First, we allow arbitrary weights on consumer surplus, firms’ profit and tariff revenue and define a more general social welfare function from (12) with:
Yi(L) = β(1
2qi2(L)) + γ( X
j∈Ni(L)
(α − c − qj(L) − tij(L)) · qji(L)) + δ( X
j∈Ni(L)
tji(L) · qij(L)).
(13) In the framework of section 2 the welfare function places equal weight on profit, consumer surplus and tariff revenue with β = γ = δ = 1. Here we assume that countries are symmetrical with respect to market size. In a political economy context we might be interested in what structures will emerge when the objective function depends only on firms’ profit which implies γ = 1 and β = δ = 0. Henceforth, social welfare is given by:
Yi(L) = X
j∈Ni(L)
(α − c − qj(L) − tij(L)) · qji(L).
One observation that can be made is that firm j’s profit in market i decreases with the number of firms that are active in market i. Let us assume i and j have a bilateral trade agreement and market i forms a bilateral link with k. This reduces country j’s welfare since welfare is given exclusively by firm profit. This observation provides the intuition for the next result.
Proposition 4.6. When countries only care about producer profit the only stable trading systems are free trade and the empty trading system.
First we investigate welfare in the empty trading system. Firm i’s profit is given by (α−c)4 2 since it only supplies to market i. When countries only care about producer profit it can be shown that starting with an empty trading system no pair of coun-tries has an incentive to form a link. The additional loss in profit due to increased competition is higher than the additional profit in the foreign market. This suggests that the empty trading system is stable.
What can also be observed is that under the complete trading system no country i, i ∈ N will sever any of its links with country j, j 6= i ∈ N , since the reduction of profits in market j is higher than the additional profit obtained due to lower competition in the domestic market. Complete proof is provided in the appendix.
Next we investigate what structures can emerge when welfare is given by consumer surplus such that β = 1, γ = δ = 0. Country i’s welfare is now:
When we consider a trading system without the global link we can show that for an arbitrary network L consumer surplus from an additional bilateral trade agreement between country i and country j with {(i, j)} 6∈ L is given by
Yi(L ∪ {{i, j}}) − Yi(L) with a bilateral link between i and j. This implies that an additional bilateral link is always profitable and countries form as many links as possible.
When countries have a multilateral trade agreement social welfare is given by:
Yi(L) = 1
2(n(α − c)
(n + 1) − (n − ˜ηi(L)) · t (n + 1) )2,
where ˜ηi(L) denotes the number of countries that are bilaterally linked with country i under MFN where i ∈ ˜ηi(L) and ˜ηi(L) =| ˜Ni(L) |.
The first derivative implies: ∂Y∂ ˜ηi(L)
i(L) = (n(α−c)(n+1) − (n− ˜(n+1)ηi(L))·t) · (n+1t ) > 0 with t < (α−c)3 . It is attractive for countries to form as many bilateral trade agreements as posssible under MFN.
Proposition 4.7. When social welfare is given by consumer surplus, global free trade is stable.
In the following we will assume that countries’ welfare is given by tariff revenue such that
Yi(L) = X
j∈Ni(L)
tji(L) · qji(L).
Since the tariff between two countries in a bilateral link is zero and between two countries in a multilateral link is t, with t > 0, we have Yi(L) = (ηi(L) − ˜ηi(L)) · t · (η α−c
i(L)+1 + (ηi(L)− ˜ηi(L))·t−(ηη i(L)+1)·t
i(L)+1 ), where ˜Ni(L) denotes the set of countries that have a bilateral trade agreement with country i under the global link, i ∈ ˜Ni(L) in the trading system L. We observe that ∂Y∂ ˜ηi(L)
i(L) < 0 such that welfare decreases with the number of bilateral links. It is therefore intuitive that the global trading system maximizes welfare and the proof can be omitted. The severance of the global link results in the empty network and tariff revenue for each country is zero. Under these considerations we can conclude:
Proposition 4.8. When countries only care about tariff revenue, the only stable trading system is the global network.
Next we combine the analysis and allow arbitrary values for β, γ and δ. Therefore, under the general welfare function as given in (13):
Proposition 4.9. Under general welfare as given by (13) the empty network, the global network and global free trade can be stable.
This result follows directly from Propositions 4.6. − 4.8..
Intuitively, for certain values of δ, countries tend to maintain as few bilateral free trade agreements as possible and maintain the global link to receive as many tariff revenues as possible. The lower δ and the higher γ countries prefer no links at all, since the additional competition will decrease firms’ profit and therefore countries will sever all their links. Intuitively, for very high values of β additional competition in the markets is profitable for consumer surplus and therefore countries will form the complete network. We can further show that a star network and the global link with a star network cannot be stable. However, the next two examples show that a
bilateral free trade agreement and a global link with a free trade agreement between one pair of countries can be stable.
Example 4.1. We set β = 12, γ = 1 and δ = 14. With (α − c) = 4 and t = 1 we get Yi({{i, j}}) = Yj({{i, j}}) = 163 > 5 = Yi(Le) = Yj(Le). Furthermore, we have for country k with k 6= i 6= j that Yk({{i, j}}) = 5 > Yk({{i, j}, {i, k}}) = 656144 and Yk({{i, j}}) = 5 > Yk({{i, j}} ∪ LG) = 6716 such that the conditions for multilateral stability of L = {{i, j}} are fulfilled.
Example 4.2. We set β = 12, γ = 12 and δ = 34. With (α − c) = 4 and t = 1 we have Yi({{i, j}} ∪ LG) = Yj({{i, j}} ∪ LG) = 24164 > 31116 = Yi(LG) = Yj(LG) and Yi({{i, j}} ∪ LG) = Yj({{i, j}} ∪ LG) = 24164 > 6418 = Yi({{i, j}}) = Yj({{i, j}}) such that neither country i nor country j has an incentive to sever the global link.
Furthermore, we can show for country k with k 6= i 6= j that Yk({{i, j}} ∪ LG) =
56
16 > Yk({{i, j}, {i, k}} ∪ LG) = 32564 and Yk({{i, j}} ∪ LG) = 5616 > Yk({{i, j}}) = 3 such that country k has no incentive to deviate. This proves that L = (LG∪ {{i, j}}) is stable.