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VEE Economics · Market Equilibrium and Adjustment · Tutorial 62 of 998

Deriving Equilibrium Effects Of Supply Shifts

Use linear market equations to calculate and explain the equilibrium effects of a supply shift while demand and both curve slopes remain unchanged.

Beginner 12 min read

What You'll Learn

  • Set up linear supply and demand equations with consistent price and quantity units
  • Derive the equilibrium price and quantity after a supply-intercept change
  • Interpret the signs of the price and quantity effects of an outward or inward supply shift
  • Use both curve slopes to determine the size of each equilibrium response
  • Check comparative-static calculations against the equilibrium equations

Why a Supply Shift Changes Both Equilibrium Price and Quantity

When supply changes but demand remains fixed, the market moves to a new intersection of the two curves. The direction of the effects is intuitive: an increase in supply tends to reduce the equilibrium price and raise the equilibrium quantity. Deriving the effects from linear equations shows exactly how large those changes are and why neither change is generally equal to the size of the supply shift.

The central task is to distinguish a change in the supply equation's intercept from a change in the quantity supplied caused by a price movement. We hold the slopes of both curves fixed, change only the supply intercept, and compare the equilibrium before and after the shift.

Definition: In a linear supply model \(Q_s=a+bP\), \(a\) is the supply intercept and \(b\) is the quantity response to price. A supply-intercept shift changes \(a\) while leaving \(b\) unchanged. If demand \(Q_d=c-dP\) is also held fixed, the new equilibrium is found where the shifted supply quantity equals demand quantity.

The Linear Model and Its Assumptions

Let \(P\) denote price and \(Q\) denote quantity. Use the direct (quantity as a function of price) equations

$$ Q_s=a+bP,\qquad Q_d=c-dP, $$

where \(b>0\) and \(d>0\). The positive supply slope means suppliers offer more as price rises; the negative demand slope means buyers demand less as price rises. The model assumes a competitive market-clearing equilibrium, a fixed supply slope, a fixed demand curve, and a supply shift represented solely by a change in \(a\). We assume the calculated equilibrium is economically relevant for the market being modeled; in particular, the equations are being used over a range where the resulting price and quantity make sense.

Suppose the supply intercept changes by \(\Delta a\), so the new supply equation is \(Q_s'=a+\Delta a+bP\). A positive \(\Delta a\) raises supply at each given price, while a negative \(\Delta a\) lowers it. The shift amount is measured in quantity units, not price units.

Deriving the Equilibrium Effects

At equilibrium, quantity supplied equals quantity demanded. Before the shift, let the equilibrium be \((P^*,Q^*)\). After the shift, call it \((P',Q')\). The derivation follows directly from the two equilibrium conditions.

1
Write the original equilibrium condition.
Set original supply equal to demand: \(a+bP^*=c-dP^*\).
2
Write the shifted equilibrium condition.
Set shifted supply equal to the unchanged demand: \(a+\Delta a+bP'=c-dP'\).
3
Subtract the original condition.
The fixed intercepts and demand intercept cancel, leaving \(\Delta a+b(P'-P^*)=-d(P'-P^*)\).
4
Solve for the price change, then the quantity change.
Rearrange for \(P'-P^*\), then use the demand equation to obtain \(Q'-Q^*\).
$$ \Delta P=P'-P^*=-\frac{\Delta a}{b+d}, \qquad \Delta Q=Q'-Q^*=\frac{d\,\Delta a}{b+d}. $$

For the quantity result, demand is unchanged, so \(Q'-Q^*=-d(P'-P^*)\). Substituting the price result gives \(\Delta Q=d\Delta a/(b+d)\). This also follows from supply: \(\Delta Q=\Delta a+b\Delta P=\Delta a-b\Delta a/(b+d)=d\Delta a/(b+d)\). Both curves therefore imply the same quantity change, as they must at the new equilibrium.

