Risk lovers and the rent over-investment puzzle

Abstract

In this paper, we prove existence and uniqueness of equilibrium in a rent-seeking contest given a class of heterogeneous risk-loving players. We explore the role third-order risk attitude plays in equilibrium and find that imprudence is sufficient for risk lovers to increase rent-seeking investment above the risk-neutral outcome. Moreover, we show that rent can be fully dissipated in a standard Tullock contest played by a large number of risk-lovers.

Appendices

Appendix 1: Proofs

1.1 Proof of Lemma 1

The slope of indifference curve q _i (y _i | $\bar{U}$) is given by

$$q_{i} '\left( {y_{i} |\bar{U}} \right) = [q_{i} \phi_{i} (I_{i} - g_{i} \left( {y_{i} )} \right) + (1 - q_{i} )\psi_{i} \left( {I_{i} - g_{i} \left( {y_{i} )} \right)} \right]g_{i}^{\prime} \left( {y_{i} } \right),$$

(14)

where

$$\phi_{i} = \frac{{u_{i}^{\prime} \left( {w + R} \right)}}{{u_{i} (w + R) - u_{i} (w)}}$$

(15)

and

$$\psi_{i} = \frac{{u_{i}^{\prime} (w)}}{{u_{i} (w + R) - u_{i} (w)}} .$$

(16)

Thus, under general conditions, the indifference curve is always upward sloping. By letting v _i : = I _i  − g _i (y _i ), we derive the second derivative of q _i (y _i | $\bar{U}$) with respect to y _i from (14) as

$$q_{i} ''(y_{i} |\bar{U}) \, = \, \left[ {q_{i} \phi_{i} \left( {v_{i} } \right) \, + \, \left( { 1- q_{i} } \right)\psi_{i} \left( {v_{i} } \right)} \right]g_{i} ''\left( {y_{i} } \right) \, + \, \left[ {q_{i} \varPhi_{i} \left( {v_{i} } \right) \, + \, \left( { 1- q_{i} } \right)\varPsi_{i} \left( {v_{i} } \right)} \right]\left[ {g_{i} '\left( {y_{i} } \right)} \right]^{ 2} ,$$

(17)

where

$$\varPhi_{i} \left( w \right) = \phi_{i} \left( w \right)\left[ {\phi_{i} \left( w \right) - \psi_{i} \left( w \right)} \right] - \phi_{i} '(w)$$

(18)

and

$$\varPsi_{i} \left( w \right) = \psi_{i} \left( w \right)\left[ {\phi_{i} \left( w \right) - \psi_{i} \left( w \right)} \right] - \psi_{i} '(w) .$$

(19)

The first term on the right-hand side of (17) is positive; however, the sign of the second term is uncertain. To guarantee that it is positive we assume (i) φ′ _i (w) ≤ φ _i (w)[φ _i (w) − ψ _i (w)] and (ii) ψ′ _i (w) ≤ ψ _i (w)[φ _i (w) − ψ _i (w)] for all w ∈ (I _i  − R,I _i ). Even though Conditions (i) and (ii) are stated in terms of φ _i and ψ _i instead of u _i , their intuition is quite simple. Using (15) and (16), we can write conditions (i) and (ii) as

$$\frac{{u_{i}^{''} (w + R)}}{{u_{i}^{\prime} (w + R)}} \le 2\frac{{u_{i}^{\prime} \left( {w + R} \right) - u_{i}^{\prime} (w)}}{{u_{i} \left( {w + R} \right) - u_{i} (w)}}$$

(20)

and

$$\frac{{u_{i}^{''} (w)}}{{u_{i}^{\prime} (w)}} \le 2\frac{{u_{i}^{\prime} \left( {w + R} \right) - u_{i}^{\prime} (w)}}{{u_{i} \left( {w + R} \right) - u_{i} (w)}}$$

(21)

for all w ∈ (I _i  − R,I _i ) respectively. Thus, both inequalities can be satisfied if

$$\frac{{u_{i}^{''} (w)}}{{u_{i}^{\prime} (w)}} \le 2\frac{{u_{i}^{''} (z)}}{{u_{i}^{\prime} (z)}}$$

(22)

for all w, z ∈ (I _i  − R,I _i  + R).

