Note that the incomes from the penalty in equilibrium exactly cover the
monitoring cost. Because the agent is held indifferent between theft and
no theft, his equilibrium profit will be the value of no theft
∏'(θ) = a 4. (10)
However, since these revenues are collected ex post, the net present value
of the game for agent i is equal to
∏<NPV' (θ) = τ-rθ) = Ri∏*(θ). (11)
1+ri
III.C. Stage Two: the Choice of Agents
At stage two, the CA sells the office. We assume that n identical poten-
tial agents compete for office by way of paying the CA up front in a first
price sealed bid auction. The best reply function of bidder i solves:
max((a-θ Ri - Bi) Pr(Bj ≤ Bi)n-1 ∀j = 1...n, j = i (12)
Bi 4b
For the uniform distribution, this implies
max(
Bi
(a
θ)
4b
Ri
B
Bi)( -i )
αj
n-1
The optimal bid is equal to
n-
1 (a
- θ)2
4b
Ri.
(13)
(14)
The CA’s expected up-front payment is given by26
f∣ r. (θn _ n - 1 (a - θ)2 pin > _ n - 1 (a - θ)2 ∩j∙∙.
E(Bi (θ)) = —--ÜTE(Ri) = n + 1 4b . (15)
26The CA’s expectation of the winner’s interest rate is the maximum R1 of the
sample of size n, which has the distribution F 1 (R)=Rn . The expected value is
therefore E(R1) = R1 RnRn-1dR = ^+. Hence, the CA’s expected income from
the sale of the office is given by E(B*(θ)) = nn+ (a-θ) .
17