7. Homework Market Penetration & Prot Maximization in Armstrong's Model Assume that 411w 1} = M'(1 e'\"'}, i = B {for Buyers}, S {for Sellers} Hence Mi is maximum number of members of side'i, also the addressable market Hence p1 = NU'Ml is the market penetration {0 5 p1 5 1} (Note: platform profit maximization can be done with respect to {U 1} or {A1} or {pi} In this homework, working with {pi} is convenient} Assume that M5 = 1000, M5 = 100, C5 = C5 =10, and A5 is such that N5 = 93 Define E = {b3 + bSUZ, i.e., E is mean benefit that members bring to the other side {i} Prove mathematically.r that there exists a critical value of b denoted by E,C , such that for a_ll b {BC the profit maximized with respect to pH {or A3 _} is negative, whereas for a_ll E 3::- Ec the optimized profit is positive. Compute 5,; {ii} If computing EC mainly algebraically proves too difficult, use numerical experimentation to estimate it. (Note: Two or three decimal points in answer is adequate} [iii] Assume that E = 0.0?5. Maximize the platform profit with respect to pB and calculate the optimal profit, and the corresponding value of pH