The production of thermal energy pollutes the atmosphere through the emission of numerous gases, and contributes to the increase in the concentration of CO₂, which causes the greenhouse effect. Suppose an electricity market characterized by the following demand and supply curves:

Demand: _d = (10-p)/2 Offer: _s = (2+p)/2

Where "Q" is the quantity and "P" is the market price. Also consider that pollution generates an "external" cost which is represented by the function:

External cost: _external() = 0.25².

a) Calculate the competitive balance of this market. b) Find the marginal social cost curve (remember that the inverse of the supply curve is the marginal cost curve). c) Calculate the optimal quantity produced from the social point of view. d) Calculate the dead weight generated in the competitive equilibrium without tax. e) Find the best Pigouvian tax that maximizes social welfare.