Solving Stray-Animal Problems by Economics Policies

Shi-Miin Liu and Hsiao-Chi Chen∗

January 2014 Abstract

Animal lives are as precious as human’s in a progressive society. This paper thus tries to investigate optimal economic policies which deal with stray animals. We construct a two-stage game to characterize interactions among the regulator, pet shops, and consumers. The regulator first chooses to subsidize or tax consumers’ purchases of companion animals, then consumers decide to buy dogs/cats from pet shops or adopt them from animal shelters or streets. Afterwards, the companion animal market is cleared. In particular, a simple Hotelling (1929) model is used to endogenize behaviors of dogs/cats purchasers and adopters. Our results show that pet animal buyers could be taxed or subsidized if the regulator aims to maximize the social welfare. However, if the regulator aims to minimize the number of stray animals or the environmental damage caused by them, taxing the purchase of dogs/cats is the regulator’s best choice. These outcomes hold no matter that the structure of the pet animal market is perfectly or imperfectly competitive.

Keywords: externality, tax, strayed dog, subsidy, perfect competition, imperfect competition JEL Classification: Q57, Q58 ∗

The authors are professors in the Department of Economics, National Taipei University, San-Shia

District, New Taipei City 23741, Taiwan, R.O.C. The correspondence author is Hsiao-Chi Chen with email address [email protected] and phone number 886-2-86741111-ext.67128.

1. Introduction Many problems are encountered and ignored in our daily lives. Stray dogs/cats are one of them. Because having no spending power and voting right, the strays usually have the least priority in the welfare concerns of the human societies. However, neglecting stray animals not only makes people suffer emotionally, but also causes financial and human-life losses. In analyzing whether adoption and low-cost spay/neuter programs are effective in reducing overpopulation of companion animals, Frank and Carlisle-Frank (2007) offer three possible utility-based reasons to minimize unwanted dogs/cats. The first one is the direct expense resulted from dog bites, traffic accidents due to evading the strays, barking noises, sanitary concerns, and other nuisances. Relevant studies using U.S. data include Sosin, et al. (1986), Sack, et al. (1996), Clifton (2002), Baetz (1992), etc. The second reason is the indirect personal pain caused by observing the plight of these pitiful animals. People thus support animals protection and their welfare by donating money (Jasper and Nelkin, 1992) or writing letters to Congressmen to urge relevant legislation (Fox, 1990). The third reason is obviously the distress of stray dogs/cats themselves. If their sufferings and deaths can be measured economically, this cost, we agree with Frank and Carlisle-Frank (2007), would probability be the greatest among the three categories. People can discuss, explore, and propose solutions for stray-animal problems from several respects. The simple catch-and-kill programs employed by many countries are proven unsuccessful, otherwise most of the stray-animal problems should have disappeared. Moreover, these programs cause huge bitterness for both the strays and people who care for them. Therefore, other ways to control the number of the strays and/or to make them have better chances being treated well are certainly needed. Because financial incentives usually have the greatest impact on human behaviors, economic means are anticipated to be the most effective methods to lessen or to solve the stray-animal problems. For instance, subsidized spay/neuter programs are promoted by many local governments. Although the usefulness of these programs in controlling animal populations is arguable (e.g., MacKay, 1993; Rush, 1985), some researchers (e.g., Frank, 2001) 1

think that the spay/neuter techniques have a powerful impact on long-term population sizes even the participation rate of pet owners is unsatisfactory. Another example is utilizing resources efficiently to educate people such that their companion animal euthanasia can be reduced (Frank, 2002). Then, sizes of the strays could decrease due to less abandoning. The developed countries value human as well as animal lives through ideas broadcasting and budgets planning. For instance, animal policemen/policewomen in the U.S. are responsible for charging animal-abuse cases and rescuing the vulnerable. Hence, how animal rights are protected and how the stray-animal problems are dealt with can reflect whether a society is progressive. To really care for animal welfare and the safety of human societies, current efforts are still insufficient even in the developed countries. Therefore, more influential and effective policies having economic consequences should be adopted by governments. This paper thus tries to propose policy solutions for the stray-animal problems through rigorous economic modeling. We hope that this bigger picture and ambition can actually make a difference and a contribution, especially when relevant studies are so few. A two-stage game is constructed to investigate optimal policies to solve or lessen the issues and negative externalities resulted from stray animals. In the first period, the regulator chooses between subsidizing and taxing the purchase of companion animals. Then, consumers decide to buy or adopt dogs/cats in the second stage. Afterwards, the companion animal market, which is perfectly or imperfectly competitive, is cleared. We use a simple Hotelling (1929) model here to endogenize the behaviors of pet animal purchasers and adopters. Our results show that pet animal buyers could be taxed or subsidized if the regulator aims to maximize the social welfare. These outcomes hold under the perfectly and imperfectly competitive markets of companion animals, under the linear and quadratic cost functions to breed dogs/cats, or under the linear and quadratic functions of environmental damage caused by strayed animals . However, if the regulator aims to minimize the number of stray animals or the associated environmental damage caused by them, taxing pet animal buyers is the only optimal choice. 2

