Lecture Notes: Determining Equilibrium Quantity and Price using Algebra

Introduction

• Topic: How to determine equilibrium quantity and price algebraically
• Useful for tests where drawing supply and demand curves is not feasible

Key Concepts

• Quantity Demanded (Qd) Equation: ( Qd = 100 - 3p )

  • Inverse relationship between Qd and price (P)
  • Law of demand: As price increases, quantity demanded decreases

• Quantity Supplied (Qs) Equation: ( Qs = 2p + 20 )

  • Positive relationship between Qs and price (P)
  • Law of supply: As price increases, quantity supplied increases

Algebraic Method for Finding Equilibrium

1. Equilibrium Condition: ( Qd = Qs )

   • At equilibrium, the quantity demanded equals quantity supplied

2. Set Equations Equal:

   • ( 100 - 3p = 2p + 20 )
   • Transition from two equations to a single equation with one unknown (P)

3. Solve for Equilibrium Price (P):

   • Collect terms: Move constants to one side, variables to the other
   • Equation simplifies to: ( 80 = 5p )
   • Divide by 5: ( p = 16 )
   • Equilibrium price ( (p^*) ) is 16

4. Find Equilibrium Quantity (Q):

   • Substitute ( p = 16 ) into either equation
   • Substitute into Demand Equation:
     • ( Qd = 100 - 3(16) = 52 )
   • Substitute into Supply Equation:
     • ( Qs = 2(16) + 20 = 52 )
   • Both yield equilibrium quantity ( (q^*) ) of 52

5. Equilibrium Point Representation:

   • Write as ( (Q, P) = (52, 16) )
   • Denote equilibrium values with asterisks: ( p^* = 16, q^* = 52 )

Conclusion

• Method allows solving for equilibrium without graphing
• Use of algebraic manipulation to find equilibrium values
• Emphasis on denoting equilibrium values to avoid confusion

Action Items

• Practice solving similar equations algebraically
• Use equilibrium notation consistently in solutions

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This concludes the notes on determining equilibrium quantity and price using algebraic methods without graphing.