These are exact changes under the stated linear model, not approximations. If supply increases, \(\Delta a>0\), then \(\Delta P<0\) and \(\Delta Q>0\). If supply decreases, \(\Delta a<0\), then \(\Delta P>0\) and \(\Delta Q<0\). The denominator \(b+d\) combines the two slopes: both curves contribute to how much price must adjust to clear the market. The demand slope \(d\) also determines the quantity response.

Key result: With \(Q_s=a+bP\), \(Q_d=c-dP\), \(b,d>0\), fixed slopes, and demand unchanged, a supply-intercept change \(\Delta a\) gives \(\Delta P=-\Delta a/(b+d)\) and \(\Delta Q=d\Delta a/(b+d)\). The formulas do not describe a demand shift, a slope change, or a simultaneous change in both curves.

Worked Applications

Worked Example: An Increase in Supply

A market has demand \(Q_d=120-2P\) and supply \(Q_s=20+3P\). A production improvement raises the supply intercept by 10 quantity units, with the demand equation and both slopes unchanged. Find the original and new equilibria, then verify the changes using the derived formulas.

Method. First solve the initial equilibrium by setting the two quantities equal:

$$ 20+3P=120-2P \quad\Longrightarrow\quad 5P=100 \quad\Longrightarrow\quad P^*=20. $$

Substitute \(P^*=20\) into supply: \(Q^*=20+3(20)=20+60=80\). Demand gives the same quantity: \(120-2(20)=120-40=80\).

The shifted supply equation is \(Q_s'=30+3P\), because \(20+10=30\). Solve for the new equilibrium:

$$ 30+3P=120-2P \quad\Longrightarrow\quad 5P=90 \quad\Longrightarrow\quad P'=18. $$

Then \(Q'=30+3(18)=30+54=84\); checking demand gives \(120-2(18)=120-36=84\). Thus \(\Delta P=18-20=-2\) and \(\Delta Q=84-80=4\).

Formula check. Here \(b=3\), \(d=2\), and \(\Delta a=10\). Therefore \(\Delta P=-10/(3+2)=-2\) and \(\Delta Q=2(10)/(3+2)=4\), matching the direct solution. A common mistake is to treat the 10-unit increase in the supply intercept as a 10-unit increase in equilibrium quantity; the market price changes, so the actual equilibrium quantity rises by only 4 units.

Worked Example: A Decrease in Supply

A market has demand \(Q_d=108-2P\) and supply \(Q_s=-12+4P\). A new regulation reduces the supply intercept by 6 quantity units, without changing demand or either slope. Find both equilibria and interpret the signs.

Method. Equate the original equations:

$$ -12+4P=108-2P \quad\Longrightarrow\quad 6P=120 \quad\Longrightarrow\quad P^*=20. $$

The initial quantity is \(Q^*=-12+4(20)=-12+80=68\). Demand confirms \(108-2(20)=108-40=68\).

The new intercept is \(-12-6=-18\), so shifted supply is \(Q_s'=-18+4P\). The new equilibrium satisfies

$$ -18+4P=108-2P \quad\Longrightarrow\quad 6P=126 \quad\Longrightarrow\quad P'=21. $$

The new quantity is \(Q'=-18+4(21)=-18+84=66\), also \(108-2(21)=66\). Hence \(\Delta P=21-20=1\) and \(\Delta Q=66-68=-2\).

Formula check. With \(b=4\), \(d=2\), and \(\Delta a=-6\), the formulas give \(\Delta P=-(-6)/(4+2)=1\) and \(\Delta Q=2(-6)/(4+2)=-2\). The higher price and lower quantity are consistent with a contraction of supply. The negative intercept in this equation is an algebraic parameter; the equilibrium quantity is positive. Do not assume that every intercept must itself be a feasible market quantity.

Worked Example: Quantity Measured in Thousands

A hypothetical market reports quantity in thousands of units. Demand is \(Q_d=240-3P\), and supply is \(Q_s=30+2P\). A supply expansion increases the intercept by 15, where 15 means 15 thousand units. Find the new price and quantity, and state the changes in the units used.

