Suppose that Y = arg max {aj + Bpj + Uij} j=0,1,...,J (1) j=0 j=0 where J is a known integer, {a;}-0 and 3 are unknown parameters, {p;};-o are known prices, and Uj are unobservable variables. Suppose that each U; follows a standard type I extreme value distribution, and that Uij and Uik are independent for all j k. 1. Use the following information to solve for 3 to within three decimal places: j X j Pj P[Y = j] 0 0 0 0.427 1 -0.5 5 0.157 2 0 10 0.157 3 1 15 0.259 The following questions require you to write code in R. We will evaluate your code for accuracy, clarity, logic, good programming practice, and efficiency. 2. Suppose that we observe a sample of realizations {Y;}_1 drawn independently and iden- tically from the data generating process (1) under the values of J and {p;}}=1 given above. Write a function that computes the log-likelihood for this sample at any conjec- tured parameter values {a;}}_1 and . You should take = 0 as given. = j=1 3. Write a function that maximizes the log-likelihood in the previous part. You should pick an appropriate optimization package for doing this, and explain why you used the package that you did. =1 4. Let {;}-1 and 8 denote the maximum likelihood estimators based on a given sample {Y}1. Use these estimators to construct an estimator of 0 Pa +33 + U1 max {a; + Bpj + Uij} j=0,1,..., J Provide an economic interpretation of 0. 5. Design, implement, and report a Monte Carlo study that clearly shows that both and are consistent and asymptotically normal as n . 6. Discuss how you would construct confidence intervals for and that have 95% coverage probability asymptotically as n . Then design, implement, and report a Monte Carlo study that clearly shows your proposed method achieves this property. The standard type I extreme value distribution is also known as the standard Gumbel distribution. Its distribution function is given by F(u) = exp(-exp(-u)).