Imperfect competition among financial serviceproviders
connecting contract theory, IO, and development
Robert M. Townsend and Victor V. Zhorin
July 30, 2014
Outline
context
I spatial separation and the geography of branch expansionI impact of financial service providers on small businesses,I financial access: intensive and extensive marginsI estimation of parameters, financial regimes, and market structuresI obstacles to trade, geographyI bank contracts in practice - an internal look
premise
I big picture - implementation with imperfect competition, regulation, welfareI getting ready for applications combining these, need to work out concepts first
theory: basic building blocks, big questions
I utility/preferences, technologyI bank surplus - programming problem, Pareto frontierI contracts - depend on obstacle, not limited a prioriI Hotelling environementI example solutions - monopoly or collusions (a realistic benchmark)
IO and mechanism design: sequential Nash (full commitment) - FI vs MH
sequential Nash vs no commitment Nash, partial commitment
adverse selection
two applications
I financial liberalizationI local incumbent (informed) vs entrants (uninformed)
conclusions/caveats: more banks, fixed costs to entry, welfare, multi-homing, informal markets, limitedcommitment, robustness
Entry and Dynamics of Bank Branches, FrameworkAssuncao et al. (2012)
Commercial banks
1986 1996
2D maps with observed heterogeneity, depending on distance of travel to branches
extensive marginsI markets: here n branchesI but with some kind of geographic aggregationI strategic gameI myopia, limited dynamicsI easy to add density of population, wealth
Credit/Insurance contractrisk contingency system in Thai banking, Townsend & Yaron
BAAC lending procedures, in practice
1 amount loaned can be repaid with regular interest
2 a credit officer goes into the field if the loan is not repaid on time
3 if justified non-repayment due to large and regional shock then the loan is restructured andrescheduled
4 interest penalty is charged if not justified non-repayment or willful default
Impact of financial service providers on householdsOLS and IV estimation, Alem and Townsend (2014)
reduced form equations come from theory - risk sharing vs financial autarkyhouseholds are running small businesses - Townsend Thai datathe cost of traveling to remote branches rationalizes several of the instruments usedBAAC helps both in consumption and investment through credit
Optimal Contracts in Theorysome examples
setupI risk-neutral bank as principal - with cost of fundsI risk-averse agent with investment projectI a function of effort, capital, shocks
full information - full commitment: contract breaks into piecesI credit at market interest r , equity and creditI full insurance with ex-post premium and indemnity, s = q − c
moral hazard - effort is unobserved, two pieces are inseparable
hidden interim information - costly state verification
adverse selectionI unobserved typesI menu of contracts, agents choose subject to truth-telling
adverse selection plus moral hazard: honesty and obedience -interact in optimal design
summary: complementary with empirical contract theory withincomplete contracts, but not putting vector of controlcharacteristics into utility, not separable
Micro foundations: contracts for household enterprise
utility of consumption computed from first principles: underlyingpreferences and technology
agents of type θ have preferences
u(c , a|θ),
where c is consumption, and a is hidden or observed effort
technology P(q|k, a, θ) is a probability to reach the output q thatdepends on agent’s type θ and the effort a exercised by an agent.
agents can borrow capital k from the intermediaries to augmenttheir labor effort. Investment can be observed or unobserved.
