3. Consider a firm that produces a single output Q using two inputs, q1 and q2. The production technology is the Cobb-Douglas:
$Q = q_1^{\frac{1}{3}} q_2^{\frac{1}{2}}$.
Let $p_1$ be the price of $q_1$, and $p_2$ be the price of $q_2$. Assume that the price of the output Q is 1.
(a) (10) Find the effect of an input price $p_2$ change on the optimal quantity, $q_1^*$, of the input good 1 using the implicit function theorem.
(b) (10) Find the effect of an input price $p_2$ change on the optimal quantity, $q_2^*$, of the input good 2 using the implicit function theorem.