Supplement for In Search of the Holy Grail: Policy Convergence, Experimentation and Economic Performance Sharun W. Mukand and Dani Rodrik

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1 Supplement or In Search o the Holy Grail: Policy Convergence, Experimentation and Economic Perormance Sharun W. Mukand and Dani Rodrik In what ollows we sketch out the proos or the lemma and propositions I and II in the paper. Lemma I: We solve or a ixed-point δ1, such that all countries in the leader cohort with δ δ 1 preer a corrupt policy, while all countries with a δ>δ1 preer to experiment. Consider irst the citizen s voting decision: in period 1, the incumbent is retained with probability π = prob(c i >ψ 1 R)= c ψr c, since c U[0, c]. Now suppose that there exists a δ 1 such that ψ 1 = prob(δ <δ 1 )=G(δ 1 ). This implies that rom the citizen s perspective the probability that the government will be retained is given by the relationship π 1 =1 RG(δ 1) c, such that associated with every δ is a corresponding π 1. Notice that this implies a negative relationship between π and δ. We now turn to the government s problem. Here a government will be indierent between experimentation or corruption i V xpmt = V corr. This latter equality is true i θση + π X r = θδ + π C [(φ 1)R + r]. Simpliying, we obtain π 1 = θδ θση (φ 1)R. Observe that the government s optimization gives a positive relationship between δ and π. For there to be a ixed-point equilibrium, we need a δ1 that simultaneously solves both the government s policy choice rule as well as the citizen s optimal voting rule. Given the continuity o the underlying unctions, since the relationship between δ and π described by the citizen s problem has a negative slope while that given by the government s problem has a positive slope, there exists a unique cut-o δ1. Proposition I: We are interested in inding whether there exist parameters such that our proposed equilibrium is satisied, where or a given history h 1 the government s decision to experiment, imitate or choose a corrupt policy is a unction o (,δ). For expositional convenience, we reproduce the payos to government in ollower cohort (z,δ ) rom imitating (V imit ), experimenting (V xpmt )or choosing a corrupt policy (V corr ), (see also Figure 1) θ L + πi [r] i a A h L (i) V k = θση + π X [r] i a j A h (ii) θδc + πc [(φ 1)R + r] i a i A c. (iii) The opportunity cost o the government s choice o a corrupt policy is increasing in δ. Thereore, in our proposed equilibrium, the probability that an observed policy (conditional on no imitation) is corrupt, is decreasing in δ. The set o all possible period one histories H 1 are divided three ways - the set o period one histories where (a) there is a sole successul leader (n L = 1), i.e. y 1 > ȳ = θδl R>y j, j 1, (b) there are two or more successul leaders, i.e n L, (c) there are no successul leaders, i.e. n L = 0. In what ollows we ocus on (a) and sketch out (b) and (c). 1

2 dv xpmt We now begin by mapping V corr and V xpmt on (V,δ) space. Observe that = r dπx < 0 dv corr and that = θδ +[(φ 1)R + r] dπc < 0. Since the citizen-voter cannot distinguish between a corrupt and an honest policy, we have π X = π C dv xpmt dv corr = π which implies that <. In order to ensure that both the downward sloping curves intersect in the relevant range on (V,δ) space (recollect that δ is bounded between δ L and δ H ), we assume (φ 1)R >θ(δl σ η). This ensures conditions on the end-points o (i) (iii) such that: V xpmt (δ = δ L )=r θση <V corr (δ = δ L )= θδ + r +(φ 1)R <V imit = r. Given the continuity o the payos and the restrictions on the end-points that we have imposed, we have an intersection between the curves V corr and V xpmt in the relevant (V,δ) space - where we denote the point o intersection ( ˆ, ˆδ). Further, notice that V imit ((i) above) is independent o δ and hence can be depicted as a horizontal line. Thereore we are now in a position to graph (i) (iii) on(v,δ) space - as illustrated in Figure 1. We are interested in obtaining the optimal policy choice or a government at z as a unction o (,δ). This is simple to obtain by mapping or each δ [δ L,δ H ] the policy a j corresponding to the V j that constitutes the upper envelope o the payos, i.e. or any δ, a j = argmax a j {V j (a j): a j {A c, A h, A h L }}(see urther below or a caveat). Keeping this in mind, observe that the payo rom imitation decreases as L increases i.e. dv imit /d L = θ L < 0. A successul leader s neighbors are deined as the set o countries whose eective distance is suiciently small such that they would preer to imitate quite irrespective o the payo rom alternative policy choices or all δ, i.e. θ L θδ +(φ 1)R + r. This simpliies to a distance L 1 = δ L 1 θ [(φ 1)R + r]. Thereore, all such neighbors o the leader imitate its policy. On the other extreme consider the cut-o or the ar-periphery. First, observe rom Figure 1 that i distance rom the leader is such that V imit V xpmt (δ = δ H ) (i.e. i θ L + r = θσ η), we either have V cor >V xpmt >V imit or δ<ˆδ or we have V xpmt >V corr >V imit or δ ˆδ. Given the continuity o the payos over the relevant space, this establishes the existence o a cut-o distance L, such that or all δ [δ L,δ H ], governments located beyond this cut-o no longer imitate - giving us the ar-periphery. Countries that lie at some intermediate distance between the neighbors and the ar-periphery (as deined above), constitute the near-periphery. As can be readily observed rom Figure 1, countries located in this region, may choose (as a unction o their δ) imitation, experimentation or corruption as their policy choices. We next check the optimality o the citizen-voter s re-election rule. Here the citizen compares the realized cost o replacing the incumbent c i, with the expected gain in rents i the policy adopted by the government is corrupt. Given the uniorm distribution or c, the re-election probability π is continuous. This is because the citizen s payo u i is decreasing continuosly in the probability that the policy is corrupt since π = prob(c i ψ R). For instance, i a A h L, then rom Bayes rule the

