Flavio Menezes
The University of
Queensland
Preliminary. Comments Welcome.
Matthew Ryan
The University of Auckland
Introduction
 Worldwide investment in PPPs jumped from $131 billion in
the early 1990s (Thomsen,2005) to $1.2 trillion dollars
globally in 2006 (World Bank PPP database).
 The popularity of PPPs linked to their perceived ability to
shift risks from the public to the private sector.
 The implications of this shift in risk are not well understood
 About 50 percent of PPPs never reach the financing stage
and, of those that do, about 50 percent are renegotiated
(Bracey and Moldovan, 2007).
Introduction (cont’d)
 In Australia, we had a couple of ‘tunnel’ bankruptcies. Administrators
took over and on-sold ‘contract’ at big losses
 Favoured interpretation: equity was by and large wiped out, debt
holders lost some, taxpayers gained. Is this right?
 Extension of the M1 Motorway in Hungary: Once completed, the project
was at risk of default as traffic was only half the amount forecast.
 The project was renationalised.
 A successor PPP contract to build the M5 highway from Budapest to
Serbia also ran into trouble for the same reason.
 Renegotiation led to the subsidisation of the toll by transfers from the
government to the concessionaire.
 Does the possibility of bankruptcy (or renegotiation to avoid bankruptcy)
affect the efficiency of the tender? Does it distort the choice of debt
versus equity?
Our Contribution
 Firms bid for a concession to build and operate a highway in a
first-price, sealed-bid auction under demand uncertainty
 Construction is bundled with operation in the tender process.
 Bids = toll to be charged.
 Bidders are able to use debt strategically in order to "hold up"
the Government.
 In low demand states, the winning firm threatens bankruptcy.
 If debt is not “too high”, the Government renegotiates and bails out
the firm.
 Motivation: understand the occurrence of bail-outs in such
arrangements, which will inform the design of better tenders.
Our Contribution: Main Results
 Our model suggests that default and re-negotiation are natural
outcomes of PPP first-price auctions.
 However, this does not result in an inefficient allocation process
 The winning bid may appear unrealistically attractive to the
Government if it fails to anticipate the "hold-up“
 Counterintuitive: More efficient firms are bailed out more often - and extract a higher expected transfer from Government -- than
less efficient firms.
 The hold-up problem may be addressed by imposing restrictions on
bidder financing.
 We also discuss Least Present Value of Revenue (LPVR) auctions as
another potential solution.
The Model: Basic Assumptions
 Bidders (risk neutral) have heterogeneous and privately
known construction costs and complete the construction
within the same timeframe; firms have common (= 0)
operating costs.
 Each firm bids the toll it will charge while it holds the
concession; The lowest bid wins, with ties resolved using
uniform randomisation.
 The winner builds the road then operates it for the
specified fixed period, charging the toll it bid into the
auction.
 For technical convenience, we ignore discounting; the
interest rate on risk-free debt is zero.
Basic Assumptions (cont’d)
 The demand for the completed road is: 𝑄 = 𝜃 − 𝑝, where 𝜃 is
a random variable.
 Suppose 𝜃 ∈ 𝜃𝐿 , 𝜃𝐻 and its distributed according to the
differentiable and strictly increasing distribution function G,
with G′=g.
𝜃𝐻
𝜃𝑑𝐺
𝜃𝐿
 Define 𝜃 =
𝜃 .
 We assume 𝜃𝐻 ≤ 2𝜃𝐿 so that equilibrium tolls do not exhaust
demand in any state. In particular, no firm will rationally bid
𝜃
more than 𝐻 .
2
Basic Assumptions (cont’d)
 Firm i has construction cost ci, drawn randomly (and
independently of other firms' costs) according to the
common distribution F, defined on the support [c, c] ,
2
where
 
0cc 
2
 F is strictly increasing and differentiable, with pdf f.
 (𝜃/2)² is the ex ante expected revenue of a monopolist.