This finding remains true under the perfectly and imperfectly competitive markets of pet animals. Our results would provide regulators in different countries effective economic means to manage the problems of stray animals. The rest of this paper is organized as follows. The model is presented in Section 2. The equilibria under both the perfectly and imperfectly competitive pet animal markets are derived in Section 3. Then, the robustness of our results in Section 3 is examined in Section 4. Finally, our conclusions are drawn in Section 5. 2. The Model We use a simple Hotelling (1929) model to characterize the behaviors of dogs/cats purchasers or adopters. Let these animal consumers be uniformly distributed over a unit location interval [0, 1]. Without loss of generality, we assume that each consumer can own one dog/cat only.1 They either buy the dog/cat from pet shops or adopt it from animal shelters or streets. Suppose that pet shops are located at end-point 0, and ¯ > 1 strays in animal shelters dog/cat shelters are located at end-point 1. There are Q and on streets at the beginning of a period. If a consumer located at point y ∈ [0, 1] buys a dog/cat from pet shops, she/he will have utility Uy = h0 − c0y − t − p,

(1)

where h0 > 0 is the happiness level from buying a dog/cat, c0 y is the cost to raise the purchased pet, t is the taxes (t > 0) paid to or the subsidies (t < 0) obtained from the regulator, and p is the purchase price of the dog/cat. In contrast, if consumer y adopts a dog/cat from animal shelters or streets, she/he paying neither price nor tax will have utility Uy = h1 − c1 (1 − y),

(2)

where h1 > 0 is her/his happiness level and c1(1 − y) is the cost to raise the adopted dog/cat. Unequal values of h0 and h1 reflect distinct happiness levels through buying 1

For the consumers owning more than one dog or cat, they can be regarded as lying in several

points of [0, 1]. Such a change will not alter our results.

3

and adopting dogs/cats, respectively. Also, we allow different values of c0 and c1, implying that the costs to raise a purchased dog/cat and a stray may not be the same. By making Uy ’s in (1)-(2) equal, we get a consumer located at x is indifferent to purchasing or adopting a dog/cat, given by h0 − c0 x − t − p = h1 − c1(1 − x) ⇔ x =

h0 + c1 − h1 − t − p . (c0 + c1)

To have x ∈ (0, 1), we impose the condition of h1 − c1 + t + p < h0 < h1 + c0 + t + p.

(3)

This condition suggests that the consumer’s happiness from buying a pet animal cannot be too large or too small. If h0 is too large, no one will adopt strays. On the other hand, the pet animal market would disappear if h0 is too small. The two extreme cases are not consistent with the real world situations. Based on the above, we have the demand function for pet shops, x=

h0 − h1 + c1 − t − p , (c0 + c1 )

(4)

and the demand function for strays from animal shelters and streets, 1−x=

h1 − h0 + c0 + t + p . (c0 + c1 )

(5)

Accordingly, a consumer’s surplus of having a dog/cat equals CS(p, x) ≡

Z

0

x

(h0 − c0 y − t − p)dy +

Z

1

x

[h1 − c1(1 − y)]dy

c1 x2 = h1 − + x[h0 − h1 + c1 − t − p] − (c0 + c1). 2 2

(6)

Pet shops are presumed to provide identical animals at a constant marginal cost c¯ > 0. These shops can compete perfectly or imperfectly. Given the pet animal demand x in (4), we can derive the associated equilibrium price (p∗ ), equilibrium quantity (q ∗), and producer surplus (π ∗) for all pet shops in a perfectly competitive market. In contrast, we use pˆ, xˆ, and π ˆ to denote the equilibrium price, equilibrium quantity, and 4

producer surplus for all pet shops in an imperfectly competitive market. Equilibria (p∗ , q ∗, π ∗) and (ˆ p, xˆ, πˆ ) will be derived in Sections 3.1 and 3.2, respectively. Next, we define the social welfare function, SW, as the sum of consumers’ surpluses, producers’ surpluses, tax revenues and negative externality values caused by stray animals. Precisely, we have

SW =

                  

¯ − (1 − x∗) + r0 x∗ + r1 (1 − x∗ )] CS(p∗ , x∗ ) + π ∗ + tx∗ − d[Q in a perfectly competitive market, ¯ − (1 − xˆ) + r0xˆ + r1 (1 − xˆ)] CS(ˆ p, xˆ) + π ˆ + tˆ x − d[Q

