Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Domestic subsidies, cooperation and coalition
formation
Arghya Ghosh and Manipushpak Mitra
University of New South Wales and Indian Statistical InstituteDecember 2, 2011

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Introduction

With reduction in tariffs, domestic subsidies attracting a lot of
attention (Bagwell and Staiger, AER, 2006)
GATT was relatively relaxed about subsidies, WTO is not
We examine the desirability of subsidies as well as possibilityof cooperation in reducing those in an oligopoly setting.

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Questions

Can unilaterally optimal subsidies be excessive from global
welfare point of view? (Part I)
2 Is [1] just a possibility or fairly robust? (Part II)
3 To what extent countries can cooperate in reducing subsidies?
(Part III)1

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

PapersPapersExport/Domestic Subsidies: Brander and Spencer (1985,
JIE)), Eaton-Grossman (1986, QJE), Dixit(1984, EJ),
Horstmann and Markusen (1986, JIE)
2 Subsidies/domestic policies in context of trade agreements:
Bagwell and Staiger (2006 AER; 2001, QJE), Lee (2011 SMU
working paper)
3 Preferential Trade Agreements: Most speakers in theconference + many more!
4 Coalition formation: To be mentioned later
1

Step backward from [2], get rid of all instruments but
domestic subsidies
Build on [1], [3], and [4] and move towards [2]Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Preliminariesn(≥ 2)-countries, each with unit mass of identical consumers
Numeraire good y and n differentiated products
Country i → firm i → variety i ∈ {1, 2, ..., n} ≡ N
Representative consumer in country i maximizes
U(q1i , q2i , ..., qni ) + yi
subject toPj∈N pji qji + yi ≤ Ii

Demand: qji = qji (p1i , p2i , ..., pni )Inverse Demand: pji = pji (q1i , q2i , ..., qni )Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

An example: Linear, Differentiated Duopoly

U(.) = aPP
1P2j∈N qji − 2j∈N qji − bj∈Nk>j qji qki

PInverse demand: pji = a − qji − b
Products arePhomogenous if b = 1,
independent if b = 0,
imperfect substitutes if b ∈ (0, 1)

k6=j qki = pji (qi )Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Production

Markets are segmented
mji : constant marginal cost of firm j to sell in market imji = m for all i, j ∈ N if there are no subsidies
Government i gives s i per unit of qii
Effectively, mji = m − s i if j = i and mji = m if j 6= i

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Profit maximization

Segmented markets + constant marginal cost ⇒ n separate
profit maximization problems for each firm.
In market i, firm j chooses qji to maximize
πji= [pji (qi ) − mji ]qjiEquilibrium outcomes for i, j ∈ N :Quantities: qji (s i )Prices: pji (s i )Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Comparative statics

RESULT 1: Let qji (s i ) and pji (s i ) respectively denoteequilibrium output and price of variety j in market i. An
increase in s i leads to higher qii . All other prices andquantities decline.
dqji (s i ))dqii (s i )
>
0,
<0
ds i
ds i
dpij (s i )dpii (s i )
<
0,
<0
ds i
ds i

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Profits, Consumer surplus and welfare
Profit in own market i : πii (s i ) = (pii (si ) − (m − s i ))qji (s i )Profit in foreign market k: πik (s k ) = (pik (sk ) − m)qik (s k )Consumer surplus:
cs i (s i ) = U(q1i (s i ), q2i (s i ), ...., qni (s i )) −X

pji (s i )qji (s i )j∈NCorresponding to s ≡ (s 1 , s 2 , ..., s n ), welfare of country i is:X
W i (s) ≡ csi (s i ) + πii (s i ) +πik (s k ) − s i qii (s i )k6=i= [U(qi (s i )) − mqii (s i ) −X
j6=ipji (s i )qji (s i )] +X
k6=iπik (s k )Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Nash subsidies

Each country i chooses s i to maximize W i (s)iFirst-order condition, dWds i(s) = 0, does not involve s k (k 6= i)⇒ for all s k , the same s i = s N (say) is the best response(Segmented markets strikes again!)Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Nash subsidies (contd.)

Nash subsidy s N is implicitly given by(pii (s N ) − m)dqii (s N ) X dpji (s N )−qji (s N ) = 0.
ds i
ds i
j6=i

Starting from pii (s i ) − m = 0, consider a small ↑ in s iiisecond-order loss: (pii − m) dq≈0
ds i

first-order gain: −⇒ pii (s N ) − m < 0

dpji
j6=i ds i qji > 0

P(1)

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Efficient subsidies

Efficient subsidy vector s ≡ {s 1 , s 2 , ..., s n } maximizesPnii=1 W (s).
Effectively n separate welfare maximization problems
Choose s i to maximize welfare in each market i:
X
cs i (s i ) + πii (s i ) +πji (s i )j6=iSymmetry ⇒ efficient s i should be the same for all i ∈ N .

