A methodological critique of Munshi's model on clientelism, public goods, and divided democracies

As social scientists, we find Soumyanetra Munshi's theoretical work on the dilemma between clientelism and public goods in divided democracies both fascinating and thought-provoking. However, this article addresses two core issues in her model. The dialogue is guided by a perspective in formal modeling that emphasizes the analytical relevance of conclusions that may appear absurd, as well as internal contradictions. We critique Munshi's model by first discussing a paradoxical outcome, followed by a logical inconsistency.

JEL classification: B21, B41, C02, C72, D72.

1 Introduction

As scholars in the field of social science, we find Soumyanetra Munshi's article, titled “Clientelism or Public Goods: Dilemma in a Divided Democracy,” to be both captivating and thought-provoking. However, we wish to address two issues we have identified within her theoretical framework. Our discussion adheres to the framework articulated by Rubinstein (2006), where one of the paramount functions of a formal model lies in generating conclusions. It is noteworthy that some of these conclusions may appear absurd, despite being derived from sound assumptions. The recognition of an absurd conclusion carries as much significance as identifying a contradiction within a mathematical model itself. In this concise paper, we engage with Munshi's work by initially examining an absurd conclusion and subsequently delving into a logical contradiction.

The paper is organized as follows. In Section 2, we present the theoretical model setup of Munshi (2022). Section 3 introduces an absurd conclusion inferred from Munshi's (2022) model. In Section 4, we present a logical inconsistency identified in the structure of Munshi's (2022) model. Finally, Section 5 presents the conclusion.

2 Preliminaries: set-up of the theoretical model

In many contemporary societies, according to Munshi (2022), the relevance of the distinction between left- and right-wing parties is diminishing compared to the distinction between elite and non-elite parties. Elite parties primarily cater to the preferences and needs of higher and middle-income groups, with a particular focus on not providing mass public goods. Non-elite individuals, who predominantly require these public goods, rely on them as high-income individuals can acquire them through market mechanisms.

Elite parties generally face a numerical disadvantage. Consequently, in order to succeed in electoral contests, elite parties can adopt one of two strategies: (1) providing the public goods demanded by non-elite individuals, even at the risk of alienating their elite support base; or (2) offering clientelistic bribes, whether in monetary form or in the form of favors and employment opportunities to a sufficient number of non-elite individuals.

2.1 Players

There is a set N = {1, 2, ..., n_N} of non-elite voters and a set E = {1, 2, ..., n_E} of elite voters. There are two political parties, the elite party A and the non-elite party B.

2.2 Timing

It is a dynamic game of two stages with perfect information. Let x_A ∈ ℝ₊ and g_A ∈ ℝ₊ be the amounts of public goods that party A offers in the first stage of the game to elite and non-elite voters, respectively. Similarly, let x_B ∈ ℝ₊ and g_B ∈ ℝ₊ be the amounts of public goods that party B offers in the first stage of the game to elite and non-elite voters, respectively. In the second stage of the game, the elite voters observe the actions x_A and x_B and decide which political party to vote for. Similarly, the non-elite voters observe the actions g_A and g_B and decide which political party to vote for.

2.3 Elite voters

Elite voter i ∈ E assigns party weights $v_{iA}^E\in {ℝ}_{+}$ and $v_{iB}^E\in {ℝ}_{+}$ to parties A and B, respectively. Political parties do not know the weights $v_{iA}^E$ and $v_{iB}^E$ such that

$\begin{array}{ll}{v}_i^E=v_{iA}^E-v_{iB}^E~U\left[\frac{b_E}{f_E}-\frac{1}{2f_E},\frac{b_E}{f_E}-\frac{1}{2f_E}\right]& (2.1)\end{array}$

is assumed to be a random variable. The probability distribution is common knowledge. The preferences of the elite voter i are defined as follows:

$\begin{array}{ll}{u}_i(x)=-{(x-x_i)}^2& (2.2)\end{array}$

such that x_i is the ideal amount of public goods for elite voters i ∈ E and x is the amount of public good offered by a political party during the election period. It is trivially verified that ∂u(x)/∂x ⋚ 0⇔x ⋛ x_i such that u_i reaches a global maximum at $x^{*}=x_i$. A voter i ∈ E chooses to vote for party A if and only if

