The protection of private property: the government as a free-rider

Abstract

This paper develops a positive theory of government that explains the cross country differences in the relative importance of publicly and privately provided protection of property. The theory focuses on the decision of the government to free-ride or not to free-ride on privately provided security. If the government chooses to free-ride on privately provided security then it shirks from the provision of a nonexcludable public good: the deterrence of stealing. The theory implies that governments are more likely to free-ride on privately provided security the less efficient is stealing in capturing private property, the less efficient is the tax system, and the more efficient is stealing in capturing government property.

Appendices

Appendix 1: Total deterrence is not an equilibrium

Assuming that $\theta \le 1-c,$ if the government does not free-ride then, from Eqs. (9) and (10) together, $Z$ is maximized with

$$\begin{aligned} \frac{\partial Z}{\partial H} >0, \ \ \ \ \ \text{ and } \ \ \ \ \ H = (F+H)^*=\frac{\theta (1-\tau )Y}{\gamma }, \end{aligned}$$

(29a)

and,

$$\begin{aligned} \frac{\partial Z}{\partial \tau }=(1-c) \left( \tau \frac{\partial Y}{\partial \tau } + Y\right) - \gamma \frac{\partial H}{\partial \tau }=0, \end{aligned}$$

(29b)

where, from Eq. (9) and $H=(F+H)^*=\theta (1-\tau )Y/\gamma$,

$$\begin{aligned} \frac{\partial Y}{\partial \tau }= \frac{\alpha }{1-\alpha }\ \frac{Y}{1-\tau }, \ \ \ \ \ \text{ and } \ \ \ \ \ \frac{\partial H}{\partial \tau }=-\theta Y/\gamma . \end{aligned}$$

Conditions (29a) and (29b) imply that, if $\theta <1-c$,

$$\begin{aligned} H^{\bullet \bullet }= \theta \ \frac{\alpha }{2 \gamma } \ Y^{\bullet \bullet }. \end{aligned}$$

(30)

and,

$$\begin{aligned} \tau ^{\bullet \bullet } = \frac{(1-\alpha )(1-c) + \theta }{1-c+\theta }, \end{aligned}$$

(31)

where, given $\theta \le 1-c,$ the level of market output if the government does not free-ride is

$$\begin{aligned} Y^{\bullet \bullet }= \left[ \frac{A \alpha (1-\tau ^{\bullet \bullet }) P^{\bullet \bullet } }{\gamma } \right] ^\frac{\alpha }{1-\alpha }= \left[ \frac{A \alpha ^2}{\gamma } \frac{1}{1+\theta /(1-c)} \right] ^\frac{\alpha }{1-\alpha }. \end{aligned}$$

(32)

Equations (10), (30) and (31) imply that, given $\theta <1-c,$ if the government does not free-ride, then it would choose $H$ large enough to deter the stealing of market output. Hence, $P^{\bullet \bullet }=1.$

Substituting Eqs. (30), (31) and (32) and $P^{\bullet \bullet }=1$ into Eq. (11), the government’s net income if $\theta \le 1-c$ and the government does not free-ride is given by

$$\begin{aligned} Z^{\bullet \bullet }=(1-\alpha ) \ (1-c) \ Y^{\bullet \bullet }. \end{aligned}$$

(33)

Given $\theta \le 1-c ,$ Eqs. (16) and (33) imply that $Z^o \ge Z^{\bullet \bullet }$ as $Y^o \ge Y^{\bullet \bullet }.$ In turn, Eqs. (15) and (32) imply that $Y^o \ge Y^{\bullet \bullet }$ obtains and the government chooses to free-ride if

$$\begin{aligned} (1-\tau ^o) P^o \ge (1-\tau ^{\bullet \bullet }) P^{\bullet \bullet }, \end{aligned}$$

which implies from Eqs. (12), (14), (31) and $P^{\bullet \bullet }=1$ that the government chooses to free-ride if

$$\begin{aligned} \frac{1}{1+\theta } \ge \frac{1}{1+\theta /(1-c)}, \end{aligned}$$

(34)

which is always satisfied for $\theta \le 1-c.$

Condition (34) will be satisfied as an inequality for any positive level of collection costs, $c>0.$ In the absence of collection costs, $c=0,$ condition (34) will be satisfied as an equality and the government’s net income would be the same whether it free-rides or not. Since free-riding is a passive activity I assume that if condition (34) holds as an equality then the government would free-ride.

