2. For the following scenarios: (i) Set the problem as an optimization problem with inequality constraints, (ii) Find the extreme values (if possible). Extra: Answer any additional question asked in the description of the problem. (a) Survivor: A consumer lives on an island where she produces two goods, as and y, according to her production possibility frontier 9:2 + y2 S 200, and she consumes all goods by herself. Her utility function is $113. She faces an environmental constraint on the total output of both goods. The environmental constraint is given by a: + y S 20. 1. Write the KKT rst order conditions ii. Find the optimal :c and y. Determine which restrictions are binding. (b) Peak load pricing: An electric company is setting up a power plant in a foreign country and it has to plan its capacity. The peak period demand for power is given by p1 = 400 q1 and the offpeak is given by p2 = 380 (12. The variable cost is constant and equal to $20 per unit (paid in both markets) and capacity costs $10 per unit which is only paid once and is used in both periods. Note that (11 + (12 S K, K > 0 being the maximum capacity. 1. Write down the Lagrangian and Kuhn-Tucker conditions for this problem. ii. Find the optimal output and capacity for this problem. iii. What's the interpretation of A1 and A2? iv. Now suppose capacity cost is 30 per unit (paid only once). Find quantities, capacity and how much of the capacity is paid for by each market (i.e. A1 and A2)? (0) Incentives: Consider the case of a parent with a kid, call him Kevin. Kevin's parents are concerned with his academic performance. Kevin's grade is a function that depends on his effort 6 2 0, 9(6), Assume that g'(e) > 0, g"(e) 0 is a policy parameter controlled by Kevin's parents. From Kevin's point of view, b is given. Using this rule, nd Kevin's optimal effort as a function of b. iii. Find the optimal b using Kevin's parents utility function