1.2 Proof of Lemma 2

Part (i) Given the optimization problem in (3), an interior solution y ^* _i satisfies the first-order condition:

$$\begin{aligned} &F\left( {y_{i}^{*} } \right)\text{ := }\frac{{Y_{ - i} }}{{\left( {y_{i}^{*} + Y_{ - i} } \right)^{2} }}[u_{i} \left( {I_{i} - g_{i} (y_{i}^{*} ) + R} \right) - u_{i} \left( {I_{i} - g_{i} (y_{i}^{*} )} \right)] \hfill \\& - \left[ {\left( {\frac{{y_{i}^{*} }}{{y_{i}^{*} + Y_{ - i} }}} \right)u^{\prime}_{i} \left( {I_{i} - g_{i} \left( {y_{i}^{*} } \right) + R} \right) + \left( {\frac{{Y_{ - i} }}{{y_{i}^{*} + Y_{ - i} }}} \right)u^{\prime}_{i} \left( {I_{i} - g_{i} \left( {y_{i}^{*} } \right)} \right)} \right]g'_{i} \left( {y_{i}^{*} } \right) = 0. \hfill \\ \end{aligned}$$

(23)

For the equivalent constrained-optimization problem given by (4) and (5), an interior solution ($y_{i}^{*},p_{i}^{*}$) satisfies the tangency condition and the constraint:

$$\frac{{Y_{ - i} }}{{\left( {y_{i}^{*} + Y_{ - i} } \right)^{2} }} - [p_{i}^{*} \phi_{i} \left( {v_{i}^{*} } \right) + \left( {1 - p_{i}^{*} } \right)\psi_{i} \left( {v_{i}^{*} )} \right]g_{i}^{\prime} \left( {y_{i} } \right) = 0$$

(24)

$$p_{i}^{*} - \frac{{y_{i}^{*} }}{{y_{i}^{*} + Y_{ - i} }} = 0 ,$$

(25)

where $v_{i}^{*}$ = I _i −g _i ($y_{i}^{*}$) and φ _i and ψ _i are defined in (15) and (16), respectively. Substituting $p_{i}^{*}$ from (25) into (24) yields

$$G\left( {y_{i}^{*} } \right)\text{ := }\frac{{Y_{ - i} }}{{\left( {y_{i}^{*} + Y_{ - i} } \right)^{2} }} - \left[ {\left( {\frac{{y_{i}^{*} }}{{y_{i}^{*} + Y_{ - i} }}} \right)\phi_{i} \left( {v_{i}^{*} } \right) + \left( {\frac{{Y_{ - i} }}{{y_{i}^{*} + Y_{ - i} }}} \right)\psi_{i} (v_{i}^{*} )} \right]g'_{i} \left( {y_{i}^{*} } \right) = 0 .$$

(26)

Note that G(y _i ) is the difference between the slope of the constraint and the slope of indifference curve at (y _i , p(y _i |Y _−i )). Since the former is decreasing (in fact, strictly decreasing when Y _−i  > 0) and the latter is strictly increasing (see Lemma 1), G(y _i ) is strictly decreasing in y _i . If Y _−i is less than a threshold (derived in part (ii) below), G(y _i ) > 0 when y _i  = 0. Moreover, we find that G(y _i ) < 0 when y _i is large enough. There consequently exists only one value $y_{i}^{*}$ such that $G\left( {y_{i}^{*} } \right) = 0$.

