The Central Idea
Let \(p\) represent price and \(q\) represent quantity.
A demand equation gives a relationship between price and quantity demanded.
A supply equation gives a relationship between price and quantity supplied.
Market equilibrium occurs when
\[ \boxed{q_d=q_s}. \]Equivalently, if both equations express price as a function of quantity, equilibrium occurs when
\[ \boxed{p_d(q)=p_s(q)}. \]Quantity Demanded
Quantity demanded is the amount buyers are willing to purchase at a particular price under the assumptions of the model.
A common linear form is
\[ \boxed{q_d=a-bp}, \]where
\[ b>0. \]As price rises, the modeled quantity demanded falls.
Quantity Supplied
Quantity supplied is the amount sellers are willing to offer at a particular price under the assumptions of the model.
A common linear form is
\[ \boxed{q_s=c+dp}, \]where
\[ d>0. \]As price rises, the modeled quantity supplied rises.
Example: Evaluate Demand
Suppose demand is
\[ q_d=500-10p. \]At price
\[ p=20, \]quantity demanded is
\[ q_d=500-10(20). \] \[ q_d=300. \]Therefore,
\[ \boxed{300\text{ units}}. \]Example: Evaluate Supply
Suppose supply is
\[ q_s=100+5p. \]At price
\[ p=20, \]quantity supplied is
\[ q_s=100+5(20). \] \[ q_s=200. \]Therefore,
\[ \boxed{200\text{ units}}. \]Equilibrium as Equality
If
\[ q_d=500-10p \]and
\[ q_s=100+5p, \]equilibrium requires
\[ 500-10p=100+5p. \]Example: Solve for Equilibrium Price
Continue with
\[ 500-10p=100+5p. \]Subtract \(100\):
\[ 400-10p=5p. \]Add \(10p\):
\[ 400=15p. \] \[ p=\frac{400}{15}=\frac{80}{3}. \]Therefore, the equilibrium price is
\[ \boxed{\$26.67\text{ approximately}}. \]Find the Equilibrium Quantity
Substitute
\[ p=\frac{80}{3} \]into either equation.
Using demand:
\[ q=500-10\left(\frac{80}{3}\right). \] \[ q= \frac{1500}{3}-\frac{800}{3} = \frac{700}{3}. \]Therefore,
\[ \boxed{233.33\text{ units approximately}}. \]Check the Equilibrium
Using supply:
\[ q_s = 100+5\left(\frac{80}{3}\right). \] \[ q_s= \frac{300}{3}+\frac{400}{3} = \frac{700}{3}. \]Demand and supply agree:
\[ \boxed{ q_d=q_s=\frac{700}{3} }. \]Equilibrium as an Intersection
Graphically, equilibrium is the intersection of the supply and demand relationships.
Algebraically, an intersection is found by setting the two expressions equal and solving.
This is the same general method used to find break-even points, crossing times, and intersections of function graphs.
Inverse Demand
A demand equation may instead express price as a function of quantity:
\[ \boxed{p_d=a-bq}, \]with
\[ b>0. \]This form is often called an inverse demand relationship.
Inverse Supply
Supply may also be written as price as a function of quantity:
\[ \boxed{p_s=c+dq}, \]with
\[ d>0. \]Example: Equilibrium from Inverse Equations
Suppose
\[ p_d=80-0.2q \]and
\[ p_s=20+0.1q. \]Set them equal:
\[ 80-0.2q=20+0.1q. \] \[ 60=0.3q. \] \[ q=200. \]Substitute:
\[ p=80-0.2(200)=40. \]Therefore,
\[ \boxed{q^*=200,\qquad p^*=\$40}. \]Notation for Equilibrium
Equilibrium values are often marked with a superscript star:
\[ \boxed{p^*} \]for equilibrium price and
\[ \boxed{q^*} \]for equilibrium quantity.
