# How do I confirm that the constraint set wL + rK < Q is convex

Suppose a firm produces Q units using labour (L) for which the wage, w>0 and the price of capital (K) is r>0. It cannot employ negative amounts of its factors: L $$\geq 0$$, K $$\geq 0$$.

The firm's constraint can be written as $$wL + rK$$ $$\leq Q$$

How does one prove that such a set is convex? We know that if a constraint is convex, some vector, z = $$\lambda x + (1-\lambda)y$$ lies within the set.

Suppose I choose $$z_1 = \lambda L_1 + (1-\lambda) L_2$$

$$z_2 = \lambda K_1 + (1-\lambda) K_2$$

where $$\lambda \exists [0,1]$$

z = L + K

$$wz_1 + rz_2 =w[\lambda L_1 + (1-\lambda) L_2] + r[\lambda K_1 + (1-\lambda) K_2]$$ $$=\lambda[wL_1 + rK_1] +(1-\lambda)[wL_2 + rK_2]$$ $$\leq \lambda Q + (1-\lambda)Q$$ $$= Q$$

Thank you.

Consider an arbitrary pair of vectors $$(L_{1},K_{1})$$ and $$(L_{2},K_{2})$$ that satisfy $$wL_{i} + rK_{i} \leq Q$$ for $$i = 1, 2$$. To show that constraint set is convex, we need to show that any convex combination of that (arbitrary) pair of vectors lies in the constraint set. Namely, for all $$\gamma \in [0,1]$$ we have
$$\begin{split} w\left(\gamma L_{1} + (1-\gamma)L_{2}\right) + r\left(\gamma K_{1} + (1-\gamma)K_{2}\right) &= \gamma \left(w L_{1} + r K_{1}\right) + \left(1-\gamma \right)\left(w L_{2} + r K_{2}\right)\\ &\leq \gamma Q + (1-\gamma)Q = Q \end{split}$$ Thus the vector, $$(L_{3},K_{3})$$, is also in the constraint set (where $$L_{3} := \gamma L_{1} + (1-\gamma)L_{2}$$, and $$K_{3} := \gamma K_{1} + (1-\gamma)K_{2}$$) and so the set is convex.