Basic Assumptions (cont’d)
 Firm i uses cash (equity) K 𝑖 and debt financing D𝑖 = c𝑖 −
K 𝑖 to fund the construction phase; firms are not cash
constrained and have access to the same competitive
market for debt.
 Firms are distinguishable only by their construction costs.
 Financing is not revealed as part of the tender process;
necessary to ensure costs are private information.
 E.g., winner finalises its finance after the auction
Basic Assumptions (cont’d)
 The value of θ is realised at the same time as revenue is earned
(not prior).
 Assumed for simplicity but it is not inconsequential: all
revenue generated over the life of the concession is available
to repay debt.
 Combined with no discounting, allows us to model the
problem as three discrete stages: the auction, the construction
phase, and the operation stage
 The entire period of the post-construction concession is
collapsed into a single period.
Endogenous financing and
renegotiation
 Assume that there is a symmetric equilibrium bidding
strategy 𝑝(𝑐𝑖 ) and define
𝑏 𝑐, θ = 𝑝 (𝑐)(𝜃 − 𝑝(𝑐)).
 The Government places value 𝑐 > 𝑇 > 0 on keeping the
current concession-holder in place;
 If 𝐷𝑖 − 𝑏 𝑐𝑖 , 𝜃 ≤ 𝑇, then renegotiation will occur with the
government covering the firm’s debt shortfall.
 Assume Nash Bargaining (𝛼 = firm’s bargaining power)
Endogenous financing and
renegotiation (cont’d)
 As θ is continuous, expect different types renegotiate in




different contingencies.
Firm i may choose debt = 𝑏 𝑐𝑖 , θ + 𝑇 but θ may depend on i.
In states worse than θ, i declares bankruptcy and the
Government will not offer a bailout, while in states better than θ
the firm will either pay its debts or else renegotiate.
If there is a non-zero probability of bankruptcy, debt servicing
costs will rise above the risk-free rate.
Given its bid and its construction cost, i chooses optimal debt.
 This determines the contingencies in which it is bankrupt and in
which it renegotiates.
 These contingencies may depend on the firm's cost type; the nature
of this potential dependence is not obvious a priori.
Optimal leverage
 If firm i bids p, it will choose a financial structure to
maximise its expected payoff contingent on winning.
 Suppose it sets its debt level at 𝐷 ∈ 0, 𝑐𝑖 .
 Expected profit contingent on being the winning bidder is
Firm is solvent
− 𝑐𝑖 − 𝐷 +
Θ1 𝑝,𝐷
Θ0 𝑝,𝐷
Interest rate on debt
𝑝 𝜃 − 𝑝 − 1 + 𝑟 𝑝, 𝐷 𝐷 𝑑𝐺 𝜃 +
𝛼 𝑇 + 𝑝 𝜃 − 𝑝 − 1 + 𝑟 𝑝, 𝐷 𝐷 𝑑𝐺 𝜃
Firm holds up the Government
Optimal leverage
Where
Θ0 𝑝, 𝐷 = {𝜃 ∈ [𝜃𝐿 , 𝜃𝐻 ] | [1 + 𝑟(𝑝, 𝐷)]𝐷 ≤ 𝑝(𝜃 − 𝑝)┊}
And
Θ1 𝑝, 𝐷 = {𝜃 ∈ [𝜃𝐿 , 𝜃𝐻 ]| 0 < [1 + 𝑟(𝑝, 𝐷)]𝐷 − 𝑝(𝜃 − 𝑝)
≤ 𝑇┊}
Note that the optimal level of debt is independent of c𝑖
given the bid p, though the latter will obviously depend on
the firm's cost type in equilibrium.
