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The sign of this derivative is negative if (14) is satis ed and positive if it is not. The proposed
perturbation involves a reduction in Hn op . If this derivative is negative, then it must be the
case that increasing Hn op by reversing this perturbation raises surplus. In that case, therefore,
Hn op = (1 􀀀 )yp. On the other hand, if the derivative is positive, then Hn op = 0.
Suppose therefore that (14) is satis ed so that Hn op = (1 􀀀 )yp. Then given our previous
results, it is easy to show that (yp; ym) = (
H
1+ ;
L
1+ ), which is the rst part of Proposition 2. On
the other hand, if (14) is not satis ed, so that Hn op = 0 then (yp; ym ) = (
H
1+ ;
L(1+ )􀀀(1􀀀 )
H
(1+)(1+ ) ),
which is the second part of Proposition 2.
9.3 Proof of Proposition 3
There are three things to establish. First, that the allocation and production plan described in
the text is a market equilibrium under the subsidy (18) and the tax (19). Second, that given
the dynamics of the equilibrium, the pet populations converge to the steady state (15). Third,
that lifetime surplus in this equilibrium exceeds that generated by the unregulated equilibrium
described in Proposition 1.
9.3.1 The plan is an equilibrium
Proving this follows the same basic steps as the proof of Proposition 1 and thus we will be brief. To
see that the proposed allocation and production plan satis es the rst condition for equilibrium,
note that the price qyp is such that r +cl 􀀀(cs 􀀀ss) = xqyp, which makes owners of pure breeds
indi erent between breeding or spaying. In addition, given that the price of young mixed breeds is
0 and the post-subsidy price of spaying is cs􀀀ss = 'cl, owners of all young pets will be indi erent
between spaying or not. The third and fourth conditions are also easily veri ed.
39
It remains to verify the second condition that new owners will choose pets in a way consistent
with the proposed equilibrium. We begin with high type new owners. Consider such an owner at
the beginning of some period t 2 and assume rst that he does not own any young pets. Let
VH(1) denote his expected equilibrium payo . Similarly, let VH(0) denote the equilibrium payo
of a high type owner at the beginning of some period t 2 who does not and cannot own a pet
that period. Under the proposed equilibrium behavior, the owner acquires a young pure breed
and thus
VH(1) = + 􀀀 qyp 􀀀 (cs 􀀀 ss) 􀀀 Ty + [f ( + ) + ( VH(1) + (1 􀀀 )VH(0))g
+(1 􀀀 )( VH(1) + (1 􀀀 )VH(0))]:
(36)
Since
VH(0) = (
1 􀀀 (1 􀀀 )
)VH(1); (37)
the equilibrium payo of the owner is
VH(1) = [ + 􀀀 qyp 􀀀 (cs 􀀀 ss) 􀀀 Ty + ( + )]
1 􀀀 (1 􀀀 )
(1 􀀀 )(1 + )
: (38)
Using this expression, (18), (19), (20) and the fact that (14) is satis ed, it is now straightforward
to show that the owner cannot achieve a higher payo by deviating from his proposed equilibrium
behaviour. If he owns young pets, then his equilibrium payo will be unchanged if they are mixed
breeds and increased by xqyp if they are pure breeds. Since the same is true for the payo s from
deviating, the arguments remain valid in this case.
Now consider low type new owners. Consider such an owner at the beginning of some period
t 2 and assume rst that he does not own any young pets. Let VL(1) denote his expected
equilibrium payo . Similarly, let VL(0) denote the equilibrium payo of a low type owner at the
beginning of some period t 2 who does not and cannot own a pet that period. Under the
prescribed equilibrium behavior, the owner either obtains a young mixed breeds or an old pet
from the shelter. In particular, then
VL(1) = 􀀀 (cs 􀀀 ss) 􀀀 Ty + [f + ( VL(1) + (1 􀀀 )VL(0))g
+(1 􀀀 )( VL(1) + (1 􀀀 )VL(0))]:
(39)
40
Since
VL(0) = (
1 􀀀 (1 􀀀 )
)VL(1); (40)
the equilibrium payo of a low type new owner is
VL(1) = [ 􀀀 (cs 􀀀 ss) 􀀀 Ty + ]
1 􀀀 (1 􀀀 )
(1 􀀀 )(1 + )
: (41)
Using this expression, (18), (19), and (20), it is now straightforward to show that the owner is
indi erent between obtaining a young mixed breed or an old pet from the shelter, and that he
cannot achieve a higher payo by deviating from his proposed equilibrium behaviour. If he owns
young pets, then they must be mixed breeds and his equilibrium payo s will be unchanged. Since
the same is true for the payo s from deviating, the arguments remain valid in this case.
9.3.2 Dynamics
We know from (21) that for all t 2, (ypt+1; ymt+1) = (
H 􀀀 ypt;
L􀀀ymt􀀀(1􀀀 )ypt): This
is a linear system of di erence equations that can be written in matrix form as
2
664
ypt+1
ymt+1
3
775
=
2
664
􀀀 0
􀀀(1 􀀀 ) 􀀀
3
775
2
664
ypt
ymt
3
775
+
2
664
H
L
3
775
:
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