θ is a multidimensional object that allows to consider different riskaversion, different risk types and different aversion to effort
a subset of θ may not be observable, hence adverse selection
Building block - standard mechanism design problemcomputed with linear programming methods
The optimal contract to maximize the bank surplus extracted from each agent:
S{u(θ)} := maximizeπ(q,c,k,a|θ)
∑θ
∑q,c,k,a
π (q, c, k, a|θ) [q − c − k]
where π(q, c, k, a|θ) is a probability distribution over the vector (q, c, k, a) given the agent’s type θ.Mother Nature/Technology Constraints:
∀{
q, k, a}∈ Q × K × A and ∀ θ ∈ Θ
∑c
π(q, c, k, a|θ) = P(
q|k, a, θ)∑
q,c
π(q, c, k, a|θ)
Incentive Compatibility Constraints for action variables:∀ a, a ∈ A× A and ∀ k ∈ K and ∀ θ ∈ Θ:
∑q,c
π (q, c, k, a|θ) u (c, a|θ) ≥∑q,c
π (q, c, k, a|θ)P(q, |k, a, θ)
P(q, |k, a, θ)u(c, a|θ)
Utility Assignment Constraints (UAC):∀ θ ∈ Θ ∑
q,c,k,a
π(q, c, k, a|θ)u(c, a|θ) = u(θ); u(θ) ≥ u0(θ)
Extensions:Truth-Telling Conditions in Adverse Selection, Dynamic Contracts
Truth-Telling Conditions (TTC) in Adverse Selection:type θ must not announce type θ′ (can add to unobserved a and k):∀ θ, θ′ ∈ Θ
∑q,c,k,a
π(q, c , k , a|θ)u(c , a|θ) ≥∑
q,c,k,a
[π (q, c , k , a|θ′) P(q, |k, a, θ)
P(q, |k, a, θ′)u(c , a|θ′)
]
Dynamic Contracts:additional state dimension - continuation utility w ,contracts π(q, c , k , a,w ′|θ), promise keeping constraint as in Karaivanovand Townsend (2014)) - method works empirically:∑
q,c,k,a,w
π(q, c , k , a,w |θ) [u(c , a|θ) + β ∗ w ] = u(θ)
Limit ourselves to static here, for now
Pareto frontier with single type
Lottery-based solutions are computed using linear programming techniques of Prescott and Townsend (1984),Phelan and Townsend (1991), specifications grounded in empirical estimations are from Karaivanov and Townsend(2014)
u(c, a) = c(1−σ(θ))
(1−σ(θ))+ χ(θ)
(1−a)γ(θ)
γ(θ)
P(q = high|k, a, θ) = p(q = high|a)f (θ)kα, α = 1/3effort and capital are complementsf (θ) maps agent’s ability θ to probability of reaching higher output, lower ability θ requires higher effort to reachthe same output at higher variance - higher production risk, here f (θ) is linear function of θ, can be non-parametric
1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20
0.2
0.4
0.6
0.8
1
1.2
1.4
Pareto frontier
utility offer
su
rplu
s
first−best
moral hazard
once frontiers are computed, can use those for market competition - but underlying contracts vary
Contracts: depend on obstaclemoral hazard vs full information, not limited a priori
Consumption, conditional on output observed (baseline)
1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Utility
MH, high output
MH, low output
FI
full information always provides full insurance contract
insurance contract under MH 6= full insurance under FI
with many (indeed continuum) of outputs, have non-linear functions- ”complex” contracts observed in reality
Contract values for heterogeneous types, full informationdifference in ability to reach assigned production quota with assigned effort in stochastic environment
(a) Surplus
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20
0.2
0.4
0.6
0.8
1
1.2
1.4
utility
Su
rplu
s
θ=0.2
θ=0.4
θ=0.6 (baseline)
θ=0.8
(b) Consumption
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20
0.2
0.4
0.6
0.8
1
1.2
1.4
utility
Co
nsu
mp
tio
n
θ=0.2
θ=0.4
θ=0.6 (baseline)
θ=0.8
(c) Labor effort
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
utility
La
bo
r e
ffo
rt
θ=0.2
θ=0.4
θ=0.6 (baseline)
θ=0.8
(d) Capital
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
utility
Ca
pita
l b
orr
ow
ed
θ=0.2
θ=0.4
θ=0.6 (baseline)
θ=0.8
Market Environment: Classical Linear Hotelling case
The agents are distributed uniformly in R1 : [0; 1] with total market mass set to one. The household cost to accessfinancial services is equal to
L ∗ |x − xi |
where x is location of the agent, xi is location of bank i , L is a spatial cost or disutility from accepting a contract.The agents of type θ at location x choose to go to bank i if contract utility from bank i satisfies participationconstraint and the real value offered is better than the one from bank i′
Vdiff = ui (θ)− L ∗ |x − xi |≥ ui′ (θ)− L ∗ |x − xi′ |≥ u0(θ)
where u0(θ) is autarky value.