Optimal leverage: the cost of debt
The firm declares bankruptcy and receives −(𝑐𝑖 − 𝐷) when is 𝜃 such that:
1 + 𝑟 𝑝, 𝐷 𝐷 − 𝑝 𝜃 − 𝑝 > 𝑇 ⇔ 𝜃 < ([1 + 𝑟(𝑝, 𝐷)]𝐷 + 𝑝² − 𝑇)/𝑝
Thus the probability of default is:
[1 + 𝑟(𝑝, 𝐷)]𝐷 + 𝑝² − 𝑇
𝐺(
)
𝑝
It follows that the cost of debt is the solution (in 𝑟) to:
𝐷
[1 + 𝑟(𝑝, 𝐷)]𝐷 + 𝑝² − 𝑇
= (1 + 𝑟)𝐷[1 − 𝐺(
)]
𝑝
+
[(1+𝑟)𝐷+𝑝²−𝑇]
𝑝
𝑝(𝜃 − 𝑝) 𝑑𝐺(𝜃).
Optimal leverage: Optimal Debt
Level
Given that the bank makes D in expected value, the firm's
expected profit is given by:
𝑝(𝜃 − 𝑝) − 𝑐𝑖 + (expected payment from the Government).
Therefore, the firm chooses D to maximise its expected transfer
payment from the Government, which is
([[1 + 𝑟(𝑝, 𝐷)]𝐷 − 𝑝(𝜃 − 𝑝)] + 𝛼(𝑇 − [[1 + 𝑟(𝑝, 𝐷)]𝐷 − 𝑝(𝜃 − 𝑝)]) 𝑑𝐺(𝜃)
Θ1 𝑝,𝐷
=
(𝛼 𝑇 + 1 − 𝛼
1 + 𝑟 𝑝, 𝐷 𝐷 − 𝑝 𝜃 − 𝑝 )𝑑𝐺(𝜃) (∗)
Θ1 𝑝,𝐷
That is, for each state 𝜃 ∈ Θ1 𝑝, 𝐷 the Government pays the amount
needed to clear the firm's debts, plus an additional transfer equal to this
The Optimal Debt Level
The Optimal Debt Level (cont’d)
Given p; we can simply find the optimal left-hand end
point for Θ₁, denoted by z
The Optimal Debt Level (cont’d)
Rather than choosing D to maximise (*), we can think of the firm choosing
𝑧 ∈ 𝜃𝐿 , 𝜃𝐻 to maximise
𝑇
𝑚𝑖𝑛{𝑧+𝑝,𝜃𝐻 }
[𝑇
𝑧
− (1 − 𝛼)𝑝(𝜃 − 𝑧)] 𝑑𝐺(𝜃)
How does optimal D vary with p?
𝑝′ > 𝑝
The higher a firm bids in the auction, the lower its expected
transfer from the Government.
More efficient firms bid lower and are
more likely to be ex-post bailed out
 Firms choose their capital structure by choosing a state (z) in
which all of 𝑇 is needed to clear their debts.
 For states θ ∈ (𝑧, 𝑧
𝑇
+ ),
𝑝
revenue is higher so not all of 𝑇 is
required for debt repayment.
 The firm can only obtain fraction α of the remainder through
bargaining.
 Once revenue is high enough to repay all debt, the firm's claims on
the Government vanish…so the expected payment that the firm can
obtain depends on how quickly revenue rises with θ; the faster
revenue increases with the demand state, the lower the expected
transfer to the firm.
 For our demand structure, lower prices reduce the rate at which
revenue increases with θ (see next).
Optimal leverage (cont’d)
 To maximise payment from the Government, a firm would choose
p=0 and borrow 𝑇; bailed out in every state and extract 𝑇 with
probability one.
 Choosing p=0 would also maximise the firm's chances of winning the
auction.
 However, it cannot be an equilibrium for all firms to bid p=0, since we
assumed that 𝑇 < 𝑐 so the winning firm would make a loss with certainty.
 Trade-off between maximising revenue from hold-up and maximising
toll revenue to be struck at a strictly positive toll.
 Since the hold-up incentive reinforces the incentive to set p low,
expect more aggressive bidding by all firms than when 𝑇 = 0.
 To quantify these trade-offs, we must ascertain the optimal choice of z
for each p > 0. The next Lemma characterises the optimal z for given
p in the case of the uniform distribution.
