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