location can be a stand in for other characteristics that make a given contact more or less attractive, heregeography is taken seriously
simplification: additive disutility
can do R2, put in roads just like on real maps - not shown here
can restrict choice by finite number of potential locations (even easier) - but we want to know ifunrestricted competition in space delivers interesting patterns
not yet including spatially different agent’s characteristics, all we need to do is to integrate over densitiesand we have that already built-in
Example solutions for monopoly (two branch collusion)
at high spatial cost all market structures offer local monopoly utility, market is not completely covered, islands of autarky
Theorem (local monopoly)The optimal spatial cost independent contract with utility u∗ can be derived from non-linear condition:
−S′u |u=u∗
S(u∗)=
1
u∗ − u0
Follows from u∗ = u0 + Lx → x =u∗−u0
L; S(u∗)dx + S′u |u=u∗ xdu = 0
10
Location
Value
X1
U(monopoly)
U(autarky)
X2
There could be as many non-interacting local monopolists as given spatial cost L allows.
Surplus and Market Share elasticity
From FOC for one branch monopoly wrt to utility promise
S ′(u)
S(u)= −µ
′(u)
µ(u)
increase in promise w decreases surplus and increase market share (if solution is interior)
1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2−35
−30
−25
−20
−15
−10
−5
0
5
Utility, ω
Ela
sticity
market share elasticity, L=0.5
market share elasticity, L=2
market share elasticity, L=4
surplus elasticity
Two Branch Monopoly: optimal contractsFull Information vs Moral Hazard
locations of branches are symmetric at [1/4; 3/4]
0 1 2 3 4 5
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Consumption
Spatial cost
co
ns
um
pti
on
first−best
moral hazard
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Labor (effort)
Spatial cost
eff
ort
first−best
moral hazard
0 1 2 3 4 51
1.5
2
2.5
Output
Spatial cost
ou
tpu
t
first−best
moral hazard
0 1 2 3 4 50
0.2
0.4
0.6
0.8
Capital borrowed
Spatial cost
ca
pit
al
first−best
moral hazard
Market Structures
collusion (multi-branch monopoly)
sequential Nash equilibrium (SNE) with full commitment tolocation and contract (business model)
full commitment to location and simultaneous Nash oncontracts (partial commitment)
simultaneous Nash in location and contracts (no commitment)
Mechanics of competition among banks: Splitting The Market
basic ingredients for all competing markets structures in equilibrium
Both banks have non-zero profit and households make a choice at the end of market processMarket share of bank 1:
z =x1 + x2
2+
u1 − u2
2L
Profits of bank 1 and 2:
P1(u1, x1|u2, x2) = S(u1)z; P2(u2, x2|u1, x1) = S(u2)(1− z);
Note: potential discontinuity as they overlap, fighting it out or separate
10
Location
Value
X1 X2Z
U1
U2
Formulation:based on endogenous ”intensity of competition” (IC)
Idea: use logits for the local relative bank offer advantage scaled by the tail mass in logits depending on the”intensity of competition” defined based on the distance of averaged bank’s offers from the maximum possibleutility. The higher the intensity of competition the more mass is in the tail, the close we are to the ”hot lava”condition. Then we integrate over all agents AND subtract the mass of agents who are outside of first bank t-pee,in autarky, or outside of second bank t-pee. Joining agents are split between the banks according to the logit.Autarky agents mass is computed precisely and subtracted from the total possible share for all points where autarkyconditions are violated.
