<p>Perhaps you are confusing two things.</p>

<p>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') &amp; \leq &amp; 
Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') \\
\\
\mu \cdot Q_2(L_2) + (1-\mu) \cdot Q_2(L_2') &amp; \leq &amp; 
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?</p>

<p>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:</p>

<p>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') &amp; \leq &amp; 
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') &amp; \leq &amp; 
Q_1(\mu \cdot L_1 + (1- \mu) \cdot L_1') \\
\\
\mu \cdot Q_2(L_2) + (1-\mu) \cdot Q_2(L_2') &amp; \leq &amp; 
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. </p>

CC BY-SA 3.0
2016-03-08T11:18:26.283Z
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?
Raul Guarini Riva
CC BY-SA 3.0
2016-03-08T15:46:18.930Z
@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
CC BY-SA 3.0
2016-03-07T23:10:56.047Z
2016-03-08T15:42:31.863Z
2016-03-08T15:42:31.863Z
Giskard
<p>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 <em>inferior</em> set in this case is convex.</p>

<p>What am I not understanding here?</p>

<microeconomics><mathematical-economics><general-equilibrium>
Why is the production possibility set convex?
Raul Guarini Riva
# Why is the production possibility set convex?

(Asked by: username_0 on 2016-03-07)

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?

**Tags:** microeconomics, mathematical-economics, general-equilibrium


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## 1 answers

### Answer by username_1 on 2016-03-07. Score: 5 | Accepted answer

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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: @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.  

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