In a market for a dirty good, the inverse demand function is given by p=100−yp=100−y, and the private marginal cost of production is given by MC=10+yMC=10+y

1)

Given that;

P=100−YPMC=10+YE=$2P=100−YPMC=10+YE=$2

At competitive equilibrium, the private marginal cost is equal to the demand equation:

100−Y=10+YY=45P=$55100−Y=10+YY=45P=$55

The equilibrium price and quantity are $55 and 45, respectively.

2)

At the social optimal level, the demand equation is equal to the social marginal cost. Social marginal cost is equal to private marginal cost plus the external cost:

SMC=PMC+E=(10+Y)+2=12+Y100−Y=12+YY=44P=$56SMC=PMC+E=(10+Y)+2=12+Y100−Y=12+YY=44P=$56

The socially optimal level of output and price are 44 and $56, respectively.

3)

The Pigovian tax is set to be equal to damage (external cost) caused by the negative externality:

Pigovian tax=External cost=$2