$\begin{array}{ll}{v}_{iA}^E+u_i(x_A)\ge v_{iB}^E+u_i(x_B)& (2.3)\end{array}$

2.4 Non-elite voters

Non-elite voter i ∈ N assigns party weights $v_{iA}^N$ and $v_{iB}^N$ to parties A and B, respectively. Political parties do not know the weights $v_{iA}^N$ and $v_{iB}^N$ such that

$\begin{array}{ll}{v}_i^N=v_{iA}^N-v_{iB}^N~U\left[\frac{b_N}{f_N}-\frac{1}{2f_N},\frac{b_N}{f_N}-\frac{1}{2f_N}\right]& (2.4)\end{array}$

is assumed to be a random variable. The probability distribution is common knowledge. The preferences of the non-elite voter i ∈ N are defined as follows:

$\begin{array}{ll}{u}_i(g)=-{(g-g_i)}^2& (2.5)\end{array}$

such that g_i is the ideal amount of public goods for non-elite voter i ∈ N, and g is the amount of public good offered by a political party during the election period. It is trivially verified that ∂u(g)/∂g ⋚ 0⇔g ⋛ g_i such that u_i reaches a global maximum at $g^{*}=g_i$ such that g_i is the optimal quantity required by non-elite voters i ∈ N. A non-elite voter i ∈ N chooses to vote for party A if and only if

$\begin{array}{ll}{v}_{iA}^N+u_i(g_A)\ge v_{iB}^N+u_i(g_B)& (2.6)\end{array}$

3 An absurd conclusion

In a first possible world, Munshi (2022) assumes that the elite party does not produce public goods for non-elite voters, i.e., g_A = 0. Although Munshi (2022) does not explicitly state the scarcity assumption, it holds that u_i(g_B)−u_i(0) > 0 if g_B > 0. Therefore, the expression (Equation 2.6) is as follows:

$\begin{array}{ll}{v}_{iA}^N-v_{iB}^N\ge u_i(g_B)-u_i(0)>0& (3.1)\end{array}$

such that $v_{iA}^N-v_{iB}^N=v_i^N$ and $u_i(g_B)-u_i(0)=2g_B·g_i-g_B^2$. Hence,

$P_A^{iN}=\text{Pr}(v_i^N\ge 2g_B·g_i-g_B^2)=\frac{1}{2}+b_N-f_N·(2g_B·g_i-g_B^2)$

is the probability that the non-elite voter i ∈ N votes for party A. The number of non-elite votes that party A expects is as follows:

$\begin{array}{ll}{V}_A^N=n_N·\left(\frac{1}{2}+b_N+f_N{g}_B^2\right)-2·f_N·g_B·{\displaystyle \sum _{i=1}^{n_N}}{g}_i& (3.2)\end{array}$

Munshi (2022, p. 491) shows that to maximize the number of votes, party B announces a policy $g_B^{*}=\overline{g}$ such that $\overline{g}=(g_1+g_2+...+g_{n_N})/n_N$ is the average demand of these non-elite voters for public goods, measured in physical units. Replacing $g_B^{*}=\overline{g}$ in Equation 3.2, the expression is as follows:

$\begin{array}{ll}{V}_A^N=n_N·\left(\frac{1}{2}+b_N-f_N{\overline{g}}^2\right)& (3.3)\end{array}$

Therefore, the number of non-elite voters $V_B^N=n_N-V_A^N$ who vote for party B is

$\begin{array}{ll}{V}_B^N=n_N·\left(\frac{1}{2}-b_N+f_N{\overline{g}}^2\right)& (3.4)\end{array}$

Why does the above lead to an absurd conclusion?

Munshi (2022) never makes clear and explicit assumptions about the behavior of voters' partisan biases, which is a serious methodological error. Indeed, Munshi (2022) does not make any assumptions about the parameters of the distribution functions (Equations 2.1, 2.4), which are behavioral parameters. When Munshi (2022) does not make explicit assumptions about behavioral variables, the reasoning remains open: anything can be said. The following corollary draws an absurd conclusion using the Munshi (2022) model set-up when parameters are added, which builds upon well-established statements of political theory and electoral behavior.

