Anticorruption and the Design of Institutions 2012/13
Lecture 3
Optimal Law
Enforcement
Prof. Dr. Johann Graf Lambsdorff
Literature
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 Becker, G.S. (1968), “Crime and Punishment: An Economic Approach,”
Journal of Political Economy, Vol. 76, 169–217.
 Becker, G.S. and G.J. Stigler (1974), “Law enforcement, malfeasance, and
compensation of enforcers,” Journal of Legal Studies, Vol. 3 (1), 1–18.
 Polinsky, M. and S. Shavell (2001), “Corruption and optimal law
enforcement,” Journal of Public Economics, Vol. 81: 1-24.
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Decision tree for a potentially corrupt businessperson
- Penalty
- Bribe
r
Pay bribe
1-r
Corrupt service
- Bribe
Do not pay a
bribe
No corrupt
service
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 The following condition states whether a risk-neutral business person will
pay a bribe:
0 < V-B-rPd  B < V-rPd
with V being the value of the corruptly provided service and r the
probability of detection. B is the value of the bribe. Detection results in
confiscation of the favor and a penalty on the side which demanded it, Pd.
 A high risk of detection, r, or severe penalties, Pd, induce businesspeople
to abstain from paying bribes. If V is sufficiently small, the calculus would
equally lead businesspeople to prefer legality.
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 A similar calculus has been derived for the public servant (see lecture
“The Economics of Corruption”). The public servant will accept a bribe if
S < (1-r)(S+B)-rPs  B > (S+Ps)r/(1-r)
with S being the official salary and Ps the penalty on the supplier of the
corrupt service.
 A bribery transaction is feasible if (S+Ps)r/(1-r) < B < V-rPd, which requires
(S+Ps)r < (1-r)V-r(1-r)Pd.
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 But the government will also recognize the costs it must devote to
detection.
 At the same time, penalties are either costly (imprisonment) or if
costless (fines) confronted with financial constraints of the convict.
 If these costs become too large, a certain level of corruption becomes
unavoidable.
 A stylized representation of this idea was suggested by Klitgaard
(Controlling Corruption: 1991).
 Assume that marginal costs of removing corruption are higher where
there is little corruption. This might be due to increased difficulty of
detection.
 Marginal costs of corruption might have any slope, positive or negative.
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The Costs of Fighting Corruption
Marginal
social
costs
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Marginal
cost of
corruption
Additional social
costs of
eradicating
corruption
Marginal cost of
removing
corruption
Optimum social
costs of
corruption
0
Optimum
quantity of
corruption
Quantity
of Corruption
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Optimal Law Enforcement
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 Polinsky and Shavell (2001) provide a formal treatment of this problem.
They investigate bribe-taking by public servants (called law enforcers in their
study). We focus on their model subsequently.
 A civilian considers acting illegally and thus harming society. If acting as an
offender (a car driver who was speeding or a constructor who procures
substandard quality) he faces detection by a public servant, for example an
inspector, with probability p and imposition of the fine f.
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 Rather than paying the fine, the public servant may be bribed by the
amount b=lf, with l determining the public servant’s bargaining power,
0<l<1. She is not confronted with penalties and risks of detection (but we
will introduce these later).
 The public servant may also falsify evidence and frame an innocent civilian,
forcing him to pay the fine f. The same bribe x=b=lf, must then be paid to
avoid the penalty.
 The probability of being detected as an innocent civilian, qp with 0<q<1, is
lower than for an offender, p, because it is difficult to falsify evidence.
 The civilian now compares the gain from committing the harmful act, g,
with the expected costs. He will prefer to offend if
g>pb-qpx=(1-q)lpf.
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 There results a critical value of the gain, ĝ , below which the civilian will
abstain from offending.
gˆ  1  q  l pf
 Bribery lowers deterrence. In a world without bribery, the full penalty f
rather than the reduced penalty lf applies. With the full penalty, f, a higher
gain to the offender would be required for offending.
 Also the size of the penalty is important. The higher the penalty, the higher
the critical value of the gain, suggesting that an infraction is less frequent.
 We can now determine social welfare.
 We define s(g) as the density of gains among individuals, s(g) is positive on
[0, ∞). We let h be the harm due to the infraction and c(p), c’(p)>0, be the
costs for detecting offenders.
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 Social welfare can be expressed as:

  g  h  s  g  dg  c  p 
ĝ
with gˆ  1  q  l pf
 A first conclusion is that increasing f is always advantageous. Increasing
fines does not increase costs for detection but raises the critical value, ĝ.