Method. Solve the original equilibrium:

$$ 30+2P=240-3P \quad\Longrightarrow\quad 5P=210 \quad\Longrightarrow\quad P^*=42. $$

The initial quantity is \(Q^*=30+2(42)=30+84=114\) thousand units. Demand confirms \(240-3(42)=240-126=114\).

After the intercept increase, supply is \(Q_s'=45+2P\), since \(30+15=45\). The new equilibrium is

$$ 45+2P=240-3P \quad\Longrightarrow\quad 5P=195 \quad\Longrightarrow\quad P'=39. $$

Then \(Q'=45+2(39)=45+78=123\) thousand units, and demand gives \(240-3(39)=123\). Thus price falls by 3 price units and quantity rises by 9 thousand units.

Formula check. Here \(b=2\), \(d=3\), and \(\Delta a=15\). The predicted changes are \(\Delta P=-15/(2+3)=-3\) and \(\Delta Q=3(15)/(2+3)=9\). The shift and quantity change are both expressed in thousands of units where applicable; changing units midway through a calculation would invalidate the comparison.

Common Mistakes and Interpretation

A frequent error is to use a formula for an inverse supply equation as though it were written in direct form. In this tutorial the supply equation is \(Q_s=a+bP\), so \(b\) is the change in quantity per unit of price. The demand equation is \(Q_d=c-dP\), so \(d\) is also a quantity-per-price slope magnitude. The denominator in the derived formulas is \(b+d\), not a sum of inverse-curve slopes.

For example, in the first worked example the wrong calculation \(\Delta P=-10/(3-2)=-10\) subtracts the demand slope magnitude from the supply slope. The correction is to add the positive magnitudes: \(b+d=3+2=5\), giving \(\Delta P=-10/5=-2\). Directly solving the new equilibrium confirms the corrected value: \(P'=18\), compared with \(P^*=20\).

Another trap is to report the intercept change as the equilibrium quantity change. When price falls after a supply expansion, suppliers move along the new supply curve toward a lower quantity than they would offer at the old price. At the same time, buyers move along the unchanged demand curve toward a higher quantity. Equilibrium is where these adjustments agree; the formula accounts for both.

These comparative-static results are useful for organizing market analysis: state which curve shifts, keep the units and slope definitions consistent, and calculate the new equilibrium rather than treating the shift itself as the equilibrium response. The results apply only under the specified linear model and a fixed demand curve; they do not by themselves account for a change in demand or in either curve's slope.

Check Your Understanding

For each problem, use the stated linear equations and keep price and quantity units consistent.

  1. A market has \(Q_d=100-2P\) and \(Q_s=20+2P\). The supply intercept rises by 8. Find the original and new equilibrium price and quantity. (Answer: Original \(P=20,\ Q=60\); new \(P=18,\ Q=64\).)
  2. Demand is \(Q_d=150-3P\), and supply is \(Q_s=30+2P\). The supply intercept falls by 10. Find the changes in equilibrium price and quantity and the new equilibrium. (Answer: \(\Delta P=2,\ \Delta Q=-6\); new \(P=26,\ Q=72\).)
  3. Let \(Q_d=200-4P\) and \(Q_s=20+P\). Supply increases by 15 quantity units through an intercept shift. Find both equilibria and verify that the quantity change agrees with demand. (Answer: Original \(P=36,\ Q=56\); new \(P=33,\ Q=68\); quantity rises by 12.)
  4. A market has \(Q_d=180-2P\) and \(Q_s=12+4P\). A supply contraction lowers the intercept by 18. Calculate the new equilibrium and explain the direction of both changes. (Answer: Original \(P=28,\ Q=124\); new \(P=31,\ Q=118\); price rises by 3 and quantity falls by 6.)
  5. Demand is \(Q_d=210-3P\) and supply is initially \(Q_s=30+2P\). After a supply shift, the observed equilibrium is \(P=34,\ Q=108\), with demand unchanged. Infer the supply-intercept change and verify it using both comparative-static formulas. (Answer: The intercept increases by 10; \(\Delta P=-2\) and \(\Delta Q=6\).)

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