Persistent Inequality, Corruption, and Factor Productivity

Elton Dusha

doi:10.1515/bejm-2018-0021

Article

Persistent Inequality, Corruption, and Factor Productivity

Elton Dusha

Published/Copyright: February 5, 2019

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From the journal The B.E. Journal of Macroeconomics Volume 19 Issue 2

Abstract

I build a model with bequests, financial frictions and corrupt bureaucrats to explain the link between corruption and inequality and its effects on productivity. Because of collateral requirements, profits are determined by wealth. If individual wealth is not publicly observed, taxation is regressive under corruption. When wealth inequality is high, corruption is more prevalent, creating persistent feedback between corruption and inequality. I calibrate the model and investigate the effect of corruption on inequality and TFP. Through regressive taxation, corruption induces wealth levels to inversely affect the productivity selection. This in turn has adverse effects on aggregate TFP.

Keywords: corruption; financial frictions; inequality; productivity; size distribution

JEL Classification: E02; E24; H2; O4

Acknowledgments

Author would like to thank Shouyong Shi for his invaluable support and patience. This paper has benefited greatly from comments from two anonymous referees, the editor and discussions with Gustavo Bobonis, Ettore Damiano, Burhan Kuruscu, Ashantha Ranasinghe and Peter Wagner. Author acknowledge financial support from the Institute for Research in Market Imperfections and Public Policy, ICM IS130002, Ministerio de Economia, Fomento y Turismo, the Anillo in Social Sciences and Humanities (project SOC 1402 on “Search models: implications for markets, social interactions and public policy”), Shouyong Shi’s Bank of Canada Fellowship and Canada Research Chair and Fondecyt project number 11140152. All remaining errors are my own.

A Appendix

A.1 A note on payoff curves

As noted above, tax rates not only affect the net profit of the entrepreneur, they also affect the entry decision. Denote by q(τ(a))=P[z>zmin(τ(a)] the probability that an individual operates given some tax rate τ(a). The utility of an individual that pays the bribe is:^[25]

U={δ{(1−τ(a))π~(z,zmin(τ))(a−b(a))+w+(a−b(a))}w/pq(τ(a))δ{(1+r)(a−b(a))+w}w/p1−q(τ(a))

If the individual decides not to participate, the utility is:

U={δ{(1−ϕ)π~(z,zmin(ϕ))a+w+a}w/pq(ϕ)δ{(1+r)a+w}w/p1−q(ϕ)

where π~(z,zmin(τ))=ληE(z1/α|z>zmin(τ(a)))−r(λ−1) and τl≤τ(a)≤ϕ. Denote by G(τ) the function:

G(τ)=q(τ){(1−τ)[ληE(z1/α|z>z(τ))−r(λ−1)]+1}+(1−q(τ))(1+r)

Using the methodology of Maskin and Riley (1984), note that expected utility is given by U=G(τ(a))[a−R(a)−θ(a,τ(a))]. Total differentiation gives dU=∂U∂τ+∂U∂R∂R∂τ=0 to get ∂R∂τ|dU=0=−∂U/∂τ∂U/∂R=(a−R(a)−θ(a,τ))G′−θτG. Given that θ(⋅,ϕ)=0 we have ∂R∂τ|τ=ϕ<0. By monotonicity of G and θ we know that ∂R∂τ is not always positive for all τ. Figure 12 is an illustration of a possible set of payoff curves for a given type a. It could be that ∂R∂τ is positive for all of the curves, or for none of them, in this case I have illustrated some that have a positive slope for some interval. Consider a case for a type a. Take two different utility levels U₁ and U₂, where U₂ > U₁. At each tax level, R₁ > R₂, which implies that the curve for U₂ lies entirely under that for U₁ for type a. Furthermore since R₁ > R₂ then we have that ∂R∂τ|π=π2<∂R∂τ|π=π1 for all τ, which gives the curves the shape in the figure below.

$Figure 12: Indifference curves for an arbitrary type a. $U_{1} < U_{2} < U_{3}$U1<U2<U3.$

Figure 12:

Indifference curves for an arbitrary type a. U1<U2<U3.