Shortage
A shortage occurs in the model when quantity demanded exceeds quantity supplied:
\[ \boxed{q_d>q_s}. \]The amount of shortage is
\[ \boxed{q_d-q_s}. \]Example: Calculate a Shortage
Let
\[ q_d=500-10p \]and
\[ q_s=100+5p. \]At
\[ p=20, \]we found
\[ q_d=300 \]and
\[ q_s=200. \]Therefore, shortage is
\[ 300-200=100. \]Thus,
\[ \boxed{100\text{ units}}. \]Surplus
A surplus occurs in the model when quantity supplied exceeds quantity demanded:
\[ \boxed{q_s>q_d}. \]The amount of surplus is
\[ \boxed{q_s-q_d}. \]Example: Calculate a Surplus
Using
\[ q_d=500-10p \]and
\[ q_s=100+5p, \]let
\[ p=40. \]Demand:
\[ q_d=500-10(40)=100. \]Supply:
\[ q_s=100+5(40)=300. \]Therefore, surplus is
\[ 300-100=200. \]Thus,
\[ \boxed{200\text{ units}}. \]Below Equilibrium Price
In the standard linear model, a price below equilibrium generally produces
\[ q_d>q_s. \]Therefore, the model indicates a shortage.
Above Equilibrium Price
In the standard linear model, a price above equilibrium generally produces
\[ q_s>q_d. \]Therefore, the model indicates a surplus.
A Shortage Function
Given demand and supply as functions of price, define
\[ D(p)=q_d(p)-q_s(p). \]Then:
- \(D(p)>0\) indicates shortage;
- \(D(p)=0\) indicates equilibrium;
- \(D(p)<0\) indicates surplus.
Equilibrium is therefore another zero-finding problem.
Example: Build the Excess-Demand Function
If
\[ q_d=500-10p \]and
\[ q_s=100+5p, \]then
\[ D(p) = (500-10p)-(100+5p). \] \[ \boxed{D(p)=400-15p}. \]Equilibrium occurs where
\[ D(p)=0. \]Constructing Demand from Two Points
Suppose demand is linear and two price-quantity pairs are known.
For points
\[ (p_1,q_1) \]and
\[ (p_2,q_2), \]the slope of quantity with respect to price is
\[ \boxed{ m_d= \frac{q_2-q_1}{p_2-p_1} }. \]A downward-sloping demand relationship has
\[ m_d<0. \]Example: Build a Demand Equation from Data
At price \(\$10\), quantity demanded is \(400\). At price \(\$30\), quantity demanded is \(200\).
Slope:
\[ m= \frac{200-400}{30-10} = \frac{-200}{20} = -10. \]Use
\[ q=-10p+b. \]Substitute
\[ (p,q)=(10,400): \] \[ 400=-100+b. \] \[ b=500. \]Therefore,
\[ \boxed{q_d=500-10p}. \]Constructing Supply from Two Points
For two supply observations,
\[ (p_1,q_1) \]and
\[ (p_2,q_2), \]use the same slope formula:
\[ \boxed{ m_s= \frac{q_2-q_1}{p_2-p_1} }. \]In a standard upward-sloping supply model,
\[ m_s>0. \]Example: Build a Supply Equation from Data
At price \(\$10\), quantity supplied is \(150\). At price \(\$30\), quantity supplied is \(250\).
Slope:
\[ m= \frac{250-150}{30-10} = \frac{100}{20} = 5. \]Write
\[ q=5p+b. \]Substitute
\[ (10,150): \] \[ 150=50+b. \] \[ b=100. \]Therefore,
\[ \boxed{q_s=100+5p}. \]Finding Equilibrium from Data-Built Models
Once the demand and supply equations have been constructed,
\[ q_d=500-10p \]and
\[ q_s=100+5p, \]solve
\[ q_d=q_s. \]The equilibrium follows from the same intersection method already developed.
Demand Intercepts
For a demand equation
\[ q_d=a-bp, \]setting
\[ p=0 \]gives
\[ q_d=a. \]This is the quantity intercept of the linear model.
The Price Intercept of Demand
Set
\[ q_d=0. \]Then
\[ 0=a-bp. \]Therefore,
\[ \boxed{ p=\frac{a}{b} }. \]This is the modeled price at which quantity demanded reaches zero.
Supply Intercepts
In
\[ q_s=c+dp, \]setting \(p=0\) gives
\[ q_s=c. \]Depending on the application, a negative intercept may be an algebraic artifact outside the economically meaningful domain rather than a literal negative quantity.
Domain Restrictions
Market models usually require
\[ p\ge0 \]and
\[ q\ge0. \]A linear formula may produce negative values outside its intended range.
