User:Feco/Temp/Ricardo

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The modern Ricardian Trade Model (also known as the Ricardian Model or the Ricardo Trade Model or Ricardo Model and CAP VARIATIONS) is based upon David Ricardo's original theory of comparative advantage. This model is the precursor for a series of more complicated international trade models.

By simplifying the world through basic assumptions, this model shows how movement from autarky to free trade leads to unequivocable gains from trade.

Model Assumptions

 * Two countries, named Home and Foreign
 * Two industries that each produce a homogenous good, either meat and rice
 * One factor of production, labor
 * No cost to transport goods between countries
 * Labor mobility between industries within countries, no labor mobility between countries
 * Perfect competition with constant returns to scale
 * Full employment
 * Labor is homogenous within countries, but there may be productivity differences between countries' labor forces
 * Production technology may differ between countries (this can contribute to differing labor productivities between countries)
 * Consumers maximize utility subject to budget constraints

Model Notation

 * Any variable relating to the country Foreign will be denoted by an asteric (*)
 * Any variable relating to an international, trade-influenced or non-autarky value will be denoted by a hat (^)
 * Any variable relating to the Meat industry will be denoted by the letter m
 * Any variable relating to the Rice industry will be denoted by the letter r
 * Any variable relating to labor will be denoted by the letter l
 * The quantity of a good will be denoted by the letter Q
 * The nominal wage in a country or industry will be denoted by the letter W
 * The real wage in a country or industry will be denoted by the symbol $$\omega$$
 * The nominal price of a good will be denoted by the letter P
 * The profit of an industry will be denoted by the symbol $$\Pi$$

Exogenous Model Variables
Several exogenous variables are necessary to construct this model. These are "given" variables that must be provided from outside the model. The model manipulates these variables to produce endogenous variables.


 * Unit labor input, the number of labor hours required to produce one unit of output. It is denoted by the letter a, with a subscript letter l to indicate labor, and a subscript letter m or letter r to denote the specific industry. Unit labor input is a function of the inherent characteristics of a country's labor force, as well as any technology used to aid production. For example, the unit labor input for the production of meat in Home is denoted as follows:


 * $$\,\!a_{lm}$$


 * The total quantity of labor available in a country is denoted as a capital L. For example, the total supply of labor in Foreign is denoted as follows:


 * $$\,\!L^*$$

The table below demonstrates the different permutations of the notation and gives sample values for each exogenous variable. These sample values will be carried through the rest of the model.

Graphical Analysis
From the exogenous variables provided, it is easy to create a production possibilities frontier (PPF) for both countries. A PPF shows all possible combinations of the two goods that each country can produce in autarky.

To calculate each country's PPF, an equation must be formulated to represent the tradeoff between producing meat and producing rice. This rate is formally known as the marginal rate of transformation. Labor is the only input to either good, so producing an extra unit of meat requires the reallocation of several labor units from the rice industry. The specific number of labor units needed depends on the unit labor input.

A two dimensional Cartesian plane is the easiest way to visualize this tradeoff, so the data must be manipulated into a Y = mX + B linear function that can be graphed. This form expresses Y as a function of m, X and B. X and Y are the axes of the graph, while m is the slope and B is the Y-intercept.

The X- and Y-axes are the quantities of meat and rice produced. Note that the mapping of each goods to an axis is arbitrary, but once the goods are assigned to an axis, it is imperative that they remain associated with that axis. In this example, meat will be assigned to the X-axis and rice will be assigned to the Y-axis.

Distance along each axis represents the quantity produced of that axis's assigned good. Thus, the goal is to express the quantity of rice (Y) as a function of the quantity of meat (X) produced. m and B will express the specific nature of the relationship between meat and rice.

