Double dividends Followed by Stock Pollutants

1. Double dividends Followed by Stock Pollutants ECON 4910

2. The Double Dividend – AKA The Holy Grail • Taxes are raised in an economy for a variety of reasons. • Much of tax revenue is raised through taxes that are distortionary. I.e. they create inefficiency.  • The hope is that environemental taxes reduces environmental problems and reduces the need for other distortionary taxes.

3. The Double Dividend and policy regime • In a command and control system, no taxes are raised. Therefore no double dividend. • The effect of command and control on the rest of the economy is uncertain. Probably increases deadweight losses elsewhere.

4. Are double dividends generally positive? • Not necessarily. A ”good” environmental tax can increase the distortionary effect of other taxes. • The problem is that other tax revenues may decrease by more than the increase in tax revenue through the envrionmental tax.

5. Illustration Taxing a polluting output

6. The tax interaction effect

7. Lessons • When attemption to reap double dividend benefits, attempt to do this in such a way that the need for distortionary taxes decreases. • If the environmental services are mostly complements, then there is a uneqiuvocal double dividend

8. Stock Pollutants and a brief lecture on Optimal Control • The problem: max∫∞U(c(t),x(t))e-rtdt subject to: dx/dt = f(c(t),x(t)), x(0) given. c is a control. x is a state variable. Both are functions of time. (I drop time notation)

9. Stock Pollutants • Problems where the dynamic nature of a pollutant must be considered. • A pollutant may persist for more than one period. • Effects may linger and be treated dynamically. • Many of the most important environmental problems today are stock pollutants. E.g. climate change.

10. Cook book solution in easy steps • Define the Hamiltonian: H= U(c,x) + λf(c,x) 2) Find optimality conditions • c = argmax H → c = c(x,λ) • dλ/dt = rλ – ∂H/∂x =rλ – ∂U/∂x – λ∂f/∂x • Insert c from into dx/dt and = dλ/dt 3) You now have two differential equations.

11. What to do with the differential equations • Compute steady states x* and λ*. • f(c(x,λ),x) = 0 and • rλ – U’x(c(x*,λ*),x*)– λf’x(c(x*,λ*),x*) = 0 • Draw a phase diagram. λ dλ/dt = 0 λ* dx/dt = 0 x* x • Paths converging to steady states are optimal

12. Optimal Control cont’d • The co-state λ(t) has an interesting economic interpretation. • If somebody at time t gave you a present of 1unit of x so that x(t) jumps to x(t) +1, then λ(t) is the value of that present at time t. • Entirely analogous to shadow prices in static theory

13. Alternative approach • Take the c(x,λ) function and differentiate it. • Gives dc/dt = c’xdx/dt + c’λdλ/dt = = c’xdx/dt + c’λ(rλ – ∂U/∂x – λ∂f/∂x) The solve c(x,λ) with respect to λ. Gives λ(c,x). • Use dc/dt and dx/dt after inserting λ(c,x) as alternative differential equations

14. Example • A firm has benefits from pollution given by: (u0 – u)2 • The pollution accumulates in nature according to dx/dt = u – δx • Damages from accumulated pollution given by –x2. • The problem: max∫∞ (-x2 - (u0 – u)2)e-rtdt subject to: dx/dt = u – δx , x(0) given

15. Solution Define the Hamiltonian: H= -x2 - (u0 – u)2 + λ(u – δx) • The value of u that maximise the Hamiltonian is given by u = u0 + ½λ • dλ/dt = rλ + 2x + δλ • dx/dt = u0 + ½λ – δx

16. Computing Steady States gives • x* = u0(r+δ)/(1+δ(r+δ)) • λ* = –2u0/(1+δ(r+δ)) • From these expressions we see (for example) • dx*/dr > 0 • dx*/dδ < 0

17. A More General Model • The problem: max∫∞ (B(u)-D(x)e-rtdt subject to: dx/dt = u – δx , x(0) given • First order conditions: • B’(u) + λ = 0 • dλ/dt = rλ + D’(x) +δλ • These can in steady state be combined into B’(u) = D’(x) (r+δ)-1

18. Dynamic Pigouvian Tax • How to implement such a solution in a market economy. • Choose a tax, but what tax. • In static models the shadow price is a good candidate. This is also the case here. • The firm maximises B(u) – τu. → B’(u) = τ. • Choose τ = λ(t) and optimum is achieved.

19. More Stock Pollutants • Stock Pollutants are much more complicated than this. • Threshold Effects • Non-linearities and problems that are not concave • Uncertainty • Catastrophic risk • All of this is very cool stuff from a scientific perspective

20. Threshold Effects – An illustration c is CO2. F is temperature