Denote by R~(a) the revenue level extracted from type a such that it gives this type the outside option at τ=τl so that G(τl)[a−R~(a)−θ(a,τl)]=G(ϕ)a. Rearranging we get

R~(a)=a(1−G(ϕ)G(τl))−θ(a,τl)

Given the above discussion, if ∂R~∂τ|τ=τl<0 for some type a, we have that ∂R∂τ<0 for all utility levels for that type, i.e. the payoff curves are strictly downward sloping. Note that ∂R~∂τ|τ=τl=(a−R~(a)−θ(a,τl))G′(τl)=G(ϕ)G(τl)aG′(τl)<0 where the last equality comes from the definition of R~. Since this result is not dependent on a, it implies that payoff curves are strictly downward sloping for all types.

Consider a case with two types, a^h and a^l where ah>al. The combinations of R and τ that give the high type the outside option (the outside option curve) lie strictly above those that give the low type the outside option if and only if κ(τ)>θa(a,τ) for all τ∈[τlϕ] where κ(τ)=1−G(ϕ)G(τ). To see why, note that the curve for type i is described by Ri=κ(τ)ai−θ(ai,τ). We need Rh−Rl>0, which implies that κ(τ)>θ(ah,τ)−θ(al,τ)ah−al. Taking limits we get the requirement κ(τ)>θa(a,τ) which sets up the following assumption:

Assumption 2

κ(τ)>θa(a,τ) for all τ and a.

Figure 13 depicts the outside option curves for the lowest and highest type.

$Figure 13: Outside option indifference curves for two types: $a^{h} > a^{l}$ah>al.$

Figure 13:

Outside option indifference curves for two types: ah>al.

A.2 Proofs

Proof.

The dependence of the function G on τ is essential for the results that follow, therefore it is useful to establish the following result ∂G∂τ<0.

G(τ) can be written as

G(τ)=(1−F(zmin(τ)))[λη(1−τ){E[z|z>zmin(τ(a))]−r}−rτ]+(1+r)=(1−F(zmin(τ)))[η(1−τ)E[z|z>zmin(τ(a))]−r]−r(1−F(zmin(τ)))τ+(1+r)

The derivative of the second part is obviously negative, so we just need to show that the first part is monotonely decreasing in τ

First we find the conditional distribution of z. Pr[z<x|z>zmin(τ)]=Pr[zmin(τ)<z<x](1−F(zmin(τ)))=F(x)−F(zmin(τ))(1−F(zmin(τ))) and the pdf is: f(z)(1−F(zmin(τ))). So that E[zi|z>zmin(τ(a))]=1(1−F(zmin(τ)))∫zmin(τ)z¯zf(z)∂z. Rearranging the terms we get:

G(τ)=η(1−τ)∫zmin(τ)z¯zf(z)∂z−r+rF(zmin(τ)).

∂∂τ∫zmin(τ)z¯zf(z)∂z=−zmin(τ)f(zmin(τ))∂zmin(τ)∂τ by the Leibniz rule.

Taking derivatives we have

∂G(zmin(τ))∂τ=−η∫zmin(τ)z¯zf(z)∂z−η(1−τ)zmin(τ)f(zmin(τ))∂zmin(τ)∂τ+rf(zmin(τ))∂zmin(τ)∂τ=−η∫zmin(τ)z¯zf(z)∂z+f(zmin(τ))∂zmin(τ)∂τ[r−η(1−τ)zmin(τ)]=−η∫zmin(τ)z¯zf(z)∂z<0

a) The proof is similar to Maskin and Riley (1984).

b) Consider a sequence of wealth types where: a_<a2<...<a¯. Here we show that <τ(ai), R(ai)>∼ai<τ(ai−1), R(ai−1)>. Suppose that this is not true, then we have that <τ(ai), R(ai)>≻ai<τ(ai−1), R(ai−1)>. Since <τ(ai−1), R(ai−1)> is optimal for type i − 1. This then implies <τ(ai−1), R(ai−1)>≽ai−1<τ(ak), R(ak)> for all k ≤ i − 1 by the assumption above and the first part of the proof. This then implies;