The quantity of a good produced (X or Y) is determined by the amount of labor devoted to its production and the unit labor input for that good. The quantity of goods produced is calculated by dividing the total labor allocated to the good by the hours of labor required to make a single good. In other words, if twenty hours of labor are spent producing meat, and a unit of meat requires four labor hours, five units of meat are produced. The quantity of meat produced, the graphical X-axis, for Home can be denoted as follows. Similar equations with the appropriate notation would indicate the quantity of rice produced at Home and the quantities of both goods produced in Foreign:


 * $$\,\! \mbox{(1)} \qquad X = Q_m = \frac{L_m}{a_{lm}}$$

The country's total labor force is the one fixed resource used to produce both goods. Use of labor in one industry deprives the other industry of that same labor. The total size of the labor force represents an upper bound on production. It is important to represent this limit mathematically by relating labor's allocation between industries to the exogenous, fixed total supply of labor. Note that the allocation of labor between sectors is endogenous and can be changed by the model's results, but the total supply of labor in each country is exogenous. Recall that each country's labor force is fully allocated to the two sectors. Thus, Home's labor force can be represented as follows. A parallel equation would indicate Foreign's labor force allocation:


 * $$\,\! \mbox{(2)} \qquad L = L_m + L_r$$

By rearranging equation (1), the quantity of labor allocated to meat can be expressed as a function of the quantity of meat produced and the unit labor input of meat:


 * $$\,\! \mbox{(3)} \qquad L_m = Q_m \times a_{lm}$$

A similar rearrangement can be performed to isolate the quantity of labor allocated to rice. Substituting equation (3) and its parallel equation for rice into equation (2) creates the equation:


 * $$\,\! \mbox{(4)} \qquad L = (Q_m \times a_{lm}) + (Q_r \times a_{lr})$$

Recall that the quantity of meat is our X-axis and the quantity of rice is our Y-axis. Thus, equation (4) can also be written as:


 * $$\,\! \mbox{(4a)} \qquad L = (X \times a_{lm}) + (Y \times a_{lr})$$

Equations (4) and (4a) can now be rearranged into Y = mX + B form for Home:


 * $$\,\! \mbox{(5)} \qquad Q_r = -(\frac{a_{lm}}{a_{lr}} \times Q_m) + \frac{L}{a_{lr}} \qquad \mbox{(5a)} \qquad Y = -(\frac{a_{lm}}{a_{lr}} \times X) + \frac{L}{a_{lr}}$$

By repeating all of the previous steps for Foreign, equations (5) and (5a) have mirrior-image equations that apply to Foreign:


 * $$\,\! \mbox{(6)} \qquad Q^*_r = -(\frac{a^*_{lm}}{a^*_{lr}} \times Q^*_m) + \frac{L^*}{a^*_{lr}} \qquad \mbox{(6a)} \qquad Y^* = -(\frac{a^*_{lm}}{a^*_{lr}} \times X^*) + \frac{L^*}{a^*_{lr}}$$

Substituting the exogenous variables into linear equations gives the necessary information to construct the PPF graphs for Home and Foreign:


 * $$\,\! \mbox{(7)} \qquad Q_r = -(\frac{10}{20} \times Q_m) + \frac{200}{20} \qquad \mbox{(7a)} \qquad Y = -(\frac{10}{20} \times X) + \frac{200}{20}$$


 * $$\,\! \mbox{(8)} \qquad Q^*_r = -(\frac{60}{30} \times Q^*_m) + \frac{180}{30} \qquad \mbox{(8a)} \qquad Y^* = -(\frac{60}{30} \times X^*) + \frac{180}{30}$$

The graph at right shows the Home and Foreign PPFs under autarky. The maximum quantity that can be produced of either good is denoted at the axis intercepts. Home is marked in blue and Foreign is marked in red. As the graph indicates, Home is able to produce more of either good than blue. This is due to Home's absolute advantage in both goods, as well as Home's larger labor force.

However, absolute advantage matters much less than comparative advantage when analyizing international trade. 

