Calculating Equilibrium After Simultaneous Shifts

Why Calculate the New Equilibrium?

When demand and supply both change, knowing the direction of each shift may not be enough to determine the new equilibrium. The result depends on the sizes of the shifts and on the slopes of the two curves. Linear equations let us calculate the new market-clearing price and quantity directly.

The key is to update both equations before solving them. A demand increase changes the demand equation; a supply decrease changes the supply equation. The new equilibrium is found where the updated quantities demanded and supplied are equal at the same price. This calculation is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Suppose market demand and supply are \(Q_d=A-BP\) and \(Q_s=C+DP\), where \(Q_d\) is quantity demanded, \(Q_s\) is quantity supplied, \(P\) is price, and \(A,B,C,D\) are constants with \(B>0\) and \(D>0\). A simultaneous shift changes the intercepts of both equations. If demand changes by \(\Delta_D\) units at every price and supply changes by \(\Delta_S\) units at every price, the updated equations are \(Q_d=(A+\Delta_D)-BP\) and \(Q_s=(C+\Delta_S)+DP\). Positive changes raise the quantity intercept; negative changes lower it.

The shift amounts are signed. For example, a decrease in supply of 5 units means \(\Delta_S=-5\), not \(+5\). Keep the original slopes in place when the problem specifies shifts of a fixed number of units at every price. Then find the price that makes the updated quantities equal and substitute that price into either equation to get the equilibrium quantity.

A General Calculation Method

Equilibrium requires \(Q_d=Q_s\). Applying that condition to the updated equations gives a direct formula for the new price:

A+\Delta_D-BP=C+\Delta_S+DP

P^*=\frac{A-C+\Delta_D-\Delta_S}{B+D}

Here \(P^*\) is the equilibrium price after the shifts. The equilibrium quantity \(Q^*\) is found by substituting \(P^*\) into either updated equation. These expressions assume the curves are linear over the relevant range, have the stated slopes, and yield an economically meaningful price and quantity. If the calculation produces an impossible value for the market described, the assumed linear equations may not apply over that range.

The calculation follows a reliable sequence:

Record the original equations.
Identify the demand and supply intercepts, the price coefficients, and the units used for quantity and price.

Apply each shift with its sign.
Add the demand shift to the demand intercept and the supply shift to the supply intercept. Do not change a slope unless the problem says it changes.

Set updated demand equal to updated supply.
Solve this equation for the new equilibrium price.

Find and check quantity.
Substitute the price into both updated equations. They must produce the same quantity.

Worked Calculations

Worked Example: Demand Rises While Supply Falls

A hypothetical market has demand \(Q_d=100-2P\) and supply \(Q_s=10+3P\), with price measured in dollars and quantity measured in units per week. Demand increases by 15 units at every price, while supply decreases by 5 units at every price. Find the original and new equilibria.

Method: update the intercepts, then equate quantities. The original equilibrium is a useful reference. Set the original equations equal:

$$ 100-2P=10+3P $$

Subtract 10 from both sides and add \(2P\) to both sides: \(90=5P\), so \(P=18\). Substitute into demand: \(Q_d=100-2(18)=100-36=64\). Supply gives \(Q_s=10+3(18)=10+54=64\), confirming the original equilibrium of 18 dollars and 64 units per week.

Now apply the simultaneous changes. The demand intercept rises from 100 to \(100+15=115\). The supply intercept falls from 10 to \(10-5=5\). Thus:

Q_d'=115-2P,\qquad Q_s'=5+3P

Set the updated quantities equal: \(115-2P=5+3P\). Subtract 5 and add \(2P\), giving \(110=5P\), so the new equilibrium price is \(P=22\) dollars. Demand at that price is \(115-2(22)=115-44=71\) units per week. Supply is \(5+3(22)=5+66=71\) units per week.

Check: both updated equations give 71 at a price of 22. The price rises from 18 to 22, and quantity rises from 64 to 71. A common mistake would be to treat the supply decrease as a positive shift; the sign check for that error appears below.

Worked Example: Opposing Shifts Leave Quantity Unchanged

A hypothetical commodity market has \(Q_d=180-3P\) and \(Q_s=12+2P\). Price is in dollars per crate and quantities are in thousands of crates per week. Demand decreases by 24 thousand crates at every price, while supply increases by 16 thousand crates at every price. Find the new equilibrium and verify whether quantity changes.

Method: adjust both intercepts and compare the resulting equilibrium. First, the original equilibrium satisfies \(180-3P=12+2P\). Therefore \(168=5P\), so \(P=33.6\) dollars. Original demand is \(180-3(33.6)=180-100.8=79.2\) thousand crates. Original supply is \(12+2(33.6)=12+67.2=79.2\) thousand crates.

Demand falls by 24, so its new intercept is \(180-24=156\). Supply rises by 16, so its new intercept is \(12+16=28\). The updated equations are:

Q_d'=156-3P,\qquad Q_s'=28+2P

Set them equal: \(156-3P=28+2P\). Subtract 28 from both sides and add \(3P\) to both sides: \(128=5P\). Thus the new price is \(P=25.6\) dollars per crate. Substitution gives demand \(156-3(25.6)=156-76.8=79.2\) thousand crates. Supply gives \(28+2(25.6)=28+51.2=79.2\) thousand crates.

Check: both equations give 79.2 thousand crates. Quantity is unchanged even though the equilibrium price falls from 33.6 to 25.6 dollars. The changes offset in quantity: the lower price raises quantity demanded along the updated demand curve, while the changed intercepts together leave the market-clearing quantity at 79.2.

