Download Lyon Coase Theorem Home-Work Problem Let there be two

Lyon
Coase Theorem
Home-Work Problem
Let there be two neighbors with a mosquito problem. Assume that the problem can be virtually
eliminated by spraying, which we will assume has no effects other than controlling the
mosquitos. If one family sprays for mosquitos, undertakes mosquito abatement, then by virtue of
location the other family will also have a reduction in mosquitos. The abatement functions will
have the following form:
A = f(E),
E = E1 + E2, fN(E) > 0 for E , (0, Eo)
fN(Eo) = 0, f(Eo) = 1
where Ei is expenditures on abatement by family i. The function f is twice continuously
differentiable and has an inverted “U” shape, with a peak height of “1.” At the peak we
have complete mosquito abatement or complete control. Let the utility functions have the
form:
Ui(yi, A)
Ui , C2, Ui strictly quasi-concave
where yi is a composite commodity of everything else, and we let its price be one. In
addition, we assume that A is a normal commodity with a negligible income effect, and
let the budget constraints be
yi + Ei - Mi = 0
a.
Define Vi(yi, E) = Ui(yi, f(E)). Show Vi(yi, E) that is strictly quasi-concave in yi
and E for yi > 0 and E , (0, Eo).
b.
Write a maximization problem that can be solved to identify the Pareto
Optimality Conditions.
c.
Identify the Pareto Optimality Conditions. Show that at the optimum
U2A
U1A
1
+ )))
] = ))
[)))
1
2
f
N
Uy
Uy
Explain this equality.
d.
Let Neighbor 2 offer to match Neighbor 1's expenditure on abatement. Show that
if the two neighbors are exactly alike then this will yield a Pareto optimum.
Explain.
e.
We now suppose that the tastes of the two neighbors are very different. At any
level of abatement, Neighbor 2 values abatement higher than does Neighbor 1;
however, both place a positive value on abatement. The two neighbors negotiate
until they reach an agreement where both are satisfied. Neighbor 2 offers to pay a
specific share, s2, of total abatement expenditures; thus, Neighbor 1's share is s1 =
1 ! s2. Following this both determine their optimal expenditures given si, E*i.
Neighbor 2 then compares E*2/s2 and E*1/s1. If E*2/s2 > E*1/s1 he offers to pay a
higher share, and if the inequality is reversed he offers a lower share.
The constrained maximization problems can be written:
Max: Ui(yi, f(Ei/si))
S.t.:
yi + Ei ! Mi = 0
However, if we define Ei := Ei/si as total expenditures given Ei and si, and define
Vi(yi, Ei) := Ui(yi, f(Ei)) then we can write the maximization problem as:
Max: Vi(yi, Ei)
S.t.:
yi + siEi ! Mi = 0
Note that in part a. above you proved that Vi is strictly quasi-concave on the
relevant domain; hence, the necessary conditions are also sufficient. Let the
solution function for Ei be given by Ei = Ei*(si, Mi). The differential equation for
this adjustment process can now be written:
ds2
))))
= N(E2*(s2, M2) ! E1*(s1, M1))
dt
where N(0) = 0 and NN > 0.
i.
We now examine the dynamics of this bargaining process to determine if
it will converge. First we examine the MEi*/Msi. Show that MEi*/Msi < 0
give that Ei* is normal.
ii.
Draw the phase diagram in the plane (s2, ds2/dt) and show that N
converges.
iii.
Show that the resulting equilibrium is a Pareto optimum.