Determining Absolute Advantage
It is easy to identify which coutry has an absolute advantage in a good; for each good, the country with the lower unit labor input has an advantage. This is fairly intuitive, because unit labor input is the measure of how much labor is used to create a single unit of output. The country that makes the same good for less inputs clearly has an advantage. In this case, Home has an absolute advantage in meat because:

$$\,\!\mbox{(9)} \qquad a_{lm} = 10 < a^*_{lm} = 60$$

In this case, Home also has an absolute advantage in rice because:

$$\,\!\mbox{(10)} \qquad a_{lr} = 20 < a^*_{lr} = 30$$

It is always better to identify absolute advantages by comparing the exogenous unit labor inputs rather than graphing the PPFs. Because a country's total labor force allows for more production of all goods, significant difference in population size can cause a country with absolute advantage to produce a numerically smaller quantity of a good. Simply relying on a PPF graph can lead to confusion in such cases.

Absolute advantage can also be analyzed by examining productivity. Productivity is the measure of units of output per hour. This is the inverse of unit labor input. So the unit labor input table can be converted to a sector productivity table by taking the inverse of each value:

In this case, the country with the higher productivity has the absolute advantage in a good. As before, Home has the absolute advantage in both meat and rice. The results from comparing productivity should always match the results from comparing unit labor input, because the two elements are mathmatically related:

$$\,\!\mbox{(11)} \qquad \frac{1}{a_{lm}} = \frac{1}{10} > \frac{1}{a^*_{lm}} = \frac{1}{60} \qquad \mbox{(12)} \qquad \frac{1}{a_{lr}} = \frac{1}{20} > \frac{1}{a^*_{lr}} = \frac{1}{30}$$

Determining Comparative Advantage
Comparative advantage is rooted in the concept of opportunity cost. In this two-good model, meat can only be gained by sacrificing rice. The specific rate of sacrifice is determined by the relative unit labor input for each good. Since Home and Foreign have different unit labor inputs, their rates of sacrifice will differ. This difference is what creates comparative advantage. It is important to note that as long as two countries' ratios of unit labor inputs differ, there will always be comparative advantage and opportunities for gains from trade.

The best way to find comparative advantage is to return to the graphical PPFs created above. Meat was arbitrarily assigned to the X-axis, and rice was assigned to the Y-axis. It is imparative that this mapping to axes remains consistent. The slope, m, of the PPF lines above can also be expressed as the quantity of meat gained for each unit of rice sacrificed. This is the ratio of X gained per sacrifice of Y. Thus, a lower slope (a shallower line) indicates that more meat is gained per sacrifice of rice.

$$\,\!\mbox{(13)} \qquad m = \frac{a_{lm}}{a_{lr}} = \frac{10}{20} < m^* = \frac{a^*_{lm}}{a^*_{lr}} = \frac{60}{30}$$

In this case, Home can gain more meat per sacrifice of rice, so Home has the comparative advantage in meat. Note that the mathematical and graphical analyses should always match. Reversing the analysis to look at rice gained per each unit of meat should always produce a comparative advantage for the other country. Since Home has the comparative advantage in meat, Foreign must have the comparrative advantage in rice.

$$\,\!\mbox{(14)} \qquad \frac{1}{m} = \frac{a_{lr}}{a_{lm}} = \frac{20}{10} > \frac{1}{m^*} = \frac{a^*_{lr}}{a^*_{lm}} = \frac{30}{60}$$

Note that the when all the terms are reversed, Foreign's ratio of unit labor inputs is lower than Home's. This indicates that Foreign has the comparative advantage in rice. Graphically, this can be shown by an exact reversal of the original PPF graph. To demonstrate, the original graph has been rotated and reflected. Foreign (red) now has a shallower slope than Home (blue). Note that the axes have been reversed as a result of the rotation, so meat is now the Y-axis and rice is now the X-axis. Thus, Foreign's shallower slope indicates it has a comparative advantage in the production of  rice, because less rice is sacrificied per unit of meat produced.

The important fact is that as long as the PPF's slopes differ, there will always be differing opportunity costs. Thus, there will always be opportunities for gains from trade.



Autarkic Price and Wage Levels
In the autarkic condition, prices and wages are determined solely by conditions within each country. Due to the simplifying assumptions of the model a very clear relationship between wages, autarkic prices and unit labor input can be determined.

