Editing Comparative advantage (section)

===Ricardian model===
The '''Ricardian model''' is a [[general equilibrium]] mathematical model of [[international trade]]. Although the idea of the Ricardian model was first presented in the ''Essay on Profits'' (a single-commodity version) and then in the ''Principles'' (a multi-commodity version) by [[David Ricardo]], the first mathematical Ricardian model was published by [[William Whewell]] in 1833.<ref>{{cite book |title= David Ricardo: Critical Assessments|last= Wood |first= John Cunningham|year= 1991 |publisher= Taylor & Francis |isbn= 9780415063807|page= 312 |url= https://books.google.com/books?id=h5GzUUfTL4sC&q=Ricardian+model&pg=PA312}}</ref> The earliest test of the Ricardian model was performed by [[Donald MacDougall|G. D. A. MacDougall]], which was published in ''[[The Economic Journal]]'' of 1951 and 1952.<ref>{{cite book |title= International Economics: A European Focus|last= Ingham |first= Barbara|year= 2004 |publisher= Pearson Education |isbn= 9780273655077|page= 22 |url= https://books.google.com/books?id=zIaS9R7HUGIC&q=international+economics}}</ref> In the Ricardian model, trade patterns depend on productivity differences.

The following is a typical modern interpretation of the classical Ricardian model.<ref>{{cite book|last1=Krugman|first1=Paul|author1-link=Paul Krugman|last2=Obstfeld|first2=Maurice|author2-link=Maurice Obstfeld|title=International Economics: Theory and Policy|year=1988|publisher=Prentice Hall|location=New York|pages=27–36|edition=2008}}</ref> In the interest of simplicity, it uses notation and definitions, such as opportunity cost, unavailable to Ricardo.

The world economy consists of two countries, Home and Foreign, which produce wine and cloth. Labor, the only factor of production, is [[labor mobility|mobile]] domestically but not internationally; there may be migration between sectors but not between countries. We denote the labor force in Home by <math>\textstyle L</math>, the amount of labor required to produce one unit of wine in Home by <math>\textstyle a_{LW}</math>, and the amount of labor required to produce one unit of cloth in Home by <math>\textstyle a_{LC}</math>. The total amount of wine and cloth produced in Home are <math>Q_W</math> and <math>Q_C</math> respectively. We denote the same variables for Foreign by appending a [[prime (symbol)|prime]]. For instance, <math>\textstyle a'_{LW}</math> is the amount of labor needed to produce a unit of wine in Foreign.

We do not know if Home can produce cloth using fewer hours of work than Foreign. That is, we do not know if <math>a_{LC}<a'_{LC}</math>. Similarly, we do not know if Home can produce wine using fewer hours of work. However, we assume Home is ''relatively'' more productive than Foreign in making in cloth vs. wine:
:<math>a_{LC}/a'_{LC}<a_{LW}/a'_{LW}.</math>

Equivalently, we may assume that Home has a comparative advantage in cloth in the sense that it has a lower opportunity cost for cloth in terms of wine than Foreign:
:<math>a_{LC}/a_{LW}<a'_{LC}/a'_{LW}.</math>

In the absence of trade, the relative price of cloth and wine in each country is determined solely by the relative labor cost of the goods. Hence the relative autarky price of cloth is <math>a_{LC}/a_{LW}</math> in Home and <math>a'_{LC}/a'_{LW}</math> in Foreign. With free trade, the price of cloth or wine in either country is the world price <math>P_C</math> or<math>P_W</math>.

Instead of considering the world demand (or supply) for cloth and wine, we are interested in the world ''relative demand'' (or ''relative supply'') for cloth and wine, which we define as the ratio of the world demand (or supply) for cloth to the world demand (or supply) for wine. In general equilibrium, the world relative price <math>\textstyle P_C/P_W</math> will be determined uniquely by the intersection of world relative demand <math>\textstyle RD</math> and world relative supply <math>\textstyle RS</math> curves.

