This article is translated from a Chinese article on my Zhihu account. The original article was posted at 2021-04-26 14:27 +0800.

The general model

The model is as follows. There are nn agents (nations), they can trade some type of good, and they use the same currency. Every agent may produce or consume the good. The benefit function of the jjth agent is BjB_j, and the cost function is CjC_j. The amount of export from the jjth agent to the kkth agent is Tj,kT_{j,k}. The amount of trade cost by the jjth agent is SjS_j. Now, we want to find the amount QjQ_j that every agent produce and the amount Tj,kT_{j,k} that every agent import from other agents. Assume that SS is only related to TT and does not depend on QQ. Also, assume that there is no externality (i.e. whenever jkj\ne k, kBj=0\partial_kB_j=0 and kCj=0\partial_kC_j=0). Also, assume that every agent is rational and with perfect information.

Now, consider the profit Πj\Pi_j of the jjth agent. Subtract the cost from the benefit, and we have

Πj=Bj ⁣(Qj+kTj,k)Cj ⁣(Qj)Sj ⁣(T).\textstyle \Pi_j=B_j\!\left(Q_j+\sum_kT_{j,k}\right)-C_j\!\left(Q_j\right)-S_j\!\left(T\right).

According to the fundamental theorem of welfare economics, TT and QQ is Pareto optimal under market equilibrium. We assume that this case happens at the stationary point of the social benefit, and the social benefit is the sum of the profit of every agent. We can then get the equations

0=QljΠj=Bl ⁣(Ql+kTl,k)Cl ⁣(Ql),l;0=Tl,kjΠj=Bl ⁣(Ql+kTl,k)Bm ⁣(Qm+kTm,k)jSjTl,m ⁣(T),l<m.\begin{align*} &0=\frac{\partial}{\partial Q_l}\sum_j\Pi_j =B_l'\!\left(Q_l+\sum_kT_{l,k}\right)-C_l'\!\left(Q_l\right),\quad\forall l;\\ &0=\frac{\partial}{\partial T_{l,k}}\sum_j\Pi_j =B_l'\!\left(Q_l+\sum_kT_{l,k}\right)-B_m'\!\left(Q_m+\sum_kT_{m,k}\right) -\sum_j\frac{\partial S_j}{\partial T_{l,m}}\!\left(T\right),\quad\forall l<m. \end{align*}

Here are n+n(n1)2n+\frac{n\left(n-1\right)}2 equations, and exactly QQ and TT have n+n(n1)2n+\frac{n\left(n-1\right)}2 degrees of freedom in total (note that TT is anti-symmetric). In principle, we are able to solve QQ and TT.

Zero trade cost

For the case where there is no trade cost, we can see that the domestic prices are all equal, and the price may be called the world price.

However, given S=0S=0, the equations above are not independent. Actually, there are only 2n12n-1 independent equations (all 2n2n components of BB' and CC' are equal). This means that, for n>2n>2, the free trade with zero trade cost is an indeterminate system.

This phenomenon looks counter-intuitive, but it is actually understandable: under zero trade cost, every two agents may trade arbitrary amount of goods under the same world price, this provides extra degrees of freedom to the model. To be specific, if (Q,T)(Q,T) is a solution to the model, then (Q,T+ΔT)(Q,T+\Delta T) is also a solution, where the anti-symmetric matrix ΔT\Delta T satisfies

kΔTj,k=0,j,\sum_k\Delta T_{j,k}=0,\quad\forall j,

where there are n1n-1 independent equations in the nn equations. Therefore, the total number of degrees of freedom in the solution of the model is


Now, the useful quantities that we can solve is the production and the net-import TjkTj,kT_j\coloneqq\sum_kT_{j,k} of every agent. Note that the net-import actually has only n1n-1 degrees of freedom because of the restriction jTj=0\sum_jT_j=0.

The middleman (re-exportation)

It is worth pointing out that the existence of the middleman or re-exportation is completely due to the presence of trade cost. Here we consider a simplified problem: there are three agents playing respectively as the producer, the retailer, and the customer. The producer does not consume (the benefit is 00); the customer does not produce (the cost and the marginal cost is infinity); and the retailer does not produce or consume. Assume that the trade between any two of them does not bring cost to the third one. Then, the social benefit is

Π=B ⁣(Tc,r+Tc,p)C ⁣(Tc,p+Tr,p)Sc ⁣(Tc,r,Tc,p)Sr ⁣(Tc,r,Tr,p)Sp ⁣(Tc,p,Tr,p).\Pi=B\!\left(T_{\mathrm c,\mathrm r}+T_{\mathrm c,\mathrm p}\right) -C\!\left(T_{\mathrm c,\mathrm p}+T_{\mathrm r,\mathrm p}\right) -S_\mathrm c\!\left(T_{\mathrm c,\mathrm r},T_{\mathrm c,\mathrm p}\right) -S_\mathrm r\!\left(T_{\mathrm c,\mathrm r},T_{\mathrm r,\mathrm p}\right) -S_\mathrm p\!\left(T_{\mathrm c,\mathrm p},T_{\mathrm r,\mathrm p}\right).