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Efficient subsidies (contd.)

Efficient subsidy s E is given by(pii (s E ) − m)dqji (s E )dqii (s E ) X
E
+
(p
(s
)
−
m)
= 0.
ji
ds i
ds i
j6=i

Starting from pii (s i ) − m = 0, consider a small ↑ in s i :iisecond-order loss: (pii − m) dq≈0
ds i

first-order gain:⇒ pii (s E ) − m > 0Pdqji
j6=i (pji − m) ds i

>0

(2)

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Nash versus efficient outcomes

Result 2: Nash subsidies are excessive, i.e. s N > s E .
Proof:(i) pii (s E ) > m > pii (s N )(ii) pii is ↓ in s i(i) + (ii) ⇔ s E < s NIntroduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Nash versus efficient outcomes

Compared to efficient outcomes, under Nash we have
Excessive subsidies: s N > s E
Lower trade volume: qjiN < qjiE ;Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Part II: Robustness, the usual suspects

Our results are robust with respect to1
2
3functional forms
mode of competition
number of firms

Just as domestic subsidy is optimal under (1)-(3) in a closed
economy setting, our results concerning excessive domestic
subsidies are also robust under (1)-(3)Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Cost heterogeneity: the new suspect in town
Uniform subsidy no longer optimal
12s ↑⇒ allocative efficiency improves⇒ production efficiency worsens

For each national variety i, suppose there are two firms:
L(ow-cost) and H(igh-cost).
qLipi − cLi + s
cH i − cL
=
=1+
qH i
pi − cH i + s
pi − cH i + ss ↑⇒ pi ↓⇒ pi + s ↑⇒ qqHL ↓ market share of low-cost typedecreases.
Nash subsidies are more likely to be excessive in a Melitz-typeworld when only L type serves the foreign market
Country i takes the productive efficiency margin into accountto the extent that its Hi steals business from Li , but it ignores
business stealing from foreign L. .

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

From s N to s E ?

Can countries agree on s E ?
To answer that we need to let the countries do something.
In trade agreements countries have bilateral instruments tij
Here each country i has only one instrument s i
Coalition formation
Each coalition chooses the subsidy level of its members tomaximize the sum of welfare of coalition members
Product market competition

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Broad brushInterested inenvironments with externalities
possible group formation to tackle the externalities
role of rules/institution

ExamplesPublic goods/bads and voluntary contribution (Public
Economics),
2 Mergers, Research joint ventures (IO)
3 Preferential/regional trade agreements (primarily on tariff
reduction)
1

Domestic subsidies, works differently from [3]. Similarity with
[1] and [2] will be clear in that later part of the talk.

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Definitions
Partition n countries into m sets of countries C1 , C2 , ..., Cm
Non-empty:Ck 6= φ,Tm
Non-overlapping: k=1 Ck = φ
SMExhaustive: k=1 Ck = N .

Each nonempty C ⊆ N is a coalition and each partition
σ = {C1 , C2 , ..., Cm } represents a particular coalition
structure.
σ(i): coalition that country i belongs to|σ(i)|: size of σ(i).
Example: N ≡ {1, 2, 3, 4, 5, 6}(C1 , C2 , C3 ) ≡ {{1}, {2, 3}, {4, 5, 6}}
σ(3) = C2 ; |σ(3)| = |C2 | = 2

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Let Σ denote the set of all possible coalition structures.
Consider a σ ∈ Σ and pick a C ∈ σ. Let c = |C | denote thenumber of countries in C . Each coalition C chooses(s 1 , s 2 , ..., s c ) to maximizeX
W C (s, σ) ≡W i (s, σ)i∈CSymmetry + segmented markets imply
1 s i is the same for all i ∈ C
2 s i is independent of coalition structure though W i depends on
subsidies offered by non-members of C .

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Stage 2: Size of coalition, subsidy and welfare
Result 3: Optimal subsidy, welfare per member lower inlarger coalition.
Let s(σ) denote the subsidy vector generated from suchcoalitional choices, s i (σ) denote the subsidy chosen bycountry i and let v i (σ) ≡ W i (s(σ), σ) denote the welfare ofcountry i. Then|σ(i)| < |σ(j)| ⇔ s i (σ) > s j (σ)⇒ v i (σ) > v j (σ)
More members in i’s coalition ⇒ more concern for πji ⇒ lowersiNote, members outside of the coalition benefit as well

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Asymmetric coalitions

Claim 1: The equilibrium coalition structure cannot have two
coalitions of same size.
Proof:
Consider i, j such that j ∈/ σ(i) but|σ(i)| = |σ(j)|.
Thens i (σ) = s j (σ)
Apply revealed preference argument
Note subsidy offered by the other coalitions do not changeIntroduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

To merge or not to merge?

