(a) [10 Points] Assume that Firm 1 and Firm 2 have constant positive marginal costs c and c correspondingly. What conditions on these marginal costs must be satisfied, so that both firms choose to produce positive quan- tities in the Cournot equilibrium? (b) [10 Points] Assume now that Firm 1 has quadratic cost function Ci(q) = qi+15q, and Firm 2 has quadratic cost function Cy(q2) = 97+30g2. What quantity will each firm produce in the Cournot equilibrium? Problem 6. [20 Points] Integration, optimization and solving sys- tems of equations Note that parts (b) and (c) do not depend on part (a) and could be completed before or after part (a). (a) [5 Points] Compute the integrals and (b) [10 Points] Now assume that H(b) and K(s) are given by b' - 05 2 H(b) = b - go 6 - 50 201 and K(8) = c B1 B1 Take the derivatives of H(b) with respect to b and of K(s) with respect to s and verify that, under the assumptions of > 0 and 81 > 0, H(b) and K(s) are concave functions. Then solve I'(b) =0 for b and K(s) = 0 for s and denote the solutions by b* and s*.Problem 5. [20 points] Cournot Model Consider a market with two firms that produce the same good. The inverse demand function for the good is given by P(Q) = 150 -Q, where () is the total output. The firms simultaneously choose how much output to produce.(c) [5 Points] Using the W for functions b'[u] and s'[c} derived in part [b] and mu) = a. + .61 (1) 31] = 0|} + .m, (2) solve for the using of parameters 50,51,09101