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

Question

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"?

Answer 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 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.