## Definition

The comparative advantage of an organization (individual, firm, or country) is in the activity that the organization can do with the maximum difference between the benefit and the opportunity cost. In other words, it is what the organization has the most advantage relative to other things the organization can do, and relative to things that others can do.

According to a neoclassical economics viewpoint, if organizations choose to work according to their comparative advantage, then social utility (and hence, social surplus) is maximized.

There are two related, but distinct, aspects of comparative advantage:

1. The law of comparative advantage itself, which offers a guide to what a particular organization should do or how certain activities should be allocated between organizations.
2. The systemic processes through which the law of comparative advantage gets implemented, i.e., the process by which each organization acquires the knowledge and incentives to perform the activities in which it has comparative advantage.

## History

The theory of comparative advantage was first formulated by David Ricardo in the beginning of the nineteenth century. Hence, it is also called the Ricardian theory of comparative advantage.

## Minimal example

### Structure of the example

In the example, there are two organizations $A$ and $B$ and two activities $M$ and $N$. Each organization has to choose one activity and each activity must be done by one organization. In other words, there are two possible allocations that need to be decided between:

• $A$ does $M$, $B$ does $N$
• $A$ does $N$, $B$ does $M$

The law of comparative advantage helps to determine which of these allocations is better (in the sense of maximizing the social utility).

### Utility-based analysis

The analysis presented here is rooted in neoclassical economics. Many features of this analysis do not make sense from the perspective of the Austrian school.

• $A$ can generate a utility of $f(A,M)$ doing $M$ and a utility of $f(A,N)$ doing $N$.
• $B$ can generate a utility of $f(B,M)$ doing $M$ and a utility of $f(B,N)$ units per hour doing $N$.

The two possible allocations give the following total social utilities

• $A$ does $M$ and $B$ does $N$: The social utility is $\! f(A,M) + f(B,N)$.
• $A$ does $N$ and $B$ does $M$: The social utility is $\! f(A,N) + f(B,M)$.

If $\! f(A,M) + f(B,N) > f(B,M) + f(A,N)$, then the first allocation is optimal. If $\! f(B,M) + f(A,N) > f(A,M) + f(B,N)$, then the second allocation is optimal. This has three equivalent interpretations:

Condition for $A$ to do $M$ and $B$ to do $N$ Verbal formulation of interpretation
$\!f(A,M) + f(B,N) > f(B,M) + f(A,N)$ We are comparing the total utility of each allocation and choosing the one with higher total utility.
$\!f(A,M) - f(A,N) > f(B,M) - f(B,N)$ Here, we are treating $N$ as the alternative activity and comparing (Between $A$ and $B$), the gain in doing $M$ over and above the opportunity cost. The organization that gets to do $M$ is the one with the maximum benefit over the opportunity cost.
$\! f(A,M) - f(B,M) > f(A,N) - f(B,N)$ Here, we are comparing the absolute advantage of $A$ over $B$ in both $M$ and $N$ and allocating to $A$ the activity where the absolute advantage is greater.

### Comparison with other rules: utility-based analysis

Comparative advantage differs from two other possible allocation rules: each organization does what it is best at and each activity is done by the organization that is best at that activity. The problem with the first allocation rule is that it may allocate both organizations to the same task, and the problem with the second allocation rule is that it may allocate both tasks to a single organization. On the other hand, the law of comparative advantage always makes sense and maximizes total output.

Note, however, that if either of those rules does give a unique allocation, then that agrees with the allocation given by comparative advantage.

These two rules are contrasted with comparative advantage below:

Informal description of rule Mathematical condition for $A$ to do $M$, $B$ to do $N$ Corresponding description of comparative advantage rule Key difference
Each organization does what it is best at $\! f(A,M) - f(A,N) > 0 > f(B,M) - f(B,N)$ $\! f(A,M) - f(A,N) > f(B,M) - f(B,N)$ Comparative advantage is only a comparative rule and does not require anything about the signs of the expressions $f(A,M) - f(A,N)$ and $f(B,M) - f(B,N)$
Each activity is done by the organization best at that activity $\! f(A,M) - f(B,M) > 0 > f(A,N) - f(B,N)$ $\! f(A,M) - f(B,M) > f(A,N) - f(B,N)$ Comparative advantage is only a comparative rule and does not require anything about the signs of the expressions $f(A,M) - f(B,M)$ and $f(A,N) - f(B,N)$

Here are some examples to illustrate how different rules would give different predictions:

Mathematical condition Numerical values that illustrate it Comparative advantage says ... Each organization doing what that organization is best at gives ... Each activity done by the organization best at it gives ...
$\! f(A,M) > f(B,M)$, $\! f(B,N) > f(A,N)$, $\! f(A,M) > f(A,N)$, $\! f(B,N) > f(A,N)$ $\! f(A,M) = 20, f(B,M) = 15$, $\! f(A,N) = 16, f(B,N) = 18$ $A$ does $M$, $B$ does $N$ $A$ does $M$, $B$ does $N$ $A$ does $M$, $B$ does $N$
$\! f(A,M) > f(B,M)$, $\! f(A,N) >f(B,N)$, $\! f(A,M) > f(A,N)$, $\! f(B,N) > f(B,M)$ $\! f(A,M) = 20, f(B,M) = 15$, $\! f(A,N) = 18, f(B,N) = 16$ $A$ does $M$, $B$ does $N$ $A$ does $M$, $B$ does $N$ This rule does not work since it would allocate $A$ with both $M$ and $N$.
$\! f(A,M) >f(B,M)$, $\! f(B,N) > f(A,N)$, $\! f(A,M) > f(A,N)$, $\! f(B,M) > f(B,N)$ $\! f(A,M) = 20, f(B,M) = 18$, $\! f(A,N) = 15, f(B,N) = 16$ $A$ does $M$, $B$ does $N$ This rule does not work since both $A$ and $B$ would choose $M$ $A$ does $M$, $B$ does $N$

## Non-minimal examples

### Real-world situations that closely approximate the minimal example

The minimal example is special in the sense that:

• Each organization has to pick exactly one activity and this is a hard constraint.
• Each activity has to be done by exactly one organization and this is a hard constraint.

Many real-world situations are fairly similar to this, though the constraints are not hard but soft as explained below:

• Each organization has to pick exactly one activity: This situation is closely approximated where, due to limitations of time or other resources, the organization can commit to only one activity. For instance, if performing each activity incurs a fixed cost of training, then it may not be possible to train for and do two different kinds of activities.
• Each activity is to be done by exactly one organization: This situation can be closely approximated if there are scarce factors of production associated with the activity, such as a specific piece of land or a specific machine, and multiple copies of it cannot be made.

An alternative way that real-world situations can have both these features is if the nature of market demand for the products of both these activities has sharply diminishing returns beyond what one organization can produce.

### Real-world situations with multiple organizations and multiple activities

In real-world situations, there is a multiplicity of organizations and a multiplicity of activities. Each organization could perform multiple activities and each activity could be performed by multiple organizations. Here, the law of comparative advantage guides how each organization should split its resources between the various activities. At a more global level, it governs which organizations perform which type of activity and in what mix.

## Key features

### Comparative advantage requires a multiplicity of both organizations and activities

The concept of comparative advantage makes sense when:

• there is more than one organization, and
• there is more than one type of activity for the organizations to choose from.

Thus, comparative advantage does not make sense in a world where there is only one organization, even if there is a multiplicity of task types to be performed. If there is only one organization, the organization chooses to do tasks based on the utility generated.

Similarly, comparative advantage does not make sense in a world where there is only one type of activity. In such a world, the organization that is best at the activity gets a priority in doing that activity, and if there is still more of the activity that needs to be done, then other organizations also engage in the activity.

In practice, when people say "organization $A$ has a comparative advantage in doing $N$" they mean comparative advantage relative to other organizations and relative to other activities, even if those other organizations and activities are not explicitly specified.

### Comparative advantage is a two-edged sword

This idea is explored in the blog post Bryan Caplan and the Zero Marginal Productivity hypothesis at the Cafe Hayek weblog by Donald Boudreaux.

A comparative advantage statement compares two (or more) organizations and two (or more) activities. Thus, a statement that "$A$ has a comparative advantage over $B$ in doing $M$ relative to $N$" has an equivalent formulation "$B$ has a comparative advantage over $A$ in doing $N$ relative to $M$." In particular, comparative advantage is not inherently competitive but has the connotations of a positive sum game.

Also, the greater the magnitude of the comparative advantage of $A$ over $B$ in doing $M$ over $N$, the greater the magnitude of the comparative advantage of $B$ over $A$ in doing $N$ over $M$.

### Comparative advantage as a sign of mixed partial

Building on the observation that comparative advantage requires a multiplicity of both organizations and activities, we also note that the direction of comparative advantage depends, roughly, on the sign of a mixed partial derivative with respect to both organizations and activities. We consider comparative advantage and the other allocation rules:

Informal description of rule What kind of partial derivative does it depend upon?
Each organization does what that organization is best at The partial derivative with respect to variation in activities, holding the organization constant.
Each activity is done by the organization best at that activity The partial derivative with respect to variation in organizations, holding the activity constant.
Comparative advantage The mixed partial derivative (this is a second-order derivative) with respect to both organizations and activities.

