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The incumbent’s preference for imperfect commitment


Using a model with forward-looking voting strategies, we examine the tax policies of public officials who maximize the weighted average of rents and benefits to their specific electoral clienteles when commitment is possible. We assume that the degree of commitment to a tax policy can be varied through its design and institutional anchoring. At the center of the analysis lies the question of the extent to which public officials restrict the policy space of future governments. On the one hand, stronger restrictions make it more difficult for political opponents to enact unwanted policy changes, but, on the other hand, they also reduce the likelihood of reelection. We show that incumbents prefer perfect commitment to the absence of any commitment and that, from the point of view of an incumbent, imperfect commitment can be superior to perfect commitment. Imperfect commitment allows incumbents to raise their reelection chances either by binding themselves and causing the opponent to deviate or binding the opponent and deviating themselves.

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  1. Self-commitment also allows politicians to avoid their own inconsistent behavior triggered by non-exponential discounting (Laibson 1997).

  2. We assume quasi-fixed deviation costs that do not depend on the extent of deviation. For example, the transaction costs of changing the law can be considered to be fixed.

  3. Our voting model combines features taken from various other models: the citizen-candidate model introduced by Osborne and Slivinski (1996) and Besley and Coate (1997), the modified Leviathan model of Edwards and Keen (1996) and the probabilistic-voting model (e.g., Persson and Tabellini 2000).

  4. To exclude some corner solutions, we assume that the Inada condition \(\lim _{x\rightarrow 0}u'(x)=\infty\) holds and that the weight politicians attach to voters’ utility is not too low, i.e., \((1-\delta )z'[r(\bar{t},\bar{t})]n/2-\delta u'(y-\bar{t})<0\).

  5. In our numerical example, the optimum tax rate is \(\tilde{t}_{i}^{i}=\max \{0,(1-\delta )y-\delta \bar{t}\}\), \(i=1,2\).

  6. For simplicity, we assume that deviation costs enter utility additively.

  7. Because politicians themselves cannot commit in the election to certain tax rates and because in the perfect symmetric setting \(EV_{1}(t_{1},t_{2},C)=EV_{2}(t_{2},t_{1},C)\), the chosen tax rates at the constitutional level are symmetrical, i.e., \(\hat{t}_{1}^{1}=\hat{t}_{2}^{2}\) and \(\hat{t}_{2}^{1}=\hat{t}_{1}^{2}\), which justifies our assumption of a random choice of one of the two candidates.


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The comments from the editor and two anonymous referees of this journal and from Andreas Mense are greatly appreciated.

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Correspondence to Matthias Wrede.

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1.1 Costs of implementing a commitment

If we introduce the costs of implementing a commitment at the constitutional level that are positively associated with the deviation costs, \(\chi (C)\), with \(\chi '>0\) and \(\chi ''\ge 0\), commitment becomes less attractive. Hence, Proposition 1 no longer holds. At the constitutional level, the ruling politician prefers no deviation costs to infinitely high deviation costs. Even finite, but large deviation costs may be inferior to no deviation costs. Compared to perfect commitment, imperfect commitment is even more preferable. Furthermore, the lowest value of deviation costs that makes at least one politician deviate is most likely the preferred choice for the incumbent.

1.2 Imperfect commitment with asymmetry in size of voter groups

We introduce asymmetry in size, \(n_{1}\ne n_{2}\), which implies asymmetry in policy choices at both stages of the game. We leave the numerical example as is, but set \(n_{i}=n/2+1\) and \(n_{j}=n/2-1\), \(j\ne i\). Figures 3 and 4 show the reelection probability and the expected utility of the politician in charge at the constitutional level if the politician is member of the larger or smaller group, respectively. A comparison of Figs. 3 and 4 demonstrates that asymmetry in group size has different effects on candidates of the larger and smaller voter group. Interestingly, the politician representing the smaller electorate faces a higher election probability than the opponent if both politicians deviate because he or she can skim off fewer rents from his or her own supporters and faces, therefore, weaker incentives to increase the tax rate for his or her constituency. Regardless, at the given level of deviation costs, both the larger and smaller party set tax rates such that the opponent, but not the incumbent, will deviate. Furthermore, in the numerical example, the politician associated with the larger group prefers imperfect commitment to perfect commitment. For a full analysis of the decisions at all levels, including the selection of politicians, under asymmetrical conditions, we have to modify our assumption of a random choice of the ruling politician at the constitutional level; however, that topic is reserved for future studies.