The model assumes perfect competition, or zero profit in industries. Zero profit means industry revenue exactly equals industry costs. For the meat industry at Home, total revenue is the quantity of meat produced times the autarkic price of meat. Since labor is the only input, total cost is the quantity of labor used times the wage rate:

$$\,\!\mbox{(15)} \qquad \Pi_m = (Q_m \times P_m) - (W_m \times L_m) = 0$$

Equation (15) can be rearranged to isolate the wage rate in the meat industry at Home:

$$\,\!\mbox{(16)} \qquad W_m = \frac{(Q_m \times P_m)}{L_m}$$

A parallel to equation (16) exisits for the rice industry at Home:

$$\,\!\mbox{(17)} \qquad W_r = \frac{(Q_r \times P_r)}{L_r}$$

Recall that the model also assumes labor mobility between industries. If wage rates differ between industries, workers will move to the higher-paying industry. Ultimately, differing wage rates are unsustainable, so it is safe to assume equal wages between industries. (The specific mechanism that causes equal wage rates is not important to this analysis). Since wages between industries are equal, equations (16) and (17) can be related:

$$\,\!\mbox{(18)} \qquad W_m = \frac{(Q_m \times P_m)}{L_m} = W_r = \frac{(Q_r \times P_r)}{L_r}$$

Equation (1) relates quantity of a good to labor unit input and quantity of labor allocated to that unit. By substituting equation (1) and its parallel for rice into equation (18), a relationship between unit labor input and autarkic price can be determined:

$$\,\!\mbox{(19)} \qquad W_m = \frac{P_m}{a_{lm}} = W_r = \frac{P_r}{a_{lr}}$$

Equation (19) can be rearranged to show that the autarkic price ratio between meat and rice is determined solely by the unit labor input ratio:

$$\,\!\mbox{(20)} \qquad \frac{P_m}{P_r} = \frac{a_{lm}}{a_{lr}}$$

Equation (19) can also be used to show the relationship between wages, autarkic prices and unit labor input.

$$\,\!\mbox{(21)} \qquad {W_m} \times a_{lm} = P_m$$

Equation (21) should make sense intuitively. With the zero-profit condition, total input costs equal output value. Since the only input is labor, the total labor cost should equal the autarkic product price. If five hours of labor are required to produce a good, the hourly wage must be one fifth of the good's price.

The key point of analysis under autarky is to show that prices and wages are solely determined by the unit labor input values. Here is the full set of autarky prices for both goods in both countries. Note that no numerical value has been associated with prices, so the price of either good is still expressed in terms of the other good:

$$\,\! \mbox{(22)} \qquad \frac{P_m}{P_r} = \frac{a_{lm}}{a_{lr}} = \frac{10}{20} \qquad \mbox{(22a)} \qquad \frac{P_r}{P_m} = \frac{a_{lr}}{a_{lm}} = \frac{20}{10}$$

$$\,\! \mbox{(23)} \qquad \frac{P^*_m}{P^*_r} = \frac{a^*_{lm}}{a^*_{lr}} = \frac{60}{30} \qquad \mbox{(23a)} \qquad \frac{P^*_r}{P^*_m} = \frac{a^*_{lr}}{a^*_{lm}} = \frac{30}{60}$$

Effects of Trade
All analysis up to this point assumed autarky, where no trade occured. Now that assumption will be removed. In the context of the model, Home and Foreign have suddenly agreed to trade meat and rice freely.

In Home, consumers suddenly face two prices for meat, their domestic autarkic price and Foreign's domestic autarkic price. Remember that prices of each good are still being expressed in relative terms of the other good. Here are the two prices Home consumers face for meat:

$$\,\!\mbox{(24)} \qquad \frac{P_m}{P_r} = \frac{a_{lm}}{a_{lr}} = \frac{10}{20} < \frac{P^*_m}{P^*_r} = \frac{a^*_{lm}}{a^*_{lr}} = \frac{60}{30} $$

In this case, the Home autarkic price is less than the Foreign autarkic price, so Home consumers have no incentive to purchase Foreign meat. Here are the two autarkic prices of rice faced by Home consumers:

$$\,\!\mbox{(25)} \qquad \frac{P_r}{P_m} = \frac{a_{lr}}{a_{lm}} = \frac{20}{10} > \frac{P^*_r}{P^*_m} = \frac{a^*_{lr}}{a^*_{lm}} = \frac{30}{60} $$

In this case, Home's autarkic price is higher than Foreign's. In situations like this, it makes more sense for Home consumers to trade their domestically-produced meat for Foreign rice. They can "buy" more rice for the same price of meat when they buy it from Foreign rather than buying it from Home.