Such values should not automatically be interpreted as meaningful market quantities.
Example: Restrict a Demand Model
Suppose
\[ q_d=500-10p. \]Nonnegative quantity requires
\[ 500-10p\ge0. \] \[ p\le50. \]Therefore, a natural nonnegative domain is
\[ \boxed{0\le p\le50}. \]Movement along a Demand Relationship
If the demand equation itself remains unchanged but price changes, the model moves to a different point on the same demand relationship.
Algebraically, the coefficients stay fixed while a new value of \(p\) is substituted.
Movement along a Supply Relationship
Likewise, if the supply equation remains unchanged and price changes, a new quantity supplied is found by substituting the new price into the same supply equation.
Shifts in Demand
A demand shift changes the demand equation itself.
For example,
\[ q_d=500-10p \]might become
\[ q_d=560-10p. \]At every given price, modeled quantity demanded is now \(60\) units higher.
Shifts in Supply
A supply shift changes the supply equation itself.
For example,
\[ q_s=100+5p \]might become
\[ q_s=160+5p. \]At every given price, modeled quantity supplied is now \(60\) units higher.
Equilibrium after a Demand Shift
Suppose supply remains
\[ q_s=100+5p, \]but demand changes to
\[ q_d=560-10p. \]Set equal:
\[ 560-10p=100+5p. \] \[ 460=15p. \] \[ p=\frac{92}{3}\approx30.67. \]Equilibrium quantity is
\[ q=100+5\left(\frac{92}{3}\right). \] \[ q=\frac{760}{3}\approx253.33. \]Therefore,
\[ \boxed{ p^*\approx\$30.67,\qquad q^*\approx253.33 }. \]Compare the Two Equilibria
Before the demand shift,
\[ p^*\approx26.67, \qquad q^*\approx233.33. \]After the increase in demand,
\[ p^*\approx30.67, \qquad q^*\approx253.33. \]In this particular linear model, the demand increase raises both equilibrium price and equilibrium quantity.
Equilibrium after a Supply Shift
Suppose demand remains
\[ q_d=500-10p, \]while supply increases to
\[ q_s=160+5p. \]Solve:
\[ 500-10p=160+5p. \] \[ 340=15p. \] \[ p=\frac{68}{3}\approx22.67. \]Then
\[ q=160+5\left(\frac{68}{3}\right) = \frac{820}{3} \approx273.33. \]Therefore,
\[ \boxed{ p^*\approx\$22.67,\qquad q^*\approx273.33 }. \]Compare a Supply Shift
Relative to the original equilibrium,
\[ p^*\approx26.67, \qquad q^*\approx233.33, \]the increased-supply model produces
\[ p^*\approx22.67, \qquad q^*\approx273.33. \]In this example, increased supply lowers equilibrium price and raises equilibrium quantity.
Parameterizing a Demand Shift
A shift can be represented by a parameter.
For example,
\[ \boxed{ q_d=(a+k)-bp }. \]The parameter \(k\) measures an additive shift in quantity demanded at every price.
Parameterizing a Supply Shift
Similarly,
\[ \boxed{ q_s=(c+h)+dp }. \]The parameter \(h\) measures an additive shift in quantity supplied at every price.
General Linear Equilibrium Formula
Let
\[ q_d=a-bp \]and
\[ q_s=c+dp, \]where
\[ b>0,\qquad d>0. \]At equilibrium,
\[ a-bp=c+dp. \]Therefore,
\[ a-c=(b+d)p. \]Thus,
\[ \boxed{ p^*=\frac{a-c}{b+d} }. \]General Equilibrium Quantity
Substitute
\[ p^*=\frac{a-c}{b+d} \]into demand:
\[ q^* = a- b \left( \frac{a-c}{b+d} \right). \]Simplifying gives
\[ \boxed{ q^*= \frac{ad+bc}{b+d} }. \]What the General Formula Shows
In
\[ p^*=\frac{a-c}{b+d}, \]equilibrium price depends on both intercept information and both slopes.
A larger demand intercept \(a\) tends to raise equilibrium price, while a larger supply intercept \(c\) tends to lower it, holding the other parameters fixed.
