The Politics of Drought Relief - Evidence from Southern India Pt 4

Appendix A A.1 Solving the Model For Drought Declaration Allocations Generalizing Voters’ Preferences to Governing Period t During the first governing period t = 1, the ruling party makes declaration allocations knowing the vote share in the second election will be 2 = vcr X 2 > 0). βic P r(σir i The probability of a voter of type i voting for the ruling party candidate in a given governing period t is t+1 P r(σir  Pt+1 s t−1 0  cr + θr (dtc )rct + ζ(vcr ) < −vcr 0 Ps=0 t+1 s t t t−1 0 > 0) = 1 s=0 cr + θr (dc )rc + ζ(vcr ) > 1 − vcr  P  0 s t t t−1 vcr + t+1 s=0 cr + θr (dc )rc + ζ(vcr ) otherwise. In all periods, I assume that the combined effect of the external shocks  and temporary shifts θr (·) and ζ(·) is within the bounds necessary to avoid a corner solution (the third case above). With this assumption, the probability of a voter of type i voting for the ruling party in the election that follows the first governing period is 2 P r(σir > 0) = 1 ϕ̄ir 0 + + θr (d1c )rc1 + ζ(vcr ) + 2cr . 2 δi and the second election vote share can be rewritten as 2 0 0 vcr = vcr + θr (d1c )rc1 + ζ(vcr ) + 2cr . 0 In the absence of declarations (rc1 = 0) and an incumbency advantage/disadvantage (ζ(vcr ) = 0), the expected vote share in the second election is simply the vote share in the prior election. Because the changes in support due to declarations and ruling-party incumbency only last one period while the common shock to voter preferences tcr is permanent, after the second 3 election occurs we have σir = ϕir + δi 2cr in governing period t = 3. Therefore, for any governing period t ∈ {1, 3, 5, ...}, σir can be thought of as voter i’s updated baseline support for the current ruling party r and can be written as t σir = ϕir + δi t X s=0 scr with 0cr = 0 and, again, tcr = 0 for t ∈ {1, 3, 5, ...}. Then, generalizing to any governing period t ∈ {1, 3, 5, ...} the probability of a voter of type i voting for the ruling party in the following election is t t+1 P r(σir 1 ϕ̄ir X s t−1 > 0) = + + cr + θr (dtc )rct + ζ(vcr ) + t+1 cr 2 δi s=0 and the vote share for the ruling party is t+1 vcr = 0 vcr + t X t−1 scr + θr (dtc )rct + ζ(vcr ) + t+1 cr . s=0 Iterating the right-hand-side of the above equation forward, I find t+1 t−1 t−1 vcr =vcr + L(θr0 , ζr0 ) + θr (dtc )rct + ζ(vcr ) + t+1 c , 0 where r was the ruling party during the governing period in t − 2. The function L accounts for the impact of temporary shifts in support for the ruling party due to incumbency and declarations,  0 0  t−3 t−2 L(θr0 , ζr0 ) = [θr0 (dt−2 + ζ(vcr 0 )] × 1{r 6= r } − 1{r = r } . c )rc Because the transitory shift in voters’ preferences due to prior declarations and incumbency is relative to the ruling party r0 , the above equation adjusts for a past positive shift for the 0 t−1 if the ruling party is the same ruling party r having increased (decreased) the vote share vcr (different) in governing period t. A.2 Solving the Ruling Party’s Allocation Decision Imposing that  is within the bounds defining the third case of the above equation for all types i ∈ I and  ∼ Φ(0, ν 2 ), the decision rule for the ruling party is     1 1 t−1 t t−1 t−1 t−1 Φ vcr − + L(θr0 , ζr0 ) + θr (d ) + ζ(vcr ) − Φ vcr − + L(θr0 , ζr0 ) + ζ(vcr ) ≥ η t , 2 2 The ruling party compares the difference in the likelihood of winning the upcoming election with a declaration to the likelihood of winning in the absence of a declaration to determine whether the resulting change in expected electoral success exceeds the marginal cost of a declaration. As described in Section 3, I will restate the above equation in terms of the ruling party’s vote margin, instead of vote share, and impose that  is distributed normally. The decision rule can be rewritten as Z t−1 1 t−1 m +L(θ 0 ,ζ 0 )+θr (dt )+ζ(vcr ) 2 rc r r t−1 1 t−1 m +L(θ 0 ,ζ 0 )+ζ(vcr ) 2 rc r r √ 1 2πν 2 exp{− 2 }d > η t . 2ν 2 In the remaining sections, I consider comparative statics for this decision rule. A.2.1 Parameter for Voter Response to Declarations and Drought I first consider the affect of marginal changes in the parameter value θr (dt ) on the return to a drought declaration. Taking the derivative of the decision rule with respect to θr (dt ), we find √ 1 2πν 2 exp{− t t−1 2 ( 12 mt−1 rc + L(θr0 , ζr0 ) + θr (d ) + ζ(vcr )) } > 0, 2ν 2 so that the marginal return is increasing in the size of the positive preference shift due to a declaration. The assumptions that drought is a negative wealth shock and voters have increasing and concave utility results in voters’ marginal utility from a declaration being greater when a drought has occurred. As stated previously, I then allow this increased marginal utility for drought-affected