3
$\begingroup$

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?

$\endgroup$
5
$\begingroup$

Perhaps you are confusing two things.

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.

$\endgroup$
  • $\begingroup$ Cool, my fault for not being so precise. In fact, Q1 and Q2 are the production of different goods in the same country, subject to the constraint of L1 + L2 =L.. I'm ok with the intuition behind it, but I would like to see a mathematical proof. Shouldn't Qw = Q1 + Q2 be a concave function and hence, determine superior convex sets? $\endgroup$ – Raul Guarini Mar 8 '16 at 11:18
  • $\begingroup$ @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. $\endgroup$ – Giskard Mar 8 '16 at 15:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.