Vdiff (x, u1, x1, u2, x2) = u1 − L ∗ |x − x1|−(u2 − L ∗ |x − x2|)
M(u1, u2) =∆
exp(umax − L/2− (u1 + u2)/2))
Total market shares are
M2(x1, u1, x2, u2) =
∫ 1
0L
(−Vdiff (x, u1, x1, u2, x2)
M(u1, u2)
)dx −
∫ 1
01
u2−L∗|x−x2|<u0dx ;
M1(x1, u1, x2, u2) =
∫ 1
0L
(Vdiff (x, u1, x1, u2, x2)
M(u1, u2)
)dx −
∫ 1
01
u1−L∗|x−x1|<u0dx ;
where
L(x) =1
1 + exp−x
The total profits for each bank are then convex and equilibrium exists per log-concavity
π1 = M1(x1, u1, x2, u2) ∗ S(u1)
π2 = M2(x1, u1, x2, u2) ∗ S(u2)
Sequential Equilibrium (Sequential SNE)
We consider a sequential game with first bank coming to locationx1.
This bank offers utility u1
For the second entrant we find optimal location x2 andtype-dependent utility offer u2 for any given choice of the firstentrant.
The first bank anticipates the entry of the second bank and itchooses its location and offer with respect to the best possibleresponse by the second bank.
For max problem pattern search or HOPSPACK solver can be used
Existence: log-concavity
IC logit formulation, ∆ = 1/10FI and MH, full commitment, smoothed by projection pursuit regression (PPR)
0 0.2 0.4 0.6 0.8 1 1.2 1.40.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.105
Spatial cost
tail
mass
FI
MH
(e) Tail mass
IC logit formulation, ∆ = 1/10FI and MH, full commitment, profit and market shares, smoothed by PPR
profits asymmetric, bank 2 has larger profits compared to bank 1
bank 2 has larger advantage in profit under FI compared to MH
the presence of MH smooth out profit difference due to flatter surplus frontier (smaller price elasticity),intensity of competition is higher under MH: the relative loss of surplus with higher price offer is smallerand is dominated by preference to increase market share
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Pro
fit
Spatial cost
bank1 (MH)bank1 (FI)
bank2 (MH)bank2 (FI)
(a) Profit
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
10
20
30
40
50
60
70
80
Spatial cost
Ma
rke
t sh
are
, %
bank1 (MH)
bank2 (FI)
bank2 (MH)
bank2 (FI)
(b) Market share
IC logit formulation, ∆ = 1/10FI and MH, full commitment, location and utility choice, smoothed by PPR
moves in locations are due to substitution effect between utility and location in profit function→ spatialseparation helps to lower utility offer leading to higher profits while preventing incumbent from beingoutcompeted completely
the central location is preferred at high spatial costs→ no local spatially separated monopoly class ofsolutions in this formulation
attempt to differentiate at high spatial cost leads to lower tail mass (lower intensity of competition) thatmakes it is easier for the entrant to poach customers from the incumbent
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Spatial cost
Lo
ca
tio
n
bank1 (MH)
bank1 (FI)
bank2 (FI, MH)
(c) Location
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
Spatial cost
Utilit
y
bank1 (MH)
bank1 (FI)
bank2 (MH)
bank2 (FI)
(d) Utility
Competition: no commitment
Nash equilibrium in case of two market entrants would be defined by {u∗1 , x∗1 , u∗2 , x∗2 } that satisfy
S(u∗2 )µ2(u∗2 , x∗2 , u∗1 , x∗1 ) ≥ S(u2)µ2(u2, x2, u∗1 , x∗1 )
S(u∗1 )µ1(u∗2 , x∗2 , u∗1 , x∗1 ) ≥ S(u1)µ1(u∗2 , x∗2 , u1, x1)
∀{u1, x1, u2, x2}
standard simultaneous Nash as in game theory literature
here: simultaneous price competition with simultaneous choice of locations
new: numerical minimax algorithm to rank order possible strategies based on ”distance to Nash” concept
requires computing out of equilibrium deviation in four continuos choice variables
both players try to maximize their profit given a candidate strategy of the opponent
algorithm converges when it finds a set of strategies with minimum deviation (ideally zero) for both players