Foreign consumers face the same relative prices—their domestic autarkic price and Home's domestic autarkic price. Thus, equations (24) and (25) can be re-written to show how Foreign's domestic prices compare to Home's:

$$\,\!\mbox{(24a)} \qquad \frac{P^*_m}{P^*_r} = \frac{a^*_{lm}}{a^*_{lr}} = \frac{60}{30} > \frac{P_m}{P_r} = \frac{a_{lm}}{a_{lr}} = \frac{10}{20}$$

$$\,\!\mbox{(25a)} \qquad \frac{P^*_r}{P^*_m} = \frac{a^*_{lr}}{a^*_{lm}} = \frac{30}{60} < \frac{P_r}{P_m} = \frac{a_{lr}}{a_{lm}} = \frac{20}{10}$$

Equations (24), (25), (24a) and (25a) all show the same thing; Home's relative autarkic price of meat and Foreign's relative autarkic price of rice are lower. Thus, there will be an international market in Home meat and Foreign rice. Like any other market, this international market will have prices. The specific price level is not important for now, but it is important to know that it exists.

It is also important to verify that each country is exporting the good in which it has a comparative advantage. Because the different autarkic prices emerge from the relative unit labor inputs, a country's comparative advantage good will always be the one it can export.

It has already been shown that there is demand for an international market in Home meat. For Home producers to have any incentive to export their meat, the international price must be higher than the Home domestic autarkic price. Conversely, for Foreign consumers to have any incentive to consume international meat, the international price must be lower than the Foreign domestic autarkic price. These conclusions lead to the following equation, and a parallel one for the rice market:

$$\,\!\mbox{(26)} \qquad \frac{P_m}{P_r} = \frac{a_{lm}}{a_{lr}} = \frac{10}{20} < \frac{\hat P_m}{\hat P_r} < \frac{P^*_m}{P^*_r} = \frac{a^*_{lm}}{a^*_{lr}} = \frac{60}{30} $$

$$\,\!\mbox{(27)} \qquad \frac{P^*_r}{P^*_m} = \frac{a^*_{lr}}{a^*_{lm}} = \frac{30}{60} < \frac{\hat P_r}{\hat P_m} < \frac{P_r}{P_m} = \frac{a_{lr}}{a_{lm}} = \frac{20}{10} $$

Equations (26) and (27) show conditions the instant trade is allowed. As consumers and producers begin reacting to the new opportunities for trade, the difference between domestic autarkic and international prices will be arbitraged away. Eventually, there will be one global price for each good. Whether that price is closer to Home's or Foreign's autarkic prices depends on relative economy sizes, differences in production and other factors. This global price for meat can be expressed in the standard ratio format:

$$\,\!\mbox{(28)} \qquad \frac{\hat P_m}{\hat P_r}$$

The global price for rice is the inverse of equation (28):

$$\,\!\mbox{(29)} \qquad \frac{\hat P_r}{\hat P_m}$$

Effects in the Export Industry
As the above analysis shows, each country has an industry that will benefit from free trade. In Home that will be the meat industry; in Foreign it will be the rice industry. In each case, the undetermined international price is higher that the autarkic domestic price. As exporters, Home's meat industry and Foreign's rice industry both experience the same effects, so it makes sense to analyze the effects from trade between the export industry and non-export industries.

Additionally, this model assumes perfect competition in industries. Thus, the existence of firms does not really matter. So the analysis can proceed by looking only at the effects on workers in the export industry. This example will focus on Home workers in the meat industry, but the same conclusions apply to Foreign workers in the rice industry.