Optimal leverage (cont’d)
Lemma 1: Let p > 0 be given. If θ is uniformly distributed,
then all firms choose 𝑧 = 𝜃𝐿 .
Proof: The firm is indifferent about which z ∈ (𝜃𝐿 , 𝜃𝐻 −
𝑇
𝑝
𝑇
)
𝑝
choose. (Recall shaded area) If 𝜃𝐿 > 𝜃𝐻 − then the firm
strictly prefers to set 𝑧 = 𝜃𝐿 . It is therefore without loss of
generality to suppose that all firms choose 𝑧 = 𝜃𝐿 .
to
Optimal leverage: Conclusion
 It follows that creditors are exposed to zero default risk, hence r
= 0, and debt satisfies D=𝑇+p(θ𝐿 − 𝑝).
 Firms which bid higher toll rates have a (weakly) lower
probability of receiving a Government bail-out, as well as a
lower expected payment from the Government.
 A firm that bids p (and chooses its debt level optimally) receives
an expected payment from the Government equal to
1
𝑇
𝑇 − 1 − 𝛼 𝑚𝑖𝑛 𝑇, 𝑝 𝜃𝐻 − 𝜃𝐿
𝑚𝑖𝑛
,1 =
2
𝑝 𝜃𝐻 − 𝜃𝐿
𝑇
1+𝛼 𝑇 2
, if 𝑇 < 𝑝
2𝑝 𝜃𝐻 −𝜃𝐻
1
− 1 − 𝛼 𝑝 𝜃𝐻 − 𝜃𝐿
2
𝜃𝐻 − 𝜃𝐿 ; and
, if 𝑇 ≥ 𝑝 𝜃𝐻 − 𝜃𝐿
Equilibrium bidding behaviour
 Given the non-differentiability in the expression above,
we must be careful about assuming differentiability of
the equilibrium bidding function.
 There are two cases in which differentiability is
plausible:
(i) when equilibrium bids are strictly increasing in c and
𝑇
bounded above by
. [Low toll equilibrium]
𝜃𝐻 −𝜃𝐿
(ii) when equilibrium bids are strictly increasing in c and
𝑇
bounded below by
. [High toll equilibrium]
𝜃𝐻 −𝜃𝐿
Low toll equilibrium
 Since hold-up incentives place downward pressure on bids, start
by searching for a low toll equilibrium.
 In such an equilibrium, all of the winning firm's profit is
obtained from exploiting the hold-up problem. If no type bids
𝑇
above
, the winning bidder is bailed out in every state.
𝜃𝐻 −𝜃𝐿
 Suppose 𝑝 is a differentiable and strictly increasing equilibrium
𝑇
𝜃𝐻 −𝜃𝐿
bidding function, with 𝑝 ≤
. Let 𝑏(𝑐, 𝜃) = 𝑝(𝑐)(𝜃 −
𝑝(𝑐)).
 If i bids 𝑝(𝑐) and chooses its debt level optimally, its expected
payoff (conditional on winning the auction) will be
1
𝑏 𝑐, 𝜃 − 𝑐𝑖 + 𝑇 − (1 − 𝛼)𝑝(𝑐) 𝜃𝐻 − 𝜃𝐿 .
2
Low toll equilibrium (cont’d)
1
2
𝑐𝑖
We define 𝛽 𝑐, 𝜃 = 𝑏 𝑐, 𝜃 + 𝑇 − 1 − 𝛼 𝑝 𝑐 𝜃𝐻 − 𝜃𝐿
So that for an efficient mechanism solves:
𝑚𝑎𝑥𝑐∈[𝑐,𝑐] [𝛽(𝑐) − 𝑐𝑖 ]𝐹ⁿ⁻¹(𝑐).
Where 𝐹 c = 1 − F c . As standard, 𝛽 𝑐 = 𝐸 𝑋|𝑋 > 𝑐 , where X is
the r.v. corresponding to the lowest of n-1 draws from F and .
