Econ 172A - Slides from Lecture 10Joel SobelOctober 21, 2010Econ 172A SobelAnnouncementsIImportant: Midterm room/seating assignments. (There arethree rooms and assigned seats.)IMidterm on October 28, 2010.IMaterial for Midterm:IProblem FormulationIGraphingIDuality (all topics)IInterpreting Excel SpreadsheetsI can ask you to “fill in the blanks” in a sensitivity table. I willnot ask you sensitivity questions like the ones I discuss intoday’s lecture.This means: material through 10-19 lecture (but today’slecture and Tuesday’s lecture practice useful skills for theexam).Econ 172A SobelExampleSimple formulation exercise.IA furniture company that makes tables and chairs.IA table requires 40 board feet of wood.IA chair requires 30 board feet of wood.IWood costs $1 per board foot and 40,000 board feet of woodare available.IIt takes 2 hours of labor to make an unfinished table or anunfinished chair.I3more hours of labor will turn an unfinished table into afinished table.I2 more hours of labor will turn an unfinished chair into afinished chair.IThere are 6000 hours of labor available.INo need to pay for this labor.Econ 172A SobelPricesThe prices of output are given in the table below:Product PriceUnfinished Table $70Finished Table $140Unfinished Chair $60Finished Chair $110Econ 172A SobelFormulationIObjective: Describe the production plans that the firm canuse to maximize its profits.IVariables: Number of finished and unfinished tables andchairs.ILet TFand TUbe the number of finished and unfinishedtables.ILet CFand CUbe the number of finished and unfinishedchairs.IRevenue:70TU+ 140TF+ 60CU+ 110CF,.ICost is 40TU+ 40TF+ 30CU+ 30CF(because lumber costs$1 per board foot).IProfit (Revenue - Cost): 30TU+ 100TF+ 30CU+ 80CF.Econ 172A SobelIThe constraints are:1. 40TU+ 40TF+ 30CU+ 30CF≤ 40000.2. 2TU+ 5TF+ 2CU+ 4CF≤ 6000.IThe first constraint says that the amount of lumber used is nomore than what is available.IThe second constraint states that the amount of labor used isno more than what is available.Econ 172A SobelSolutionIExcel finds the answer to the problem to be to construct onlyfinished chairs (1333.333).IIt is crazy to produce13chair.IThis means that the LP formulation is not literally correct.INow: We pretend that fractional chairs are ok.ILater: We will introduce methods to deal with the constraintthat variables should be integers.IThe profit is $106,666.67.Econ 172A SobelSensitivity questionsIWhat would happen if the price of unfinished chairs went up?ICurrently they sell for $60.IAllowable increase in the coefficient is $50, it would not beprofitable to produce them even if they sold for the sameamount as finished chairs.IIf the price of unfinished chairs went down, then certainly youwouldn’t change your solution.IWhat would happen if the price of unfinished tables went up?IThe allowable increase is greater than 70.IEven if you could sell unfinished tables for more than finishedtables, you would not want to sell them.IHow could this be?IAt current prices you don’t want to sell finished tables.IHence it is not enough to make unfinished tables moreprofitable than finished tables, you must make them moreprofitable than finished chairs.IDoing so requires an even greater increase in the price.Econ 172A SobelMore QuestionsIWhat if the price of finished chairs fell to $100?IThis change would alter your production plan: it involves a$10 decrease in the price of finished chairs and the allowabledecrease is only $5.IComplete answer: resolve problem.IThe best thing to do is specialize in finished tables, producing1000 and earning $100,000.IIf you continued with the old production plan your profit wouldbe 70 × 133313= 93, 33313, so the change in production planwas worth more than $6,000.Econ 172A SobelIHow would profit change if lumber supplies changed?IThe shadow price of the lumber constraint is $2.67.IThe range of values for which the basis remains unchanged is 0to 45,000.IThis means that if the lumber supply went up by 5000, thenyou would continue to specialize in finished chairs, and yourprofit would go up by $2.67 × 5000 = $10, 333.IIf lumber supply increased by more, you run out of labor andwant to reoptimize.IIf lumber supply decreased, then your profit would decrease,but you would still specialize in finished chairs.Econ 172A SobelIHow much would you be willing to pay an additionalcarpenter?ISkilled labor is not worth anything to you.IYou are not using the labor than you have.IYou would pay nothing for additional workers.ISuppose that industrial regulations complicate the finishingprocess, so that it takes one extra hour per chair or table toturn an unfinished product into a finished one. How wouldthis change your plans? (This problem differs from theoriginal one because the amount of labor to create a finishedproduct increases by one unit.)IYou cannot read your answer off the sensitivity table, but a bitof common sense tells you something.IThe change cannot make you better off.ITo produce 1,333.33 finished chairs you’ll need 1,333.33 extrahours of labor.IYou do not have that available.ISo the change will change your profit.IUsing Excel, it turns out that it becomes optimal to specializein finished tables, producing 1000 of them and earning$100,000.Econ 172A SobelNew ActivityIThe owner of the firm comes up with a design for a beautifulhand-crafted cabinet.IEach cabinet requires 250 hours of labor (this is 6 weeks offull time work) and uses 50 board feet of lumber.ISuppose that the company can sell a cabinet for $200, wouldit be worthwhile?Econ 172A SobelOne ApproachIChange problem by adding an additional variable and anadditional constraint.IThe coefficient of cabinets in the objective function is 150,which reflects the sale price minus the cost of lumber.IThe final value increases to 106,802.7211.IThe solution involved reducing the output of unfinished chairsto 1319.727891 and increasing the output of cabinets to8.163265306.IYou could not have guessed these figures in advance, but youcould figure out that making cabinets was a good idea.Econ 172A SobelUsing Sensitivity AnalysisIValue the inputs to the production of cabinets.ICabinets require labor, but labor has a shadow price of zero.IThey also require lumber.IThe shadow price of lumber is $2.67, which means that eachunit of lumber adds $2.67 to profit.IHence 50 board feet of lumber would reduce profit by $133.50.ISince this is less than the price at which you can sell cabinets(minus the cost of lumber), you are