1. A 2U15 report by the music industry estimated the revenue lost to the industry every year from illegal downloading. In this problem we will derive some of the estimates that may have gone into thur calculation (approximately). First, start with the individual consumer's problem. Suppose a typical consumer has a yearly entertainment budget of I that they can allocate between music downloads (D) and other forms of entertaimnent (E) Consumer preferences are characterized by a utility function UH), E). (a) Write an exprmsion for the consumer's budget constraint as a function of their enter tainment budget and the prices of music downloads (PD) and other entertainment (PE). (b) Write the consumer's constrained optimization problem in Lagrangian form. (Note: You do not need to solve it or derive rst order conditions.) Suppose the solution to the consumer's optimization problem yields an individual demand curve of (c) Calculate the consumer's income elasticity of demand for music downloads. (d) Is music a luxury good, necessity good, inferior good, or none of those? Explain.It costs $1 per song to download music legally. Suppose the effective cost of downloading a song illegally is $0.1 per song, after accounting for the risk of getting caught. Further, imagine a typical consumer has an entertainment budget of / = 900, and there are 100 million consumers that illegally download music in the market. (e) The graph below depicts the typical consumer's demand curve for music. On this graph or one you draw, show the consumer surplus one gains from illegally downloading music. (f) Does downloading music satisfy the Law of Demand? Explain. (g) Calculate the aggregate demand curve for illegal downloaders as a function of Pp if there are 100 million downloaders and each downloader has a budget of / = 900. (Hint: 90005 = 30).(h) A naive commentator claims that since there are 15 billion illegal downloads every year and each song would cost $1 to download legally, illegal downloading must be costing the industry Our Lil:-- in revenue. Explain in words why this value is incorrect. (i) How much revenue would the industry actually earn every year if all the illegal down- loaders switched to legal downloads? (j) To calculate total surplus in the music industry, we would need a measure of producer surplus. Could we use the lost revenue we calculated in part (i) as a measure of the producer surplus lost to illegal downloading? Why or why not