Thomas Piketty, Paris School of Economics

Academic year 2012-2013

Course Notes:

Optimal corrective taxation of externalities:

a simple numerical example

Continuum of agents i in [0;1]

Two goods: non-energy goods c and energy goods e

Identical utility function: U = U(c,e,E) = (1-α)log(c) + αlog(e) – λlog(E)

With: c = individual c consumption

e = individual e consumption

E = aggregate e consumption = negative externality (e.g. global warming)

Linear production function (full substitutability): everybody supplies one unit of labor, and labor can be used to produce linearly c or e

Aggregate budget constraint: C + E < Y = 1

Laissez-faire equilibrium:

Max U(c,e,E) under c+e<y=1

→ c = (1-α)y  &  e = αy

Say, α = 20% & 1-α=80% : in the absence of corrective taxation, we spend 20% of our ressources on energy (20% of the workforce works in the energy sector, etc.)

Social optimum:

Max U(C,E,E) under C+E<Y=1

→ C = (1-α)Y/(1-λ)  & E = (α-λ)Y/(1-λ)

Say, α = 20% & 1-α=80% & λ=10%: given the global warming externality , we should only be spending 11% of our ressources on energy

How to implement the social optimum? A corrective tax tE on energy consumption financing a lump sum transfer equals to tE:

Max U(c,e,E) under c+pe<y

With : p =1+t  & y =1+tE

→ c = (1-α)y   &  e = αy/p

→ Optimal corrective tax : α/p = (α-λ)/(1-λ)  

I.e. p = 1+t = α(1-λ)/(α-λ) = 180%  

Say, α = 20% & 1-α=80% & λ=10%: we need a tax rate t=80% to correct the global warming externality; in effect, consumers pay their energy 80% higher than production costs; they keep spending 20% of their budget on energy, but 80%/180% = 45% of these spendings are paid to the government in energy taxes; i.e. 9% of national income Y goes into energy taxes, and everybody receives a green dividend equals to 9% of national income; in effect, the size of the energy sector is divided by two