<p>To "specialize completely" means a country only produces one of the two goods.</p>

<p>Given your firms' (there seem to be four, one for each good in each country) profit maximization problems their optimality conditions are
$$
p_i \cdot \alpha_i^j \leq w^j
$$
with equality if $l_i^j &gt; 0$. From this, you can easily derive what parameter combinations of $p_i$ and $\alpha_i^j$ will result in conditions when a country specializes completely. </p>

<p>I am not sure what you mean by 'fail', and the indices in your workforce notation seem a little off. Perhaps <a href="https://economics.stackexchange.com/questions/8354/ricardos-theory-of-comparative-advantage">my detailed discussion of the Ricardian model</a> can be of benefit. Beware it is not the exact same model as yours. For example, prices are not given exogenously, but they result from a supply and demand equilibrium.</p>

CC BY-SA 3.0
2016-03-05T08:12:08.913Z
2017-02-03T20:27:21.373Z
2017-04-13T12:51:38.103Z
Giskard
<p>I am considering the Ricardian model. </p>

<p>There is an economy with two countries $S,T$. Two goods $a,b$. Each country has linear production technology with labor ($\ell$) as only factor of production. 
$$y_i^j=\alpha_i^j\ell$$
where $i \in \lbrace a,b\rbrace$ and $j \in \lbrace S,T \rbrace$.</p>

<p>$$\ell_S + \ell_T = 1$$</p>

<p>The firm's maximization problem is </p>

<p>$$\max_{\lbrace \ell \rbrace} \pi^j_i = \alpha_i^j\ell p_i-w^j \ell$$</p>

<p>There are 3 cases to this problem</p>

<p><strong>Case 1:</strong> $p_i^j \alpha_i^j &lt;w^j$</p>

<p>$\ell = 0$ </p>

<p>$y_i^j = 0$</p>

<p><strong>Case 2:</strong> $p_i^j \alpha_i^j &gt; w^j$</p>

<p>$\ell = 1$ </p>

<p>$y_i^j = \alpha_i^j$</p>

<p><strong>Case 3:</strong> $p_i^j \alpha_i^j = w^j$</p>

<p>$\ell \in [0,1]$ </p>

<p>$y_i^j = [0,\alpha^j_i]$</p>

<p><strong>My Question</strong> </p>

<p>Apparently, cases 1 and 2 aren't valid. I'm not sure why. <strong>Can someone explain?</strong></p>

<p>The logic is supposed to be </p>

<blockquote>
  <p>"if there is complete specialization then $\ell_S = 1, \ell_T = 0, y_T = 0$ or $L_S = 0, L_T = 1, y_S = 0$, so therefore cases 1 and cases 2 fail" </p>
</blockquote>

<p>I just don't understand this explanation at all and would like a better explanation. What does it mean to "specialize completely"? </p>

CC BY-SA 3.0
2016-03-05T08:12:45.823Z
I think the work constraint is wrong and it should be $l_a^j + l_b^j = 1$ instead.
Giskard
<international-trade><international-economics>
In the Ricardian model, why are two out of 3 cases invalid?
Stan Shunpike
# In the Ricardian model, why are two out of 3 cases invalid?

(Asked by: username_0 on 2016-03-05)

I am considering the Ricardian model.

There is an economy with two countries $S,T$. Two goods $a,b$. Each country has linear production technology with labor ($\ell$) as only factor of production.
$$y\_i^j=\alpha\_i^j\ell$$
where $i \in \lbrace a,b\rbrace$ and $j \in \lbrace S,T \rbrace$.

$$\ell\_S + \ell\_T = 1$$

The firm's maximization problem is

$$\max\_{\lbrace \ell \rbrace} \pi^j\_i = \alpha\_i^j\ell p\_i-w^j \ell$$

There are 3 cases to this problem

**Case 1:** $p\_i^j \alpha\_i^j <w^j$

$\ell = 0$

$y\_i^j = 0$

**Case 2:** $p\_i^j \alpha\_i^j > w^j$

$\ell = 1$

$y\_i^j = \alpha\_i^j$

**Case 3:** $p\_i^j \alpha\_i^j = w^j$

$\ell \in [0,1]$

$y\_i^j = [0,\alpha^j\_i]$

**My Question**

Apparently, cases 1 and 2 aren't valid. I'm not sure why. **Can someone explain?**

The logic is supposed to be

> "if there is complete specialization then $\ell\_S = 1, \ell\_T = 0, y\_T = 0$ or $L\_S = 0, L\_T = 1, y\_S = 0$, so therefore cases 1 and cases 2 fail"

I just don't understand this explanation at all and would like a better explanation. What does it mean to "specialize completely"?

**Tags:** international-trade, international-economics

> **Comments:**
> **username_1**: I think the work constraint is wrong and it should be $l_a^j + l_b^j = 1$ instead.  

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## 1 answers

### Answer by username_1 on 2016-03-05. Score: 1

To "specialize completely" means a country only produces one of the two goods.

Given your firms' (there seem to be four, one for each good in each country) profit maximization problems their optimality conditions are
$$
p\_i \cdot \alpha\_i^j \leq w^j
$$
with equality if $l\_i^j > 0$. From this, you can easily derive what parameter combinations of $p\_i$ and $\alpha\_i^j$ will result in conditions when a country specializes completely.

I am not sure what you mean by 'fail', and the indices in your workforce notation seem a little off. Perhaps [my detailed discussion of the Ricardian model](https://economics.stackexchange.com/questions/8354/ricardos-theory-of-comparative-advantage) can be of benefit. Beware it is not the exact same model as yours. For example, prices are not given exogenously, but they result from a supply and demand equilibrium.


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