Corollary 1. An Absurd Conclusion—In a liberal democracy without clientelism and without constraints on the decision to vote, the median voter votes for the elite party if

1. they prefer the elite party, b_N/f_N > 0.

2. they do not receive public goods from the elite party, g_A = 0.

3. they do not receive payment for their vote from the elite party, c₁+c₂+...+c_nN = 0.

4. they receive public goods from the non-elite party, g_B > 0.

In a uniform distribution, the mean is equal to its median, and consequently, the median voter is located in the median of the voter distribution, that is, at the point b_N/f_N. If f_N > 0 and b_N > 0, it holds that the median of non-elite voters ideologically prefer party A to party B. From the above, we have that the inequality

$\begin{array}{ll}{V}_A^N=n_N·\left(\frac{1}{2}+b_N-f_N{\overline{g}}^2\right)\ge 0\iff \frac{1}{2f_N}+\frac{b_N}{f_N}\ge {\overline{g}}^2& (3.5)\end{array}$

geometrically corresponds to the case illustrated by Figure 1.

Figure 1

Figure 1. Uniform distribution of partisan preferences among non-elite voters. Source: own elaboration.

The square of the average demand for public goods by non-elite voters is inevitably less than the maximum value of the distribution. This holds true under the condition that f_N → 0, indicating a high variance in partisan biases among non-elite voters, and reflecting the fragmented nature of the non-elite social group in terms of ideology. In contrast, the elite constitutes a small, exclusive, and cohesive group wielding significant political, social, and economic influence. In stark contrast, the non-elite voters exhibit heterogeneity, low social integration, and limited access to centers of power. With minimal political autonomy, the median voter among the non-elite is highly susceptible to the influence of the ruling class (Stokes et al., 2013; Holcombe, 2021). Although the elite itself is not a homogeneous social group in terms of identities and interests, its cooperative processes revolve around a clear and defined axis: the preservation of privilege.

Due to their lack of privileges and the broad diversity within the social group, non-elite voters often lack awareness regarding the benefits of social cooperation, resulting in a fragmented scenario (Schwander, 2019; Wren and McElwain, 2011; Inglehart, 1997). Consequently, the formation of cohesive political networks becomes challenging within this context (Daby, 2021; Graziano, 1976).

The crux of the matter lies in the presence of effective class cohesion, where each individual within the social group enjoys equitable access to scarce resources, both political and economic. This equitable access ensures the stability of the social group. However, given the lack of symmetrical access to resources among the general public, achieving effective cohesion becomes challenging, leading to persistent social disarticulation (Stokes, 2007; Häusermann and Schwander, 2012; Hicken, 2007; Robinson and Verdier, 2003). Consequently, the expression (Equation 3.5) holds true. Furthermore, comparing the expressions (Equations 3.3, 3.4) if f_N → 0 we have that

$\begin{array}{l}{V}_A^N>V_B^N\iff n_N·\left(\frac{1}{2}+b_N-f_N{\overline{g}}^2\right)\\ >n_N·\left(\frac{1}{2}-b_N+f_N{\overline{g}}^2\right)\iff b_N-f_N{\overline{g}}^2>0\end{array}$

This situation arises when the median voter, paradoxically, not only votes for their own oppressor but also strictly prefers them over the political party that represents their interests more effectively. While such scenarios may be conceivable in non-liberal political regimes characterized by high levels of citizen alienation, they are highly implausible within a liberal and democratic institutional framework. Unless explicit institutional constraints are in place that compel such behavior, which is not the case, such outcomes are unlikely to occur.

The non-elite political party (B) is assumed to have broad ideological and identity connections with the majority of non-elite voters; consequently, if political party B provides public goods to non-elite voters, the electoral support of non-elite voters for party B should be expected to be in the majority. On the other hand, although non-elite voters can vote for elite parties, it is not feasible for the elite party to obtain the majority support of non-elite voters in elections without the supply of public goods or payment for votes. Without the supply of public goods or monetary payments, the majority support of the non-elite voters for the elite party is unfeasible and, therefore, rationally absurd. A scenario of the majority support of non-elite voters for an elite party will necessarily be mediated by a strategy of rapprochement through the provision of goods and services (Thachil, 2014a,b).