 But there will be wealth constraints that limit the size of the fine. Let w0 be
the wealth of the offender. This marks the upper limit of the fine. The
optimal fine is then the maximal possible fine f*=w0.
 The optimal probability of detection, p, is confronted with two
countervailing effects. A higher p is costly, but it also increases ĝ , which
reduces the harm from the infraction.
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 Corruption in the form of bribery or framing is harmful because it inhibits
deterrence, (Becker and Stigler 1974: 5). This result is valid irrespective of
the size of the harm, h.
 Assuming h to be rather small results in optimal enforcement costs to be
low and p small. Once introducing corruption, deterrence would be lower as
compared to a situation without corruption. As a consequence, the optimum
level of enforcement must increase. This increase in enforcement costs runs
counter to public welfare and proves the adverse welfare effect of
corruption.
 We learn from the model, that any government will weigh the costs of
enforcement, considering also the potential corruption of its own public
servants, against the reduction of harmful acts.
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 Another approach to avoiding bribery and framing by public servants
would be to confront this behavior, not just the civilian’s infraction, with
penalties and the risk of detection. This has been suggested by Becker and
Stigler (1974) and implemented in the model by Polinsky and Shavell (2001).
 Assume that the bribe is detected with probability q resulting in the bribe
transaction being undone and the fine fB being imposed on the offender and
the public servant each. The fine f is not collected. For the sake of simplicity
fB is equal for both actors. The public servant is assumed to be endowed
with the same level of wealth, enabling her to pay the same fine. We now
drop the public servant’s option to frame the civilian.
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 Now, an offender who is caught will prefer to pay a bribe only if
(1-q)b+qfB<f.
 The public servant will accept a bribe if
(1-q)b-qfB>0.
 A bribe is feasible only if
qfB/(1-q)<b<(f-qfB)/(1-q)
 A bribe will be paid if qfB<(f-qfB)  2qfB<f, which we assume subsequently.
 The public servant can now achieve the fraction l of the total surplus,
f-2qfB.
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 The bribe must now compensate for the public servant’s expected penalty
q(fB+b) and provide her with the fraction l of the total surplus f-2qfB. We
thus obtain
b = l(f-2qfB)+q(fB+b)  b = [l(f-2qfB)+qfB]/(1-q).
 A civilian will now offend if the probability of detection multiplied by the
costs of the bribe strategy are lower than his gain. The costs of the bribe
strategy are (1-q)b in case of non-detection and qfB in case of detection.
g>p(b(1-q)+qfB)=p[l(f-2qfB)+qfB]+pqfB
=p[lf+2(1-lqfB]
 An increase in fB increases the critical value of ĝ . The optimum penalty is
thus the maximum penalty, fB=f=w0.
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 Social welfare can be expressed as:

  g  h  s  g  dg  c  p   c (q)
with gˆ  p  l f  2 1  l  qf 
gˆ
B
B
 The optimal probability of detection, q, is confronted with two
countervailing effects. A higher q is costly, but it also increases ĝ , which
reduces the harm from the infraction.
 The government has two instruments for deterring the infraction. It can
employ more public servants so as to increase p, or it can employ more
prosecutors so as to increase q.
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Appendix
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Exercise
1) Let the harm h be 1.5 and the distribution function be s(g)=0.5 for all g
with 0≤g ≤2. Determine and solve the integral.
2) Let the harm h be 1 and the distribution function be s(g)=1-g/2 for all g
with 0≤g ≤2.
a) Determine and solve the integral.
b) Let ĝ =pf due an absence of bribery and c(p)=p2. How can you determine
the optimal p?
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Appendix
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3) Let the harm h be € 1500; the gain g that individuals obtain from
committing the harmful act be distributed uniformly between € 0 and €
2000; the enforcement expenditure c required to detect violators with
probability p be € 10 000p2. The wealth of offenders w is € 10 000.
a) Determine the first-best outcome when disregarding enforcement costs!
b) Determine the maximum social welfare considering enforcement costs if
corruption is absent!
c) Determine the maximum social welfare with corruption (bribery and
framing). Let the bargaining power of the public servant l be 0.7 and the
ratio of the probability that an innocent individual could be framed to the
probability that an offender is detected, q, be 0.3.
d) The government can now detect bribery with the probability q and
enforcement costs 5000q2. Let l  0.49 and q = 0. The probability p can be
shown to remain as in question c). Determine the optimal q!
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