Fig. 3
figure 3

Election probability and expected utility of Politician 1 at the constitutional level if \(n_1>n_2\)

Fig. 4
figure 4

Election probability and expected utility of Politician 1 at the constitutional level if \(n_1<n_2\)

1.3 Imperfect commitment with common interest and uncertainty

We introduce a public good, G, and uncertainty about its costs. The well-behaved utility function of voter i is \(u(y_{i}-t_{i},G)\) and the rent net of public expenditure is \(r(t_{1},t_{2},G,\kappa _{k})=n_{1}t_{1}+n_{2}t_{2}-\kappa _{k}G\), with \(\kappa _{k}\), \(k=l,h\) indicating the per-unit cost of public-good production, where \(\kappa _{l}<\kappa _{h}\). For simplicity, we assume an additively separable voter utility function \(u(y_{i}-t_{i},G)\). We modify the politician’s utility accordingly. The probability of the costs being equal to \(\kappa _{k}\) is \(q_{k}\), \(i=l,h\). Public good costs are revealed at the beginning of the public policy level, such that voters and politicians know the true costs. To account for the fact that uncertainty requires some flexibility, we assume that the incumbent can commit to tax rates only, but not to public good provision. The public good is the residual parameter that always is determined by the politician at the policy level. Hence, for given tax rates \(t_{1}\) and \(t_{2}\) at the policy level, the politician maximizes his or her utility by solving

$$\begin{aligned} \max _{G}v_{i}(t_{1},t_{2},G,\kappa _{k},\alpha _{i})=(1-\delta _{i})z[r(t_{1},t_{2},G,\kappa _{k})]\alpha _{i}+\delta _{i}u(y_{i}-t_{i},G),\quad i=1,2\,. \end{aligned}$$

The first-order condition for an interior solution for \(\alpha _{i}=1\),

$$\begin{aligned} -(1-\delta )\kappa _{k}\,z'[r(t_{1},t_{2},G,\kappa _{k})]+\delta \partial u(y-t_{i},G)/\partial G=0,\quad i=1,2\,, \end{aligned}$$

determines \(G=G_{i}(t_{1},t_{2},\kappa _{k})\) with

$$\begin{aligned}&\frac{\partial G_{i}}{\partial \kappa _{k}}=\frac{z'(1-\delta )-z''(1-\delta )\kappa _{k}G_{i}}{z''(1-\delta )\kappa _{k}^{2}+\delta (\partial ^{2}u/\partial G_{i}^{2})}<0\,,\\&\frac{\partial G_{i}}{\partial t_{i}}=\frac{\partial G_{i}}{\partial t_{j}}=\frac{z''(1-\delta )\kappa _{k}n/2}{z''(1-\delta )\kappa _{k}^{2}+\delta (\partial ^{2}u/\partial G_{i}^{2})}>0 \end{aligned}$$

for \(n_{i}=n_{j}=n/2\). If the politician determines the tax rates at the policy level, he or she will tax the opponent’s electorate prohibitively. We also assume, as before, that interest in the own supporting group is sufficiently strong such that \(0\le \tilde{t}_{i}^{i}<\tilde{t}_{i}^{j}=\bar{t}\).