Next, we define

$$\Delta (y_{i}^{ * } ) \, : = u_{i} (I_{i} - g_{i} (y_{i}^{ * } ) \, + R) - u_{i} (I_{i} - g_{i} (y_{i}^{ * } )).$$

(27)

Since u′ and u″ are strictly positive, we know that ∆(y _i ) is strictly positive and strictly decreasing in y _i . Given the definitions of G(y _i ) and ∆(y _i ), we can write the first-order condition in (23) as

$$F(y_{i}^{ * } ) \, = G(y_{i}^{ * } )\varDelta (y_{i}^{ * } ) \, = \, 0.$$

(28)

The properties of G(y _i ) and ∆(y _i ) described above imply that F(y _i ) is strictly decreasing for all $y_{i} \le y_{i}^{*}$ and that $y_{i}^{*}$ is the only value of y _i such that the first-order condition holds. Thus we can say that F(y _i ) ≥ (≤) 0 if and only if y _i  ≤ (≥) $y_{i}^{*}$; it follows that Eu _i in (3) is strictly quasiconcave in y _i . Since budgets do not constrain each player’s investment, the only possible corner solution would be at zero, which occurs only when Y _−i is so large that $y_{i}^{*}$ satisfying the first-order condition is negative. If this is the case, then we know that a positive solution to the first-order condition in (3) does not exist because F(y _i ) < 0 for all y _i  ≥ 0, so player i’s optimal decision is setting y _i  = 0.

Part (ii) We find the status quo level of player i’s utility with no investment and no chance of winning to be u _i (I _i ). The constrained optimization problem yields an interior solution, i.e., $y^{*}$ > 0 given Y _−i , whenever the slope of the indifference curve q _i (y _i | $\bar{U}$ = u _i (I _i )) at y _i  = 0 is less than the slope of the constraint p(y _i |Y _−i ) at y _i  = 0. Since $q_{i} '(y_{i} |\bar{U})$) is given by (14) and $p^{\prime} \left( {y_{i} |Y_{ - i} } \right) = \frac{{Y_{ - i} }}{{\left( {y_{i} + Y_{ - i} } \right)^{2} }}$, we have $q_{i} '(y_{i} |\bar{U} = u_{i} \left( {I_{i} } \right)) \, < p'\left( {y_{i} |Y_{ - i} } \right)$ at y _i  = 0 if and only if

$$\psi_{i} (I_{i} )g_{i}^{\prime} \left( 0 \right) < \frac{1}{{Y_{ - i} }} .$$

(29)

By letting $\kappa_{i} = { 1}/[\psi_{i} (I_{i} )g_{i} '\left( 0 \right)]$, we find that $y^{*}$ > 0 if and only if Y _−i  < κ _i .

Part (iii) Define player i’s share function, the probability that player i wins the contestgiven his optimal investment as s _i ($Y^{*}$) where $Y^{*}$ =  (Y _−i ) + Y _−i . Since $y_{i}^{*}$(0) > 0, $Y^{*}$ = 0 is never an equilibrium and we do not need to define s _i (0). Using the constraint (25), we rewrite the tangency condition (24) as

$$\frac{{1 - p_{i}^{*} }}{{Y^{*} }} - [p_{i}^{*} \phi_{i} \left( {v_{i}^{*} } \right) + \left( {1 - p_{i}^{*} } \right)\psi_{i} \left( {v_{i}^{*} )} \right]g_{i}^{\prime} \left( {y_{i} } \right) = 0,$$

(30)

which can be rearranged as

$$p_{i}^{*} = \frac{{[1 - \psi_{i} \left( {v_{i}^{*} } \right)g_{i}^{\prime} \left( {y_{i}^{*} } \right)Y^{*} ]}}{{\left[ {1 - \psi_{i} \left( {v_{i}^{*} } \right)g_{i}^{\prime} \left( {y_{i}^{*} } \right)Y^{*} } \right] + \phi_{i} \left( {v_{i}^{*} } \right)Y^{*} }} .$$

(31)

Therefore, we have

$$s_{i} (Y^{*} ) = \hbox{max} \left\{ {\frac{{[1 - \psi_{i} \left( {v_{i}^{*} } \right)g_{i}^{\prime} \left( {y_{i}^{*} } \right)Y^{*} ]}}{{\left[ {1 - \psi_{i} \left( {v_{i}^{*} } \right)g_{i}^{\prime} \left( {y_{i}^{*} } \right)Y^{*} } \right] + \phi_{i} \left( {v_{i}^{*} } \right)Y^{*} }},0} \right\} .$$

(32)

