then substitute the equilibrium price into either schedule to find equilibrium quantity.
For a useful general case, let demand be \(Q^d=a-bP\) and supply be \(Q^s=c+dP\), where \(b>0\) and \(d>0\). The demand curve slopes downward in this quantity-form representation, while the supply curve slopes upward. If the resulting equilibrium has a nonnegative price and quantity within the relevant market range, it is:
When a shock changes one intercept, the other schedule is held fixed for that step. If demand’s intercept changes by \(\Delta a\), the equilibrium changes by \(\Delta P=\Delta a/(b+d)\) and \(\Delta Q=d\Delta a/(b+d)\). If supply’s intercept changes by \(\Delta c\), the changes are \(\Delta P=-\Delta c/(b+d)\) and \(\Delta Q=b\Delta c/(b+d)\). These formulas describe intercept shifts while the slopes remain unchanged; they should not be applied unchanged if a shock alters a slope or changes the form of a schedule.
Why Step Effects Add Up
Let the successive equilibrium prices be \(P_0,P_1,\ldots,P_n\), and let the corresponding quantities be \(Q_0,Q_1,\ldots,Q_n\). The change at step \(j\) is \(P_j-P_{j-1}\) for price and \(Q_j-Q_{j-1}\) for quantity. Adding the price changes cancels every intermediate price: \(P_1-P_0+P_2-P_1+\cdots+P_n-P_{n-1}=P_n-P_0\). The same cancellation holds for quantity.
This telescoping identity is the key check on a sequential calculation: the sum of the recorded step effects must equal the final-minus-initial change. It does not mean every shock has the same effect regardless of the market’s starting schedules. For fixed-slope linear curves with intercept shifts, the formulas above give each shock’s effect directly. If slopes or curve shapes also change, an individual shock’s measured effect can depend on which other changes have already occurred. The total change between the same initial and final equilibria remains the final equilibrium minus the initial equilibrium.
Worked Example: Supply Shifts Before Demand
A market initially has demand \(Q^d=110-2P\) and supply \(Q^s=14+2P\). First, a production improvement raises the supply intercept from 14 to 20. Later, a change in preferences lowers the demand intercept from 110 to 100. Find the initial, intermediate, and final equilibria, the effect of each shock, and the total change. Report price and quantity changes as final minus initial for each step.
Method. Solve demand equal to supply at the initial schedules, then change only supply’s intercept and solve again. Finally, change only demand’s intercept and solve the last equilibrium. This avoids treating the two shifts as if they occurred simultaneously.
Initial equilibrium. Set \(110-2P=14+2P\). Then \(96=4P\), so \(P_0=24\). Substitution gives \(Q_0=110-2(24)=110-48=62\); supply gives \(14+2(24)=14+48=62\). The initial equilibrium is \((P_0,Q_0)=(24,62)\).
After the supply shift. Demand remains \(110-2P\), and supply is now \(20+2P\). Set \(110-2P=20+2P\). Then \(90=4P\), so \(P_1=22.5\). Quantity is \(110-2(22.5)=110-45=65\), and supply gives \(20+2(22.5)=20+45=65\). The intermediate equilibrium is \((22.5,65)\). The supply-shift step effect is \(\Delta P_1=22.5-24=-1.5\) and \(\Delta Q_1=65-62=+3\).
After the demand shift. Supply remains \(20+2P\), and demand is now \(100-2P\). Set \(100-2P=20+2P\). Then \(80=4P\), so \(P_2=20\). Quantity is \(100-2(20)=60\), and supply gives \(20+2(20)=60\). The final equilibrium is \((20,60)\). The demand-shift step effect is \(\Delta P_2=20-22.5=-2.5\) and \(\Delta Q_2=60-65=-5\).
Total change and check. Price changes by \(20-24=-4\), and quantity changes by \(60-62=-2\). Adding step effects gives \(-1.5+(-2.5)=-4\) for price and \(3+(-5)=-2\) for quantity. Both totals match the initial-to-final comparison.
Worked Example: Reversing the Order of the Same Shocks
Use the same initial market, \(Q^d=110-2P\) and \(Q^s=14+2P\), and the same final schedules, \(Q^d=100-2P\) and \(Q^s=20+2P\). This time, the demand intercept falls first and the supply intercept rises second. Find both step effects and verify the total change.
Initial equilibrium. As before, \(110-2P=14+2P\) gives \(P_0=24\) and \(Q_0=62\). Record this as the starting point.
Demand shift first. Demand becomes \(100-2P\), while supply remains \(14+2P\). Equating gives \(100-2P=14+2P\), so \(86=4P\) and \(P_1=21.5\). Quantity is \(100-2(21.5)=100-43=57\); supply gives \(14+2(21.5)=14+43=57\). The intermediate equilibrium is \((21.5,57)\), so this step changes price by \(21.5-24=-2.5\) and quantity by \(57-62=-5\).