the Fan-Glicksberg fixed point theorem: we have continuity, not quazi-concavity
or embrace ε-neighborhood equilibrium ( and we can report what ε is numerically)
robust to change in parameters
IC logit formulation, ∆ = 1/10FI, full commitment and no commitment
intensity of competition is naturally higher with full commitment since incumbent is under higher threat ofbeing annihilatedno commitment tail mass and intensity of competition varies less over the spatial costs range due tosymmetry of entrantsthere is no dip in intensity of competition as in full commitment because there is no spatial separationunder no commitment at any spatial cost
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
Spatial cost
tail
ma
ss
full commitment
no commitment
Competition:full commitment/no commitment, IC logits, ∆ = 1/10
the absence of entry advantage lowers profits for both banks under no commitment
profits are smoothed out under no commitment over the large range of spatial costs
profits under no commitment are exactly the same for both banks (magenta line), market is split 50-50
(a) Profit
0 0.5 1 1.5 2 2.5 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Spatial cost
Pro
fit
no commitment
bank1 − full commitment
bank2 − full commitment
(b) Market Share, %
0 0.5 1 1.5 2 2.5 30
10
20
30
40
50
60
70
80
Spatial cost
Mark
et share
, %
bank1,2 − no commitment
bank1 − full commitment
bank2 − full commitment
Competition:full information with full commitment/no commitment, logits, IC logits, ∆ = 1/10
utility offers are higher under no commitment at low-medium spatial costswelfare can be strictly better under no commitment at low-medium spatial costs!
For no commitment Nash at center location with fixed tail mass the optimal utility is at
S′(u)
S(u)= −
1
2 M
(c) Utility offer choice
0 0.5 1 1.5 2 2.5 32.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
Spatial cost
Utilit
y
bank1,2 − no commitment
bank1 − full commitment
bank2 − full commitment
Competition: partial commitment
common assumption in IO literature: ex-ante investment withex-post price competition, similar to Doraszelski and Pakes(2007)
here - sequential choice of locations with second-stagesimultaneous price competition
first mover advantage
second entrant separates to go for the market segment at themargin
second entrant makes higher offer, gets lower surplus andlower market share due to inferior location
Comparative staticsfull information, partial commitment, M = .00001
metric: distance to Nash (black line at the bottom) indicates goodconvergence properties
notice: Betrand limit at L→ 0 for no logits (and small logit tailmass)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Spatial cost
Pro
fit
bank1
bank2
distance to Nash
(a) Profit
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4
0.5
0.6
0.7
0.8
0.9
1
Spatial cost
Consum
ption
bank1
bank2
(b) Consumption
Adverse selection, two unobserved types
Option for banks to offer type dependent contractsUtility Assignment Constraints (UAC) - type dependent offer from bankis set at utility level u(θ) :∀ θ ∈ Θ ∑
q,c,k,a
π(q, c , k , a|θ)u(c , a|θ) = u(θ)
For two typesθ ∈ {θ1, θ2}
truth-telling constraints (TTC) can be combined with UAC (given offerof utility per type) to result in explicit bounds for TTC:
u(θ2) ≥∑
q,c,k,a
[π (q, c , k , a|θ1)
P(q, |k , a, θ2)
P(q, |k , a, θ1)u(c , a)
]
u(θ1) ≥∑
q,c,k,a
[π (q, c , k , a|θ2)
P(q, |k , a, θ1)
P(q, |k , a, θ2)u(c , a)
]
Adverse selection, two unobserved typescontract properties
Safer types have stricter larger expected output produced at alleffort levels, riskier type have larger probability of failure
Utility offers for types are close due to TTC, but contractcharacteristics and surplus are completely different!
