2. (30 points) Suppose that a perfectly competitive firm uses three inputs 1, k, and m with input prices w,r, and v, respectively, to produce output q, which it sells at price p. Its production function is given by: q = 12.5(lk)5 + In 10m. : = (. a) (10 points) Derive the first-order condition with respect to input m using the limit definition of the derivative. Hint: lim n00 ? (1 + )" = lim(1 +n)ze n-0 b) (10 points) Given that the optimal amount of all inputs is positive, check whether the second-order conditions hold. c) (10 points) Calculate the total differential of the first-order conditions to sign the effect of: 2 2 v on m*; i) von k*; 111) r on m"; iv) r on k". 2. (30 points) Suppose that a perfectly competitive firm uses three inputs 1, k, and m with input prices w,r, and v, respectively, to produce output q, which it sells at price p. Its production function is given by: q = 12.5(lk)5 + In 10m. : = (. a) (10 points) Derive the first-order condition with respect to input m using the limit definition of the derivative. Hint: lim n00 ? (1 + )" = lim(1 +n)ze n-0 b) (10 points) Given that the optimal amount of all inputs is positive, check whether the second-order conditions hold. c) (10 points) Calculate the total differential of the first-order conditions to sign the effect of: 2 2 v on m*; i) von k*; 111) r on m"; iv) r on k