Unknown Demand Parameter
Suppose demand is
\[ q_d=a-8p \]and supply is
\[ q_s=40+4p. \]If equilibrium price is known to be
\[ p^*=20, \]then
\[ a-8(20)=40+4(20). \] \[ a-160=120. \] \[ a=280. \]Therefore,
\[ \boxed{a=280}. \]Unknown Supply Parameter
Suppose demand is
\[ q_d=600-12p \]and supply is
\[ q_s=c+6p. \]If equilibrium occurs at price
\[ p=25, \]then
\[ 600-12(25)=c+6(25). \] \[ 300=c+150. \] \[ c=150. \]Therefore,
\[ \boxed{c=150}. \]Unknown Slope from Equilibrium Information
Suppose
\[ q_d=500-10p \]and
\[ q_s=100+dp. \]If equilibrium price is
\[ 25, \]then
\[ 500-10(25)=100+25d. \] \[ 250=100+25d. \] \[ 150=25d. \] \[ d=6. \]Therefore,
\[ \boxed{d=6}. \]Price as the Independent Variable
In equations such as
\[ q_d=500-10p, \]price is treated algebraically as the input and quantity as the output.
This is convenient for evaluating quantities at a specified market price.
Quantity as the Independent Variable
In inverse equations such as
\[ p_d=80-0.2q, \]quantity is the input and price is the output.
Both representations can describe the same linear relationship after algebraic rearrangement.
Converting Demand to Inverse Demand
Suppose
\[ q=500-10p. \]Solve for \(p\):
\[ 10p=500-q. \] \[ p=50-0.1q. \]Therefore,
\[ \boxed{p=50-0.1q}. \]Converting Supply to Inverse Supply
Suppose
\[ q=100+5p. \]Solve for \(p\):
\[ 5p=q-100. \] \[ p=0.2q-20. \]Therefore,
\[ \boxed{p=0.2q-20}. \]Checking the Same Equilibrium in Inverse Form
Set
\[ 50-0.1q=0.2q-20. \] \[ 70=0.3q. \] \[ q=\frac{700}{3}. \]Then
\[ p=50-0.1\left(\frac{700}{3}\right) = \frac{80}{3}. \]This matches the equilibrium found from the original quantity equations.
Supply and Demand Tables
A table can help compare the two relationships at several prices.
\[ \begin{array}{c|c|c|c} p & q_d & q_s & \text{market condition}\\ \hline 20 & 300 & 200 & \text{shortage}\\ \frac{80}{3} & \frac{700}{3} & \frac{700}{3} & \text{equilibrium}\\ 40 & 100 & 300 & \text{surplus} \end{array} \]Market Equilibrium Is a Model Outcome
The algebraic equilibrium is the point where the stated supply and demand equations intersect.
Real markets may include taxes, regulation, inventories, contracts, strategic behavior, changing expectations, and many other effects.
This lesson studies the algebraic structure of the simplified model rather than claiming that every market behaves exactly this way.
Checking Equilibrium by Substitution
After finding \(p^*\), compute both
\[ q_d(p^*) \]and
\[ q_s(p^*). \]They should be equal.
If they differ, recheck the algebra.
Checking with Direction
In the standard model, if price is moved below equilibrium while the curves are unchanged, demand should exceed supply.
If price is moved above equilibrium, supply should exceed demand.
This provides a quick qualitative check.
Checking the Domain
A mathematically valid intersection can still lie outside the meaningful domain.
For example, a negative equilibrium price may signal that the chosen linear equations are not appropriate over the stated range.
Applied solutions should satisfy
\[ p^*\ge0 \]and ordinarily
\[ q^*\ge0. \]Common Error: Setting Price Equal to Quantity
Equilibrium does not mean
\[ p=q. \]It means quantity demanded equals quantity supplied, or equivalently the supply and demand price functions agree at the same quantity.
Common Error: Adding Supply and Demand
Equilibrium is found by setting the two relationships equal.
It is not found by adding
\[ q_d+q_s. \]Common Error: Confusing Shortage with Surplus
Shortage means
\[ q_d>q_s. \]Surplus means
\[ q_s>q_d. \]Subtract the smaller quantity from the larger one to report the magnitude.
Common Error: Ignoring Which Variable Is on Which Axis
Some equations write quantity as a function of price, while others write price as a function of quantity.
Before solving, identify what each variable represents and which quantity the equation outputs.