We find that s _i is a continuous function of $Y^{*}$ for $Y^{*}$ > 0 and that 1 is its upper bound. In addition, as $Y^{*}$ approaches 0, s _i converges to 1. From part (ii) we know that $y^{*}$ = 0 if Y _−i  ≥ κ _i, where κ _i  = 1/[ψ _i (I _i )g _i ′(0)]. Thus we can say that s _i  = 0 if $Y^{*}$ ≥ κ _i , which is consistent with the numerator of (32). To show (d) we totally differentiate the system of Eqs. (24) and (25) with respect to Y _−i .

$$\frac{{dp_{i}^{*} }}{{dY_{ - i} }} = \frac{{ - \frac{{Y_{ - i} }}{{Y^{*4} }} - \frac{{y_{i}^{*} }}{{Y^{*2} }}\left[ {p_{i}^{*} \phi_{i} \left( {v_{i}^{*} } \right) + (1 - p_{i}^{*} )\psi_{i} \left( {v_{i}^{*} } \right)} \right]g_{i}'' \left( {y_{i}^{*} } \right) + \frac{{y_{i}^{*} }}{{Y^{*2} }}[p_{i}^{*} \phi^{\prime}_{i} \left( {v_{i}^{*} } \right) + \left( {1 - p_{i}^{*} } \right)\psi^{\prime}_{i} \left( {v_{i}^{*} )} \right]\left[ {g_{i}^{\prime} \left( {y_{i} } \right)} \right]^{2} }}{{q_{i}{''} \left( {y_{i}^{*} |u_{i}^{*} } \right) - p{''} (y_{i}^{*} |Y_{ - i} )}}$$

(33)

$$\frac{{dy_{i}^{*} }}{{dY_{ - i} }} = \frac{{\frac{{y_{i}^{*} - Y_{ - i} }}{{Y^{*3} }} + \frac{{y_{i}^{*} }}{{Y^{*2} }}\left[ {\phi_{i} \left( {v_{i}^{*} } \right) - \psi_{i} \left( {v_{i}^{*} } \right)} \right]g_{i}^{\prime} \left( {y_{i}^{*} } \right)}}{{q_{i}^{\prime\prime} \left( {y_{i}^{*} |u_{i}^{*} } \right) - p^{\prime\prime} (y_{i}^{*} |Y_{ - i} )}}$$

(34)

where $u_{i}^{*}$ is the optimal EU corresponding to $y_{i}^{*}$. Since $q_{i}^{\prime\prime} (y_{i}^{*} |u_{i}^{*} ) \ge 0$ and

$$p^{\prime\prime} \left( {y_{i}^{*} |Y_{ - i} } \right) = \frac{{ - 2Y_{ - i} }}{{\left( {y_{i}^{*} + Y_{ - i} } \right)^{3} }} < 0 ,$$

(35)

the denominator of each equation is always positive. Consider the numerator of (33). The middle term is nonnegative because u _i is convex and g _i is concave. Thus $\frac{{dp_{i}^{*} }}{{dY_{ - i} }} < 0$ if

$$\frac{{y_{i}^{*} }}{{Y^{*2} }}[p_{i}^{*} \phi^{\prime}_{i} \left( {v_{i}^{*} } \right) + \left( {1 - p_{i}^{*} } \right)\psi^{\prime}_{i} \left( {v_{i}^{*} )} \right]\left[ {g_{i}^{\prime} \left( {y_{i} } \right)} \right]^{2} < \frac{{Y_{ - i} }}{{Y^{*4} }}.$$

(36)

Substituting g _i ′(y _i ) from (24) and $\frac{{y_{i}^{*} }}{{Y^{*} }}$ from (24) in (36) yields

$$p_{i}^{*} (1 - p_{i}^{*} )[p_{i}^{*} \phi^{\prime}_{i} \left( {v_{i}^{*} } \right) + \left( {1 - p_{i}^{*} } \right)\psi^{\prime}_{i} \left( {v_{i}^{*} )} \right] < \left[ {p_{i}^{*} \phi_{i} \left( {v_{i}^{*} } \right) + \left( {1 - p_{i}^{*} } \right)\psi_{i} \left( {v_{i}^{*} } \right)} \right]^{2} .$$