Worked Example: Two Increases with Different Effects

A hypothetical market has demand \(Q_d=220-4P\) and supply \(Q_s=16+2P\), with price in dollars and quantity in units per day. Demand increases by 12 units at every price, and supply increases by 6 units at every price. Calculate the new equilibrium and the changes in price and quantity.

Method: solve the updated equations, then compare with the original equilibrium. Initially, \(220-4P=16+2P\), so \(204=6P\) and \(P=34\) dollars. The original quantity is \(220-4(34)=220-136=84\) units; supply confirms \(16+2(34)=84\).

The updated demand intercept is \(220+12=232\), and the updated supply intercept is \(16+6=22\). Therefore \(Q_d'=232-4P\) and \(Q_s'=22+2P\). Equating them gives \(232-4P=22+2P\), hence \(210=6P\), so \(P=35\) dollars.

The new demand quantity is \(232-4(35)=232-140=92\) units. The new supply quantity is \(22+2(35)=22+70=92\) units. Relative to the original equilibrium, price changes by \(35-34=1\) dollar and quantity changes by \(92-84=8\) units.

Check: both updated curves give 92 units at 35 dollars. Although both intercepts rise, the price rises here because the demand intercept increases by more than the supply intercept. The result depends on the signed size of both changes, not just on the fact that both curves shift.

Why the Formula Works

The price formula follows directly from the market-clearing condition. Starting with updated demand equal to updated supply, move the intercepts to the left and the price terms to the right:

(A+\Delta_D)-(C+\Delta_S)=BP+DP=(B+D)P

Since \(B>0\) and \(D>0\), the denominator \(B+D\) is positive. Dividing by it yields \(P^*=(A-C+\Delta_D-\Delta_S)/(B+D)\). The demand shift enters with a plus sign and the supply shift with a minus sign because an increase in the supply intercept raises the quantity available at any given price, reducing the price needed to clear the market.

The same calculation also gives a concise way to check changes relative to the original equilibrium. With slopes unchanged, subtract the original equilibrium price from the new one:

\Delta P^*=P_{\mathrm{new}}^*-P_{\mathrm{old}}^* =\frac{\Delta_D-\Delta_S}{B+D}

To obtain the quantity change, substitute the price change into demand. The demand intercept changes by \(\Delta_D\), while the price change affects quantity by \(-B\Delta P^*\):

\Delta Q^*=\Delta_D-B\Delta P^* =\frac{D\Delta_D+B\Delta_S}{B+D}

These change formulas are checks, not replacements for correctly updating and solving the equations. They apply when the shifts alter intercepts by fixed amounts and leave both slopes unchanged.

Common Sign Error and Final Checks

Return to the first example. The demand change was \(\Delta_D=+15\), and the supply change was \(\Delta_S=-5\). The correct new price is:

P^*=\frac{100-10+15-(-5)}{2+3} =\frac{110}{5}=22

A common incorrect calculation treats the supply decrease as \(+5\), leading to a mistaken supply equation \(Q_s=15+3P\). Equating that equation with the updated demand gives \(115-2P=15+3P\), so \(P=20\) and quantity \(75\). But those values do not use the stated supply decrease. At \(P=20\), the correctly updated supply is \(5+3(20)=65\), not 75; demand is \(115-2(20)=75\). Since the quantities differ, this cannot be equilibrium.

Write decreases as negative changes before updating intercepts.
Keep price and quantity units attached to the final result; do not confuse a quantity stated in thousands with individual units.
Verify the final quantity in both updated equations. A mismatch indicates an arithmetic or equation setup error.
Do not round intermediate values if a calculation produces a non-integer equilibrium; round only the final reported result to the precision requested.

Key takeaway: For simultaneous shifts in linear demand and supply, update both intercepts using signed changes, set the new quantities equal, and verify the resulting quantity in each equation. The relative sizes of the shifts and the slopes determine the new equilibrium.

Check Your Understanding

For each problem, update both equations before solving. Treat all stated quantity changes as changes at every price.

A market has \(Q_d=90-2P\) and \(Q_s=6+2P\). Demand increases by 10 units and supply increases by 6 units. Find the new equilibrium price and quantity. (Answer: \(P=22\), \(Q=56\).)
A market has \(Q_d=150-3P\) and \(Q_s=6+2P\). Demand decreases by 18 units and supply decreases by 6 units. Find the new equilibrium. (Answer: \(P=26.4\), \(Q=52.8\).)
A market has \(Q_d=240-5P\) and \(Q_s=15+3P\). Demand increases by 20 units while supply decreases by 9 units. Find the new equilibrium and verify quantity using both equations. (Answer: \(P=31.75\), \(Q=101.25\).)
Quantities are measured in hundreds of units. A market has \(Q_d=320-4P\) and \(Q_s=20+2P\). Demand decreases by 28 hundred units and supply increases by 14 hundred units. Find the new equilibrium, stating the quantity in hundreds of units. (Answer: \(P=43\), \(Q=120\) hundred units.)
A market has \(Q_d=180-2P\) and \(Q_s=30+3P\). Demand increases by 25 units and supply decreases by 10 units. Find the original and new equilibria, then state the changes in price and quantity. (Answer: Original \(P=30, Q=120\); new \(P=37, Q=131\); price rises by 7 and quantity rises by 11.)

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