Common Error: Treating a Shift as a Movement along the Same Curve
Changing price within an unchanged equation moves to another point on the same relationship.
Changing an intercept or slope changes the relationship itself.
Common Error: Accepting Negative Quantities
A linear model may produce
\[ q<0 \]outside its useful range.
Negative physical market quantity is ordinarily not meaningful, so restrict the model's domain.
Common Error: Forgetting Units
A price may be measured in dollars per item, while quantity may be measured in units, thousands of units, or another scale.
Interpret coefficients using the units stated in the problem.
A Reliable Procedure for Market Equilibrium
- Identify the demand equation.
- Identify the supply equation.
- Confirm that both are written using compatible variables.
- Set quantity demanded equal to quantity supplied.
- Solve for equilibrium price or quantity.
- Substitute back to find the other equilibrium value.
- Verify that both equations give the same result.
- Check that price and quantity lie in the meaningful domain.
A Reliable Procedure for Shortage and Surplus
- Substitute the stated price into demand.
- Substitute the same price into supply.
- Compare the two quantities.
- If \(q_d>q_s\), compute shortage \(q_d-q_s\).
- If \(q_s>q_d\), compute surplus \(q_s-q_d\).
- If they are equal, the price is an equilibrium price.
A Reliable Procedure for Shifts
- Write the original supply and demand equations.
- Identify which equation changes.
- Write the new equation explicitly.
- Resolve the equilibrium using the new pair of equations.
- Compare old and new \(p^*\) and \(q^*\).
- Describe the change only for the model actually given.
Practice 1: Evaluate Demand
Given
\[ q_d=600-12p, \]find quantity demanded at
\[ p=25. \]Practice 2: Evaluate Supply
Given
\[ q_s=120+6p, \]find quantity supplied at
\[ p=25. \]Practice 3: Find Equilibrium
Using Practices 1 and 2, find equilibrium price and quantity.
Practice 4: Shortage
For the same equations, determine the shortage or surplus at
\[ p=20. \]Practice 5: Surplus
For the same equations, determine the shortage or surplus at
\[ p=35. \]Practice 6: Inverse Equilibrium
Find equilibrium for
\[ p_d=90-0.3q \]and
\[ p_s=30+0.1q. \]Practice 7: Build Demand from Two Points
A linear demand relationship passes through
\[ (p,q)=(10,500) \]and
\[ (30,300). \]Find the demand equation in the form \(q=a-bp\).
Practice 8: Build Supply from Two Points
A linear supply relationship passes through
\[ (p,q)=(10,100) \]and
\[ (30,220). \]Find the supply equation.
Practice 9: Equilibrium from Constructed Models
Using Practices 7 and 8, find equilibrium price and quantity.
Practice 10: Demand Shift
Demand is
\[ q_d=500-10p, \]supply is
\[ q_s=100+5p, \]and demand shifts to
\[ q_d=620-10p. \]Find the new equilibrium.
Practice 11: Supply Shift
Demand is
\[ q_d=500-10p, \]and supply shifts from
\[ q_s=100+5p \]to
\[ q_s=190+5p. \]Find the new equilibrium.
Practice 12: Unknown Demand Intercept
Demand is
\[ q_d=a-8p, \]supply is
\[ q_s=80+4p, \]and equilibrium price is \(\$30\). Find \(a\).
Practice 13: Convert to Inverse Demand
Rewrite
\[ q_d=400-8p \]as price as a function of quantity.
Practice 14: Feasibility
Demand is
\[ q_d=200-4p. \]Find the nonnegative price domain of this linear demand model.
Practice 15: Identify the Error
A student says equilibrium for
\[ q_d=400-5p \]and
\[ q_s=100+10p \]occurs where
\[ p=q. \]Explain the error and find the correct equilibrium.