4 A logical inconsistency

In Propositions 1 and 2 of Munshi's paper we have detected not only a methodological problem in terms of units of measure, but also in the use of ex-post parameters not defined in the model assumptions. Both the problem of units of measurement and the introduction of ex-post variables lead the reasoning toward a logical contradiction. In subsections 1 and 2 we reconstruct the reasoning based on the variables used to formulate propositions 1 and 2 of Munshi (2022). In subsection 4.3, the logical inconsistency that is logically inferred from the propositions of Munshi (2022) is presented.

4.1 Elite voters

Good x is the public good consumed primarily by elite voters. Additionally, party B does not produce goods for elite voters, i.e., x_B = 0. Therefore, given utility function (Equation 2.2) and expression (Equation 2.3), elite voter i votes for party A if

$\begin{array}{l}{v}_i^E\ge x_A^2-2x_A·x_i\end{array}$

such that $v_i^E=v_{iA}^E-v_{iB}^E$. Hence, given the density function (Equation 2.1), it holds that

$\begin{array}{lll}{P}_A^{iE}& =& \text{Pr}(v_i^E\ge x_A^2-2x_A·x_i)=\frac{1}{2}+b_E-f_E·(x_A^2-2x_A·x_i)\end{array}$

is the probability that elite voter i ∈ E will vote for party A such that

$\begin{array}{ll}{V}_A^E=n_E·\left(\frac{1}{2}+b_E-f_E·x_A^2\right)+2·f_E·x_A·{\displaystyle \sum _{i=1}^{n_E}}{x}_i& (4.1)\end{array}$

is the number of votes that party A expects to get from elite voters. The elite party must choose the amount x_A of public goods with which it reaches the maximum number of expected votes such that

$\begin{array}{l}\frac{\partial V_A^E}{\partial x_A}=-2·f_E·n_E·x_A+2f_E·{\displaystyle \sum _{i=1}^{n_E}}{x}_i=0\end{array}$

if and only if, $x_A^{*}=\overline{x}$ such that $\overline{x}=(x_1+x_2+...+x_{n_N})/n_N$ is the average demand of these elite voters for public goods, measured in physical units. In consequence,

$\begin{array}{ll}{V}_A^E=n_E·\left(\frac{1}{2}+b_E+f_E·{\overline{x}}^2\right)& (4.2)\end{array}$

4.2 Non-elite voters

In this set-up, Munshi (2022) assumes that the elite party provides public goods for non-elite voters, i.e., g_A > 0. Given the utility function (Equation 2.5) and expression (Equation 2.6), the non-elite voter i votes for party A if

$\begin{array}{l}{v}_i^N\ge {(g_A-g_i)}^2-{(g_B-g_i)}^2\end{array}$

such that $v_i^N=v_{iA}^N-v_{iB}^N$. Both parties provide amounts of the public good consumed primarily by non-elite voters. Therefore, voter i votes for party A if

$\begin{array}{l}{v}_i^N\ge g_A^2-g_B^2-2·g_i·(g_A-g_B)\end{array}$

Therefore, given the density function (Equation 2.4), we have that

$\begin{array}{l}{P}_A^{iN}=\text{Pr}(v_i^N\ge g_A^2-g_B^2-2·g_i·(g_A-g_B))\\ =\frac{1}{2}+b_N-f_N·\left(g_A^2-g_B^2-2·g_i·\left(g_A-g_B\right)\right)\end{array}$

is the probability that non-elite voter i ∈ N will vote for party A such that

$\begin{array}{ll}{V}_A^N=n_N·\left(\frac{1}{2}+b_N-f_N·\left(g_A^2-g_B^2\right)\right)+2f_N·\left(g_A-g_B\right){\displaystyle \sum _{i=1}^{n_N}}{g}_i& \left(4.3\right)\end{array}$

is the number of votes that party A expects to obtain from non-elite voters. Party A must solve the decision problem about the quantity g_A that maximizes $V_A^N$ such that

