Risk lovers and the rent over-investment puzzle

Paan Jindapon; Christopher A. Whaley

doi:10.1007/s11127-015-0270-y

Public Choice

July 2015, Volume 164, Issue 1–2, pp 87–101 | Cite as

Risk lovers and the rent over-investment puzzle

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Paan JindaponEmail author
Christopher A. Whaley

Article

First Online: 20 June 2015

Received: 26 September 2014

Accepted: 09 June 2015

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3 Citations

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.

Keywords

Prudence Risk lovers Rent seeking Contest

JEL Classification

C72 D81 D72

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Notes

Acknowledgments

We thank Klaus Abbink, Paul Pecorino, Ray Rees, Harris Schlesinger, Richard Watt, and seminar participants at University of Alabama, University of Canterbury, and the 2013 Public Choice Society Annual Meeting for valuable comments. We acknowledge Culverhouse College of Commerce for financial support.

Appendix 1: Proofs

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).

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).

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 _i ^H denotes income with the rent and z _i ^L is income without it, with all other variables being defined previously. The author incorrectly claims u _i(z _i ^H ) − u _i(z _i ^L ) 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.

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Paan Jindapon
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- 2
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Christopher A. Whaley
- 1

1.University of AlabamaTuscaloosaUSA
2.Bangkok UniversityBangkokThailand

Risk lovers and the rent over-investment puzzle

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