The political economy of voluntary public service

Epple and Romano (1996) identify corresponding equilibrium preferences in a setting where majority rule determines the quality of a public good (e.g., public education) that will be implemented and financed with a proportional income tax. In their model, individuals with moderate income favor a relatively high level of quality for the public good. In contrast, individuals with the lowest and the highest incomes prefer lower levels of quality. The preference of low-income individuals reflects their limited wealth. The preference of high-income individuals reflects in part their equilibrium consumption of a substitute good (e.g., private education).

Mulligan et al. (2004), among others, observe that majority voting often fails to accurately reflect the intensities of voters’ preferences, and so may systematically favor some policies (or groups of voters) over others. Other studies (e.g., Mulligan 2015; Koch and Birchenall 2016) demonstrate that voluntary service often is more efficient than mandatory service. Our analysis complements these studies by identifying conditions under which majority rule favors mandatory service even though voluntary service is more efficient.

Because the adequate jury pool constraint binds under an optimal VJS policy, an individual who opts in is certain to serve and an individual who opts out is certain not to serve. Therefore, when the individual with \(c\,=\,{\widehat{c}}\) opts in under the identified VJS policy, his expected welfare is \(w-\,{\widehat{c}}\,=\left[ \,\frac{ N\,-\,T}{N}\,\right] {\widehat{c}}-\,{\widehat{c}}\,=-\,\frac{T}{N }\,{\widehat{c}}\). This is precisely the expected welfare he secures if he opts out under VJS (\(\,-\,F\,=\,-\,\frac{T}{N}\,{\widehat{c}}\)) and his expected welfare under MJS.

This is the case because when g(c) is the uniform density, the intervals \(\left[ \,{\underline{c}},\frac{1}{2}\left( {\underline{c}}+{\widehat{c}}\,\right) \,\right]\) and \(\left[ \,\frac{1}{2} \left( {\underline{c}}+{\widehat{c}}\,\right) ,{\widehat{c}}\, \right]\) contain the same (expected) number of individuals, as do the intervals \(\left[ \,{\widehat{c}}\,,\frac{1}{2}\left( {\widehat{c}}\,+ {\overline{c}}\right) \,\right]\) and \(\left[ \,\frac{1}{2} \left( {\widehat{c}}\,+{\overline{c}}\right) , {\overline{c}}\,\right]\).

It can be shown that the expected welfare gain under VJS relative to MJS is proportional to \(A_{W}-A\) when g(c) is the uniform density. (See the Observation that follows the proof of Proposition 4 in the Appendix.) Therefore, given the symmetric gains and losses from VJS around \(c_{1}\) and \(c_{2}\) and the equal likelihood of all c realizations, a majority will prefer VJS to MJS if and only if \(A\le A_{W}\) when g(c) is the uniform density.

g(c) is strictly log concave if \(\log (g(c))\) is a strictly concave function of c . If g(c) is strictly log concave, then G(c) is strictly log concave, so \(\frac{d}{dc}(\frac{g(c)}{G(c)})<0\) for all \(c\in \left[ \,{\underline{c}} ,{\overline{c}}\,\right]\). Many common density functions are strictly log concave, including the uniform, normal, exponential, and logistic densities. See Bagnoli and Bergstrom (2005) for additional discussion and analysis. Many studies in the public choice literature (e.g., Caplin and Nalebuff 1988) focus on settings where relevant density functions are log concave.

Recall from Lemma 1 that individuals with \(c\in (c_{1},c_{2})\) prefer MJS to VJS. Therefore, if \(G(c_{2})-G(c_{1})>\frac{1}{ 2}\) when \(A=A_{w}\), more than half of individuals prefer MJS to VJS when the two jury systems secure the same expected welfare (i.e., when \(A=A_{w}\)).

Mize et al. (2007) report there were 148, 558 state jury trials and 5, 463 federal jury trials in the U.S. in 2006. Table 2 takes the number of trials to be 1000, for expositional ease. If the values of N and T in Table 2 were to be multiplied by \(k>0\), the values of \(A_{m}\) and \(A_{w}\) would also be multiplied by k. The entries in the other columns in the table would not change.

This conclusion reflects numerical solutions that consider all values of \(\alpha =\beta \in (\,1,25\,]\) in increments of 1, all values of \(N/T\in \left( 1,2\,\right]\) in increments of 0.001, all values of \(N/T\in \left[ \,2,10\,\right]\) in increments of 0.01, all values of \(N/T\in \left[ \,10,1000\,\right]\) in increments of 1, and all values of \(N/T\in \left[ \,1,000,10,000\,\right]\) in increments of 10.

