Ricardo's theory of comparative advantage

Question

The aim of this question is to better explore the mathematical economics model behind Ricardo's theory of comparative advantage and the claims that can be made based on this model. This seems necessary because of this question and some of the answers given to it. Following is a description of the model as I understand it.

Suppose there are two countries (England and Portugal, denoted by E and P) producing two goods (good x and good y). The only input needed to produce these goods is labor. The economics of scale are constant in both industries of both countries and the labor requirement to produce good $m$ (where $m \in \left\{x,y\right\}$) in country $i$ is denoted by $a_{m,i}$. Without loss of generality I will assume that England enjoys comparative advantage in producing the good $x$, meaning $$ \frac{a_{x,E}}{a_{y,E}} < \frac{a_{x,P}}{a_{y,P}}. $$ Let us denote the size of the English labor pool by $L_E$, that of the Portuguese by $L_P$. Denote the produced quantities by $\left(q_{x,E},q_{y,E}\right), \left(q_{x,P},q_{y,P}\right)$. So England may choose any production $\left(q_{x,E},q_{y,E}\right)$ as long as $$ q_{x,E} \cdot a_{x,E} + q_{y,E} \cdot a_{y,E} \leq L_E. $$ The income of a country is determined by the value of its production. Denote the equilibrium prices by $p_x, p_y$.

Now follow some assumptions that do result in some loss of generality, but I think these are necessary for simple illustrations.
Assume that the English public's consumption (denoted by $c_{m,E}$) preferences of said goods, $\left(c_{x,E},c_{y,E}\right)$, are described by symmetric a Cobb-Douglas utility function. Assume the same for Portugal, where Portuguese consumption is denoted by $\left(c_{x,P},c_{y,P}\right)$.

A discussion of how England can maximize her profit is probably useful, but I seek to focus on the following claims and whether they are true given this model:

1. England will always produce good $x$ (in some positive quantity).
2. England will never produce good $y$.
3. England will never export good $y$.

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In case you are wondering: No, this is not a homework question. But because you do not have to take my word for this I tried to phrase the question in accordance with the current guidelines for exercises.

Answer 1

The solution concept used in Ricardo's modell is the competitive equilibrium. Let the set of countries $N$ be defined as $N = \left\{E,P\right\}.$ (England, Portugal) Then the competitive equilibrium is a vector $$ \left(p,\left(q_{x,i},q_{y,i}\right)_{i\in N},\left(c_{x,i},c_{y,i}\right)_{i\in N}\right), $$ where $p$ is the equilibrium price ratio of the goods $x$ and $y$, so $p = \frac{p_x}{p_y}$, and $\left(q_{x,i},q_{y,i}\right)$ and $\left(c_{x,i},c_{y,i}\right)$ are the production and consumption vectors of country $i$. The equilibrium vector has the following properties:

1. Profit maximization:
   Each country choose a production that maximizes her profit given her production capabilites and the equilibrium price ratio. The set of possible production vectors $T_i$ is defined by the size of the countries labor pool $L_i$ and the labor requirements $a_{x,i},a_{y,i}$ as defined in the question $$ q_{x,i} \cdot a_{x,i} + q_{y,i} \cdot a_{y,i} \leq L_i. $$ A production vector $\left(q_x^i,q_y^i\right)$ is profit maximizing if $$ \max\limits_{(x,y)\in T_i} p \cdot x + y = p \cdot q_{x,i} + q_{y,i}. $$
2. Utility (welfare) maximization:
   The vector $(c_{x,i},c_{y,i})$ maximizes country $i$'s utility if $$ \max\limits_{p \cdot x + y \leq p \cdot q_{x,i} + q_{y,i}} U_i(x,y) = U_i(c_{x,i},c_{y,i}). $$
3. The good markets are in equilibrium, i.e. in the market of each good demand equals supply (technically only the values are equal, but here, i.e. with Cobb-Douglas preferences, prices are always positive in equilibrium, so there is no difference). The equations for these are $$ \sum\limits_{i\in N} q_x^i = \sum\limits_{i\in N} c_x^i \hskip 20pt \sum\limits_{i\in N} q_y^i = \sum\limits_{i\in N} c_y^i. $$