1
𝑏 𝑐, 𝜃 = 𝐸 𝑋|𝑋 > 𝑐 − 𝑇 +
1 − 𝛼 𝑝 𝑐 𝜃𝐻 − 𝜃𝐿
2
To find 𝑝 𝑐 , it is necessary to solve
𝑝(𝜃 − 𝑝) + 𝑇 −
𝛬(𝑐) − 𝑇 (**)
1
2
1 − 𝛼 𝑝 𝑐 𝜃𝐻 − 𝜃𝐿 = 𝛬(𝑐) ⇔ 𝑝(𝜃 − 𝑝) =
1
2
Where 𝜃 = 𝜋𝜃𝐻 + (1 − 𝜋)𝜃𝐿 and 𝜋 = 𝜋 − (1 − 𝛼).
Low toll equilibrium: Conclusion
We can solve (**) provided 𝜃 ≥ 0
and
𝜃
2
2
≥ 𝑐 − 𝑇. The solution is
valid provided 𝑝 𝑐 ≤
𝑇
𝜃𝐻 −𝜃𝐿
.
These conditions will be met if
weak demand or weak bargaining
power for the concessionaire. In
particular, a low toll can partially
offset the effects of a weak
bargaining position for the firm.
High toll equilibrium
Let 𝑝 be a differentiable and strictly increasing equilibrium bidding
𝑇
𝑇
function, with 𝑝 𝑐 ≥
. Since no type bids below
and 𝑝
𝜃𝐻 −𝜃𝐿
𝜃𝐻 −𝜃𝐿
is strictly increasing, the winning firm is solvent with positive
probability. If firm i bids 𝑝 𝑐 , its expected payoff (conditional on
winning the auction) is
(1 + 𝛼)𝑇 2
𝑏 𝑐, 𝜃 − 𝑐𝑖 +
.
2𝑝(𝑐) 𝜃𝐻 − 𝜃𝐿
The presence of a hold-up problem gives stronger incentives to lower
the toll price than in the low-toll equilibrium. Once the toll reaches
𝑇
, marginal (hold-up) incentives for further toll reductions are
𝜃𝐻 −𝜃𝐿
1
constant at (1 − 𝛼) 𝜃𝐻 − 𝜃𝐿 .
2
High toll equilibrium (cont’d)
In this case we have
(1 + 𝛼)𝑇 2
𝑏 𝑐, 𝜃 = 𝐸 𝑋|𝑋 > 𝑐 −
2𝑝(𝑐) 𝜃𝐻 − 𝜃𝐿
To find 𝑝 𝑐 , it is necessary to solve
𝑝(𝜃 − 𝑝) +
2 𝜃𝐻 −𝜃𝐿
(1+𝛼)𝑇 2
1+𝛼 𝑇 2
2𝑝 𝑐 𝜃𝐻 −𝜃𝐿
(***)
= 𝛬(𝑐) ⇔ 𝑝(𝑝2 − 𝜃𝑝 + 𝛬(𝑐)) =
High toll equilibrium (conclusion)
For existence we require strong demand, high returns from holdup and a strong bargaining position for the firm.
Mixed equilibrium
 The model with a uniform θ distribution also admits
equilibria that are mixtures of the "low toll" and the
"high toll" variety.
 In these "mixed" equilibria, the bidding function is
non-differentiable at a toll equal to
𝑇
𝜃𝐻 −𝜃𝐿
.
 It resembles the "low toll" equilibrium below this value
and the "high toll" equilibrium above.
Summary
 Firms choose debt levels strategically to hold-up the Government;
winning firm threatens default with positive probability.
 Under our demand structure, a firm that bids lower is able to extract a
higher expected bail-out from the Government.
 If the equilibrium bidding function is strictly increasing, more
efficient firms make higher demands on the public purse (and hence
charge very low tolls).