<τ(ai−1), R(ai−1)>≻ai<τ(ak), R(ak)> for all k ≤ i − 1. Now consider a scheme that keeps the same τ(ai) for all i but increases R to R~=R+δ for all k ⩾ i. For δ small enough, <τ(ai−1), R(ai−1)>≺ai<τ(ak), R~(ak)>. Therefore there exists another contract that gives the bureaucrat higher wealth. This is a contradiction. Therefore:

<τ(ai), R(ai)>∼ai<τ(ai−1), R(ai−1)>. By this line of argument, the lowest level a_ gets no surplus so that <τ(a_), R(a_)>∼a_<ϕ,0> which gives us b.

c) We need to show that the optimal menu for the bureaucrat to offer is: <R~(a∗), τl>, <0,ϕ> where a∗=arg⁡maxa(1−H(a))(κa−θ(a,τl)) and R~(a)=κa−θ(a,τl) is the revenue that gives type a the outside option at τ=τl, and κ(τl)≡1−G(ϕ)G(τl). We do this in three steps:

Step 1: Show that τ∈{τl,ϕ} in any optimal contract. Suppose that there exists an optimal menu that offers a contract <R(a),τ(a)> where τ∈(τlϕ) for some types a∈[a_a¯]. Consider first the contract offered to type ā. Given the fact that this is the highest type and increasing revenue does not affect the (2) and (1) constraints of types lower than ā, τ(a¯)=τl. Now consider the contracts offered to the next two lower types, a_k and a_k−1. Suppose that τ(ak),τ(ak−1)∈(τlϕ). Note that given the result in part a, τ(ak)∈[τl,τ(ak−1)]. Consider a small ε increase in τ(a_k) such that ε∈ϵ where ϵ is such that ∂R∂τ|dU=0 is constant for all ε ∈ ϵ. Since the (1) and (2) constraints for all a < a_k are unaffected by this small change in τ(a_k), the only revenue that is affected is that of type a_k and ā. Denote by Δai the change in revenue for type a_i. Given that ∂R∂τ|dU=0<0 we have that Δa¯>0 and Δak<0. Now since <R(a),τ(a)> is optimal it must be that this increase in τ(a_k) results in an aggregate loss in revenue so that |Δa¯|<|Δak|. If that is the case, given that Δa¯ is constant and that the indifference curves are strictly downwards sloping, ∃ε∈ϵ such that |Δa¯|>|Δak| if we reduce τ(ak) by ε. Therefore τ(ak)∈{τl,τ(ak−1)}. If that is the case, one can make the same argument to show that τ(ak−1)∈{τl,τ(ak−2)} and so on until the last type that is included in the contract â. Since â will receive his outside option, it is easy then to see that τ(a^)∈{τl,ϕ} which implies the desired result.

From now on, when we refer to a contract, it is implied that τ=τl whenever R ≠ 0.

Step 2: Show that for any optimal contract R(a)∈[R~(a_) R~(a¯)] ∀a∈[a_ a¯] where R~(a) denotes the revenue that gives type a the outside option at τ=τl. (R~(a)=κa−θ(a,τl)). Suppose that R(a)<R~(a_) for some a>a_, then setting R(a)=R~(a_) would still violate neither (2) or (1) and increase revenue so that R(a)<R~(a_) is not optimal.

On the other side of the segment, suppose R(a)>R~(a¯) for some a_<a<a¯. then R(a) = 0 given that it violates the (1) condition for type a. By setting R(a)=R~(a¯) the bureaucrat can increase revenue so that R(a)>R~(a¯) is never optimal.

Step 3: Here we finally prove the main result.

Suppose that there exists another menu <R(a),τl> where a∈A⊆[a_ a¯] that maximizes revenue for the bureaucrat, where R(a)∈[R~(amin) R~(amax)] as per step 2 where amin=minA and amax=maxA. Also R(a)≠R(a∗) ∀a∈A where a^∗ is as defined in the proposition. Denote by Ri≡min(R(a)) for all a ∈ A. Then by the monotonicity of θ there exists an a~ such that Ri=R~(a~). Take any a^⩾a~. The payoff to this type is given by