voters to translate into a more positive shift in preferences for the ruling party, or θr (1) > θr (0). This combined with the above result suggests that the marginal return to a drought declaration is higher for drought-affected areas. Further, because the ruling party will provide declarations to the constituencies with a marginal return above the marginal cost ηt , this indicates that declaration likelihood will be higher in drought-affected areas. A.2.2 Ruling Party’s Vote Margin The ruling party’s vote margin in the election that precedes the governing period is a strong signal for how the ruling party will perform in the next election. Taking the derivative of the marginal return to a declaration with respect to the vote margin results in the equation  t t−1 2 ( 12 mt−1 1 rc + L(θr0 , ζr0 ) + θr (d ) + ζ(vcr )) √ } exp{− 2ν 2 2 2πν 2 t−1 2  ( 12 mt−1 rc + L(θr0 , ζr0 ) + ζ(vcr )) − exp{− } . 2ν 2 t−1 )], which is the This derivative equals 0 at a vote margin m̄ = −[θr (dt ) + 2L(θr0 , ζr0 ) + 2ζ(vcr return-maximizing vote margin for allocating a declaration. Because the marginal return to providing a declaration is single-peaked, the return to allocating a declaration is declining for vote margins larger than or smaller than the return-maximizing margin. In turn, this suggests that the likelihood of receiving a drought declaration is also declining for increasingly higher, or lower, vote margins. According to the model θr > 0 and so imposing no incumbency advantage/disadvantage (ζ = 0) and no lagged declaration impacts, the return-maximizing vote margin will occur at a negative prior vote margin, m̄ = θr (dt ) < 0. This simplified scenario is shown in Figure A.1 in the case that a drought occurs and the case in which no drought occurs in t. In this scenario, for a given magnitude of the vote margin, |mt−1 cr |, the return to allocating a declaration will be higher in non-aligned constituencies where mt−1 cr < 0. Also, as long as the the return maximizing vote margin, or the shift in support due to a declaration, is not too large, the return to a declaration will be decreasing in the absolute value of the ruling party’s prior vote margin (a linear approximation to the relationship shown in the figure). The model imposes that the shift in support is in fact less than 10 percent. A.2.3 Incumbency Advantage/Disadvantage The most decisive impact of the incumbency effect is the discontinuity it creates in the marginal return to a declaration at a vote margin of 0. As shown above in the equation for the ruling party’s decision rule, the probability of winning the election in t + 1 changes discontinuously at 0 due to the fact that in places previously won (mt−1 cr > 0) support for the ruling party is increased by ζ while in places previously lost (mt−1 cr < 0) support is decreased by ζ (with ζ > 0 representing an incumbency advantage). This results in a discontinuity in the marginal return to a declaration, as shown in Figure A.2, that is negative (positive) moving from aligned to non-aligned constituencies in the case of an incumbency advantage (disadvantage) relative to a smooth curve in the case of no incumbency effect. Another way to visualize this discontinuity, as shown in Figure A.2, an incumbency advantage (disadvantage) takes the baseline curve for the marginal return to a declaration and shifts it towards (away from) a vote margin of 0 separately for aligned and non-aligned constituencies. Also shown in the figure, identifying this discontinuity in the marginal return to a drought declaration may not be possible using the empirically observable declaration likelihood. For the scenarios shown in the figure, a constituency with a vote margin close to 0 will receive a declaration if it is aligned or non-aligned (because the ruling party targets competitive constituencies). Therefore, the discontinuity will only be estimated in empirical specifications if there are situations where the marginal cost η t is high enough that aligned areas are favored over non-aligned areas (or vice-versa) even when elections are close; such a scenario can be seen in the figure if we consider a ruling incumbency advantage (e.g., zeta = .1) depicted with a green dashed line and a marginal cost η t > .12, so a horizontal line just above the blue dotted line in the figure. Figure A.2 also shows how the return-maximizing vote margin varies with an incumbency disadvantage, incumbency advantage, and no incumbency effect. Comparing an incumbency advantage to the results with no incumbency effect, the return-maximizing vote margin will occur for a more negative number −[θr (dt ) + 2ζ] in aligned constituencies and a less negative number −[θr (dt ) − 2ζ] (or positive number if 2ζ > θr ) in non-aligned constituencies. An