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\begin{document}

\title{Online Appendix for ``The Distributive Effects of Corruption''}


\author{Mircea Popa}

\maketitle
\textbf{Proof of proposition 1}: Write the value functions of the
government in recursive form (the Bellman equations) for the equilibrium
path of the cooperative situation described by the strategies above,
as well as for the best one-stage deviation ($OSD$) by the government:
\begin{eqnarray}
V_{COOP}^{g}(M_{t,i}) & = & R_{t,i}-c+\delta\left[(1-p)E\left[V_{COOP}^{g}(M_{t+1,o})\right]+pE\left[V_{COOP}^{g}(M_{t+1,i\prime})\right]\right]\\
V_{OSD}^{g}(M_{t,i}) & = & R_{t,i}+\delta\left[(1-p)E\left[V_{PUN}^{g}(M_{t+1,o})\right]+pE\left[V_{PUN}^{g}(M_{t+1,i\prime})\right]\right]
\end{eqnarray}
Here $V_{COOP}^{g}(M_{t,i})$ is the value function of the government
on the equilibrium path in the cooperative equilibrium described by
the strategies above, in stages where the government is matched with
an insider player, $V^{g}(M_{t,o})$ is the value function of the
government when matched with an outsider player, $V_{OSD}^{g}(M_{t,i})$
is the value of the best one-stage deviation by the government, and
$V_{PUN}^{g}(M_{t,i})$ is the value function of the government when
play has switched to the punishment prescribed by the strategies described
above. The reason future instances of the value function are written
in expectation is because these values might depend on the $\alpha$
parameters drawn in future stages, and those parameters are random.
The subgame perfect equilibrium condition is that no profitable one
shot deviations exist for the government:
\begin{equation}
\forall t,\forall i\,\,\,\, V_{COOP}^{g}(M_{t,i})\geq V_{OSD}^{g}(M_{t,i})
\end{equation}
This condition will have to hold with equality, as there is no need
for citizens to pay more than enough rents to the government according
to condition \emph{ii)} (the remaining income is incorporated in citizens'
utilities as the private good). To see what values of the rent parameter
lead to the condition holding, compute $V_{COOP}^{g}:$ First, take
the expected value of both Bellman equations in (1-2). Doing this
will lead to equilibrium conditions involving the expected value of
the rent payment, $E[R_{i,t}]$, so individual behavior by the citizens
will have to correspond to a probability law that leads to a certain
expected value for the rent, If it is further assumed that citizens
use symmetric strategies, it is also possible to denote all $i$ players
by a single subscript, rather than using $i$ and $i\prime$. Using
the fact that $E[E[X]]=E[X]$ and the linear property of the expected
value, the Bellman equations now become:
\begin{eqnarray*}
E\left[V_{COOP}^{g}(M_{t,i})\right] & = & E\left[R_{t,i}\right]-c+\delta\left[(1-p)E\left[V_{COOP}^{G}(M_{t+1,o})\right]+pE\left[V_{COOP}^{G}(M_{t+1,i})\right]\right]\\
E\left[V_{OSD}^{G}(M_{t,i})\right] & = & E\left[R_{t,i}\right]+\delta\left[(1-p)E\left[V_{PUNISH}^{G}(M_{t+1,o})\right]+pE\left[V_{PUNISH}^{G}(M_{t+1,i})\right]\right]
\end{eqnarray*}
Then from the first equation:
\begin{equation}
E\left[V_{COOP}^{g}(M_{t,i})\right](1-\delta p)=E\left[R_{t,i}\right]-c+\delta(1-p)E\left[V_{COOP}^{G}(M_{t+1,o})\right]
\end{equation}
The value function when being paired with an outsider player in the
cooperative equilibrium can be derived from its own set of two Bellman
equations, again written in expectation, and again assuming symmetric
strategies by $O$ players:
\begin{eqnarray}
E\left[V_{COOP}^{g}(M_{t+1,o})\right] & = & E\left[R_{t+1,o}\right]-c+\nonumber \\
 &  & +\delta\left[(1-p)E\left[V_{COOP}^{g}(M_{t+2,o})\right]+pE\left[V_{COOP}^{g}(M_{t+2,i})\right]\right]\\
E\left[V_{OSD}^{g}(M_{t+1,o})\right] & = & E\left[R_{t+1,o}\right]+\nonumber \\
 &  & +\delta\left[(1-p)E\left[V^{g}(M_{t+2,o})\right]+pE\left[V^{g}(M_{t+2,i})\right]\right]
\end{eqnarray}
The value functions following a deviation by the government here do
not carry any subscript regarding the strategy used by players who
are matched in those periods, because once stage $t+2$ has been reached
following a defection by the government, for the citizens this event
lies in the same information set as reaching stage $t+2$ following
cooperative behavior by the government. Therefore, the citizens have
to behave identically in the two histories, as they do not know which
one they are in. The government will have unique best responses to
whatever behaviors the citizens engage in, so it always has to be
that $E\left[V_{COOP}^{g}(M_{t+2,o})\right]=E\left[V^{g}(M_{t+2,o})\right]$
and \foreignlanguage{english}{$E\left[V_{COOP}^{g}(M_{t+2,i})\right]=E\left[V^{g}(M_{t+2,i})\right]$.}

