Please help me with question d and e

1. Mutitasking and ownership vs performance contracting. Con- sider the Pepsi-Crown example discussed in class, with the same notation, except that now Crown makes two types of investments: P = wg+w1u1 +1r2sg, K = To +116\" +ryga2, Crown's costs are is? and is: Suppose that cl is modifying bottles to t the bottling machines and a2 is labeling. Coke and Pepsi have exactly the same bottling machines, so that in = 71 > {1. But the more that Crown makes the label look like a Pepsi logo, the less it looks like a Coke logo, and vice versa: 1r: > I} > 72 [the idea is that positive values of a; generate more \"Pepsi-like\" logos, negative values more Coke-like; Crown's cost is g, no matter what sign a; has); in fact, for simplicity assume 72 = -.-72. Finally, assume that so > \"yo (e.g., the Pepsi plant is located closer to Crown's factory than is the Coke plant). a) What are the rst-best values of cl and a2, i.e., those that maximize the joint surplus of Pepsi and Crown (i.e. P e? ag)? b) Suppose Crown owns the plant. 'What level of modifying and labeling will Crown choose? What is the total surplus generated? c) Answer the same questions as in (b) for the case of Pepsi ownership. d) Which ownership structure do you expect will emerge? Why? e) Suppose that a \"bottle quality index\" Q, which is veriable in court, becomes available. Crown and Pepsi contemplate signing a contract in which Crown's payment will be conditioned on Q, i.e. Crown will be paid .9 +bQ for the bottles it produces, where s is a xed payment (that may be negative) and b is chosen to optimize the joint surplus of Crown and Pepsi. Suppose that Q = qlul + qzaz + e, where e is a random variable with mean zero, and q], q; are both positive. 1) Under what conditions will Crown and Pepsi prefer contracting on Q to allowing hold-up and renegotiation under the ownership structure you found in (d)? (That is, derive an inequality relating 131,132, 771,113 under which they prefer the performance contract). ii) Interpret the expression you found in (i). For instance, do you expect the performance contract to be more or less likely to be used when Q is very sensitive to label quality but not very responsive to bottle modication, while Pepsi's prots depend signicantly on how well the bottles t their machines but not so much on Crown's labeling activity? Explain