Claim 2: Consider S(mall), B(ig ) ⊂ N where |S| < |B|.
If |B| < 2|S|, members in both coalitions are better off under a
new coalition S ∪ B irrespective of the differentiation
parameter b ∈ (0, 1).
If |B| > 2|S|, members in smaller coalition S are better off bystaying separate for b < 34

Thus the result is ”if and only if” for b ∈ (0, 34 )For the remainder of talk we focus on b ∈ (0, 34 )

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Ideal world

Ideally, an equilibrium coalition structure should be
Merger proof: no two coalitions have incentive to merge
togetherSplit proof: no individual country has an incentive to split
from a coalition

By definition
Grand coalition is merger proof since no further merger ispossible
Coalition structure of all singleton is Split proof since nofurther split is possible

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Non-existenceN ≡ {1, 2, 3, 4, 5}Merger proof: N ; ({1}, {2345})
But neither of the above is split-proof
Consider unilateral splitting by country 5
The group it splits from remains at least 3 times large (after
splitting)than {5}So, splitting gives higher payoff to country 5

Split proof rules out grand coalition almost by definition(except for small n), while merger proof always involve grandcoalition.
We look at the set of merger proof coalition structures; so
grand coalition remains a possibility
Select one that survives an extensive-form based
charaterization.

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Towards selectionn = 1 ⇒ (i)Nn = 2 ⇒ (i)Nn = 3 ⇒ (i)Nn = 4 ⇒ (i)N ; (ii)({1}, {234})
n = 5 ⇒ (i)N; (ii)({1}, {2345})
n = 6 ⇒ (i)N ; (ii)({1}, {23456})
n = 7 ⇒ (i)N ; (ii)({1}, {234567}, (iii)({12}, {34567})
n = 8 ⇒ (i)N;(ii)({1}, {2345678}), (iii)({12}, {345678})
Observe: Grand coalition forms for n = 1, 2, 3, 5, 8, ....

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Fibonacci sequence and Fibonacci decomposition
Fibonacci numbers form a sequence defined by the following
recursion relation: F0 = 1, F1 = 2 andFk+2 = Fk+1 + Fk
Let F = {F1 = 1, F2 = 2, F3 = 3, F4 = 5, F5 = 8, F6 =13, F7 = 21, F8 = 34, . . .} denote the set of Fibonacci
numbers.
The Fibonacci decomposition of any positive integer m, is aunique (in some sense minimal) expression of the formm = Fp(1) + . . . + Fp(L)where Fp(l) ∈ F for all l ∈ {1, . . . , L} and p(l + 1) ≥ p(l) + 2
for all l ∈ {1, . . . , L − 1}.
To illustrate take the number of attendees and find thedecomposition.

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

A characterization
Claim: Fibonacci decomposition of n is merger proof for all n.
Proof:
Fibonacci decomposition of n does not contain consequent
Fibonacci numbers (as it would not have been minimal then)e.g., 18 = 13 + 5 (skips 8), 27 = 21 + 5 + 1(skips 13,8,3,2)
Adjacent (non-adjacent) numbers in Fibonacci sequence differ
by less (more) than a factor of two
Fk+2= Fk+1 + Fk < Fk+1 + Fk+1 < 2Fk+1= Fk+1 + Fk > Fk + Fk > 2Fk

Tying Back: A coalition of size Fk always merges with acoalition of size Fk+1 but not with the one with size Fk+2 orgreater.

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Sequential formation of coalitions in presence of
externalities

Technique
Bloch (GEB, 1996)
Ray and Vohra (JET 1997, GEB 1999)

Applications
Bloch (Rand,1995), Ray and Vohra (2001)

Similar ideas in simultaneous form games (Yi, 1996 JIE; ?
GEB )Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Game 1: A bargaining game
One country (say 1) first proposes a coalition C and eachprospective member responds
If all accept, the coalition is formed; and the procedure isrepeated among the remaining firms and country with lowest ichosen as the new initiator
If one of the prospective members of C rejects, then she
initiates the next proposal and so on...
Countries while accepting or rejecting makes a prediction
regarding complementary coalitions(which in equilibrium must
be correct)
Coalition structure forms when all agreements conclude
If bargaining continues forever, countries get zeroIntroduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

The bargaining game (contd.)
Restrictions
Unanimity
Once a coalition is formed, it cannot change its
composition(restrictive)
3 Sequential formation
1
2Richness
Forward looking nature of the game
Payoffs in our game derived from a game with a optimal policy
choice and market competition.