$\frac{\partial V_A^N}{\partial g_A}=-2g_A·f_N·n_N+2f_N{\displaystyle {\sum }_{i=1}^{n_N}{g}_i=0}$

if and only if, $g_A^{*}=\overline{g}$ such that $\overline{g}$ is the average demand of these non-elite voters for public goods, measured in physical units. In consequence,

$\begin{array}{ll}{V}_A^N=n_N·\left(\frac{1}{2}+b_N\right)& \left(4.4\right)\end{array}$

Therefore, by adding the quantities (Equations 4.2, 4.4) we have that

$\begin{array}{ll}{V}_A^P=\frac{1}{n_N+n_E}\left(n_E·\left(\frac{1}{2}+b_E+f_E·{\overline{x}}^2\right)+n_N·\left(\frac{1}{2}+b_N\right)\right)& \left(4.5\right)\end{array}$

is the total number of votes that the elite party obtaints from both types of voters if it offers public goods to non-elite voters.

4.2.1 Non-elite voters

Munshi (2022, p. 493) considers another possible world in which party A, instead of promising public goods (g_A) during elections to get votes, buys the votes of non-elite voters. Given the utility function (Equation 2.5), the non-elite voter i votes for party A if

$\begin{array}{l}{v}_{iA}^N+c_i\ge v_{iB}^N+u_i\left(g_B\right)\iff v_i^N\ge g_i^2-{\left(g_B-g_i\right)}^2-c_i\end{array}$

such that $v_i^N=v_{iA}^N-v_{iB}^N$ and the term c_i ∈ ℝ₊is a monetary payment, measured in money units, which the elite party pays to each of its non-elite voters i ∈ N, such that $c=\sum _{i=1}^{n_N}{c}_i$ is total expenses in clientelism of the elite party. The non-elite party (B) is the only one that promises public goods during elections that are primarily consumed by non-elite voters. Therefore, given the density function (Equation 2.4), we have that

$\begin{array}{l}{P}_A^{iN}=\text{Pr}\left(v_i^N\ge g_i^2-{\left(g_B-g_i\right)}^2-c_i\right)=\frac{1}{2}+b_N-f_N·\left(2g_i{g}_B-g_B^2-c_i\right)\\ \end{array}$

is the probability that non-elite voter i ∈ N will vote for party A such that

$\begin{array}{ll}{V}_A^N=n_N·\left(\frac{1}{2}+b_N-f_N{\overline{g}}^2\right)+f_N{\displaystyle \sum _{i=1}^{n_N}}{c}_i& \left(4.6\right)\end{array}$

is the expected number of votes that is maximized by promising a quantity $g^{*}=\overline{g}$ of public goods to non-elite voters. Therefore, adding the quantities (Equations 4.2, 4.6) we have that

$\begin{array}{ll}{V}_A^C=\frac{1}{n_N+n_E}·\left(n_E·\left(\frac{1}{2}+b_E+f_E·{\overline{x}}^2\right)+n_N·\left(\frac{1}{2}+b_N-f_N{\overline{g}}^2\right)+f_N{\displaystyle \sum _{i=1}^{n_N}}{c}_i\right)& \left(4.7\right)\end{array}$

is the total number of votes that the elite party obtaings from both types of voters ir it buys votes from non-elite voters.

4.3 Logical inconsistency

Proposition 1. Munshi (2022, p. 493). $V_A^C>V_A^P$ iff $c>n_N·{\overline{g}}^2$.

The inequality $c>n_N·{\overline{g}}^2$ compares expenses c, measured in money units, with physical quantities $n_N·{\overline{g}}^2$, an expression with scarce theoretical interpretation value. This inequality does not have any methodologically valid interpretation, a situation that cannot be simply solved by future empirical researchers by establishing non-theoretical bridging assumptions in applied work (Davis, 2005). This is because as asserted by Sraffa, measurement in theory requires absolute precision, while statistical measurement is only approximate (Hagemann, 2020; pp 199).