G(τl)(a^−R(a^)−θ(a^)), while the payoff given by R~(a~) is G(τl)(a~−R~(a~)−θ(a~)). Since a~≤a^ then θ(a~)⩽θ(a^) and R(a^)⩾R(a~), which implies that R(a^) violates the (2) condition for type â. This in turn implies that the optimal contract choice for type â is R~(a~). Since the choice of â was arbitrary, this is true for all a⩾a~. In that case the revenue to the bureaucrat is: (1−H(a~|amin≤a≤amax))(κa~−θ(a~,τl)) where a ∈ A. However, since we assumed that R(a)≠R(a∗) then it is clear that the bureaucrat can increase revenue by offering R(a∗) to all a⩾a∗ which proves the result.

d) Note that the FOC is: −h(a)(κ(τl)a−θ(a,τl))+(1−H(a))(κ(τl)−θa(a,τl))=0. For an interior solution to exist, given the monotonicity of θ, the FOC must be positive at the lower bound a_, which gives us the desired result. If this condition fails to hold then the solution is a=a_.

e) Suppose ρ is increasing in a. Consider a mean preserving spread with cdf H~(a)=H(a+ξ) where ξ has mean zero. Denote by a~=arg⁡maxa(1−H~(a))(κa−θ(a,τl)). Suppose a~<a^, then it must be that f′(a~)=f′(a^) given that f′′=0. Also, since f′>0 then f(a~)<f(a^). By the FOC, we have that 0=f′(a~)−ρ(a~)f(a~)=f′(a^)−ρ(a^)f(a^)⇒ρ(a^)f(a^)=ρ(a~)f(a~) which implies that ρ(a~)>ρ(a^) which is a contradiction. □

A.2.1 Aggregation

Aggregation without corruption:

The aggregate labour demand equation is:

Ld=λ((1−α)w)1/αE(z1/α|z>zmin(τ))E(a)(1−F(zmin(τ)))

which, together with a normalized labour supply L^s = 1 implies:

w=(1−α)(λE(a)∫zmin(τ)z¯z1/αf(z)∂z)α

Note that ∂w∂zmin<0; a higher entry threshold reduces labour demand at the extensive margin while leaving the intensive margin unaltered.

Each entrepreneur’s entry decision depends on the aggregate state through the wage rate w. In order for aggregate outcomes to be consistent with individual decisions, it must be that each individual facing the threshold z_min results in an aggregate threshold that is consistent. More precisely, the existence of such a threshold requires that the equation

zmin=φ(zmin)=[rα(1+τλ(1−τ))(λE(a)∫zminz¯z1/αf(z)∂z)1−α]α

has a fixed point.^[26] It is relatively straightforward to show that φ′<0. Taking the limits of the function as z_min approaches the boundaries, we get

limzmin→z_φ(z)=(rα(1+τλ(1−τ))[λE(a)E(z1/α)]1−α)α>0 and limzmin→z¯φ(z)=0 which guarantees the existence of a fixed point.

Given the above setup, wealth levels evolve according to:

at+1={(1−γ)[(1−τ)π(at,zt)+wt+at]if zt≥zmint(1−γ)[wt+(1+r)at]else

Aggregation with corruption:

Aggregate demand is given by:

Ld=λ((1−α)w)1/α{[1−F(zmin(τl))]E[z1/α|z>zmin(τl)]E[a−κ(τl)a∗|a>a∗][1−H(a∗)]+E[z1/α|z>zmin(ϕ)]E[a|a<a∗]H(a∗)[1−F(zmin(ϕ))]}

where the first part of the expression is the aggregate labour demand of those who are above the threshold a^∗ and are therefore able to lower their tax rates. After some transformation:

Ld=λ((1−α)w)1/α[I(z1/α,z(τl),z¯)I(a−κ(τ)a∗,a∗,a¯)+I(z1/α,z(ϕ),z¯)I(a_,a∗)]

where I(y1,y2,y3)=∫y2y3y1g(x)∂x and g(⋅) is the appropriate pdf. The labour demand equation implies the following equation for wages and thresholds in the aggregate:

w=(1−α)[λI(z1/α,z(τl),z¯)I(a−κ(τ)a∗,a∗,a¯)+λI(z1/α,z(ϕ),z¯)I(a_,a∗)]αzmin(τl)=rτlλ(1−τl)(rα[λI(z1/α,zmin(τl),z¯)I(a−κ(τ)a∗,a∗,a¯)+λI(z1/α,zmin(ϕ),z¯)I(amin,a∗)]1−α)αzmin(ϕ)=rϕλ(1−ϕ)(rα[λI(z1/α,zmin(τl),z¯)I(a−κ(τ)a∗,a∗,a¯)+λI(z1/α,zmin(ϕ),z¯)I(amin,a∗)]1−α)α

