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Recitation 4

Market Equilibrium and Elasticity

Concepts

Beer Market Setup

Demand: \( Q^D = 100 - 6P \)

Supply: \( Q^S = 10P \)

(Quantities in thousands per month)

Exercise 1: Movement Along a Curve

Suppose the price of beer rises from $6.25 to $7.00.

  1. Does the demand curve shift?
  2. Does the supply curve shift?
  3. What changes: quantity demanded, quantity supplied, or both?
  4. Calculate the new quantity demanded.

  1. No.
  2. No.
  3. Both quantity demanded and supplied change.
  4. \( Q^D = 100 - 6(7) = 58 \) (thousands per month)

Exercise 2: Demand Shift Identification

For each scenario, state: (i) which curve shifts, (ii) direction.

  1. A new study says moderate beer drinking is healthy.
  2. The price of wine falls.
  3. Student income falls in a recession.
  4. UNC enrollment increases by 20%.

  1. Demand shifts right.
  2. Demand shifts left.
  3. Demand shifts left.
  4. Demand shifts right.

Exercise 3: Supply Shift Identification

For each scenario, state: (i) which curve shifts, (ii) direction.

  1. Automation lowers production costs.
  2. Government imposes a $1 tax per six-pack.
  3. Several breweries shut down.

  1. Supply shifts right.
  2. Supply shifts left.
  3. Supply shifts left.

Exercise 4: Elasticity of Demand

Recall the elasticity formula:

\[ E_D = \frac{dQ^D}{dP} \times \frac{P}{Q} \]

  1. Find \( \frac{dQ^D}{dP} \).
  2. Compute \( E_D \) at \( P^* = 6.25, Q^* = 62.5 \).
  3. Compute \( E_D \) at \( P = 10 \).
  4. Compute \( E_D \) at \( P = 2 \).
  5. At what price is demand unit elastic?

  1. \( \frac{dQ^D}{dP} = -6 \)
  2. \( E_D = (-6) \times \frac{6.25}{62.5} = -0.6 \)
  3. At \( P = 10 \): \( Q = 40 \), \( E_D = (-6) \times \frac{10}{40} = -1.5 \)
  4. At \( P = 2 \): \( Q = 88 \), \( E_D = (-6) \times \frac{2}{88} = -0.136 \)
  5. Unit elastic when \( E_D = -1 \): \[ -6 \times \frac{P}{100 - 6P} = -1 \] \[ 6P = 100 - 6P \] \[ P = \frac{100}{12} = 8.33 \]

Exercise 5: Supply Shift with New Equation

A supply shock gives the new supply equation:

\[ Q_2^S = 10P - 20 \]

  1. Find the new equilibrium using original demand.
  2. Compute \( E_S \) at the new equilibrium, where \( E_S = \frac{dQ^S}{dP} \times \frac{P}{Q} \).

  1. Set demand equal to new supply: \[ 100 - 6P = 10P - 20 \] \[ 120 = 16P \] \[ P = 7.5, \quad Q = 55 \]
  2. \( \frac{dQ^S}{dP} = 10 \), so \[ E_S = 10 \times \frac{7.5}{55} = 1.36 \]