Study Notes for Concepts in Mathematics: Linear Demand and Supply Curves

Concepts in Math

Math 170

Course Information

• Spring 2026

• Institution: Salve Regina University

Linear Demand and Supply Curves

Overview of Demand and Supply

• In a free-market economy, consumer demand for a specific commodity varies with its unit price.

• Definition of a free market: An economic system where prices of goods and services are determined by supply and demand, represented by interactions between sellers and buyers.

• The relationship between unit price and quantity demanded is described using a demand equation.

• The graphical representation of the demand equation is known as a demand curve.

Law of Demand

• States: If the price increases (↑) then demand decreases (↓), and if the price decreases (↓), demand increases (↑).

• Example: If candy bars are inexpensive, consumer purchase quantity increases significantly.

Demand Function

Definition of Demand Function

• A demand function is mathematically expressed as: $p = f(x)$ where:

  • p: Unit price

  • x: Quantity of the commodity

• Typically characterized as a decreasing function concerning x.

• Meaning: As x increases, p decreases.

Characteristics of a Linear Demand Function

• A linear demand function is a linear equation concerning x and p.

• Both x and p take only positive values.

• The graph is a straight line indicating a negative slope, representing the demand curve located in the first quadrant.

Example of Demand Function

• Case: Sentinel iPod™ alarm clock

  • Quantity demanded (48,000 units) when unit price = $8.

  • Quantity demanded (32,000 units) when unit price = $12.

  • Find the demand equation, unit price for quantity demanded of 40,000 units, and quantity demanded at $14.

Example Solution a) Find the Demand Equation

• Let:

  • p = unit price of iPod alarm clock (in dollars)

  • x = number of units demanded (in thousands)

• Points on the demand curve:

  • At p = 8, x = 48 → Point (48, 8)

  • At p = 12, x = 32 → Point (32, 12)

• Demand equation is linear (a straight line).

Example Solution b) Unit Price for 40,000 units

• For x = 40 (40,000 units), the demand equation yields p = $10.

Example Solution c) Quantity Demanded at $14

• For p = 14, the demand equation yields a demand of 24,000 iPod alarm clocks.

Another Example

• Case: Designer dorm room rug (52 in. x 87 in.)

  • Quantity demanded = 500 when unit price = $100.

• For each $20 decrease in price, quantity demanded increases by 500 units.

• Goal: Determine demand equation and sketch its graph.

Example Solution for Rug

• Let:

  • p = price of the rug (in dollars)

  • x = quantity demanded when price is p.

• Point on demand curve: (500, 100).

• Slope based on price decrement: $m = \frac{500}{20} = 25$

• Demand equation: $p = -25x + 100$.

Linear Supply Curves

Overview of Supply Function

• There exists a relationship in competitive markets between unit price and the quantity of a commodity available.

• Definition of competitive market: Many buyers and sellers exist.

• This relationship is described using a supply equation, with its graphical representation known as a supply curve.

Law of Supply

• Example: If a candy bar sells for $10, then many units will be supplied.

• If the price drops to $0.25, the supply would be considerably less, as substantial profit is not possible.

Definition of Supply Function

• A supply function is described mathematically as: $p = f(x)$ where:

  • p: Unit price

  • x: Number of units of the commodity

• Generally characterized as an increasing function of x.

• Meaning: As x increases, p also increases.

Characteristics of a Linear Supply Function

• A linear supply function corresponds to a linear equation in p and x.

• The graph of the supply function features a straight line with a positive slope, existing in the first quadrant.

Example of Supply Equation

• Given supply equation: $4p - 5x = 120$

  • Interpretation: p in dollars, x in units of 100.

• Goals: sketch curve and determine marketed units at $55.

Example Solution a) Sketching the Curve

• For the p-intercept, set x = 0 to yield p = 30.

• For the x-intercept, set p = 0 yielding x = –24.

• Solve for p:
  $p = \frac{5x}{4} + 30$.

Example Solution b) Marketed Units at $55

• Substitute p = 55 into the supply equation.
  $4(55) - 5x = 120$

• Solve for x to determine marketed units:
  $x = 20$

• Thus, marketed units are 2000 (given x is in units of 100).

Market Equilibrium

Concept of Market Equilibrium

• Market equilibrium occurs when the price is sufficiently high: consumer demand decreases, and it becomes low, affecting supplier interest negatively.

• Equilibrium: The price point where buyers and sellers reach agreement.

• Market equilibrium is achieved when quantity produced equals quantity demanded.

Properties of Market Equilibrium

• At equilibrium, market supply meets demand stabilizing prices.

• Equilibrium quantity: The quantity produced.

• Equilibrium price: The price corresponding to equilibrium quantity.

Geometrical Interpretation

• Graphically, it is where the demand curve intersects the supply curve.

• At point (x0, p0), both supply and demand conditions are satisfied:

  • $x_0$ is equilibrium quantity.

  • $p_0$ is equilibrium price.

Equilibrium Examples

Example Scenarios

• If a buyer wants to buy a candy bar for $1, but the seller wants to sell it for $5, no transactions occur until an agreement is reached.

Example: ThermoMaster's Thermometer

• Demand equation: $5x + 3p - 30 = 0$ (where p = price, x = quantity demanded in thousands).

• Supply equation: $52x - 30p + 45 = 0$ (where x is quantity supplied).

• Goal: Calculate equilibrium quantity and price.

Example Solution Steps

1. Solve the system of equations:
   $5x + 3p - 30 = 0$ \and$52x - 30p + 45 = 0$.

Using Substitution for Solutions

• Express p in terms of x:
  $3p = -5x + 30$ yields $p = -\frac{5}{3}x + 10$.

• Substitute into second supply equation to find x:
  $52x + 50x - 300 + 45 = 0$.

Equilibrium Quantity and Price Results

• Final results yield equilibrium quantity of 2500 (thousands) and price $5.83 per thermometer.

Fair Trade Coffee Example

• Demand curve: $P = -5.6Q_d + 25$

• Supply curve: $P = 0.4Q_s + 1$

• Calculate equilibrium price P = $2.6 and equilibrium quantity 4 units.

Conclusion

• Understanding linear demand and supply curves is pivotal for analyzing market behavior, producing insights on price elasticity and strategic pricing in realistic scenarios. The equilibrium point forms the basis of how markets function efficiently and sustainably.