1. Optimization with Constraints 3. Consider the following equation: f($,y) = 10$0'75y'25 subject to the constraint 100 3a: 4y = 0 Suppose :1: > 0 and y > 0. Solve for the optimal point (22*,y'). . Expected Values a. Which of the following random variables has the greatest Expected Value? i ii iii iv . A lottery where you win $1, 000 with probability 0.00001 or lose $10 with probability 0.99999 '. f (X ) = 5X + 1, where X is a lottery where you win $1 with probability 0.5 or lose $1 with probability 0.5 '. A lottery where you win $10 with 25% probability and 0 otherwise. . A lottery where you win $0.50 with certainty. 2 Monopoly 1. Using the Markup Rule. In some situations, it can be very difficult to infer what marginal cost actually is. However, the markup rule can be very valuable in providing an estimate.Suppose there is a market with a monopolist. They face demand from the inverse demand curve: Market 1: p(q) = (12 (a) Solve for the elasticity for the good. (b) What is Marginal cost at this elasticity? i. 49:2 ii. 0.5:; iii. 0.5 iv. 0.5!}2 v. None of the Above 2. Government Policies and Monopoly. In lecture we solved the monopoly model without friction. We will now expand this mindset to think what the outcomes of typical government policy interventions would be. Suppose a monopolist has a cost function C(q) = 2q2. They face demand from a market with an inverse demand curve: 10(q) = 18 39* (a) Solve for the optimal quantities and prices in the market. What is the rm's total prot at equilibrium? (b) Suppose the government of Market 1 puts a lump sum tax of $5 on the rm. What is the rm's total prot at the new equilibrium? (c) Suppose the government of Market 1 removes the lump sum tax and puts a per unit tax of $0.50 on each unit sold. What is the rm's total prot at the new equilibrium? ((1) Suppose the government of Market 1 removes the per unit tax and puts a price ceiling of $3. What is the rm's total prot at the new equilibrium? \fAnd has Marginal cost M C = 10. What is the deadweight loss incurred by the monopoly? Round your answer to the nearest hundredth. (a) 1155.63 (b) 2311.25 (1:) 117.5 (d) 43