(7)

in an imperfectly competitive market,

where r0 and r1 are the probabilities of consumers abandoning their dogs/cats after ¯ − (1 − x∗) + r0x∗ + r1 (1 − x∗ )] and buying and adopting them, respectively; and [Q ¯ − (1 − xˆ) + r0 xˆ + r1 (1 − xˆ)] are the numbers of stray animals in the society at the [Q end of the period. Here parameter d > 0 represents the marginal damage resulted from stray animals. We will first analyze a linear environmental damage. Then, a quadratic environmental damage function will be discussed in the extension section. A two-stage game is constructed to characterize the interactions among the regulator, consumers, pet shops, and animal shelters. In the first period, the regulator announces a tax or a subsidy to maximize the social welfare. Given the tax or subsidy, consumers decide to purchase dogs/cats from pet shops or to adopt them from animal shelters or streets. Then, the pet animal market is cleared. Since our game has complete information, its subgame perfect Nash equilibria (hereafter SPNE) can be derived by the backward induction method as follows.

3. The Equilibria under Different Market Structures

The SPNE under perfectly and imperfectly competitive pet markets are derived in the ensuing subsections.

3.1. A Perfectly Competitive Pet Animal Market 5

Suppose that the market supply function of dogs/cats is p(x) = α + βx with α > 0, β > 0, and α > c¯.2 Recall that c¯ > 0 is the constant marginal production cost of all pet shops. Together with the market demand for x defined in (4), we can get the equilibrium price and quantity of pets, α(c0 + c1 ) + β(h0 + c1 − h1 − t) and (c0 + c1 + β) (h0 + c1 − h1 − t − α) . = (c0 + c1 + β)

p∗ =

(8)

x∗

(9)

At (p∗ , x∗ ), the producer’s surpluses of all pet shops are π ∗ = (p∗ − c¯)x∗ ,

(10)

the consumer’s surpluses in (6) become CS(p∗ , x∗) = h1 −

c1 (x∗ )2 + x∗ [h0 + c1 − h1 − t − p∗ ] − (c0 + c1 ), 2 2

and the social welfare function in (7) changes to SW = h1 −

c1 (x∗)2 ¯ − 1 + r1 + x∗ (1 + r0 − r1)]. + x∗(h0 + c1 − h1 − c¯) − (c0 + c1 ) − d[Q 2 2

The regulator will choose t∗ to maximize the social welfare. The associated first-order and second-order conditions are respectively ∂SW ∂t 2 ∂ SW ∂t2

−1 [h0 + c1 − h1 − c¯ − x∗(c0 + c1 ) − d(1 + r0 − r1 )] and (11) (c0 + c1 + β) −(c0 + c1) = < 0. (12) (c0 + c1 + β)2 =

Since the social welfare function is strictly concave in t implied by (12), we can derive t∗ from (11) as follows. t∗ = 2

(c0 + c1 + β)[¯ c + d(1 + r0 − r1)] (h0 + c1 − h1 )β −α− . (c0 + c1 ) (c0 + c1)

This supply function can be derived from m identical pet shops who own the same production

function and maximize their profits simultaneously. An example can be found in Jehle and Reny (2011, p. 167).

6

At (t∗ , p∗ ), condition (3) can be reduced to3 h1 + c¯ − c1 + d(1 + r0 − r1 ) < h0 < h1 + c0 + c¯ + d(1 + r0 − r1).

(13)

Moreover, under condition (13), we have x∗ > 0 and p∗ > 0.4 These results can be summarized below. Proposition 1 : Suppose condition (13) holds. Then the SPNE under a perfectly competitive pet animal market are (h0 + c1 − h1 )β (c0 + c1 + β)[¯ c + d(1 + r0 − r1 )] −α− c0 + c1 c0 + c1 (h0 + c1 − h1 )β (c0 + c1 + β)[¯ c + d(1 + r0 − r1 )] ≥ (≤) 0 iff α + ≤ (≥) , c0 + c1 c0 + c1 [h0 + c1 − h1 − c¯ − d(1 + r0 − r1)] > 0, and = c0 + c1 β[h0 + c1 − h1 − c¯ − d(1 + r0 − r1 )] = α+ > 0. c0 + c1

t∗ =

x∗ p∗

Proposition 1 derives the conditions under which the regulator would subsidize or tax the purchases of dogs/cats. For instance, if the supply of dogs/cats is small (large α), the equilibrium price of the pet animal market becomes large. Then, it is optimal for the regulator to subsidize pet animal buyers to increase the surpluses of consumers and producers. Similarly, if the happiness of raising purchased dogs/cats (h0 ) is large or the happiness of raising strayed dogs/cats (h1 ) is small, it is optimal for the regulator to subsidize the purchases of pet animals. In contrast, if the marginal damage of strayed dogs/cats (d) is large, it is optimal for the regulator to impose taxes on the purchases of dogs/cats so that the consumers are willing to adopt more strays. Similarly, if the probability of abandoning purchased dogs/cats (r0) is large or the probability of abandoning adopted dogs/cats (r1) is small, it optimal for the regulator to tax the purchases of pet animals. Moreover, the optimal tax (t∗) owns the ensuing properties. 3

By simple calculations, we can show that condition x∗ > 0 is equivalent to h0 > h1 + c¯ − c1 +

d(1 + r0 − r1 ), while x∗ < 1 is equivalent to h0 < h1 + c0 + c¯ + d(1 + r0 − r1 ). 4

Condition (13) implies p∗ = α +

β[h0 +c1 −h1 −¯ c−d(1+r0 −r1 )] c0 +c1

7

> 0.

