The production of batteries for eBikes is polluting and causes a negative externality

whose marginal costs are estimated at 2q. The inverse demand function and the

supply function for the monopolist who produces these batteries is, respectively, p = 3 - q

and p = 2q

 

By imposing a tax on the production of batteries, the government wants to ensure that

the social optimum is reached. The government defines it

social optimum as that equilibrium outcome in which, given the market structure,

the externality is completely internalized. However, she has doubts between a specific

tax and an ad valorem tax.

 

Calculate both (the specific tax and the ad valorem tax) where

the monopolist produces a quantity corresponding to it

social optimum as defined by the government. Submit your

answers clearly.

Socially optimum price = $2.40 and quantity = 0.60

To generate this combination of price and quantity, either of the following will do the job:

Specific tax of 2Q 

OR

Ad Valorem tax of $1.20

Step-by-step explanation

Without government intervention and recognition of the negative externality, the

market equilibrium occurs when demand = supply.

Demand: P = 3-Q, Supply: P=2Q

P =3-Q = 2Q or

3Q = 3,

Q = 1, 

P = 3-1 = 2

Market price will be $2 and quantity will be 1.

With the externality, if the firms had recognized and internalized it,

their supply would have been P = 2Q +2Q = 4Q

In that case, equilibrium would have been where 3-Q = 4Q or

5Q = 3

Q = 0.60

P = 3-0.60 = $2.40

Thus, the government could put a specific tax of 2Q which would have brought about

the socially optimum quantity of 0.60 and a price of $2.40

Instead of the specific tax, the government could place an ad valorem tax of X.

Thus the firms' supply curve would be P = X +2Q

We know that the socially optimum Q =0.60 and P = $2.40.

Therefore $2.40 = X + 2(Q) = X + 2(0.60) = X + $1.20

Thus X = $2.40 -$1.20 = 

$1.20

Consequently, if a flat tax of $1.20 is imposed,

the supply would be P= 1.20+2Q

Equilibrium would occur when demand equals supply or when

3-Q = 1.20 +2Q or

3Q = 1.80

Q = 0.60 and

P = 3-Q = $2.40.

Thus the socially optimum price of $2.40 and quantity of 0.60 would be realized.

Conclusion:

Specific tax of 2Q would generate the socially optimum price pf $2.40 and quantity of 0.60.

Ad valorem tax of $1.20 will also produce the same results and achieve the socially optimum P and Q.