Download 1 QUESTION 1: TRANSBOUNDARY POLLUTION WITH EMISSIONS

Kennedy: Theory of Environmental Regulation (Draft 1)
QUESTION 1: TRANSBOUNDARY POLLUTION WITH EMISSIONS TRADING
(45 minutes). Recall the climate change problem from Project 3, in which countries chose
optimal policies with respect to abatement and defensive action. Here we will simplify
that problem by excluding defensive action but we will extend consideration to more than
two countries, and introduce emissions trading among those countries.
Consider a setting with n countries in which country i has fixed output yi . The labour
cost of producing yi is
(Q1.1)
c( yi , xi ) = ωxi2 yi
where ω > 0 is a fixed productivity parameter, and xi ∈ [0,1] is a choice variable that
determines the cleanliness of production. In particular, production of yi using process xi
generates emissions
(Q1.2)
ei = (1 − xi ) y i
Damage to country i from climate change is
(Q1.3)
d i = δEyi
where
n
(Q1.4)
E = ∑ ei
i =1
and δ > 0 is a fixed damage parameter. Note that ω and δ are the same for all countries
here. Output is the only potential source of heterogeneity.
1.1 The Global Planning Problem
Suppose all countries have the same output. Thus, yi = y ∀i . Then the global planning
problem involves choosing x (identical across countries) to minimize global cost:
(Q1.5)
min n(ωx 2 y + δEy )
x
where
(Q1.6)
E = n(1 − x) y
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Kennedy: Theory of Environmental Regulation (Draft 1)
(a) Find the solution to this planning problem, denoted x * . Identify a condition on δ that
ensures an interior solution.
(b) Now suppose that the countries are heterogeneous with respect to output. Let μ and
σ 2 denote the mean and variance of output respectively. Write down the global costminimization problem.
(c) Write down the first-order conditions to this problem, and confirm that they solve for
(Q1.7)
xi* =
δnμ
∀i
2ω
How much does country i emit at this global optimum? What are aggregate emissions at
this global optimum, denoted E * ?
1.2 The Non-Cooperative Equilibrium
(d) Write down the choice problem over xi for country i in the non-cooperative game,
using the notation
(Q1-8)
E −i = ∑ e j
j ≠i
(e) What is the best-response function for country i in terms of xi ? Why is it independent
of E−i ?
(f) What is the non-cooperative equilibrium (NCE) level of emissions for country i? Let
êi denote this level of emissions.
(g) Confirm that aggregate emissions in the NCE are equal to
(Q1-9)
δn( μ 2 + σ 2 )
Eˆ = nμ −
2ω
Why is Ê decreasing in σ 2 ?
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Kennedy: Theory of Environmental Regulation (Draft 1)
1.3 Emissions Trading
Now suppose that these countries agree to a treaty under which aggregate emissions are
fixed, and countries can trade emissions amongst themselves. Each country promises to
emit only as much as the permits they hold. The initial allocation to country i is its NCE
level of emissions. Thus, the aggregate supply of permits is Ê . Note that the treaty does
not reduce aggregate emissions, but they are now fixed by agreement.
(h) Let q denote the price of permits. Use (Q1.2) to express xi for country i in terms of its
emissions, and write down the choice problem for country i as a choice over emissions.
(Remember that its own choice of emissions now has no impact on aggregate emissions,
which are fixed at Ê by the treaty).
(i) Find emissions for country i as a function of q, and derive the aggregate demand for
permits.
(j) Find the equilibrium price of permits, denoted q~ .
(k) Derive a threshold value of yi , denoted y , such that countries with yi > y are permit
buyers and countries with yi < y are permit sellers.
(l) Suppose all countries are identical. Explain why q~ > 0 but there is no trade.
(m) Now suppose that by some miracle, all countries agree that the global supply of
permits should be E * rather than Ê . What is the equilibrium price of permits, denoted
q * ? Explain why q * is independent of σ 2 .
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