Lemma 1 : Suppose condition (13) holds. Then we have ∂t∗ ∂r0

(i)

=

∂t∗ ∂h0

(ii)

d(c0 +c1 +β) (c0 +c1 )

=

−β (c0 +c1 )

(iii)

∂α

= −1 < 0 and

(iv)

∂t∗ ∂c0

=

(v)

∂ c¯

=

∂t∗ ∂h1

< 0 and

∂t∗

∂t∗

∂t∗ ∂r1

> 0 and

∂t∗ ∂d

=

=

=

β (c0 +c1 )

< 0,

> 0,

(c0 +c1 +β)(1+r0 −r1 ) (c0 +c1 )

β[h0 +c1 −h1 −¯ c−d(1+r0 −r1 )] (c0 +c1 )2 (c0 +c1 +β) (c0 +c1 )

−d(c0 +c1 +β) (c0 +c1 )

> 0 and

∂t∗ ∂c1

> 0, =

−β[h1 +c0 +¯ c+d(1+r0 −r1 )−h0 ] (c0 +c1 )2

< 0, and

> 0.

When pet buyers are more likely to abandon the animals, the regulator should raise the taxes or lower the subsidies to discourage their purchases of dogs/cats because ∂x∗ ∂t

=

−1 (c0 +c1 )

< 0 implied by (4). In contrast, if dogs/cats adopters are more likely to

abandon the strays, the regulator ought to lower the taxes or raise the subsidies to deter their adoption based on

∂(1−x∗ ) ∂t

=

1 (c0 +c1 )

> 0 implied by (5). That is what Lemma

1(i) shows. When the happiness from buying dogs/cats increases, it is optimal for the regulator to encourage the purchase of pet animals by lowering the taxes or raising the subsidies so that consumers’ surpluses would increase. Alternatively, if adopting dogs/cats brings more happiness, it is optimal for the regulator to promote the strayanimal adoption by raising the taxes or lowering the subsidies. That is what Lemma 1(ii) states. When the market supply decreases (i.e., the rise of α), the equilibrium quantity of pet animals will fall. Then lowering the taxes or increasing the subsidies can raise the equilibrium quantity, as well as the surpluses of consumers and producers.5 Oppositely, if the marginal externality of strayed dogs/cats (d) is large, the regulator will raise the taxes or lower the subsidies to reduce the number of the strays. That is the content of Lemma 1(iii). Finally, when the costs of raising purchased dogs/cats (c0 ) or breeding them (¯ c) rise, the regulator should promote the stray-animal adoption by increasing the taxes or lowering the subsidies. In contrast, if the costs of raising adopted animals (c1 ) rise, the regulator should promote the purchase of dogs/cats by lowering the taxes or increasing the subsidies. These are shown by Lemma 1(iv)-(v).

3.2. An Imperfectly Competitive Pet Animal Market 5

Equations (8)-(10) imply

∂p∗ ∂t

=

−β (c0 +c1 +β)

< 0,

∗

c¯) ∂x ¯. ∂t < 0 by α > c

8

∂x∗ ∂t

=

−1 (c0 +c1 +β)

< 0, and

∂π∗ ∂t

=

∂p∗ ∗ ∂t x

+ (p∗ −

In this section, we consider a Cournot oligopoly market with J, J ≥ 2, identical pet animal shops. Suppose that these shops have the same cost function c(q j ) = c¯q j , j = 1, 2, . . . , J. Given the inverse demand function of dogs/cats, p = (h0 + c1 − h1 − t) − (c0 + c1)(

PJ

j=1

q j ) implied by (4), firm j’s profit function is

π j = [(h0 + c1 − h1 − t) − (c0 + c1 )(

J X

q j )]q j − c¯q j , j = 1, 2, . . . , J,

j=1

where h0 + c1 − h1 − t > c¯ is needed. Applying the standard arguments used in the Cournot model, we can get that all shops would supply the same amount of pet animals at equilibrium, i.e., q 1∗ = q 2∗ = . . . = q J∗ = qˆ =

h0 + c1 − h1 − t − c¯ > 0. (c0 + c1)(J + 1)

(14)

Accordingly, the equilibrium price of pet animals equals pˆ =

h0 + c1 − h1 − t + J c¯ , (J + 1)

(15)

the equilibrium quantity of pet animals is xˆ = J qˆ =

J (h0 + c1 − h1 − t − c¯) (c0 + c1 )(J + 1)