So the condition that $V_{COOP}^{g}(M_{t+1,o})\geq V_{OSD}^{g}(M_{t+1,o})$
leads to $E\left[V_{COOP}^{g}(M_{t+1,o})\right]\geq E\left[V_{OSD}^{g}(M_{t+1,o})\right]$,
which then leads to: 
\begin{equation}
E\left[R_{t+1,o}\right]-c\geq E\left[R_{t+1,o}\right]
\end{equation}
which is never true as long as $c>0$. Therefore cooperation cannot
be sustained in stages where the government meets outside players,
so the value function of the government when matched with an outside
player is:
\begin{equation}
E\left[V_{COOP}^{g}(M_{t+1,o})\right]=\frac{\delta p}{1-\delta p}E\left[V_{COOP}^{g}(M_{t+2,i})\right]
\end{equation}
Putting this back in the value function for cooperating with an insider:
\begin{equation}
E[V_{COOP}^{g}(M_{t,i})]=\frac{1-\delta(1-p)}{1-\delta}\left(E\left[R_{t,i}\right]-c\right)
\end{equation}
The value function for the best one stage deviation is given by capturing
the entire rent payment now, which triggers the punishment future
matches with insider players. Therefore a stream of zero payments
follows defection by the government, and 
\begin{equation}
E\left[V_{OSD}^{g}(M_{t,i})\right]=E\left[R_{t,i}\right]
\end{equation}
The no-deviation condition has to hold: $E\left[V_{COOP}^{g}(M_{t,i})\right]=E\left[V_{OSD}^{g}(M_{t,i})\right]$,
so it follows that:
\begin{eqnarray}
E\left[R_{t,i}\right] & = & \frac{1-\delta(1-p)}{1-\delta}\left(E\left[R_{t,i}\right]-c\right)\\
\implies E[R_{t,i}] & = & \frac{c(1-\delta(1-p))}{\delta p}
\end{eqnarray}
Using symmetric strategies for all players, it follows that a constant
rent payment by any insider player keeps the government in the cooperative
equilibrium, and that payment is: 
\begin{equation}
R_{t,i}=\frac{c(1-\delta(1-p))}{\delta p}
\end{equation}
For citizens, the value functions in the cooperative equilibrium are
simpler:
\begin{eqnarray}
V_{COOP}^{i}(M_{t,i}) & = & \alpha_{i}G_{i}-R_{t,i}+w+\delta V^{i}(\neg(M_{t+1,i}))\\
V_{OSD}^{i}(M_{t,i}) & = & w+\delta V^{i}(\neg(M_{t+1,i}))
\end{eqnarray}
Future periods, in which $i$ is not matched, have the value function
$V_{i}(\neg(M_{t+1,i}))$. So the cooperation condition $V_{COOP}^{i}\geq V_{OSD}^{i}$
reduces to: 
\begin{equation}
\alpha_{i}\geq\frac{c(1-\delta(1-p))}{\delta p}
\end{equation}
\\