Practice Answers
Practice 1
\[ q_d=600-12(25). \] \[ q_d=600-300. \] \[ \boxed{300}. \]Practice 2
\[ q_s=120+6(25). \] \[ q_s=120+150. \] \[ \boxed{270}. \]Practice 3
Set demand equal to supply:
\[ 600-12p=120+6p. \] \[ 480=18p. \] \[ p=\frac{80}{3}\approx26.67. \]Then
\[ q=120+6\left(\frac{80}{3}\right). \] \[ q=120+160=280. \]Therefore,
\[ \boxed{ p^*\approx\$26.67,\qquad q^*=280 }. \]Practice 4
At \(p=20\),
\[ q_d=600-240=360, \] \[ q_s=120+120=240. \]Therefore,
\[ 360-240=120. \] \[ \boxed{120\text{ unit shortage}}. \]Practice 5
At \(p=35\),
\[ q_d=600-420=180, \] \[ q_s=120+210=330. \]Therefore,
\[ 330-180=150. \] \[ \boxed{150\text{ unit surplus}}. \]Practice 6
\[ 90-0.3q=30+0.1q. \] \[ 60=0.4q. \] \[ q=150. \]Then
\[ p=90-0.3(150). \] \[ p=45. \]Therefore,
\[ \boxed{q^*=150,\qquad p^*=\$45}. \]Practice 7
Slope:
\[ m= \frac{300-500}{30-10} = -10. \]Write
\[ q=-10p+b. \]Using \((10,500)\):
\[ 500=-100+b. \] \[ b=600. \]Therefore,
\[ \boxed{q_d=600-10p}. \]Practice 8
Slope:
\[ m= \frac{220-100}{30-10} = \frac{120}{20} = 6. \]Write
\[ q=6p+b. \]Using \((10,100)\):
\[ 100=60+b. \] \[ b=40. \]Therefore,
\[ \boxed{q_s=40+6p}. \]Practice 9
\[ 600-10p=40+6p. \] \[ 560=16p. \] \[ p=35. \]Then
\[ q=600-10(35)=250. \]Therefore,
\[ \boxed{p^*=\$35,\qquad q^*=250}. \]Practice 10
\[ 620-10p=100+5p. \] \[ 520=15p. \] \[ p=\frac{104}{3}\approx34.67. \]Then
\[ q= 100+ 5\left(\frac{104}{3}\right) = \frac{820}{3} \approx273.33. \]Therefore,
\[ \boxed{ p^*\approx\$34.67,\qquad q^*\approx273.33 }. \]Practice 11
\[ 500-10p=190+5p. \] \[ 310=15p. \] \[ p=\frac{62}{3}\approx20.67. \]Then
\[ q= 190+ 5\left(\frac{62}{3}\right) = \frac{880}{3} \approx293.33. \]Therefore,
\[ \boxed{ p^*\approx\$20.67,\qquad q^*\approx293.33 }. \]Practice 12
At equilibrium price \(30\),
\[ a-8(30)=80+4(30). \] \[ a-240=200. \] \[ a=440. \]Therefore,
\[ \boxed{a=440}. \]Practice 13
\[ q=400-8p. \] \[ 8p=400-q. \] \[ p=50-\frac18q. \]Therefore,
\[ \boxed{ p_d=50-0.125q }. \]Practice 14
Require
\[ q_d\ge0. \] \[ 200-4p\ge0. \] \[ p\le50. \]With nonnegative price,
\[ \boxed{0\le p\le50}. \]Practice 15
Equilibrium requires quantity demanded to equal quantity supplied, not price to equal quantity.
\[ 400-5p=100+10p. \] \[ 300=15p. \] \[ p=20. \]Then
\[ q=400-5(20)=300. \]Therefore,
\[ \boxed{p^*=\$20,\qquad q^*=300}. \]What to Remember
A common linear demand model is
\[ \boxed{q_d=a-bp}, \]while a common linear supply model is
\[ \boxed{q_s=c+dp}. \]Equilibrium occurs where
\[ \boxed{q_d=q_s}. \]For the general linear pair,
\[ \boxed{ p^*=\frac{a-c}{b+d} }. \]A shortage occurs when
\[ q_d>q_s, \]and a surplus occurs when
\[ q_s>q_d. \]The most important habit is to treat equilibrium as an intersection problem and to distinguish changes in price along a fixed relationship from changes in the relationship itself.
Where We Go Next
Supply-and-demand models use equations to describe how price and quantity relate. Many everyday financial problems instead adjust a base amount by percentages: taxes are added, discounts are subtracted, commissions are earned, and markups transform cost into selling price.
The next lesson, Taxes, Discounts, Commissions, and Markups, develops those percentage-based applications.