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Problem 1. Utility maximization. (52 points) In this exercise, we consider a standard maximization problem with an unusual utility function. The utility function is u(x, y) = VI + vy. The price of good z is p, and the price of good y is py- We denote income by M, as usual, with M > 0. This function is well-defined for x > 0 and for y > 0. From now on, assume : > 0 and y > 0 unless otherwise stated. 1. Compute du/Or and Ou/02r. Is the utility function increasing in r? Is the utility function concave in x? (3 points) 2. The consumer maximizes utility subject to a budget constraint. Write down the maximization problem of the consumer with respect to a and y. Explain briefly why the budget constraint is satisfied with equality. (Hint: you can use the answer in point 1) (5 points) 3. Write down the Lagrangean function. (2 points) 4. Write down the first order conditions for this problem with respect to r, y, and A. (4 points) 5. Solve explicitly for r* and y* as a function of pr, py, and M. (8 points) 6. Do the solutions for a* and y* satisfy the positivity constraint, that is, a* > 0 and y* > 0?.(2 points) 7. Are these points maxima of the problem above? Check that the determinant of the bordered Hessian is positive at r*, y*, and A*. (8 points) 8. Use the expression for a* that you obtained in point 6. Differentiate it with respect to M, that is, compute or* /OM. Is this good a normal good? (4 points) 9. We are now interested in the sign of Or*/Op,- That is, we would like to know if the demand function is downward sloping. (at higher prices, a lower quantity of the good is demanded). Argue, using the answer that you gave in point 9 and something you learnt in class, that we know the sign of Or* /Op,. What is this sign? (8 points) 10. Can you guess the solutions for r* and y* for the following maximization problem? Develop your argument. (8 point) max (vi + vy) (1) s.t.pix + pyy = MProblem 1: A system is being designed. The inter-arrival times of customers are expected to be exponentially distributed with mean 1/) = 50 msec. Three options are considered as illustrated in Figure 1. (a) One single-server queue with infinite buffer space. The service times are exponentially distributed with mean 1/ = 20 msec. (b) Two single-server queues, each with infinite buffer space. Customers are randomly dispatched to each queue with an equal probability. The service times are exponen- tially distributed with mean 1/ = 40 msec at each server. (c) One two-server queue with infinite buffer space. The service times are exponentially distributed with mean 1/ = 40 msec at each server. Find the response time in each option using queueing analysis.Problem 2: Consider a single-server queue with infinite buffer space (a) Consider the situation . The inter-arrival time is a constant and is given by 1 sec . The service time required by each customer is always 0.5 sec 2 What is the mean waiting time per customer? (b) Consider the situation . The inter-arrival time is exponentially distributed with mean 1 sec . The service time required by each customer is exponentially distributed with mean 0.5 sec What is the mean waiting time per customer? (c) Compare the answers of (a) and (b), what conclusions can you draw?Problem 3: An Internet Service Provider has 4 dial-up ports. Connection requests follow a Poisson arrival process with mean arrival rate of 3 requests per hour. The session duration of each connection request is exponentially distributed with a mean of 1.5 hours. What is the probability that a connection request will be rejected? 3