# Why is the production possibility set convex?

I'm studying International Trade and came across this question: Suppose a specific factor model where the global production of something is given by $Q_{w}=Q_{1}(.) + Q_{2}(.)$, with $Q_{1}, Q_{2}$ strictly increasing on all arguments and also strictly concave. So $Q_{w}$ will also be concave. As so, the superior sets of the level curves should be convex. But what I see is that the inferior set in this case is convex.

What am I not understanding here?

If $Q_1$ and $Q_2$ denote the production of goods 1 and 2 in a single country and you are in the space defined by them given $L_1 + L_2 = L$ then it is indeed the inferior set (the production possiblity set defined by the feasible $(Q_1,Q_2)$ pairs) that is convex. Formally this means that for all feasible $L_1,L_2,L_1',L_2'$ labor allocations you have \begin{eqnarray*} \mu \cdot Q_1(L_1) + (1-\mu) \cdot Q_1(L_1') & \leq & Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') \\ \\ \mu \cdot Q_2(L_2) + (1-\mu) \cdot Q_2(L_2') & \leq & Q_2(\mu \cdot L_2 + (1- \mu) \cdot L_2'). \end{eqnarray*} As this is equivalent to saying $Q_1$ and $Q_2$ are concave this is true given the condition that you describe. In this situation the function $Q_1 + Q_2$ has no meaning at all. What are we adding up? Ducks and cars? What will be the unit of measurement here?
But if $Q_1$ and $Q_2$ represent the production of a single good in countries 1 and 2 given the amount of labor allocated and the space is $L_1, L_2$, the curves being levels of $Q_w$, then it is the upper level curves that are convex. You can see this straight away because the marginal products of $L_1$ and $L_2$ are decreasing hence their rate of teachnical substitution is 'decreasing' as well. Formally:
By definition the function $Q_w(L_1,L_2)$ is concave if for all $L_1,L_2,L_1',L_2'$ and for all $\mu \in [0,1]:$ \begin{eqnarray*} \mu \cdot Q_w(L_1,L_2) + (1-\mu) \cdot Q_w(L_1',L_2') & \leq & Q_w(\mu \cdot L_1 + (1- \mu) \cdot L_1',\mu \cdot L_2 + (1- \mu) \cdot L_2'). \end{eqnarray*} We know that the functions $Q_1$ and $Q_2$ are concave, meaning \begin{eqnarray*} \mu \cdot Q_1(L_1) + (1-\mu) \cdot Q_1(L_1') & \leq & Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') \\ \\ \mu \cdot Q_2(L_2) + (1-\mu) \cdot Q_2(L_2') & \leq & Q_2(\mu \cdot L_2 + (1- \mu) \cdot L_2'). \end{eqnarray*} Adding these up we get \begin{eqnarray*} \mu \cdot \left(Q_1(L_1) + Q_2(L_2)\right) + (1-\mu) \cdot \left(Q_1(L_1') + Q_2(L_2')\right) \leq \\ \leq Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') + Q_2(\mu \cdot L_2 + (1- \mu) \cdot L_2'). \end{eqnarray*} And since $$Q_w(L_1,L_2) = Q_1(L_1) + Q_2(L_2)$$ we have \begin{eqnarray*} \mu \cdot \left(Q_1(L_1) + Q_2(L_2)\right) + (1-\mu) \cdot \left(Q_1(L_1') + Q_2(L_2')\right) \leq \\ \leq Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') + Q_2(\mu \cdot L_2 + (1- \mu) \cdot L_2') \\ \\ \mu \cdot \left(Q_w(L_1,L_2)\right) + (1-\mu) \cdot \left(Q_w(L_1',L_2')\right) \leq \\ \leq Q_w(\mu \cdot L_1 + (1- \mu) \cdot L_1',\mu \cdot L_2 + (1- \mu) \cdot L_2'). \end{eqnarray*} which is what we set out to prove.
• @RaulGuarini I think there is a fundamental misunderstanding as $Q_w = Q_1 + Q_2$ has no meaning in the situation you describe. (How do you add up different goods?) I gave formal math proofs anyway. – Giskard Mar 8 '16 at 15:46