Game Theoretical Discussion of Free Trade

In a previous article, I outlined how it is easy to prove mathematically that quotas reduce overall welfare, because the cost to consumers overwhelms the benefits gained by domestic firms. Given this fact, and that the benefits are more spread out, one would initially expect that free trade policies would be universal, at least in democracies. But this is hardly the case. A look at the news over the past decade is evidence of the mixed record of the world when it comes to free trade. Steel tariffs, French cheese protection, and import substitute industrialization are all examples of protectionism trumping protectionism. In this article I will argue that this empirical result can be explained by the distributional effects of free trade and their interactions with the different domestic institutions of nations. I will approach this explanation in a game theoretical context.

What was obvious from the previous article was that the total consumer welfare loss was greater than the total producer welfare gain. However, with almost every good that is produced, the number of consumers greatly outnumbers the number of producers. As a result, the per person consumer welfare loss will be far less than the per firm producer welfare gain. In other words, protectionism concentrates benefits and spreads out costs, and free trade spreads out benefits and concentrates costs.

As a result, consumers face a free rider problem. Any consumer lobbying group will then have to overcome steep cooperation problems. Because any one consumer has a strong incentive to defect, as his single deviation will not usually change the outcome of any lobbying effort. Alternatively, if a person is the only person paying the lobbying cost, even if victory is achieved, the benefits to that individual will likely not even come close to the lobbying costs.

This means that consumers face a prisoner’s dilemma. It can be modelled by:

All Other Consumers
Individual Consumer Lobby Not Lobby
Lobby T-L,N(T-L) -L,0
Not Lobby T,N(T-L) 0,0

Because this is a multiple person game, I have aggregated all consumers besides the test case being examined. As a result, this model does not appear to be a prisoner’s dilemma. However, as can be seen, for any individual consumer not lobby dominates lobby regardless of the actions of everyone else acting as a unit. It can then be reasoned that not lobbying will be the equilibrium for everyone, and the consumers are stuck in a position where they would be all better off if they could insure cooperation.

For firms, the situation is much more likely to be a stag hunt. Individual contributions to lobbying will make a difference in the end result, even if all other firms are assumed to be acting as a single unit. This is even more likely to be the case when a country has regional electoral districts that subdivide the country, like US Congressional Districts. Smaller sub-regions will have fewer competing firms in an industry, and politicians elected from these regions are more likely to care about these firms because they employ a large portion of the local population.

Empirically, there are many implications of the model I presented before. The first is that countries with many electoral districts will be more likely to undertake protectionist policies, and countries with large districts, like Israel, will be more likely to favor free trade. Additionally, politicians elected from broader bases of supporters will be more likely to support free trade. This is likely why the president of the United States has historically supported free trade more than individual congressmen.

In a future article, I will explore the implications that information access has on free trade.


The Case for Free Trade: Maximizing Total Welfare


One of the most contentious issues in the world today is that of free trade. For centuries, governments have undertaken efforts to protect domestic industries by erecting trade barriers. What is interesting is that a simple model shows that the gains of free trade far outweigh the gains of protection. So given that a government is acting rationally to raise the total wellbeing of the nation as a whole, it should easily decide to remove tariffs, quotas and other protections. In this article, I will prove that in a simplified example using a supply and demand model, there are high costs to protectionism and therefore net gains when a country pursues free trade.

In order to prove that the gains of free trade are greater than the cost, I will model a situation where the good in question is apples. To simplify the problem, I will aggregate the supply and demand functions. I will examine a single country, which I will call Protania, which under free trade will have an equilibrium where its’ domestic demand for apples is supplied by both domestic producers and foreign producers. Finally, I will assume that the domestic firms are inefficient with respect to foreign firms, either due climate reasons, technological lags, or some other reason. Because of this, it follows that it is safe to assume that these foreign firms supply apples to their home countries.  As a result, Protania’s producers do not export any apples. So the domestic demand of foreign countries can be left out of the following thought experiment.

Under free trade conditions, with no quotas or restrictions, here are the demand and supply curves, and the equilibrium point:

Aggregate Foreign and Domestic Supply(AS): image001

Foreign Supply[1]image003

Domestic Supply:image005

Aggregate Domestic Demand (AD): image007

Equilibrium Quantity and Price under free trade (EFT): image009

A strong measure of welfare of the different groups are consumer and producer surplus.

Consumer surplus (under free trade):  image011

Producer surplus: image013

Domestic Producers’ Surplus[2]image015

Foreign Producers’ Surplus: image020

One of the simplest protectionist policy tools is the use of quota on imported goods. In the free trade equilibrium within my example, it is easy to see that foreign producers are earning a higher surplus, and that domestic producers could benefit from a quota system that protects them from more efficient foreign competition.

Let us suppose that the government initiates a quota. In order to assume a reasonable threshold for this quota, it is important to note that a government will likely allow some level of foreign imports. In other words, the quota will not be zero. However, the government would not initiate the quota unless it hoped to have some effect on production domestically, and as a result it will set the quota on imported apples at a level that is lower than the number of apples imported in equilibrium under free trade. In simpler terms, this means that the quota must be greater than zero but less than 31.5 apples. For the sake of simplicity, I will arbitrarily pick 15 as the maximum number of apples allowed to be imported.

This quota will change the supply function of the foreign firm into a piecewise function that is defined as  image021 and q=15 when quantity is greater than 15. In order to apply this effect to the greater picture, I integrated the foreign producer’s  supply equation with the domestic producer’s, to achieve the following supply equation:



The demand equation remains the same, and the resulting equilibrium occurs on the second interval of the supply equation, at the point where quantity is 37.5 apples and the price is $8 an apple. As can be seen, the price has risen dramatically, and the quantity has fallen significantly. But this was expected, as any reduction in supply must result in higher prices thus less quantity. The interest part is whether consumer and producer surplus have changed.

Consumer Surplus:  image027

Producer Surplus:  image029

Domestic Producers’ Surplus[3]image031

Foreign Producers’ Surplus: image033

Total Surplus with quota=$369.63


Free Trade Quota System Net Change
Total Producer Surplus $81 $144.63 +$63.63
Total Foreign Producer Surplus $56.7 $87.93 +$31.23
Domestic Producer Surplus $24.3 $56.7 +$32.4
Total Consumer Surplus $324 $225 -$99
Total Surplus $405 $369.63 -$35.37


Given the two policy options, the above table shows that free trade maximizes total welfare. Further, if it is accepted that there are more consumers of a good than producers, than it is also true that free trade also spreads this welfare most evenly among the most people. Intuitively, the qualification of consumers being more numerous than producers is true for almost all markets. Regardless of whether a government values equality or efficiency, free trade seems to be the best route. It would appear that in almost every case, we would expect governments to pick free trade.  The paradox is that governments choose protection more often than not. This is where cooperation problems are powerful explanatory models. In a future article, I will go more in depth into the strategic interactions behind free trade lobbying.

[1] As mentioned earlier, domestic supply and foreign supply equations are modified so as to be much less efficient at producing. This means that for a given price, foreign firms produce much more than domestic firms.

[2] 13.5 is found by finding the quantity that the domestic firm will produce at the given price of $5.60 in the overall market.

[3] 22.5 is the quantity produced by the domestic firms with the equilibrium price of $8.