Mize et al. (2007, p. 8) observe that even though many individuals are summoned to perform jury service each year, “less than 1 percent of the adult American population” is actually empaneled on a jury. Consequently, it may be reasonable to assume that, on average, individuals who are summoned to perform jury service spend one day in this capacity. The assumption that an individual who is summoned to perform jury service forfeits one day’s income may be most relevant for self-employed individuals.

Salem and Mount (1974, Table II) report that the U.S. income distribution in the 1960s was reasonably well captured by a Gamma density with parameters of this magnitude. (The Gamma density is \(g(c)=\frac{1}{\Gamma (\alpha )}\left[ \lambda ^{\alpha }e^{-\lambda c}c^{\alpha -1}\right]\) for \(c>0\), where \(\Gamma (\alpha )\equiv \int \limits _{0}^{\infty }e^{-x}x^{\alpha -1}dx\).) Singh and Maddala (1976) report that the U.S. income distribution generally is better approximated by a Gamma density than by the two other densities most commonly employed for this purpose–the Pareto density and the log normal density.

The mean of the identified Gamma distribution is \(\frac{\alpha }{\lambda }= \frac{20,000}{3}\). This mean is multiplied by 7.5 (\(=50,000/[\,20,000/3\,] \,\)) to generate a mean annual income of \(\$50,000\).

If the average opportunity cost declines from \(\$200\) to \(\$100\) in this setting, majority rule implements MJS when VJS would secure a higher level of expected welfare if \(\frac{A}{N}\in (\$9.45,\$11.79)\). Under VJS, individuals with annual incomes below \(\$8,540\) volunteer to perform jury service. When \(A=A_{m}\), individuals who perform jury service are paid \(\$19.58\) whereas those who opt out of jury service pay \(\$14.58\) to do so. If \(A=\frac{1}{2}\,A_{m}\), then individuals who perform jury service are paid \(\$24.31\) whereas those who opt out of jury service pay \(\$9.85\).

If the variance declines from 100 to 50 in this setting, majority rule implements MJS when VJS would secure a higher level of expected welfare if \(\frac{A}{N}\in (\$1.38,\$1.65)\). Under VJS, individuals with annual incomes below \(\$48,167\) volunteer to perform jury service. When \(A=A_{m}\), individuals who perform jury service are paid \(\$162.40\) whereas those who opt out of jury service pay \(\$30.28\).

A more complete treatment of these concerns would quantify and account explicitly for the social value of fairness, jury accuracy, and trial by peers.

The adequate jury pool requirement is not constraining when \({\overline{p}}\le 1-\frac{T}{N}\). This setting is of particular interest because it allows the welfare-maximizing VJS policy to be determined by factors other than those that underlie the welfare-maximizing VJS policy analyzed in Sects. 3 and 4.

This conclusion reflects numerical solutions that consider all values of \(\alpha \in \left[ \,1,15\,\right]\) in increments of 1, all values of \(\beta \in \left[ \,\alpha ,25\,\right]\) in increments of 1 (except \(\alpha =\beta =1\)), and all values of \({\overline{p}}\in \left[ \,0.1,\,0.9\,\right]\) in increments of 0.01. Bose et al. (2019) prove analytically that when g(c) is a symmetric, piecewise linear density with an inverted-V shape, majority rule always favors MJS over the modified VJS policy (so \(A_{m}<A_{w}\)). Bose et al. (2019) also show that the extent to which majority rule favors MJS when g(c) is V-shaped can be mitigated to some extent when \({\overline{p}}\) is small. This is the case because as \({\overline{p}}\) declines, the welfare gains and losses from VJS become less sensitive to c (because the individuals with the highest c’s who request exemption from jury service are not ensured of exemption). Consequently, any deficiency of majority voting in fully reflecting the intensity of preferences for VJS is ameliorated.

Bowles (1980, p. 371) notes that the costs of jury service “lie mainly in the value of production lost to the economy as a result of the absence of jurors from work.” Koch and Birchenall (2016) identify conditions under which social welfare increases when a voluntary military enlistment policy replaces a draft, thereby allowing the most productive individuals to continue to earn high wages and pay high taxes.