Econ 172A, Fall 2008: Problem Set 2Instructions: Due: November 13, 2008, in class (no late papers).1. This is formulation problem that appeared on the midterm (with a few minor changes).A local company can produce three products, A, B, and C. The company can sell up to 3000units of Product A, up to 2000 units of Product B, and up to 2000 units of Product C. Eachunit of Product C uses 2 units of A and 3 units of B. Products A and B can be producedfrom either Process I or Process II (or combinations of these two processes). In Process I theCompany can produce two units of A and three units of B for $6. In Process I I, the companycan produce one unit of A and two units of B for $5. The unit prices for the products are $5for A, $4 for B, and $25 for C. The quality levels of each product are: A, 8; B, 7; C, 6. Theaverage quality level of the units sold must be at least 7.We defined variables: Let xibe the number of units of product i sold for i = A, B, C. Let Ljbe the level Proce ss j is operated, for j = I and II and arrived at this formulation:max 5xA+ 4xB+ 25xC− 6LI− 5LIIsubject to xA+ 2xC− 2LI− LII≤ 0xB+ 3xC− 3LI− 2LII≤ 0−xA+ xC≤ 0xA≤ 3000xB≤ 2000xC≤ 2000x, L ≥ 0I changed the direction of the third inequality to write the problem in standard form.(a) Solve the problem using Excel.(b) What happens to the solution and the value if the price of Product B goes down to 3?(c) What happens to the solution and the value if the price of Product C goes down to 5?(d) How does the solution of the problem change if the minimum average quality increasesto 7.5?(e) How do profits change if the capacity to produce B increases to 4000 (from 2000)?(f) How do profits change if the capacity to produce C increases to 4000 (from 2000)?(g) Suppose that the average quality had to be 7.5 instead of 7. How would the solutionchange?2. I s olved a version of a linear programming problem using Excel. I attach the answer reportand the sensitivity report from Excel. In these reports, I replaced several values with questionmarks (???). Your job is to replace these question marks with the correct information. I havenot given you enough information to reconstruct the problem. You should fill in the missingvalues using your knowledge of Excel, duality theory, and complementary slackness. You maynot have sufficient information to complete the table. If you cannot determine some of themissing numbers, then say so. If you can fill in a value, then explain what permitted you todo so. Can you determine the final value of the problem? If so, what is it?3. The California Park Authority controls two tracts of land. Tract 1 consists of 300 acres andtract 2, 100 acres. Each acre of tract 1 can be used for redwo od trees, as a wilderness preserve,or for hunting. Each acre of tract 2 can be used for redwood trees, as a wilderness preserve,1or for camping. The capital (in hundreds of dollars) and labor (in worker-days) required tomaintain one acre of each tract and the profit (in thousands of dollars) per acre for eachpossible use of land are given in the table below. Capital of $150,000 and 200 worker-days oflabor are available. I formulated the problem of maximizing profit as a linear programmingproblem under the assumption that the park authority can choose to leave portions of eachtract unused (using no resources and earning no profit).Capital Labor ProfitTract 1 Redwood 3 .10 .20Wilderness 3 .20 .04Hunting 4 .20 .05Tract 2 Redwood 1 .05 .06Wilderness 30 5.00 .09Camping 10 1.01 1.10I arrived at the following formulation:max .2xR+ .04xW+ .05xH+ .06yR+ .09yW+ 1.1yCsubject to 300xR+ 300xW+ 400xH+ 100yR+ 3000yW+ 1000yC≤ 150000.1xR+ .2xW+ .2xH+ .05yR+ 5yW+ 1.01C≤ 200xR+ xW+ xH≤ 300yR+ yW+ yC≤ 100xRxWxHyRyWyC≥ 0In the formulation, there are six variables. xidenotes the number of acres of the first tractallocated to use i (R stands for redwood; W for wilderness; and H for hunting). The yiare thecorresponding quantities for tract 2. The first constraint captures the limitation on capital;the second constraint states that no more than 200 worker-days of labor are available; thethird and fourth constraints bound the amount of land in each tract; and the final line statesthat variables are non-negative.I solved this problem using Excel. The output follows this problem. Use the output to answerthe questions below. Answer the questions independently (so that a change described in onepart applies only to that part).(a) What is the profit maximizing way to allocate the tracts? (Give units.)(b) What is the profit associated with the profit maximizing allocation? (Give units.)(c) An economics professor owns land adjacent to tract 1. She offers to sell the land to thestate for $200 per acre. The land can be used just like the land in tract one. Would thestate increase its profit by buying the land? Explain.(d) Answer the previous question assuming that the professor owned land adjacent to tract2.(e) A state legislator proposes to devote some parts of the land to bungie jumping. Bungiejumping requires no capital and 5 worker-days of labor per acre whether it is done intract 1 or in tract 2. How high would the profit need to be for the state to benefit fromorganizing bungie jumping in tract 1? How high would the profit need to be for the stateto benefit from organizing bungie jumping in tract 2?(f) If the profit from hunting in tract 1 increased to $100 an acre, how would the solutionand total profit change?2(g) If the profit from camping in tract 2 dec reased by $25 an acre, how would the solutionand the total profit change?(h) Suppose rich wilderness lovers move to California and raise the profit associated withwilderness use in either tract to $250 per acre. How would the solution and total