(16)

by (14), and the equilibrium total profit of the J shops is π ˆ = J [ˆ p − c¯]ˆ q=

J (h0 + c1 − h1 − t − c¯)2 . (c0 + c1 )(J + 1)2

(17)

Given (ˆ p, xˆ, πˆ ) in (15)-(17), the regulator’s social welfare function in (7) becomes ¯ − (1 − xˆ) + r0 xˆ + r1 (1 − xˆ)] SW = CS(ˆ p, xˆ) + π ˆ + tˆ x − d[Q (ˆ x)2 c1 ¯ − 1 + r1 + xˆ(1 + r0 − r1 )]. (c0 + c1 ) − d[Q = h1 − + xˆ(h0 + c1 − h0 − c¯) − 2 2 The associated first-order and second-order conditions are respectively ∂SW ∂t 2 ∂ SW ∂t2

−J [h0 + c1 − h1 − c¯ + J t − d(1 + r0 − r1 )(J + 1)] and(18) (c0 + c1 )(J + 1)2 −J 2 = < 0. (19) (c0 + c1 )(J + 1)2

=

9

Again, equation (19) implies that the social welfare is a strictly concave function of t. ˆ Hence solving (18) will give us the optimal t, d(1 + r0 − r1)(J + 1) − (h0 + c1 − h1 − c¯) tˆ = . J ˆ can be reduced to (13). Under condition (13), As in Section 3.1, condition (3) at (ˆ p, t) we have xˆ > 0 and pˆ > 0.6 These results are summarized below. Proposition 2 : Suppose condition (13) holds. Then the SPNE under an imperfectly competitive pet animal market are d(1 + r0 − r1 )(J + 1) − (h0 + c1 − h1 − c¯) tˆ = J ≥ (≤) 0 iff h0 + c1 ≤ (≥) h1 + c¯ + d(1 + r0 − r1 )(J + 1) 1 xˆ = J q¯ = [(h0 + c1 − h1 − c¯) − d(1 + r0 − r1)] > 0, and (c0 + c1 ) 1 pˆ = [(h0 + c1 − h1 + c¯(J 2 − 1)) − d(1 + r0 − r1 )(J + 1)] > 0. J (J + 1) Proposition 2 specifies the conditions under which the regulator will tax or subsidize dogs/cats buyers. For instance, when the happiness from purchasing dogs/cats (h0 ) or the cost of raising adopted strays (c1 ) is large, it is optimal for the regulator to subsidize the purchase of dogs/cats to enhance the surpluses of consumers and producers. In contrast, if the happiness from raising adopted strays, the cost of breeding dogs/cats, the marginal damage caused by the strays, the probability of abandoning purchased dogs/cats, and the number of pet animal shops are large, or the probability of abandoning the adopted strays is small, then the regulator should tax the purchase of dogs/cats. Moreover, optimal tˆ has the ensuing properties. Lemma 2 : Suppose condition (13) holds. Then we have ∂ˆ t = d(J+1) ∂r0 J ∂ tˆ −1 (ii) ∂h0 = J
0, ∂h1

> 0 and 0 and

< 0,

Condition h0 > h1 + c¯ − c1 + d(1 + r0 − r1 ) suggests x ˆ > 0, while pˆ > 0 is implied by h0 + c1 −

h1 − tˆ + J c¯ > h0 + c1 − h1 − tˆ − c¯ > 0 based on (13).

10

∂ˆ t = (1+r0 −rJ1 )(J+1) > 0, ∂d ∂ tˆ ∂ˆ t (iv) ∂c = 0 and ∂c = −1 < 0, and J 0 1 ˆ ∂ˆ t 0 −r1 ) (v) ∂∂c¯t = J1 > 0 and ∂J = (h0 +c1 −h1 −¯Jc)−d(1+r 2

(iii)

> 0.

All intuitions behind Lemma 2 are the same as those behind Lemma 1 except for

∂ˆ t ∂c0

and

∂ˆ t . ∂J

First, under a perfectly competitive pet animal market, Lemma 1(iv)

shows that the regulator will increase the taxes or lower the subsidies if the cost of raising purchased dogs/cats (c0 ) rises. However, in an imperfectly competitive pet animal market, the optimal subsidies or taxes are not affected by c0 . Changing c0 will 2

influence the social welfare directly by varying the term of − (c0 +c21 )(ˆx) in consumer’s surplus in (6), and indirectly through varying pet animal demand xˆ. Based on the first-order condition, we find that the direct and the indirect effects will cancel each ˆ other out.7 Thus, c0 does not affect t. Second, Lemma 2(v) shows that the regulator will raise the taxes or lower the subsidies if there are more pet animal shops. This result is absent in a perfectly competitive pet animal market. More pet animal shops mean more dogs/cats supply. Thus, the equilibrium price of pet animals will fall, but the equilibrium quantity will increase. This in turn implies that fewer stray animals will be adopted, and more negative externalities will be caused by them. To lower these externalities, the regulator should raise the taxes or lower the subsidies to promote the adoption of stray animals.