Pay-off relevant state at any stage of the game: ongoingproposal, the structure already formed
Consider Markov strategies and find Markov perfectequilibrium

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Game 2: Choice-of-coalition size game
Denote the game by ∆(n, v )n: number of countries;
v is the vector of welfare valuations vi0 s where i ∈ N .

Country i is ranked i-th in the order of moves.
The game ∆(n, v ) is as follows:
Country 1 starts the game and chooses an integer
c1 ∈ {1, . . . , n}.
Country c1 + 1 then moves and chooses c2 ∈ {1, . . . , n − c1 }.
Country c1 + c2 + 1 chooses c3 from {1, . . . , n − c1 − c2 }...

The game continues until
PM the sequence of integers(c1 , . . . , cM ) satisfies m=1 cm = n.
Find the subgame-perfect equilibrium of this game
Equilibrium coalition structures are identical upto a
permutation of countries in Game 1 and Game 2.
Outcome of ∆(n, v ) is a decomposition of the number n.

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Strategy in the game of choice of coalition size

A strategy τi for player i in the game ∆(n, v ) is a mappingfrom the set Σi−1 to the set of integers {1, . . . , n − i + 1}.
For any coalition structure σi−1 of the first i − 1 countries,
country i chooses a coalition size τi (σi−1 ).
Note that all players need not be called to announce coalition
sizes in the gameFor any strategy profile τ , a single coalition structure σ(τ ) is
formed and country i receives a payoff of vi (σ(τ )).

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Subgame-perfect equilibrium and equilibrium Coalition
Structure

A strategy profile τ ∗ is a subgame perfect equilibrium if andonly if ∀ i ∈ N , ∀ σi−1 ∈ Σi−1 and ∀ τi ∈ {1, . . . , n − i + 1},∗ )) ≥ v (σ(τ (σ∗vi (σ(τi∗ (σi−1 ), τ−iii i−1 ), τ−i )).
A coalition structure σ(τ ∗ ) generated by a sub game perfectequilibrium τ ∗ is called an equilibrium coalition structureIntroduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Equilibrium Coalition Structure: partial characterization
Result 5: Assume quadratic utility function
and Cournot
competition among firms. For b ∈ 0, 43 ,the unique numerical equilibrium coalition structure for anyn(≥ 2) is the Fibonacci decomposition for n, andgrand coalition forms in equilibrium if and only if n is a
Fibonacci number.

What about b ∈ (3/4, 1)?
Ongoing: Last attempt at recurrence relationships (inparticular, Lucas sequence) before we turn to simulations.
The fact that calculations are falling in line with recurrence
relationships is not completely coincidental..
Welfare expressions are quadratic
Fibonacci and few other popular sequences in number theory
are second-order difference equations

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

Lessons from coalition formation game and beyond
Fibonacci

Endogenous Majority: There always exists a coalition withmajority of members
Free-riders: Almost always, there will always be someGrand Coalition: Always a subsequence where grandcoalition forms (things fall in place)
Things not efficient, but may not be too far off
Plurilateral arrangements

Introduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

1. Why are we concerned about subsidies now?

Requires multiple instruments or at least trade cost
Does t ↓⇒ s N (t) − s E (t) ↑?
More importantly, does t ↓⇒ W (s E (t)) − W (s N (t)) ↑?
At some level, yes, if one compare the two extremes(prohibitive tariff and trade)
Is it possible that Nash subsidies were insufficient before but
excessive now?
2 If there is a choice of choosing one instrument first, and one
later, is this the order — tariff negotiations first, subsidies
next?
1

Models with trade cost + heterogeneity show some promiseIntroduction

The Model

Optimal Subsidies

Coalition formation

Additional Questions/Concerns

2. Entry
Tricky; in presence of entry, subsidy is not necessarily positiveThree things are at work
1
2
3

Allocative efficiency
Production efficiency
Variety

Subsidy improves [1] and usually [3] as well, but worsens [2] inany environment where N ↑ leads to q ↓ [business-stealing
effect]Homogenous products Cournot: Optimal subsidy is zero forlinear demand
Entry in oligopoly trade models rare, more so in context oftrade agreements
Venables (1985, JIE), Horstman and Markusen (1986, JIE),
Bagwell and Staiger (forthcoming 2010, JIE)
Our conjecture: results will depend on the particular way theentry is modelled — entry of new varieties or entry of new
firms for a given variety