Appendix 2: Stealing of government revenue

The representative household maximizes its net income, as given by Eq. $(1^*)$, by choosing its time allocation to its seven possible activities, subject to $y=Al^\alpha$ Eqs. (3), (4), (27) and (28), taking $\tau$, $w$, $Y$, $R$, $S$, $F$, $H$, and $G$ as given. In addition to conditions (5) to (8), the first-order Kuhn-Tucker conditions of this problem are

$$\begin{aligned} \frac{\partial U}{\partial s}= (1-c) \tau y \frac{\partial \hat{q}}{\partial s} -\gamma \le 0, \;\;\;\;\;\; s \ge 0, \;\;\;\;\;\; \frac{\partial U}{\partial s} s = 0; \end{aligned}$$

(35)

and,

$$\begin{aligned} \frac{\partial U}{\partial g}= w - \gamma \le 0, \;\;\;\;\;\; g \ge 0, \;\;\;\;\;\; \frac{\partial U}{\partial g} g = 0, \end{aligned}$$

(36)

where, from Eq. (28),

$$\begin{aligned} \frac{\partial \hat{q}}{\partial s} = \frac{\phi G}{(G+\phi s)^2}. \end{aligned}$$

In a symmetric equilibrium in pure strategies in which every household behaves identically: $l=L$, $y=Y$, $r=R$, $s=S$, $f=F$, $h=H$, $g=G$, $p=1-q=P$ and $\hat{P}=1-\hat{q}$.

Assuming, as before, that $\gamma$ is high enough such that non-negativity constraint on home production does not bind, condition (36) implies that the household supply of publicly paid guards for the protection of the government’s tax revenue is perfectly elastic at $w=\gamma$. The household’s choice of $l$ and $r$ are still represented by Eqs. (9) and (10). In addition, the household’s choices of $s$ from condition (35) implies that the equilibrium value of $\hat{P}$, the fraction of tax revenue not lost to stealing, as a function of $G$, the resources allocated to protect tax revenue, is given by

$$\begin{aligned} \hat{P}= \left\{ \begin{array}{ll} \displaystyle \sqrt{ \frac{\gamma G}{ \phi (1-c) \tau Y }} &{} \text{ if }\; G < G^*,\\ 1 &{} \text{ if }\; G \ge G^*. \end{array} \right. \;\;\;\;\;\; \text{ where } \;\;\;\;\;\; G^*=\frac{\phi (1-c) \tau }{\gamma } Y. \end{aligned}$$

(37)

The number $G^*$ is the level of security of tax revenue that deters stealing of tax revenue and is defined such that $G \ge G^*$ results in $S=0$ because $\partial U/\partial s<0$ from condition (35).

The government maximizes its net income, as given by Eq. $(2^*)$, by making irreversible choices of $\tau ,$ $H$ and $G$ subject to Eqs. (5) to (10) and (35) to (37).

Solving the government’s problem using Eq. (37) we find that the government chooses

$$\begin{aligned} G= min \left\{ \ \phi , \ \frac{1}{4 \phi } \ \right\} \ \frac{1-c}{\gamma } \ \tau Y, \end{aligned}$$

(38)

and, substituting Eq. (38) into Eq. (37), that the equilibrium fraction of tax revenue net of collection costs retained by the government after losses to theft is

$$\begin{aligned} \hat{P}= min \{\ 1 , \ 1/2\phi \ \}. \end{aligned}$$

(39)

Equations (38) and (39) together imply that if the efficiency of stealing in capturing government property is low, in particular if $\phi \le 1/2,$ then the government chooses $G=G^*$ such that stealing of government revenue is deterred, $\hat{P}=1.$ But, if the efficiency of stealing in capturing government property is high, $\phi > 1/2$, then $G < G^*$ and the government loses the fraction $1/2\phi$ of its tax revenue to stealing, $\hat{P}=1/2\phi .$

Equation (38) also tells us that the ratio of resources allocated by the government to protect its own tax revenue to its tax revenue, $\gamma G/ (1-c) \tau Y,$ is only a function of $\phi ,$ the efficiency of stealing in capturing government property, and independent of the tax rate and whether the government chooses to free-ride or not to free-ride on privately provided security.

Substituting Eqs. (38) and (39) into Eq. $(2^*)$, the government’s problem after taking into account the choice of $G,$ is given by

$$\begin{aligned} \max _{\tau , H} Z= (1-c) B(\phi ) \tau Y - \gamma H. \end{aligned}$$

(11*)

subject to (5) to (10), where

$$\begin{aligned} B(\phi )\equiv 1 - \phi \ \ \ \text{ if } \ \ \ \phi \le 1/2, \ \ \ \text{ and } \ \ \ B(\phi ) \equiv 1/4\phi \ \ \ \text{ if } \ \ \ \phi > 1/2. \end{aligned}$$

Given that the ratio $\gamma G/(1-c) \tau Y$ does not depend on whether the government chooses to free-ride or not, Eqs. (12) to (25) still describe the equilibrium except that $(1-c) B(\phi )$ appears now in place of $1-c.$ Since $B^{\prime }(\phi ) < 0,$ the effects of $c$ and $\phi$ on the government’s decision to free-ride or not to free-ride are isomorphic.