More specifically, we saw that the specific factor in the sector whose relative price increases will gain, and that the specific factor in the other sector (whose relative price decreases) will lose. We also saw that the welfare changes for the mobile factor are ambiguous.

Measuring gains vs. losses would be difficult as welfare is inherently subjective. A better way is to see if those who benefit can compensate others who lose and still be better off.

Aggregate gains from trade can be shown as followed:
In a country that cannot trade, the output of a good must equal its consumption. If DC is consumption of cloth and DF consumption of food, then in a closed economy, Dc=Qc and Df=Qf
With trade the value of consumption still needs to be equal but the mix of goods can be different

The left hand side is the level of food imports in the domestic economy which is determined by the relative price of cloth and the economy’s cloth exports. Eq. 4.7 is therefore the budget constraint
BC slope: Consuming one less unit of cloth gives you Pc extra which can buy you Pc/Pf units of food.
Before trade if an economy produces at a point like 2, after trade it can achieve anything within the shaded region. The set of feasible points are then where the BC is in the shaded area.
BC slope: Consuming one less unit of cloth gives you Pc extra which can buy you Pc/Pf units of food.
Before trade if an economy produces at a point like 2, after trade it can achieve anything within the shaded region. The set of feasible points are then where the BC is in the shaded area.

Gains from trade:
In the absence of trade the economy would have to consume what it produced on the production possibility frontier. An economy that trades can consume more of both goods and hence it may be possible in principle to give each individual more of both goods, making everyone better off.