4. Extensions

In this section, the previous frameworks are extended in two directions. First, we examine whether Propositions 1 and 2 remain true if the regulator is assumed to minimize the number of stray animals, or to minimize the environmental damage caused by them. Second, we consider a quadratic cost function for pet animal shops 7

Equation (18) suggests that ˆt will meet the first-order condition of h0 − h1 + c1 − (c0 + c1 )ˆ x − c¯ − ˆ

c) 1 +c1 −t−¯ d(1 + r0 − r1 ) = h0 − h1 + c1 − (c0 + c1 ) J(h(c00−h − c¯ − d(1 + r0 − r1 ) = 0, which is independent +c1 )(J+1)

of c0 .

11

or a quadratic environmental damage function. Then, we can infer the robustness of our results.

4.1. Minimizing the Number of Stray Animals

Suppose that the regulator wants to minimize the number of strayed dogs/cats, instead of maximizing the social welfare. The game structure here is the same as that in Section 3.1 except that the regulator now choose t˜ to solve the problem of mint

¯ − (1 − x∗) + r0 x∗ + r1(1 − x∗) Q

s.t.

−t ≤ t ≤ t¯

(20)

under a perfectly competitive pet animal market. As in Section 3.1, the regulator is allowed to subsidize or tax the purchase of dogs/cats. Thus, t˜ can be negative or positive. To make sure the existence of the solution, we impose an upper bound for ¯ − (1 − x∗ ) + r0 x∗ + r1 (1 − ¯ and subsidy (−t). Define Lagrangian function L = Q tax (t) x∗) − λ1 [t + t] + λ2 [t − t¯]. The associated first-order conditions are ∂L ∂x∗ ∂L = (1 + r0 − r1 ) − λ1 + λ2 ≥ 0, t · = 0, ∂t ∂t ∂t ∂L ∂L = −t − t ≤ 0, λ1 · = 0, and ∂λ1 ∂λ1 ∂L ∂L = t − t¯ ≤ 0, λ2 · = 0. ∂λ2 ∂λ2 Equation (9) implies

∂x∗ ∂t

=

−1 (c0 +c1 +β)

(21) (22) (23)

< 0. There are three possible solutions. First,

if ˜t ∈ (−t, t¯), we have λ1 = λ2 = 0. Then,

∂L ∂t

=

−(1+r0 −r1 ) c0 +c1 +β

< 0, which contradicts

the condition in (21). Second, if t˜ = −t, then we have λ1 ≥ 0 and λ2 = 0 by (22)(23). Accordingly,

∂L ∂t

=

−(1+r0 −r1 ) c0 +c1 +β

− λ1 < 0, which contradicts the condition in (21)

again. Third, if t˜ = t¯, then we have λ1 = 0 and λ2 ≥ 0 by (22)-(23). Consequently, ∂L ∂t

=

−(1+r0 −r1 ) c0 +c1 +β

+ λ2 . By letting λ2 =

(1+r0 −r1 ) c0 +c1 +β

> 0, we obtain the unique solution,

i.e., t˜ = ¯t. It implies that the regulator will choose to tax pet shops, and will set the tax rate as high as possible. On the other hand, if the regulator aims to minimize the environmental damage caused by stray animals, it will choose an optimal tax or 12

subsidy to solve the problem of mint

¯ − (1 − x∗) + r0 x∗ + r1 (1 − x∗ )] d[Q

s.t.

−t ≤ t ≤ t¯.

(24)

It is easy to see that the solutions of problem (24) are the same as those of problem (20). Moreover, the same arguments can be applied to the imperfectly competitive pet animal market because equation (16) suggests

∂x ˆ ∂t

=

−J (c0 +c1 )(J+1)

< 0. These results are

summarized below. Proposition 3 : Suppose that condition (13) holds and the regulator aims to minimize the number of stray animals or the environmental damage caused by them. Then it is optimal for the regulator to set the taxes as high as possible no matter the structure of the pet animal market is perfectly or imperfectly competitive.

4.2. Pet Animal Shops Having Quadratic Cost Functions In this section, we examine the robustness of our previous results assuming that pet animal shops have quadratic, instead of linear, cost functions. Similar conclusions can be obtained if we replace linear environmental damage functions by quadratic ones.8 Thus, we present the outcomes for quadratic cost functions only. Suppose that the cost function of pet animal shops is 2c¯ x2 under a perfectly competitive market. Given (x∗, p∗ ) in (8)-(9), the regulator’s social welfare function becomes SW = h1 −

c1 (x∗ )2 ¯ − 1 + r1 + x∗ (1 + r0 − r1)]. + x∗(h0 + c1 − h0) − (c0 + c1 + c¯) − d[Q 2 2

The associated first-order and second-order conditions for optimal ˇt are ∂SW ∂t 2 ∂ SW ∂t2 8

−1 [h0 + c1 − h1 − d(1 + r0 − r1 ) − (c0 + c1 + c¯)x∗ ] = 0, (25) (c0 + c1 + β) −(c0 + c1 + c¯) = < 0. (c0 + c1 + β)2 =

Under a quadratic environmental damage function, the regulator may still subsidize or tax the

purchase of dogs/cats. However, the impacts of model’s parameters on optimal taxes or subsidies become uncertain except for the parameter of market supply. These results are available upon request.