[[File:World relative supply and demand in the classical Ricardo model of one-factor international trade between two countries.svg|thumb|upright=1.3|The demand for cloth relative to wine decreases with the relative price of cloth in terms of wine; the supply <math>RS</math> of cloth relative to wine increases with relative price. Two relative demand curves <math>RD_1</math> and <math>RD_2</math> are drawn for illustrative purposes.]]
We assume that the relative demand curve reflects substitution effects and is decreasing with respect to relative price. The behavior of the relative supply curve, however, warrants closer study. Recalling our original assumption that Home has a comparative advantage in cloth, we consider five possibilities for the relative quantity of cloth supplied at a given price.

* If <math>\textstyle P_C/P_W = a_{LC}/a_{LW}<a'_{LC}/a'_{LW}</math>, then Foreign specializes in wine, for the wage <math>P'_W/a'_{LW}</math> in the wine sector is greater than the wage <math>P'_C/a'_{LC}</math> in the cloth sector. However, Home workers are indifferent between working in either sector. As a result, the quantity of cloth supplied can take any value.
* If <math>\textstyle P_C/P_W < a_{LC}/a_{LW}<a'_{LC}/a'_{LW}</math>, then both Home and Foreign specialize in wine, for similar reasons as above, and so the quantity of cloth supplied is zero.
* If <math>\textstyle a_{LC}/a_{LW}<P_C/P_W < a'_{LC}/a'_{LW}</math>, then Home specializes in cloth whereas Foreign specializes in wine. The quantity of cloth supplied is given by the ratio <math>\textstyle \frac{L/a_{LC}}{L'/a'_{LW}}</math> of the world production of cloth to the world production of wine.
* If <math>\textstyle a_{LC}/a_{LW}<a'_{LC}/a'_{LW}<P_C/P_W</math>, then both Home and Foreign specialize in cloth. The quantity of cloth supplied tends to infinity as the quantity of wine supplied approaches zero.
* If <math>\textstyle a_{LC}/a_{LW}<a'_{LC}/a'_{LW}=P_C/P_W</math>, then Home specializes in cloth while Foreign workers are indifferent between sectors. Again, the relative quantity of cloth supplied can take any value.

[[File:Consumption possibilities in the classical Ricardo model of one-factor international trade between two countries.svg|thumb|upright=1.3|The blue triangle depicts Home's original production (and consumption) possibilities. By trading, Home can also consume bundles in the pink triangle despite facing the same productions possibility frontier.]]
As long as the relative demand is finite, the relative price is always bounded by the inequality
:<math> a_{LC}/a_{LW}\leq {P_C/P_W}\leq {a'_{LC}/a'_{LW}}.</math>

In autarky, Home faces a [[production–possibility frontier|production constraint]] of the form
:<math> a_{LC}Q_C+a_{LW}Q_W\leq L,</math>
from which it follows that Home's cloth consumption at the production possibilities frontier is
:<math>Q_C=L/a_{LC}-(a_{LW}/a_{LC})Q_W</math>.

With free trade, Home produces cloth exclusively, an amount of which it exports in exchange for wine at the prevailing rate. Thus Home's overall consumption is now subject to the constraint
:<math>a_{LC}Q_C+a_{LC}(P_W/P_C)Q_W\leq L</math>
while its cloth consumption at the ''consumption possibilities'' frontier is given by
:<math>Q_C=L/a_{LC}-(P_W/P_C)Q_W\geq L/a_{LC}-(a_{LW}/a_{LC})Q_W</math>.

A symmetric argument holds for Foreign. Therefore, by trading and specializing in a good for which it has a comparative advantage, each country can expand its consumption possibilities. Consumers can choose from bundles of wine and cloth that they could not have produced themselves in closed economies.

There is another way to prove the theory of comparative advantage, which requires less assumption than the above-detailed proof, and in particular does not require for the hourly wages to be equal in both industries, nor requires any equilibrium between offer and demand on the market.<ref>{{cite book |title= Mathematical methods for Economic analysis|last= L. |first= G.|year= 2021 |url= https://bookdown.org/praxeolab/mathematics-for-economics/chapter1.html}}</ref> Such a proof can be extended to situations with many goods and many countries, non constant returns and more than one factor of production.