A firm wants to find the level of output q that maximizes her profits (q, p) =

A firm wants to find the level of output q that maximizes her profits π(q, p) = pq − 1 2 q 2, where p > 0 is the price, which is exogenously determined. 

(a) Without solving explicitly for q ∗ (p), use the Envelope Theorem to come up with an expression for dπ∗ dp, where π ∗ is the value function. 

(b) Now solve for q ∗ (p) and use your result from (a) to determine the sign of dΠ dp. What is the interpretation? 

(c) Write now the value function π ∗ (p) and calculate its derivative with respect to p. How does your result compare to what you found in point (a), where you applied the envelope theorem