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2−0.2
0
0.2
0.4
0.6
0.8
1
1.2
utility offer for risky type
su
rplu
s
risky type
safe type
1.5 2 2.5 3 3.50.2
0.4
0.6
0.8
1
Consumption
utility offer
co
ns
um
pti
on
risky type
safe type
Adverse selection equilibrium
a menu of contracts offered
utility offers for types has to be in close range to satisfy TTC
due to surplus elasticity bad types get better utility value from beingin adverse selection at the expense of bank profit, for good typesTTC don’t bind
computing: tracing utility of one type, finding utility for related typevia surplus optimality
Welfare implications of financial liberalization experiment
real value computed for risky and safe households at all locations
local two-branch monopoly at fixed locations [1/4; 3/4]
interstate banking allowed under:a) full information competition on contracts; b) adverse selection competition on contracts;
full commitment Nash on prices only
assume one of existing branches with location 3/4 is bought by a challenger;
adverse selection contracts are intertwined (red and blue lines overlap): more welfare for risky type atmedium spatial costs where banks need to attract good types from autarky - results in better contract forriskier types
(a) Full Information
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
spatial cost
valu
e
total household value
competitive risky
competitive safe
monopoly risky
monopoly safe
(b) Adverse Selection
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
spatial cost
valu
e
total household value
competitive risky
competitive safe
monopoly risky
monopoly safe
Local information advantage and type segmentation
Full commitment on contracts and locations with banks operatingin different information regimes with sequential entry
The first entrant (a local bank or an incumbent) comes to the areaand it studies his customers long enough to gain full informationabout their true types establishing relationships in the process (notmodeled)
The first entrant anticipates that there will another player in thearea (a global bank or a challenger) and it commits to location andcontract menu based on established relationships trying to prevent achallenger from taking over his market share
The global bank doesn’t have information advantage that the firstentrant possesses and he operates in adverse selection regime
The global bank already knows both location and contracts offeredby the local bank, so potentially it is capable to undercut thecompetitor if such strategy is profit optimal.
Local Information Advantage: Profits, Locations, andContracts
both local and global bank are left with non-zero profit even at zerospatial costs
overall, at most spatial costs the local bank makes larger profit andhe keeps close to central location
at large spatial cost both are in local monopoly condition
at low spatial costs the local bank increases the gap between offersfor riskier and safer types
the local bank raises utility offer for good clients
the global bank outcompetes local bank on riskier types
the global bank loses all good types since it can’t match the offerfrom the local bank due to TTC binding
the local incumbent don’t compete for riskier types, it postsarbitrary low offer, because the global challenger would raise itsoffer for both safe and riskier types attempting to outcompete, inresult, the local incumbent would lose more on safe type than itgains on riskier types
Local Information Advantage:market segmentation, no logits
(a) Market shares of bank1 (FI)
0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
Spatial cost
Ma
rke
t sh
are
, %
bank1, FI, risky type
bank1, FI, safe type
(b) Market shares of bank2 (AdS)
0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
Spatial cost
Ma
rke
t sh
are
, %
bank2, AdS, risky type
bank2, AdS, safe type
FI vs FI and AdS vs AdS competition shows no such effect
at low spatial costs the incumbent gets exactly 100% of good(safer) types
the challenger gets exactly 100% of bad (riskier) types
Conclusions
more entrants
fixed costs to entry
welfare implications
multi-homing
informal markets
limited commitment
robustness
References
Alem, M. and R. M. Townsend (2014). An evaluation of financial institutions: Impact on consumption andinvestment using panel data and the theory of risk-bearing. Annals Issue of the Journal of Econometrics.
Assuncao, J., S. Mityakov, and R. Townsend (2012). Ownership matters: the geographical dynamics of BAAC andcommercial banks in Thailand. Working paper.