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By solving (25), we get c¯(h0 + c1 − h1 ) + d(1 + r0 − r1 )(c0 + c1 + β) β(h0 + c1 − h1 ) −α− (c0 + c1 + c¯) (c0 + c1 + c¯) β(h0 + c1 − h1 ) c¯(h0 + c1 − h1 ) + d(1 + r0 − r1 )(c0 + c1 + β) ≥ (≤) 0 iff α + ≤ (≥) . (c0 + c1 + c¯) (c0 + c1 + c¯)

tˇ =

This implies that the regulator may still subsidize or tax the purchase of dogs/cats even the production cost function of pet animals becomes nonlinear. At tˇ, condition (3) reduces to h1 − c1 + d(1 + r0 − r1 ) < h0 < h1 + c0 + c¯ + d(1 + r0 − r1 ).

(26)

Under (26), the comparative statics results of tˇ are summarized below. Lemma 3 : Suppose that conditions (26) holds. Then we have ∂ˇ t ∂ˇ t 0 +c1 +β) 0 +c1 +β) = d(c > 0 and ∂r = −d(c < 0, ∂r0 (c0 +c1 +¯ c) (c0 +c1 +¯ c) 1 c−β) c) ∂ tˇ ∂ˇ t (ii) ∂h = (c0(¯+c ≥ (≤) 0 iff c¯ ≥ (≤) β and ∂h = (c0(β−¯ ≥ c) +c1 +¯ c) 0 1 +¯ 1 ∂ˇ t ∂ tˇ 1 +β)(1+r0 −r1 ) (iii) ∂α = −1 < 0 and ∂d = (c0 +c(c > 0, c) 0 +c1 +¯ ∂ tˇ 0 −r1 )+h1 −h0 −c1 ] (iv) ∂c = (¯c−β)[d(1+r ≥ (≤) 0 iff c¯ ≤ (≥) β, (c0 +c1 +¯ c )2 0 c−h0 +d(1+r0 −r1 )] ∂ tˇ (v) ∂c = (¯c−β)[h1 +c(c0 +¯ ≥ (≤) 0 iff c¯ ≥ (≤) β, and c )2 1 0 +c1 +¯ ˇ 1 −h0 −d(1+r0 −r1 ] (vi) ∂∂c¯t = (c0 +c1 +β)[h(c00+c > 0. +c1 +¯ c )2

(i)

(≤) 0 iff c¯ ≤ (≥) β,

Lemma 3 shows that the impacts of r0 , r1 , α, d and c¯ on optimal taxes or subsidies remain true qualitatively when production cost functions of pet animal shops become quadratic. The intuitions are the same as those behind Lemma 1. However, the impacts of h0 , h1 , c0 and c1 on optimal taxes or subsidies become ambiguous under quadratic cost functions. Since the intuitions behind parts (iv) and (v) are similar to that behind part (ii), we give explanations for part (ii) only. When the happiness from raising purchased dogs/cats (h0) enlarges, the demand for pet animals will increase. Accordingly, equilibrium price and quantity of the market will rise, and the costs to breed them expand too. Under linear production cost functions, the former effect always dominates the latter, making equilibrium profits of pet animal shops negatively related to the taxes or the subsidies.9 However, the story 9

If c(x) = c¯x, we have

∂π∗ ∂t

∗

= (p∗ − c¯) ∂x ∂t +

∂p∗ ∂t x

14

< 0 by p∗ > c¯.

changes if the production cost functions become quadratic. Under the circumstance, the equilibrium price of pet animals increases at rate β as the purchased amount of dogs/cats rises, while the production costs expand at rate c¯. Thus, equilibrium profits of pet animal shops may increase or decrease with rising taxes.10 If c¯ > β, the effect of higher production costs dominates. Hence, it is optimal for the regulator to raise the taxes or lower the subsidies to mitigate the market demand for pet animals. In contrast, if c¯ < β, the effect of higher production costs is not significant. Therefore, it is optimal for the regulator to lower the taxes or raise the subsidies to promote the purchase of dogs/cats. On the other hand, if the happiness from raising strayed animals (h1) increases, the demand for pet animals will fall. Then, the equilibrium price and quantity of pet animals will reduce, and their total production costs will decrease at an increasing rate. If c¯ > β, the effect of smaller production costs dominates. By lowering the taxes or raising the subsidies, the regulator may stimulate some demand for pet animals under a low-level production cost of them. Thus, the social welfare will rise due to higher consumers’ and producers’ surpluses. In contrast, if c¯ < β, smaller production costs resulted from falling demand for pet animals is not significant. Then it is optimal for the regulator to discourage the purchase of dogs/cats by raising the taxes or lowering the subsidies. Under the circumstance, the social welfare will increase due to the diminishing environmental damage caused by strayed animals.