Let us examine what these properties imply. The set $T_i$ is a triangle. As goods have positive value in equilibrium so all the labor is used up and the production vector is chosen from the production possibility frontier. Which industry ($x$ or $y$) can employ labor more lucratively? In industry $x$ a unit of labor produces value $\frac{p_x}{a_{x,i}}$. Similarly the value produced in industry $y$ is $\frac{p_y}{a_{y,i}}$. If $\frac{p_x}{a_{x,i}} > \frac{p_y}{a_{y,i}}$ only good $x$ is produced, if $\frac{p_x}{a_{x,i}} < \frac{p_y}{a_{y,i}}$ only good $y$ is produced, if $\frac{p_x}{a_{x,i}} = \frac{p_y}{a_{y,i}}$ it does not matter how labor is allocated among the industries as long as all labor is used. So the profit maximizating quantities are $$ \left(q_{x,i},q_{y,i}\right) = \left\{ \begin{array}{cc} \left(\frac{L_i}{a_{x,i}},0\right) & \frac{a_{x,i}}{a_{y,i}} < p \\ \alpha \cdot \left(\frac{L_i}{a_{x,i}},0\right) + (1 - \alpha) \cdot \left(0,\frac{L_i}{a_{y,i}}\right) & \frac{a_{x,i}}{a_{y,i}} = p \\ \left(0,\frac{L_i}{a_{y,i}}\right) & \frac{a_{x,i}}{a_{y,i}} > p. \end{array} \right. $$ The optimum condition for the utility maximization problem is $$ MRS_i(c_{x,i},c_{y,i}) = \frac{c_{y,i}}{c_{x,i}} = p. $$ Because the utility functions in England and Portugal have the same form we can take this further. From $$ \frac{c_y^A}{c_x^A} = p = \frac{c_y^P}{c_x^P}. $$ we get $$ \frac{c_{y,A}}{c_{x,A}} = p = \frac{c_{y,P}}{c_{x,P}}. $$ we get $$ c_{y,P} = \frac{c_{y,A}}{c_{x,A}} \cdot c_{x,P}. $$ Using this $$ \frac{c_{y,A}+c_{y,P}}{c_{x,A}+c_{x,P}} = \frac{c_{y,A}+\frac{c_{y,A}}{c_{x,A}} \cdot c_{x,P}}{c_{x,A}+c_{x,P}} = \frac{c_{x,A}}{c_{x,A}} \cdot \frac{c_{y,A}+\frac{c_{y,A}}{c_{x,A}} \cdot c_{x,P}}{c_{x,A}+c_{x,P}} = \frac{c_{y,A} \cdot c_{x,A} + c_{y,A} \cdot c_{x,P}}{c_{x,A} \cdot \left(c_{x,A}+c_{x,P}\right)}. $$ so $$ \frac{c_{y,A}+c_{y,P}}{c_{x,A}+c_{x,P}} = \frac{c_{y,A} \cdot c_{x,A} + c_{y,A} \cdot c_{x,P}}{c_{x,A} \cdot \left(c_{x,A}+c_{x,P}\right)} = \frac{c_{y,A}}{c_{x,A}} = p. $$ What this says is that the relative demand ($\frac{c_{y,i}}{c_{x,i}}$) is not only equal to the price ratio for individual countries but also the relative aggregate world demand is equal to the price ratio. (Again, this is only true if the individual countries have Cobb-Douglas utility functions with identical parameters.) We now have a relatively easy way to find the equilibrium price ratio: we calculate relative aggregate supply. As aggregate supply equals aggregate demand in equilibrium, relative aggregate supply will equal relative aggregate demand, and as we have just shown it will also equal $p$. We get relative supply from the profit maximizing productions of the individual countries. Let us first discuss the aggregate of the profit maximizing productions, which I will denote by $(q_x,q_y)$. So $(q_x,q_y) = (q_{x,E} + q_{x,P},q_{y,E} + q_{y,P})$ which means $$ (q_x,q_y) = \left\{ \begin{array}{cc} \left(\frac{L_E}{a_{x,E}} + \frac{L_P}{a_{x,P}},0\right) & \frac{a_{x,E}}{a_{y,E}} < \frac{a_{x,P}}{a_{y,P}} < p \\ \left(\frac{L_E}{a_{x,E}} + \alpha \cdot \frac{L_P}{a_{x,P}} , (1 - \alpha) \cdot \frac{L_P}{a_{y,P}} \right) & \frac{a_{x,E}}{a_{y,E}} < p = \frac{a_{x,P}}{a_{y,P}} \\ \left(\frac{L_E}{a_{x,E}} , \frac{L_P}{a_{y,P}} \right) & \frac{a_{x,E}}{a_{y,E}} < p < \frac{a_{x,P}}{a_{y,P}} \\ \left(\alpha \cdot \frac{L_E}{a_{x,E}}, (1 - \alpha) \cdot \frac{L_E}{a_{y,E}} + \frac{L_P}{a_{y,P}} \right) & \frac{a_{x,E}}{a_{y,E}} = p < \frac{a_{x,P}}{a_{y,P}} \\ \left(0, \frac{L_E}{a_{y,E}} + \frac{L_P}{a_{y,P}} \right) & p < \frac{a_{x,E}}{a_{y,E}} < \frac{a_{x,P}}{a_{y,P}} . \end{array} \right. $$ The relative aggregate supply is the ratio $\frac{q_x}{q_y}$. It is perhaps best described by this image:

The relative aggregate demand is the ratio $\frac{c_x}{c_y}$. As we have discussed $\frac{c_y}{c_x} = p$ so $$ \frac{c_x}{c_y} = \frac{1}{p}. $$ As a result one can draw the relative aggregate demand in the previous graph as hyperbole. The intersection with the relative aggregate supply curve will give yield the equilibrium price ratio and will also yield information about the production of individual countries. Where this intersection occurs depends on the parameters $L_E,L_P,a_{x,E},a_{y,E},a_{x,P},a_{y,P}$. I will distinguish between three types of equilibria, each represented in the following figure:

In the 1. equilibrium the price ratio is $p = \frac{a_{x,E}}{a_{y,E}} < \frac{a_{x,P}}{a_{y,P}}$. Thus Portugal specializes and only produces good $y$, but England does not specialize but produces both good $x$ and $y$. Producing either good gives her the same value. The exact equilibrium quantities England produces are determined by the value the aggregate demand curve takes at price $p$, because $$ \frac{q_x}{q_y} = \frac{c_x}{c_y} = \frac{1}{p} $$ and $$ q_x = q_{x,E} + q_{x,P} = q_{x,E} + 0 \hskip 20pt q_y = q_{y,E} + q_{y,P} = q_{y,E} + \frac{L_P}{a_{y,P}}. $$ In this case England will still not achieve a net export of good $y$. The preferences tell us that Portugal will consume both goods $x$ and $y$. But the only way it can pay for the goods $x$ consumed is by trading some of its goods $y$, so Portugal, not England, will be a net exporter of good $y$ while England is a net exporter of good $x$.
In the 2. equilibrium both countries specialize: England produces only good $x$, Portugal produces only good $y$. This is usually presented as the textbook case.
The 3. equilibrium is like the 1. equilibrium, but here England specializes and produces only good $x$ while Portugal does not specialize and produces both.

So to answer my original questions:
Given that England has a comparative advantage in producing good $x$,

1. England will always produce good $x$.
2. England may also produce good $y$.
3. England will never export good $y$.

Answer 2

Two suppositions (that are probably wrong) are implied by your introduction:

• Linearity of the production functions in the two countries
• Free trade (as in, trade at no cost)

Using those, and with two assumptions (that are induced by a Cobb-Douglas, but far less restrictive):

1. Good $x$ and $y$ enjoy some degree of substitution in both countries' aggregate demand
2. Good $x$ and $y$ cannot be substituted completely one for another in any of the two countries (ie no matter what quantity of $x$ the consumers are offered, they will not give up there last unit of $y$),

then we can conclude:

1. England will always produce good $x$ or not produce anything. If England did produce good $y$ but not good $x$, then that good $y$ could be bought in Portugal for a quantity of $x$ easier to produce than $y$, except if Portugal produced only good $y$. This is impossible because of the limited substitution across both economies. TRUE

   2. No, in case all production in Portugal is dedicated to $y$, then there may or may not be an additional production of $y$ in England because the global equilibrium requires "more" $y$ than Portugal can produce. Not that in real life, the residual production of $y$ or $x$ in each country should be linked to local advantages for a residual part of that demand, rather than to the above argument. FALSE

   3. They could not. For this you would need Portugal not to be able to substitute that $y$ locally, that is, Portugal only producing only $y$. In this situation Portugal couldn't export $x$, because they wouldn't be producing it, and so, would not import $y$. TRUE