To workers and consumers, real income matters much more than nominal wages or prices. Real wages are calculated by dividing nominal wages by a nominal price. In this model, that nominal price can be either the price of meat or the price of rice. In autarky, Home's meat workers' real wage in terms of meat is:

$$\,\!\mbox{(30)} \qquad \omega_{m,m} = \frac{W_m}{P_m}$$

Recall equation (19), which shows the relationship between nominal wage, price and unit labor input. Substituting that equation into equation (30) gives the relationship between meat workers' real wage in terms of meat and meat workers' unit labor input:

$$\,\!\mbox{(31)} \qquad \omega_{m,m} = \frac{1}{a_{lm}}$$

Home's meat workers' real wage in terms of rice is:

$$\,\!\mbox{(32)} \qquad \omega_{m,r} = \frac{W_m}{P_r}$$

Equation (19) again can be substituted in order to show the relationship between nominal wage, price and unit labor input. Substituting equation (19) into equation (32) gives the relationship between meat workers' real wage in terms of rice and meat workers' unit labor input: $$\,\!\mbox{(33)} \qquad \omega_{m,r} = \frac{P_m}{P_r} \times \frac{1}{a_{lm}}$$

Recall from equation (26) that the unknown international meat price is higher than Home's domestic meat price. Recall equations (31) and (33), which show Home meat workers' real wage in terms of meat and rice, respectively:

$$\,\!\mbox{(26)} \qquad \frac{P_m}{P_r} < \frac{\hat P_m}{\hat P_r} \qquad \mbox{(31)} \qquad \omega_{m,m} = \frac{1}{a_{lm}} \qquad \mbox{(33)} \qquad \omega_{m,r} = \frac{P_m}{P_r} \times \frac{1}{a_{lm}}$$

Equation (31) expresses meat workers' real wage in terms of meat. Equation (26), which shows the increase from autarkic to international prices, has no impact on equation (31). Intuitively, this makes sense. The increase in meat workers' nominal wage is exactly offset by the increased price of meat, so real income in terms of meat does not change. Equation (26) does have an impact on equation (33) because relative prices are part of it. Substituting the new, international price level into equation (33) gives:

$$\,\!\mbox{(34)} \qquad \hat \omega_{m,r} = \frac{\hat P_m}{\hat P_r} \times \frac{1}{a_{lm}}$$

Since the international price ratio is higher than the autarkic price ratio, meat workers' real wage in terms of rice is higher when trade occurs. Because real wages in terms of meat did not change, and real wages in terms of rice did change, Home workers in the meat industry are unambiguously better off under free trade than under autarky. This conclusion can be generalized to apply to the rice workers in Foreign, or the workers in any export sector under this model.

Effects in the Non-Export Industry
To examine the effects of trade on the non-export industry, Home's rice industry will be analyzed. This analysis also applies to Foreign's meat industry, or more generally, any industry that is at a comparative disadvantage after trade begins in this model.

The same process that produced equations (30) and (31) for Home's meat workers can be used to produce the real wage in terms of rice for Home's rice workers:

$$\,\!\mbox{(35)} \qquad \omega_{r,r} = \frac{1}{a_{lr}}$$

The same process that produced equations (32) and (33) for Home's meat workers can be used to produce the real wage in terms of meat for Home's rice workers:

$$\,\!\mbox{(36)} \qquad \omega_{r,m} = \frac{P_r}{P_m} \times \frac{1}{a_{lr}}$$

Recall from equation (27) that the unknown international rice price is lower than Home's domestic rice price. Recall equations (35) and (36), which show Home rice workers' real wage in terms of rice and meat, respectively:

$$\,\!\mbox{(27)} \qquad \frac{\hat P_r}{\hat P_m} < \frac{P_r}{P_m} \qquad \mbox{(35)} \qquad \omega_{r,r} = \frac{1}{a_{lr}} \qquad \mbox{(36)} \qquad \omega_{r,m} = \frac{P_r}{P_m} \times \frac{1}{a_{lr}}$$

Model Takeaways

 * Trade occurs due to comparative advantage. In this model, comparative advantage is due only to differing productivity that can arise from a labor force's inherent characteristcs or from the technology used by a labor force.
 * Everyone benefits from trade