\noindent \textbf{Proof of proposition 2:} Write the value functions
of the government on the equilibrium path and the best one stage deviation
for the government: 
\begin{eqnarray}
V_{COOP}^{g}(M_{t,i}) & = & R_{i,t}-c+\delta\left[(1-q)E\left[V_{COOP}^{G}(M_{t+1,o})\right]+qE\left[V_{COOP}^{G}(M_{t+1,i})\right]\right]\\
V_{OSD}^{G}(M_{t,i}) & = & R_{i,t}+\delta\left[(1-q)E\left[V_{PUN}^{G}(M_{t+1,o})\right]+qpE\left[V_{PUN}^{G}(M_{t+1,i})\right]\right]
\end{eqnarray}


The government's conditions for maximizing income while being in this
cooperative equilibrium are
\begin{eqnarray}
(t,i) & = & \left(t,\arg\max_{j}V_{COOP}^{g}(C_{t,j}),\mbox{ s.t. }j\in\mathfrak{M}_{t}\right)\\
V_{COOP}^{g}(M_{t,i}) & \geq & V_{OSD}^{g}(M_{t,i})
\end{eqnarray}


(The maximization problem is set up here only with respect to $i$
players, as in equilibrium an $o$ player is never chosen, by the
same logic as in the previous section.) The optimization condition
for the government is similar to that in case 1 but with the addition
of the problem of choosing among the citizens which have been matched,
that is of choosing at each $t$ the one $i$ (or $o$) for which
the value function conditions have to hold. Since the forms of the
two relevant value functions are identical to those in case 1, except
for replacing $p$ with $q$, the same derivation leads to the following
cooperative equilibrium condition for the government:
\begin{equation}
E\left[R_{t,i}\right]\geq\frac{c(1-\delta(1-q))}{\delta q}
\end{equation}


The condition is not yet reduced to equality, because there might
be more than one insider player willing to pay a bribe. To see which
$i$ player is chosen, write the value functions for the citizens:
\begin{eqnarray}
V_{COOP}^{i}(C_{t,i}) & = & \alpha_{i}G_{i}-R_{t,i}+w+\delta V^{i}(\neg(M_{t+1,i}))\\
V_{OSD}^{i}(C_{t,i}) & = & w+\delta V^{i}(\neg(M_{t+1,i}))\\
V^{i}(\neg C_{t,i}) & = & V_{OSD}^{i}(C_{t,i})
\end{eqnarray}
The set of relevant value functions now includes the last one, for
the situation in which player $i$ is not chosen at time $t$. Then
the cooperation condition for citizens (which is the same as the condition
for being willing to participate in a cooperative equilibrium altogether
by being chosen as partner by the government) is: 
\begin{equation}
\alpha_{i}\geq R_{i}
\end{equation}


In case more than one citizen is matched, it is clear that the one
with the highest $\alpha_{i}$ level will become the partner of the
government official, and will offer a rent payment (just slightly
above) the maximum rent payment of the second-ranked drawn citizen.

With the given strategies, the government's problem of choosing the
player to cooperate with in each period to maximize its value function
for cooperation (equation 19) is reduced to choosing the player who
can pay the highest rent each period, as there is nothing to be gained
by choosing a player who can pay less.

The citizens in $I$ then solve: 
\begin{eqnarray}
 &  & \arg\max_{R_{i}}\alpha_{i}-R_{i}\nonumber \\
\mbox{subject to} &  & \alpha_{i}\geq R_{i}\mbox{ and }R_{i}\geq R_{j}\,\,\,\,\forall j\in\mathfrak{M}_{t}
\end{eqnarray}
 This leads for the player who has $\alpha^{(1)}$ at time $t$ to
set:
\begin{equation}
R_{t,i}=\alpha^{(2)}
\end{equation}

\end{document}
