Application of Intermediate Value Theorem for General Equilibrium Here's a problem restated from Ross Starr's General Equilibrium Theory.


  Consider a two-commodity economy with an excess demand function $Z(p)=(Z_1(p),Z_2(p))$. The price space is $p \in P = \left \{ p | p \in \mathbb{R}^{2}, p \geq 0, p_1 + p_2 =1 \right \}$. Let $Z(p)$ be continuous, bounded and fulfill Walras' Law as an equality, that is $p_1Z_1(p)+p_2Z_2(p)=0$. Assume $Z_1(0,1)&gt;0$, $Z_1(1,0)&lt;0$, $Z_2(0,1)&lt;0$, $Z_2(1,0)&gt;0$.
  Use the intermediate value theorem and Walras' Law to show that the economy has a competive equilibrium. That is, demonstrate that there is a price vector $p* \in P$ so that $Z(p*)=(0,0)$.


And I have a hint: Characterize $Z(p)$ as $Z(\alpha , 1- \alpha)$ for $0 \leq \alpha \leq 1$. Use the intermediate value theorem to find $0 \leq \alpha \leq 1$ so that $Z_1(\alpha , 1- \alpha)=0$. Then apply Walras' Law.

I'm having problem finding $\alpha$, how could I find it?
A stronger hint: Write $p_2=1-p_1$, so that $Z(p)=Z(p_1,1-p_1)$. Use the conditions $Z_1(0,1)&gt;0,Z_1(1,0)&lt;0$ etc. and the intermediate value theorem to argue that there exists a $p_1^*\in(0,1)$ such that $Z_1(p_1^*,1-p_1^*)=0$.