Question

Albany-Berkeley Clinker (ABC), LLC is the monopoly provider of ready mix concrete in East Bay. The factory can produce a bag of concrete for a constant marginal cost of $2. Monthly demand for concrete is equal to LaTeX: Q^d = 20,000 - 4,000P Q d = 20 , 000 − 4 , 000 P . Suppose that each bag of concrete causes $2 worth of social damages due to air pollution from the ABC plant. Now suppose that the government imposes the Pigouvian prescription and sets a tax on the concrete equal to $2 per bag. Calculate total welfare (including consumer surplus, producer surplus, the externality and tax revenue). Food for thought: Did welfare go up when the tax was imposed? Is this surprising? ( please hand-draw graph and show work)

a.

CS =

PS =

Externality =

Tax Revenue =

Total Welfare =

b. What is the optimal tax rate per bag of concrete?

c. Calculate total welfare if the tax is set to the optimal rate you found in the prior question.

Answer 1

Solution:

Demand: Qd = 20,000 - 4,000*P

Demand is same as the social marginal benefit (SMB), which is same as private marginal benefit (PMB), as well.

Private marginal cost (PMC) of firm = $2 (which is constant)

Due to externality, social marginal cost (SMC) = PMC + externality cost = 2 + 2 =$4

With Pigouvian tax of $2, private marginal cost goes up by $2 (since now for each bag, an extra cost o f$2 is incurred in form of cost), so after taxation, PMC = SMC = $4

Without taxation, initial or most optimum (though not efficient, due to externality) market equilibrium was achieved where PMB(=P) = PMC = $2

So, initial optimal quantity of bags: Q = 20,000 - 4,000*2 = 12,000 bags

So, consumer surplus = (1/2)*(Q)*(5 - P) (why 5, because 5 is the vertical intercept (20000/4000 = 5))

CS = (1/2)*(12000)*(5 - 2) = $18,000

Producer surplus = 0 (since marginal cost = price, so no surplus produced for producers)

External cost (due to production not at efficient level, due to negative externality) = externality cost*Q = 2*12000 = $24,000

So, initial total welfare = CS + PS - externality = 18000 + 0 - 24000 = -$6,000

Now, with taxation:

New optimal level of quantity : PMB(=SMB) = PMC' (=SMC)

So, Q = 20000 - 4000*4 = 4,000 bags

New consumer surplus, CS' = (1/2)*(Q')*(5 - P')

CS' = (1/2)*(4000)*(5 - 4) = $2,000

Producer surplus is still 0 (as PMC' = P')

Tax revenue = tax*quantity = 2*4000 = $8,000

Externality = 0 (since, now externality cost has been converted into tax, which is revenue for government, and efficient output level of 4000 bags is being produced). But now deadweight loss due to taxation is being incurred.

New total welfare = CS' + PS' + Tax revenue - externality

= 2000 + 0 + 8000 - 0 = $10,000

Clearly, with taxation, social welfare has increased. This must not be entirely surprising as with taxation, production of harmful chemical has reduced to an extent that externality cost has reduced considerably, and a part has been converted into revenue for the government.

INITIAL (BEFORE TAX) SITUATION:

Initial equilibrium at point E, Consumer surplus is blue triangle: triangle FBE, producer surplus = 0, with no tax, tax revenue = 0, externality = rectangle ABED (part of blue and entire grey area)

NEW (AFTER TAX) SITUATION:

New equilibrium at point C, rest as denoted in figure (producer surplus and externality = 0)