In the Ricardian model, why are two out of 3 cases invalid?

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"?

• I think the work constraint is wrong and it should be $l_a^j + l_b^j = 1$ instead. Commented Mar 5, 2016 at 8:12

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.