Supply shift second. Keep demand at \(100-2P\), and change supply to \(20+2P\). Then \(100-2P=20+2P\), so \(80=4P\) and \(P_2=20\). Quantity is \(100-40=60\), also \(20+40=60\). This step changes price by \(20-21.5=-1.5\) and quantity by \(60-57=+3\).
Check. The total changes are \(20-24=-4\) in price and \(60-62=-2\) in quantity. The step effects sum to \(-2.5+(-1.5)=-4\) and \(-5+3=-2\). The intermediate equilibrium differs from the preceding example because the order differs. Here, because the curves retain the same slopes and the shocks are intercept shifts, the effect associated with each particular shift is unchanged.
Worked Example: A Demand Increase Followed by a Supply Decrease
A market has demand \(Q^d=96-3P\) and supply \(Q^s=12+P\). Demand then shifts outward to \(Q^d=108-3P\). After that, supply shifts inward to \(Q^s=4+P\). Calculate every equilibrium and compare the two step effects with the total change.
Initial equilibrium. Set \(96-3P=12+P\). Then \(84=4P\), so \(P_0=21\). Quantity is \(96-3(21)=96-63=33\), which agrees with supply: \(12+21=33\).
After demand increases. Equate \(108-3P\) and \(12+P\): \(96=4P\), so \(P_1=24\). Quantity is \(108-3(24)=108-72=36\); supply gives \(12+24=36\). The first step effect is \(24-21=+3\) in price and \(36-33=+3\) in quantity.
After supply decreases. Now equate \(108-3P\) and \(4+P\). Then \(104=4P\), so \(P_2=26\). Quantity is \(108-3(26)=108-78=30\), and supply gives \(4+26=30\). The second step effect is \(26-24=+2\) in price and \(30-36=-6\) in quantity.
Check. From the initial to the final equilibrium, price changes by \(26-21=+5\) and quantity by \(30-33=-3\). The step effects give \(3+2=5\) and \(3+(-6)=-3\), respectively. The checks confirm both the intermediate calculations and the final comparison.
Common Mistake: Rounding an Intermediate Equilibrium
A frequent error is to round an intermediate price to a whole number, or to make an arithmetic slip while solving, then use that incorrect value to calculate the next step. For example, after supply changes in the first worked example, the equation is \(110-2P=20+2P\). An incorrect calculation might report \(P=22\) and \(Q=66\). But substitution exposes the problem: demand at \(P=22\) is \(110-44=66\), while supply is \(20+44=64\). These quantities are not equal, so \((22,66)\) is not an equilibrium.
The correct equation gives \(90=4P\), so \(P=22.5\). Substituting gives \(Q^d=110-45=65\) and \(Q^s=20+45=65\). Keep exact values through all steps, and check every proposed equilibrium in both schedules. Round only if the requested final presentation calls for it.
Another trap is to report only the total change when asked for the effects of successive shocks. The total can be correct while the intermediate equilibrium and step effects are missing or wrong. A clear solution records each equilibrium as an ordered pair \((P,Q)\), labels each shock, and calculates every step as “new equilibrium minus immediately preceding equilibrium.”
Check Your Understanding
For each problem, find the requested intermediate or final equilibria and report changes as new minus old. Check quantities using both schedules.
- A market starts with \(Q^d=84-2P\) and \(Q^s=12+P\). Supply shifts to \(18+P\), then demand shifts to \(78-2P\). Find all three equilibria, both step effects, and the total change. (Answer: Equilibria \((24,36)\), \((22,40)\), \((20,38)\); step effects \((-2,+4)\), \((-2,-2)\); total change \((-4,+2)\).)
- Initially, \(Q^d=150-3P\) and \(Q^s=30+2P\). Demand shifts to \(165-3P\), then supply shifts to \(20+2P\). Find the equilibrium after each stage and the total changes. (Answer: Equilibria \((24,78)\), \((27,84)\), \((29,78)\); step effects \((+3,+6)\), \((+2,-6)\); total change \((+5,0)\).)
- A market begins with \(Q^d=120-2P\) and \(Q^s=24+2P\). Demand shifts to \(108-2P\), followed by supply shifting to \(30+2P\). Calculate each equilibrium and separate the two effects. (Answer: Equilibria \((24,72)\), \((21,66)\), \((19.5,69)\); step effects \((-3,-6)\), \((-1.5,+3)\); total change \((-4.5,-3)\).)
- For \(Q^d=90-2P\) and \(Q^s=10+2P\), supply first shifts to \(14+2P\), and then demand shifts to \(102-2P\). Find the initial and final equilibria and each step effect. (Answer: Equilibria \((20,50)\), \((19,52)\), \((22,58)\); step effects \((-1,+2)\), \((+3,+6)\); total change \((+2,+8)\).)
- A market starts with \(Q^d=132-4P\) and \(Q^s=12+2P\). Demand falls to \(120-4P\), and afterward supply falls to \(6+2P\). Find the equilibrium path and verify the sum of the step effects. (Answer: Equilibria \((20,52)\), \((18,48)\), \((19,44)\); step effects \((-2,-4)\), \((+1,-4)\); total change \((-1,-8)\).)