(37)

Conditions (i) and (ii) in the proof of Lemma 1 guarantee that (37) holds, so $\frac{{dp_{i}^{*} }}{{dY_{ - i} }} < 0$ for all Y _−i ∈ (0,κ _i ). Now consider (34). Substituting $q_{i}^{\prime\prime} (y_{i}^{*} |u_{i}^{*} )$ from (17) and $p^{\prime\prime} (y_{i}^{*} |Y_{ - i} )$ from (35) into (34) yields $\frac{{dy_{i}^{*} }}{{dY_{ - i} }} > - 1$. Given the definition of $Y^{*}$ above, it follows that $\frac{{dY^{*} }}{{dY_{ - i} }} = \frac{{dy_{i}^{*} }}{{dY_{ - i} }} + 1$.

Since $\frac{{dy_{i}^{*} }}{{dY_{ - i} }} > - 1$, then $\frac{{dY^{*} }}{{dY_{ - i} }} > 0$. Given $\frac{{dp_{i}^{*} }}{{dY_{ - i} }} < 0$, it follows that $\frac{{ds_{i} }}{{dY_{ - i} }} < 0$ for all $Y^{*}$ ∈ (0,κ _i ).

1.3 Proof of Proposition 2

Using a generalization of Hermite–Hadamard’s inequality given by Theorem 5.11 in Pe˘carić et al. (1992), we have

$$\left( \frac{1}{n} \right)u^{\prime}\left( {I - x + R} \right) + \left( {\frac{n - 1}{n}} \right)u^{\prime}\left( {I - x} \right) < \frac{n}{2R}\mathop \smallint \limits_{I - x}^{I - x + 2R/n} u^{\prime}\left( w \right)dw$$

(38)

when u′′′ < 0. Using Steffensen’s inequality given by Theorem 6.19 in Pe˘carić et al. (1992), we find

$$\mathop \smallint \limits_{I - x}^{I - x + 2R/n} u^{\prime}\left( w \right)dw < \frac{2}{n}\mathop \smallint \limits_{I - x}^{I - x + R} u^{\prime}\left( w \right)dw$$

(39)

when u″ > 0. Thus, (38) and (39) jointly imply

$$\left( \frac{1}{n} \right)u^{\prime}\left( {I - x + R} \right) + \left( {\frac{n - 1}{n}} \right)u^{\prime}\left( {I - x} \right) < \frac{1}{R}\mathop \smallint \limits_{I - x}^{I - x + R} u^{\prime}\left( w \right)dw$$

(40)

and the bracketed term in (9) is greater than 1. Proposition 2 clearly follows.

Appendix 2: A note on equilibrium in Yamazaki (2009)

In the proof of Lemma 1 found in Yamazaki (2009), the author attempts to confirm the convexity of an indifference curve G by examining its first partial derivative with respect to y _i :

$$\frac{{\partial G_{i} }}{{\partial y_{i} }} = \frac{{g_{i}^{\prime} (y_{i} )}}{{u_{i} \left( {z_{i}^{H} } \right) - u_{i} \left( {z_{i}^{L} } \right)}}\left[ {u_{i}^{\prime} \left( {z_{i}^{L} } \right) + \frac{{\bar{U} - u_{i} \left( {z_{i}^{L} } \right)}}{{u_{i} \left( {z_{i}^{H} } \right) - u_{i} \left( {z_{i}^{L} } \right)}}(u_{i}^{\prime} \left( {z_{i}^{H} } \right) - u_{i}^{\prime} \left( {z_{i}^{L} } \right))} \right] ,$$

(41)

where z ^H _i denotes income with the rent and z ^L _i is income without it, with all other variables being defined previously. The author incorrectly claims u _i (z ^H _i ) − u _i (z ^L _i ) to be non-increasing in y _i when the difference is actually increasing in y _i under the assumption of risk aversion. Therefore we cannot conclude that indifference curves are convex and the existence of a unique equilibrium under DARA breaks down.