At the constitutional level, incumbent i determines taxes by solving

$$\begin{aligned} \max _{t_{1}^{i},t_{2}^{i}}\,\, EV_{i}(t_{1}^{i},t_{2}^{i},C)&:=\sum _{k=l,h}q_{k}\left\{ p_{i}(t_{1}^{i},t_{2}^{i},\kappa _{k},C) \left[ v_{i}({t_{1}^{i}}^{*},{t_{2}^{i}}^{*},G_{i} ({t_{1}^{i}}^{*},{t_{2}^{i}}^{*},\kappa _{k}),\kappa _{k},1)-C^{*}\right] +\right. \\&\quad \left. [1-p_{i}(t_{1}^{i},t_{2}^{i},\kappa _{k},C)]v_{i}({t_{1}^{j}}^{*},{t_{2}^{j}}^{*},G_{j}({t_{1}^{j}}^{*},{t_{2}^{j}}^{*},\kappa _{k}),\kappa _{k},0)\right\} \,, \end{aligned}$$

where \({t_{l}^{m}}^{*}={t_{l}^{m}}^{*}({t_{1}^{i}},{t_{2}^{i}},\kappa _{k},C)\) and \(C^{*}=C^{*}({t_{1}^{i}},{t_{2}^{i}},\kappa _{k},C)\), \(i,l,m=1,2\).

If deviation costs are zero, \({t_{j}^{i}}^{*}=\tilde{t}_{j}^{i}(\kappa _{k})\), \(C^{*}=0\), and \(p_{i}(t_{1}^{i},t_{2}^{i},\kappa _{k},C)=1/2\), \(i,j=1,2\), \(k=l,h\). If deviation costs are prohibitively high, \({t_{j}^{i}}^{*}=t_{j}^{i}\), \(C^{*}=0\) and \(p_{i}(t_{1}^{i},t_{2}^{i},\kappa _{k},C)=1/2\), \(i,j=1,2\). Hence, Proposition 1 may not hold if \(\tilde{t}_{j}^{i}(\kappa _{l})\ne \tilde{t}_{j}^{i}(\kappa _{h})\); otherwise, Proposition 1 still holds. If tax rates determined at the policy level vary with the public good’s costs, the incumbent may benefit considerably from flexibility, but otherwise he or she still prefers prohibitively high deviation costs (perfect commitment) to no deviation costs (no commitment). Intermediate deviation costs allow incumbents to increase their reelection chances by either binding themselves and causing the opponent to deviate or binding the opponent and deviating themselves. Hence, as in the benchmark model, from the incumbent’s point of view, imperfect commitment can be superior to perfect commitment.

To demonstrate the possible outcomes, we again use a numerical example. The voters’ utility function is \(u=\gamma \ln (y_{i}-t_{i})+(1-\gamma )\ln G\), which implies that the politician in power determines the public good’s supply such that \(G(t_{1},t_{2},\kappa _{k})=n(t_{1}+t_{2})(1-\gamma )\delta /[2(1-\gamma \delta )\kappa _{k}]\). If the politician deviates at the policy level, he or she chooses tax rates \(\tilde{t}_{i}^{i}=\max [0,y(1-\gamma \delta )-\bar{t}\gamma \delta ]\) and \(\tilde{t}_{j}^{i}=\bar{t}\), \(i=1,2\) and \(j\ne i\). The parameter values in the numerical example are \(n=5\), \(y=2\), \(\gamma =0.6\), \(C=0.15\), \(\kappa _{l}=0.75\), \(\kappa _{h}=1.25\) and \(q_{l}=1-q_{h}=0.5\). Figure 5 shows the expected utility of the politician in charge at the constitutional level for \(\delta =0.2\) and \(\delta =0.8\), respectively. In any case, the incumbent sets the tax rates that induce only one politician to deviate.

Fig. 5
figure 5

Expected utility of Politician 1 at the constitutional level for \(\delta =0.2\) (top) and \(\delta =0.8\) (bottom)

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Wrede, M. The incumbent’s preference for imperfect commitment. Public Choice 180, 285–300 (2019).

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