incumbency disadvantage would have the opposite effect. Again, assuming the incumbency effect is not too large in magnitude, the marginal return to a drought declaration will increase in electoral competition. The figure also displays the marginal cost of providing a declaration, η t , which the ruling party considers in its decision rule; again, η t represents the marginal cost in the N th constituency and so it will shift depending upon the shift in the marginal return to a declaration caused by the incumbency effect. In the remaining section, I impose an incumbency advantage and consider the impact of prior declarations and prior incumbency advantage on the relationship between the marginal return to a drought declaration and the ruling party’s vote margin. A.2.4 Accounting for Prior Incumbency Effects As shown in the equation for the ruling party’s decision rule above, the prior incumbency effects shift the value of the marginal return over the distribution of the vote margin and therefore t−1 )]. the return-maximizing vote margin for the ruling party, m̄ = −[θr (dt ) + 2L(θr0 , ζr0 ) + 2ζ(vcr An incumbency advantage following the election t − 3 temporarily increased support for the 0 then ruling party r in election t − 1 for aligned constituencies; therefore, to estimate support 0 for the ruling party r in t the vote share/margin must be adjusted downward if r = r and upward otherwise (with opposite effects for non-aligned constituencies). While here I assume an incumbency advantage, in general the prior incumbency effect depends on: (1) whether an incumbency advantage or disadvantage exists, (2) the party identity of the ruling party in each period, and (3) whether the constituency was aligned after the t − 3 election. These variations, however, can be collapsed into two cases: in some constituencies the ruling party r will have been benefited from the incumbency effect in election t − 1 and in others the current ruling party was disadvantaged. Figure A.3 plots the ruling party’s decision rule, for example parameter values, for the case where there is no prior incumbency effect and the cases where a constituency was advantaged or disadvantaged by a prior incumbency effect (assuming no prior declaration effect). As shown, the inclusion of prior incumbency effects shifts the return-maximizing vote margin along the distribution of prior ruling party vote margins. This prior-incumbency-induced shift reduces the marginal return to providing a declaration for some margins mt−1 cr and increases it for others. Further, the prevailing marginal cost of a declaration ηt , shown in the figure, assumes all constituencies fall into one of the given cases and, given these parameter values, happens to be the same in each case. In reality, however, different constituencies will be either advantaged or disadvantaged by past incumbency effects and so the marginal cost η t will depend on the observed marginal returns to a declaration in the observed constituencies. A.2.5 Accounting for Prior Declaration Effects Next, I assume no prior incumbency effect and focus on the variation induced by prior declaration effects. The effect of a prior declaration depends of course on whether a constituency received a declaration and whether or not the ruling party is the same in both elections. A dec0 laration in t − 2 temporarily increased support for the ruling party r relative to the opposition party in election t − 1; and so, to assess support for the ruling party r in governing period t 0 the vote share/margin needs to be adjusted downwards if r = r and upwards otherwise in constituencies that previously received a declaration. Figure A.4 plots the ruling party’s decision rule for the case where there was no prior declaration and the cases where a prior declaration occurred and either advantaged or disadvantaged the current ruling party in election t − 1 (imposing no prior incumbency effect). Again, the marginal cost to providing a declaration η t assumes all constituencies fall into one of the 3 cases shown and happens to be the same marginal cost in each case for the given set of parameter values. However, the marginal cost will depend on the number of constituencies that did in fact receive a declaration in t − 2 and whether it benefited or disadvantaged the ruling party r in election t − 1. Figure A.1: The Marginal Return to Drought Declaration by Ruling Party’s Vote Margin Note: This figure depicts the marginal return to a drought declaration and marginal cost to a declaration, as described in Equation 2, for different possible values of the ruling party’s (first election) vote margin. For exposition, I fix cr ∼ N (0, .04). This figure assumes no incumbency effects and no shifts in support due to