### Dependence of comparative advantage on relative value

Determining what allocation of activities to organizations is in keeping with the law of comparative advantage requires some way of measuring and comparing the value generated by different types of activities. In particular, a change in the exchange value between two types of activities can affect the direction of comparative advantage. We consider comparative advantage and the two alternative rules:

Informal description of rule What kind of notion of value does this depend upon?
Each organization does what that organization is best at It should be possible to compare (ordinally) the value of different types of activities that the organization can do. The best activity choice is invariant under order-preserving transformations of the mapping from activities to their value measurement for each fixed organization.
Each activity is done by the organization best at that activity It should be possible to compare (ordinally) the value of the same activity done by different types of organizations. The best activity choice is invariant under order-preserving transformations of the mapping from organizations to the value measurement for each fixed activity.
Comparative advantage This requires a quantitative (cardinal) measurement of the value across organizations and activities. A change in the cardinal measurements, even if it preserves the relative ordering of individual activities, can change the comparisons of the sums and differences, and hence affect the direction of comparative advantage.

### Comparative advantage does not create reasons to exist

Comparative advantage makes sense for organizations that already have an independent reason to exist and have to select between various activities to perform. It is not a rationale for the creation of new organizations or for freezing existing organizations into rigid structures.

In particular, comparative advantage does not justify the existence of countries, but rather, makes a broad statement about what activities different countries' economies should specialize in. However, greater gains may be achieved through the migration of people between countries if some countries provide a legal and economic environment that results in much greater productivity for the migrants than the countries the migrants came from.

### Comparative advantage in a market economy

In a market economy, the relative value of activities is determined through their exchange value, i.e., how they trade relative to one another (which may be measured through money or through barter). Thus, trade and exchange two related purposes:

• It determines the direction of comparative advantage, by determining the exchange value through the free buying and selling of goods and the price mechanism.
• It helps implement the comparative advantage by creating incentives for each organization to stick to its comparative advantage.

### Comparative advantage in a planned economy

In a planned economy:

• To determine the direction of comparative advantage, a suitable exchange value needs to be determined. In the absence of a price mechanism and free buying and selling of goods, this exchange value must be based on explicit calculations and assumptions, such as a utilitarian calculus.
• To implement the comparative advantage, explicit rewards, punishments, and incentives needs to be created to substitute for the natural incentives that would arise in a market economy.

According to the critique of central planning based on the economic calculation problem (made explicit by Ludwig von Mises and later by Friedrich von Hayek), it is difficult for a planner to have the requisite local knowledge to compare the values of different activities and hence determine the direction of comparative advantage.

## Comparative advantage, division of labor, and specialization

Further information: comparative advantage and specialization

The basic definition and formulation of comparative advantage does not depend on any assumption about greater gain in skills due to specialization. Comparative advantage suggests a division of labor even without the benefits of skill gains that may arise from specialization.

There is, however, an important way in which comparative advantage promotes skill gains through further specialization. Going back to the minimal example, if comparative advantages allocates activity $M$ to organization $A$ and activity $N$ to organization $B$, then, at the margin, organization $A$ has an incentive to improve in activity $M$, but does not have an incentive to improve in activity $N$ (because even with some improvement, $A$ would still have a comparative advantage doing $M$). Similarly, $B$ has an incentive to improve in $N$ but not in $M$.

## References

### Online articles explaining comparative advantage

Article Author Location Comment
Comparative Advantage: An Economics By Topic Detail Lauren F. Landsburg Library of Economics and Liberty Excerpts from and links to historically important writings (as well as recent online writings) on the subject
Comparative Advantage: Free Trade Benefits High-Paid U.S. Workers Dwight R. Lee The Freeman Discussion of comparative advantage (basic theory) and application to international trade
A Brief History of the Concept of Comparative Advantage Morgan Rose Library of Economics and Liberty Historical overview, and current context (as of 2001)
Comparative Advantage Donald Boudreaux Concise Encyclopedia of Economics, available online at the Library of Economics and Liberty Explanation with mathematical example of comparative advantage plus trade
Treasure Island. The Power of Trade. Part I Russell Roberts Library of Economics and Liberty A short story illustrating comparative advantage and trade.
Ricardo's Difficulty Idea Paul Krugman personal web space at MIT Discussion of why comparative advantage is hard to grasp
The Theory of Comparative Advantage -- Overview Steven Suranovic internationalecon.com An explanation of comparative advantage, its relation with and distinction from the concept of advantageous trade, and its interpretation.