Proposition 2. Munshi (2022, 494). When both clientelism and public goods provision is equally costly, then $V_A^C>V_A^P$ iff $\frac{p}{\overline{g}}>\lambda ·n_N$.

The parameter λ ∈ [0, 1] is the percentage of non-elite voters that having received clientelistic payments from party A, do not vote for this party. The parameter p ∈ ℝ₊ is the price of one unit of public good, such that p·g is the total cost of producing g units of public good (Munshi, 2022: 494). According to Munshi: This can roughly be interpreted (proposition 2) to mean that the cost of $\overline{g}$ is greater than the cost of clientelism (Munshi, 2022: 494). This proposition states that $p/\overline{g}$ is the cost of $\overline{g}$ and λ·n_N is the cost of clientelism. We deduce the inconsistency here in corollary 2 below.

Corollary 2. If $p/\overline{g}$ is the cost of $\overline{g}$ and $\overline{g}>1$ we have a logical inconsistency.

Proof. We will work on the two possible meanings of Munshi expression, because it is not clear from the article if cost is total cost or average cost. Let us reason by contradiction, each one of the cases, and let us assume that there is no logical inconsistency.

Case 1. $p/\overline{g}$ is the average cost of producing $\overline{g}$ units of public good—If $\left(p/\overline{g}\right)·\overline{g}$ is the total cost of producing $\overline{g}$ units and p·1 is the total cost of producing 1 unit then $\overline{g}=1$ and $\overline{g}>1$, which is a contradiction.

Case 2. $p/\overline{g}$ is the total cost of producing $\overline{g}$ units of public good — If the cost of producing a unit of public good is p and $\overline{g}>1$ then the cost of producing $\overline{g}$ units is strictly greater than the total cost of producing one unit, i.e., $p/\overline{g}>p$. Hence, $\overline{g}<1$ and $\overline{g}>1$, which is a contradiction.

Therefore, it has been shown that if $p/\overline{g}$ is the cost of producing $\overline{g}$ units then we arrive at a logical inconsistency.

5 Conclusion

We welcome the addition of Munshi (2022) to the scholarship on clientelism in elections and the lack of public goods provision by parties that win following that strategy. Such a topic and approach are of fundamental importance of understanding the problems of democracies in different countries and cultures. However, we have detected two problems in this article, first, we have revealed a problem of tautological reasoning, the non-elite voters of this model already have a partisan affiliation to the Elite party. Therefore, they do not need to be swayed to it by either expenses in public goods or clientelistic payments by the Elite party, an absurd conclusion; and second, we have demonstrated the incomparability of the units of measurement in clientelism versus public goods and in their corresponding cost valuation. These two features are at the core of Munshi's argument and the narratives of selected cases.

In the first feature discussed, we highlighted that in the expression for the number of non-elite individuals who vote for the Elite party, there is a generalizability problem. The parameters can be chosen in such a way that a higher number of non-elite voters simply prefer the Elite party. Public goods expenses or clientelistic payments are not needed to sway these voters; they were already “affiliated” with the Elite party. In the second, we detected that the expressions requires the impossibility of comparing physical units of public goods against monetary amounts of clientelistic expenses. Furthermore, the definition of the monetary cost of producing public goods is not clear, we do not know if these public goods are purchased from a private provider or produced by the government, in which case we would need to know if the marginal costs are increasing or constant (Rubinstein, 2006; Friedman, 1966). We prove that both a marginal and an average cost interpretations contradict themselves, invalidating the important inference in Munshi's paper that public goods provision is more costly than clientelism.

We do not intend to imply that we do not highly value Munshi's work; on the contrary, we build upon it as a framework for developing new applied theory and informing an empirical research agenda. The article addresses a topic of fundamental importance, and the structure of the model merits further exploration and improvement. Our aim is simply to highlight certain analytical features that, in our view, require deeper analysis to ensure conceptual soundness.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

AC: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. HG: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. JM: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. No hay clientelismo que dure cien años ni municipio que lo resista: política pública territorial (Project code: PRY-153), funded by the Vicerrectoría de Investigaciones y Posgrados, Universidad de Caldas, Manizales, Colombia.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

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