3 citizen-voter knows or sure that the policy is honest - and hence ψ 0 π I = 1. Furthermore since the voter observes the location o the government s policy choice a j, he can make an assessment o the likelihood o whether the policy is corrupt or honest. In particular, i a government who s distance rom the leader is ˆ (the corresponding to ˆδ in igure 1), chooses to not imitate, it will perceived by the voter to have chosen a corrupt policy with probability one. This in turn will imply that π 0 (since c R). Thereore, or a relatively small L, we have V corr >V imit since θδ θ L + r. In contrast, when L is suiciently large the inequality will reverse itsel. Now we sketch out alternative histories with two or more successul leaders, i.e. n L > 1. Without loss o generality, suppose there are two leaders z L1 and z L. There are two possible sub-cases. First, is the possibility that none o the countries in the ollower cohort lies simultaneously within a distance L o both the leaders z Li. In this case, the preceding analysis carries over and z s decision problem is as i there is a single leader. However, we note that L (n L =)> L (n L = 1) since higher δ countries are more likely to imitate, thereby lowering the average δ (and hence π) o those who do no imitate. In diagrammatic terms, this is equivalent to a downward shit o the V corr,v xpmt curves - increasing L. Observe that or any F (z), we can always choose a θ large enough so as to ensure that this downward shit o the V corr and V xpmt is not so large so as to make imitation the dominant strategy or all countries in F (z). Second, is the possibility that z lies between and within a distance L o two or more countries z L1 and z L, i.e. z L1 z L < L. In this case, a ollower country s payo is maximized by imitating the leader with whom it has the least eective distance and there is no experimentation by any o the countries located between these two successul leaders. Finally, i no successul leaders emerged in the leader cohort, then decision making in the ollower cohort is identical to that in the leader cohort. Proposition II: In what ollows we begin by considering the history h 1, with n L = 1. Eicient discipline implies that imitation o a leader enhances private sector national income, either by reducing the costs o corruption or the cost o experimentation. Thereore, or disciplining to be eicient or any country z (δ, ), it is the case that V imit θδ c V corr. The latter is true when, θ il + πi + π C[(φ 1)R + r] L {(δ 1 θ (πc [(φ 1)R + r]+r)}. Now recollect that eicient imitation occurs when L σ η, where σ η <δ L. Thereore, or the inormational externality to cause eicient disciplining, the ollowing two conditions must be met:(a) there exists a non-empty set o countries which are disciplined into imitating a leader or political reasons, but would not do so otherwise, and (b) or a subset o these countries, private-sector income increases due to the reduced corruption that results because o greater disciplining. Now or (a) to hold, parameters must be such that V imit >V corr i.e. i δ + 1 θ {r(1 πc ) π C (φ 1)R} L. On the other hand, or (b) to 3

4 hold, parameters must be such that θ L > θδ R. It is easy to check that both (a) and (b) are satisied or a suiciently low bound on δ L and r. We now demonstrate that there exist parameters in the near-periphery, that display ineicient disciplining. There are two possibilities: (a) countries that would otherwise have preerred to experiment with an honest policy (i.e. those with high δ) are ineiciently disciplined into imitating a leader primarily or political reasons; (b) countries that would have chosen corrupt policies, are disciplined into adopting honest policies, but the loss in imitating a policy o a distant country is greater than the gain due to a reduction in corruption. First, we describe parameters such that (a) is satisied: i.e. V imit V xpmt i ση + r θ (πi π X ) L. Since πi = 1 and π X < 1, we have ineicient imitation or a large enough r (recollect that e <σ η ). (b) Now consider the possibility o ineicient disciplining o corrupt governments. For this to be true, distance L has to be such that V imit V corr, even though θ L < θδ R. To see that parameters exist that satisy these conditions, observe that V imit V corr δ + 1 θ [r(πi π C ) π C (φ 1)R] L. Imposing the above condition implies that a suicient condition is i δ + 1 θ [r(πi π C ) π C (φ 1)R] L >δ + R θ. Simpliying we obtain the ollowing r(π I π C ) >R(1 + p C (φ 1)) where π I = 1 and π C < 1. It is easy to check that this is true or ego rents that are suiciently large - a restriction consistent with others in the paper. In order to show that expected income in the near-periphery is lower than in the ar-periphery, examine Figure 1. From the preceding paragraph, there is ineicient imitation i V imit >V xpmt, even though such imitation results in a decrease in private national income as compared to experimenting, i.e. even though θ < θση. Recollect that imitation is or political reasons (i.e. r) when a government imitates despite the eective distance L being greater than e = σ η. Now suppose that such a government lies at some > e. Observe that or any country located at such a, political considerations ensure that the country ineiciently imitates, i.e. V imit ( ) >V xpmt ( ). Then there is ineicient imitation by all countries with δ [ δ, δ H ]. Observe that all countries with δ in this range but with L > L will witness an increase in income as all governments in this region preer to experiment. Since L >σ η or countries in this region, private national income rom experimentation is greater than imitation. This implies a higher expected income or countries in this region. Finally, it is easy to check that or countries in the ar-periphery, none o the countries are disciplined into enacting honest policies. Those with a high δ, such that V xpmt >V corr, would have chosen honest policies in any case. However, since experimentation is being carried out in this region, some countries will achieve a very high income, while others will do poorly, where the variance in incomes is given by ση. 4

5 Government Payos V imit ( L =0)=r Deadweight loss V cor (δ L ) =-θδ + (φ-1) R+r δ^ V xpm (δ L ) = -θσ + r V corr V xpmt V imit ( ) = -θ σ η -θδ Figure 1: Equilibrium Policy Choices o the Incumbent