 We computed the equilibrium bid function when θ is uniformly
distributed and verified that it is strictly increasing. I
 Firms never actually go bankrupt, but each firm i credibly threatens
default in an interval of states of the form 𝜃𝐿 , 𝜃 ∗ 𝑖 for some 𝜃 ∗ 𝑖 ∈
𝜃𝐿 , 𝜃𝐻 .
∗
 When 𝜃 = 𝜃𝐻 firm i threatens default in all states, so its return on
𝑖
equity comes entirely from Government transfers.
Summary (cont’d)
 Depending on parameters, we may observe:
 "low toll" equilibrium, in which toll bids are so low that
𝜃 ∗ 𝑖 = 𝜃𝐻 for every firm;
 "high toll" equilibrium in which 𝜃
∗
𝑖
< 𝜃𝐻 for every firm; or
 a "mixed equilibrium" in which more efficient firms are
solvent in some states while less efficient firms threaten
default in all states.
Summary (cont’d)
 Weak demand or weak concessionaire bargaining power
encourage "low toll" equilibria, while strong demand or
strong bargaining power encourage "high toll" equilibria.
 The link between demand and tolls is natural, though
the correlation between bargaining strength and
equilibrium toll levels may seem counter-intuitive.
 The latter arises because toll reductions are a partial
substitute for bargaining strength.
 Under our demand structure, lower tolls allow the firm to
credibly threaten bankruptcy in more states and thereby
increase the returns from hold-up.
Conclusion
 We model the optimal financial structure and equilibrium
bidding behaviour and show that:
 The auction remains efficient, but
 bids are lower than they would be if all bidders were equity
financed, and
 the more efficient the winning firm, the more likely it is to
require a Government bail-out and the higher the expected
transfer it extracts from the Government through hold-up.
 These insights complement the work of Engel, Fischer and
Galetovic (2001) on LPVR tenders.
Conclusion
 A LPRV tender allocates the contract to the bidder with the
lowest present value of expected revenue; the concession
contract will remain in place until the firm recovers its bid.
 EFG argue that, unlike fixed term contracts, the LPVR
mechanism results in an optimal allocation of risk -- firms are
risk-averse in their model.
 bidders bid a "guaranteed" PVR -- not quite fully guaranteed,
as demand may be so low that the winning firm holds the
concession forever without earning the PVR.
 The Government sets the toll rate for each demand state
before bids are taken. This toll rate reflects optimal
congestion pricing, plus optimal risk-sharing given that the
PVR is not guaranteed in every state.
Conclusion
 EGF (2001, 2008) assume that is possible to find a mechanism that
avoids further renegotiation by fully transferring the risk of default to
the government

Fhe firm does not default even if the revenues are not sufficient to service
debt in a particular period.
 EFG (2001) have two models -- one with and one without
"commitment".
In the absence of commitment, the Government must ensure nonnegative PVR in every state (not just in expectation).
 They treat this as an extra constraint in the design of the optimal contract.

Conclusion
 In each case, the Government effectively renegotiates in advance
 It determines the toll and concession duration as a function of the state,
so there is no need to re-negotiate ex post.
 The Government bears the default risk (under no commitment) through
agreeing up front to a state-contingent toll/transfer and concession
duration.
 EFG consider a more complete contract than that available in our
model.
 We assume that it is politically infeasible for governments to commit
to a transfer/subsidy/toll increase even though, in practice,
governments may want to avoid default

As renegotiation is often inevitable, we explore its implications for the
efficiency of the PPP tender and the choice of the capital structure of the
firm.
Conclusion
 Another distinction from EFG (2001, 2008) is that we do not
consider differences in risk attitudes. Instead, we look at the
nature of the risk.
 In our model the government would like to shift the risk of
default to the firm.
 This might provide yet an alternative motivation for the
introduction of a LPVR auction.
 The use of a LPVR tender, which could allow the firm to
change either the duration of the contract or the price, would
result in a greater ability for the firm to manage risk.
 Our future research plans includes a characterisation of
equilibrium behaviour in LPVR auctions when the capital
structure of the firm is endogenous.