5. Conclusions

Being nice to animals, especially dogs/cats, is not only human, but also has huge economic values. Many problems and costs existent in our living environments can be resolved if we are able to find ways to control stray animals. This paper thus tries to seek possible policy solutions from three criteria of the regulator. Our models have two features. First, we apply game theoretical models to characterize the interactions among the regulator, pet shops, and consumers. Second, we endogenize consumers’ behaviors of purchasing and adopting dogs/cats. We discover that the goals pursued 10

If c(x) = c2¯ x2 , the sign of

∂π∗ ∂t

∗

= [βx∗ + p∗ − c¯x∗ ] ∂x ∂t can be positive or negative.

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by the regulator will affect the equilibrium results. If the regulator aims to maximize the social welfare, it may tax or subsidize the purchase of dogs/cats. These findings hold no matter that the market structure of pet animals is perfectly or imperfectly competitive, that the cost function to breed pet animals is linear or quadratic, or that the environmental damage caused by strayed animals is in linear or quadratic form. However, if the regulator aims to minimize the number of stray animals or the environmental damage caused by them, taxing pet animal buyers is the only optimal choice. Similarly, this finding remains true under both the perfectly and imperfectly competitive markets of pet animals. Moreover, we investigate how model’s parameters affect the equilibrium subsidies or taxes. Under a perfectly competitive pet animal market, we find that the regulator will raise the taxes or lower the subsidies when the probability of abandoning purchased dogs/cats increases, the probability of abandoning adopted dogs/cats falls, the happiness from raising the purchased dogs/cats decreases, the happiness from raising stray dogs/cats rises, the cost to raise purchased dogs/cats enlarges, the cost to raise stray dogs/cats reduces, the cost to breed dogs/cats becomes higher, the market supply of pet animals diminish, or the environmental damage caused by stray animals expands. Most of these results remain true when the pet animal market changes from perfectly to imperfectly competitive. However, these comparative static outcomes may differ when the cost functions to produce pet animals or the environmental damage functions become nonlinear. Finally, since this paper focuses on taxing or subsidizing the purchases of pet animals, it would be interesting to explore alternative policy instruments in the future. For instance, we can investigate the policy of subsidizing consumers’ adopting strayed animals under the same criteria used by this model. By doing so, we can see which policy instrument will be more effective in achieving the goals of the regulator. References Baetz, A. (1992). “Why We Need Animal Control?” First National Urban Animal 16

Management Conference, Brisbane. Clifton, M. (2002). “Animal Control Is People Control.” Animal People, May. Fox, M. W. (1990). “Inhumane Society: The American Way of Exploiting Animals.” St. Martin’s Press, New York. Frank, J. (2001). The economics, ethics, and ecology of companion animal overpopulation and a mathematical model for evaluating the effectiveness of policy alternatives. Unpublished doctoral dissertation, Rensselaer Polytechnic Institute, Troy, NY. Frank, J. (2002). “The Actual Contribution and Potential Contribution of Economics to Animal Welfare Issues.” Society and Animals, 10(4): 421-428. Frank, J. and P. Carlisle-Frank (2007). “Analysis of Programs to Reduce Overpopulation of Companion Animals: Do Adoption and Low-cost Spay/Neuter Programs Merely Cause Substitution of Sources?” Ecological Economics, 62(3-4), 740-746. Hotelling, H. (1929). “Stability in Competition.” The Economic Journal, 39(153): 41-57. Jasper, J. M. and D. Nelkin (1992). “The Animal Rights Crusade: The Growth of a Moral Protest.” The Free Press, New York. Jehle, G. A. and P. J. Reny, Advanced Microeconomic Theory, 2011, Third Edition, Addison Wesley. MacKay, C. A. (1993). “Veterinary Practitioners’ Role in Pet Overpopulation.” Journal of the American Veterinary Medical Association, 202 (6), 918-920. Sacks, J. J., M. Kresnow, and B. Houston (1996). “Dog Bites; How Big a Problem?” Injury Prevention, 2(1):52-54. Rush, R. I. (1985). “City of Los Angeles Animal Care and Control.” In A. K. Wilson and A. N. Rowan (Eds.), Proceedings of a workshop on animal control (pp. 55-58). 17

Boston: Tufts Center for Animals. Sosin, D. M., J. J. Sacks, and R. W. Sattin (1986). “Causes of Nonfatal Injuries in the United States.” Accident Analysis and Prevention, 24:685-687.

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