prior incumbency or prior declarations. The plot is split by a vertical line where the ruling party’s vote margin is equal to zero: non-aligned constituencies fall to the left of the line while aligned constituencies fall to the right of the line. The figure plots the marginal return in the case where no drought has occurred (θr (0) = .08, solid black line) and a drought has occurred (θr (1) = .1, dashed blue line). The return-maximizing vote margin, m̄(dt ), in each case is highlighted by a vertical line and point on the curve. The marginal cost to providing a declaration, η t , is shown as a dashed red line. According to the model, areas with a vote margin at which the marginal return to a declaration exceeds the marginal cost will receive a declaration. Figure A.2: The Marginal Return to Drought Declaration with Incumbency Effects Note: This figure plots the marginal return to a drought declaration curve, as described by the left-hand side of Equation 2, for different possible values of the ruling party’s (first election) vote margin; likewise, the figure includes the marginal cost of a declaration (left-hand side of Equation 2) represented by horizontal lines in the figure. This figure assumes θr (1) = θr (0) and no shifts in support due to prior incumbency or prior declarations. For exposition, I consider 2,001 points along the ruling party’s vote margin distribution and fix cr ∼ N (0, .04), and θr = .08, and N t = 800. The plot is split by a vertical line where the ruling party’s vote margin is equal to zero: non-aligned constituencies fall to the left of the line while aligned constituencies fall to the right of the line. The figure plots the marginal return and associated marginal cost in the case with no incumbency advantage/disadvantage (ζ = 0, solid black line), an incumbency advantage (ζ = .1, dashed green line), and an incumbency disadvantage (ζ = −.05, dotted blue line). Figure A.3: The Marginal Return to Drought Declaration with Prior Incumbency Effects Note: This figure plots the marginal return to a drought declaration, as described by the left-hand side of Equation 2, for different possible values of the ruling party’s (first election) vote margin; likewise, the figure includes the marginal cost of a declaration (left-hand side of Equation 2) represented by a horizontal line in the figure. This figure assumes θr (1) = θr (0) and no shifts in support due to prior declarations. For exposition, I consider 2,001 points along the ruling party’s vote margin distribution and fix cr ∼ N (0, .04), θ = .08, ζ = .05 (incumbency advantage), and N t = 800. The plot is split by a vertical line where the ruling party’s vote margin is equal to zero: non-aligned constituencies fall to the left of the line while aligned constituencies fall to the right of the line. The figure plots the marginal return and associated marginal cost in the case with no prior (or lagged) incumbency effect (solid black line), an incumbency effect that benefited the current ruling party in the past election (dashed green line), and an incumbency effect that disadvantaged the current ruling party in the past election (dotted blue line). With the given parameters and assuming all constituencies fall into one of the three cases, the marginal cost happens to be the same in each case and is therefore only plotted once (dashed red line). Figure A.4: The Marginal Return to Drought Declaration with Prior Declaration Effects Note: This figure plots the marginal return to a drought declaration, as described by the left-hand side of Equation 2, for different possible values of the ruling party’s (first election) vote margin; likewise, the figure includes the marginal cost of a declaration (left-hand side of Equation 2) represented by a horizontal line in the figure. This figure assumes θr (1) = θr (0) and no shifts in support due to prior incumbency. For exposition, I consider 2,001 points along the ruling party’s vote margin distribution and fix cr ∼ N (0, .04), θ = .08, ζ = .05 (incumbency advantage), and N t = 800. The plot is split by a vertical line where the ruling party’s vote margin is equal to zero: non-aligned constituencies fall to the left of the line while aligned constituencies fall to the right of the line. The figure plots the marginal return in the case with no prior (or lagged) incumbency effect (solid black line), a declaration that benefited the current ruling party in the past election (dashed green line), and a declaration that disadvantaged the current ruling party in the past election (dotted blue line). With the given parameters and assuming all constituencies fall into one of the three cases, the marginal cost happens to be the same in each case and is therefore only plotted once (dashed red line).