Question

Suppose the demand curve for aluminum is Q = 800 – 10P, where P is the price per unit of aluminum and Q measures millions of units of aluminum. The private marginal cost of producing aluminum is MC = Q + 20, while the external marginal cost of producing aluminum is $10.

1. Find the equilibrium price and quantity when the firm ignores the cost of the externality;
2. Find the equilibrium price and quantity when the cost of the externality is incorporated into the firm’s marginal cost function

Answer 1

Demand equation: Q = 800 - 10P

Inverse demand equation is: P = 80 - 0.1Q

Private Marginal Cost = PMC = Q + 20

When firm ignores the cost of externality, equilibrium is given by:

Demand = PMC

80 - 0.1Q = Q +20

1.1Q = 60

Q = 54.54545

P = Q + 20 = 64.54545

Hence, private equilibrium price P = 64.54545

Private equilibrium quantity is Q = 54.54545

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The externality marginal cost = $10

Hence, social marginal cost = PMC + externality cost

SMC = Q +20 +10

SMC = Q +30

Now, the equilibrium price and quantity when the cost of the externality is incorporated into the firm’s marginal cost function is:

Demand = SMC

80 - 0.1Q = Q +30

1.1Q = 50

Q = 45.4545

P = Q + 30 = 75.4545

Hence, when the cost of the externality is incorporated into the firm’s marginal cost function, equilibrium price = 75.4545 and equilibrium quantity is Q = 45.4545