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Prove the following result in the derivation of the asymptotic optimality of the bid-price control policy. Let X be a random variable with finite first and second order moments, i.e., E[X] = µ < +∞, E[(X − E[X])^2] = σ^2 < +∞. Let x+ = max{x, 0}. Prove the following: ∀m ∈ R,
E[(X − m)+] ≤ (√ (σ2 + (m − µ)^2) − (m − µ))/2.
Hint: You can without loss of generality let m = 0
![E[(X - m)+] ≤
√σ² + (m − µ)² - (m - µ)
-
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F82214613-ec36-4dd8-8d54-892cd4bff3e3%2Fbe925e9f-1d4e-4603-b0c6-be129fe7626b%2F84itlfk_processed.jpeg&w=3840&q=75)
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- The Revealed Preference Theory was formulated by- (A) Lionel Robbins (C) Joan Robinson (B) Paul Samuelson (D) Alfred MarshallFour companies (Firm 1, 2, 3 and 4) are producing a product for the market. Each company will decide the number of products produced. You are given that • . Each firm can choose its qi. Given the quantites produced by four companies (denoted by 91, 92, 93, 94 respectively), the market price of the product is P = 400 - 91-92-93 - 94. Cost of producing one product is 20 (*Note: So the total cost for producing qi units of product is 20qi. Firm 1 and Firm 2 will first decide the quantities produced simultaneously at the beginning. After knowing 91 and 92, Firm 3 and Firm 4 will decide the quantities produced simultaneously. (a) State the strategic profiles of each firm. (b) Find all possible subgame perfect equilibrium for this games. Provide full justification to your answer.Marbella has 101 residents. All wear the same fancy clothes and each has the same utility function, u(m, b, B) = m + 14b − b2 − B/50, where m is the amount of macaroni (in kilograms) that he or she eats per day, b is the number of hours that he or she spends on the beach per day, and B is the total number of person-hours spent per day on the beach by other residents of Marbella. Each has an income of $10 per day and macaroni costs $1 per kilogram. The city council is considering a law that would limit the amount of time that any person can spend on the beach. How many hours per day should they allow in order to maximize the utility of a typical Marbellite? A. 6 B. 9 C. 7 D. 10 E. They could not possibly be made better off by legislation that limits their freedom to choose.
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