A similar argument to the one made in the case without corruption establishes the existence of a fixed point for both zmin(τl) and zmin(ϕ).

A.2.2 Notes on calibration

The calibrated economy (Sweden) has a probability of being caught of 1. Therefore calibration for Sweden simply means reproducing the model without corruption in order to calibrate the economy to the parameters in Table 1. Note that for this step the only calibrated parameter is the within generation productivity variance, for which I use the percentage of entrepreneurship as reported by GEM. This is simply the calibration of the economy in Section 2.1 where τ = ϕ. The steps involved are the following:

Simulate a Pareto wealth distribution (the initial distribution is not important)
Discretize through Tauchen an AR(1) process with st. dev σ and persistent parameter ρ (σ to be calibrated, ρ is taken from the data)
Wage w here is endogenous and has to be found through the following process:
1. the threshold is:
  zmin(ϕ)=(rη(1+ϕλ(1−ϕ)))α.
  where η=α(1−αw)1−αα, ϕ, r, λ and α are given in Table 1.
2. Guess a wage w which gives η which gives zmin(ϕ).
3. One then can compute E(z1/α|z>zmin(ϕ)) and F(zmin(ϕ)).
4. then compute L^d:
  Ld=λ((1−α)w)1/αE(z1/α|z>zmin(ϕ))E(a)(1−F(zmin(ϕ)))
5. calculate a new wage that clears the labour market where supply is normalized to 1.
6. repeat until the difference between the guess and the outcome is equal.
The calibrated parameter here is σ. In order to find it I target the percentage of entrepreneurship in the population (Table 1).
Repeat the exercise above for a large number of periods until you get stationary distributions for wealth and productivity.

The calibration question is: If Sweden were a more corrupt economy, how would its key variables of interest respond? In order to do this the task is to first find the wealth threshold for each period given a cost function θ. Therefore the first task is to pick a probability of being caught and calibrate the cost function. Thus the algorithm for computing the steady state is:

Pick the probability of being caught that corresponds to the relative position in the corruption index.
For each period t compute the variables of interest (a^∗, τ_l, bribe etc). Note that the steps of finding the wage here include the wealth threshold so the steps 3(a–f) above are:
1. Guess a wage w which gives η which gives:
  zmin(ϕ)=(rη(1+ϕλ(1−ϕ)))αzmin(τl)=(rη(1+τlλ(1−τl)))α
2. One then can compute E(z1/α|z>zmin(ϕ)) and F(zmin(ϕ)) and E(z1/α|z>zmin(τl)) and F(zmin(τl)).
3. Labour demand here is:
  Ld=λ((1−α)w)1/α∗{[1−F(zmin(τl))]E[z1/α|z>zmin(τl)]E[a−κ(τl)a∗|a>a∗][1−H(a∗)]+E[z1/α|z>zmin(ϕ)]E[a|a<a∗]H(a∗)[1−F(zmin(ϕ))]}
4. calculate a new wage that clears the labour market where supply is normalized to 1.
5. repeat until the difference between the guess and the outcome is equal.
Since that the probability of being caught in the model is given by:
θ(a∗,τ)=ωb(a∗)
and given that a^∗ is computed for each period, one can calculate the v₁ that corresponds to the desired probability. (note that the cost function is θ(a,τ)=(ϕ−τ)2(v1a+v2)
Repeat steps 1–3 for different positions in the corruption index. These are reported in the calibration section.

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Published Online: 2019-02-05

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Articles in the same Issue

https://doi.org/10.1515/bejm-2018-0021

Keywords for this article

corruption; financial frictions; inequality; productivity; size distribution