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Appendix: Two Branch Monopoly (collusion):Full Information vs Moral Hazard: locations of branches are symmetric at [1/4; 3/4]
(a) Profit
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Spatial cost
Pro
fit
first−best
moral hazard
(b) Profit maximizing utility offer
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
Spatial cost
Uti
lity
first−best
moral hazard
Appendix: Full commitment with counterfactuals
Convex out of equilibrium response, no logits
the first entrant makes an error in his choice of location
out of equilibrium profit functions are convex for L ∈ (L,∞) -perfect identification for locations and utilities
at higher spatial costs the optimal location lies in continuousinterval x1 ∈ (x , 1
2 ), eventually becomes local monopoly
(a) Profit at L = 1.8
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.25
0.3
0.35
0.4
0.45
0.5
X1
Pro
fit
bank 1
bank 2
MAX bank 1 profit
(b) Profit at L = .5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x1
Pro
fit
bank 1
bank 2
Appendix: No commitment with counterfactuals
Nash surface, logits
we get simultaneous Nash solution
we graph out of equilibrium Nash surface at all possible strategies of bank 1 deviating from Nash
also confirm that we indeed have true Nash by computing this in the space of all strategies of bank 2
fully convex with logits
at larger spatial cost the spatial separation strategy becomes profit optimal
fig. (a) below illustrates typical Nash equilibrium with middle locations, stable over a large range of spatialcosts and tail mass values M, fig (b) show a ridge in the surface that causes spatial separation under largerspatial costs, results are general for M > 0, L > 0 domain
(a) L = 0.5,M = 0.1 (b) L = 2,M = 0.3
Appendix: Costly entry, multiple entry, multi-stage entry
SNE with entry cost
Entry cost cE = 0.3, Spatial Cost L ∈ [0, 1) (entry cost absorbed)
bank location utility offer market share profit1 center max to cover entry 50% 02 center max to cover entry 50% 0
Entry cost cE = 0.3, Spatial Cost L = 2 (deterrence of entry)
bank location utility offer market share profit1 center 2.44 (to prevent entry) 83% 0.182 border 2.44 (doesn’t enter) 0% 0
SNE for three entrants
SNE with simultaneous branch opening
Appendix: computing Nash equilibria:Minimax rank ordering
We propose the following technique to order by rank all possiblestrategies using the metric we call ”distance to Nash”. Both banks enterthe market simultaneously and use a strategy set of G = {u∗1 , u∗2 , x∗1 , x∗2 }with payoff
P1(G ) = S(u∗1 )µ1(u∗2 , x∗2 , u∗1 , x∗1 );P2(G ) = S(u∗2 )µ2(u∗2 , x
∗2 , u∗1 , x∗1 )
At strategy set G in Nash equilibrium there exist no deviating strategysets G1 = {u1, u
∗2 , x1, x
∗2 }, G2 = {u∗1 , u2, x
∗1 , x2} that would provide higher
profit for any of the deviating banks. We can define and compute thefollowing metrics
d(G ,G1) = max(P1(G1)− P1(G ), 0),P1(G1) = S(u1)µ1(u∗2 , x∗2 , u1, x1)
d(G ,G2) = max(P2(G2)− P2(G ), 0),P2(G2) = S(u2)µ2(u2, x2, u∗1 , x∗1 )
Appendix: computing Nash equilibria:Minimization of maximum deviation
First step: we compute P1(G ) and P2(G ) for a trial strategy set of G .Then, in the second stage we solve
maximizeG1
d(G ,G1) s.t P1(G ) > 0 : ∀G1
maximizeG2
d(G ,G2) s.t P2(G ) > 0 : ∀G2
We denote the solution of those maximization problems as d(G ,G1) andd(G ,G2). Then we compute distance to Nash as
d(G ,G1,G2) = d(G ,G1) + d(G ,G2)
In the final stage we solve for
GNash := argminG
d(G) : ∀{G , G1, G2}.
At Nash equilibrium GNash the solution of this two-step optimization problem d(GNash) = 0.At all other strategies this function is strictly positive and well-defined.All possible strategies can be rank-ordered by their ”distance from Nash” even when there is no true Nashequilibrium.
Appendix: Computing solutions to Max problems
general structure: many-layered embedded optimization with linear programming based lotteries at theinnermost basic level and non-linear bank profits optimization on top
for lottery formulation and codes see Alex Karaivanov web page http://www.sfu.ca/ akaraiva/me4.html
the lottery formulation of contract problem for agents is solved using LP solver Gurobi(http://www.gurobi.com/) via MATLAB interface
SIMD processor (http://en.wikipedia.org/wiki/SIMD) optimized arrays with Kronecker tensor products areused throughout to make computations highly efficient
bank profit optimization is done using a globally convergent Augmented Lagrangian Pattern Searchalgorithm
non-linear solver KNITRO (http://www.ziena.com/threealgs.htm) with multi-start option is used forrobustness checks