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Lecture 5: Objective Equations AGEC 352 Fall 2012 – September 10 R. Keeney Linear Programming Linear ◦ All of the functions are linear in the variables Constraint inequalities (feasible space) Objective equation Programming ◦ Nothing to do with computer code ◦ Comes from a term in planning ◦ First application was in military procurement and distribution Review Feasibility In lecture 4 we developed a graphical approach to the feasible space ◦ Boundary defined by the linear constraints Inequalities create a region on either side of a constraint that is feasible (half-space) Combine the inequalities using the most restrictive in any area of the graph The shape of the feasible space is the key in linear programming ◦ Linear segments create kinks or corner points Level Curves An equation in 3 variables is impossible to graph in 2 dimensions ◦ In economics, we use the level curve Y Z X Level Curve Example Y Production Function Y = f(X1 | X2) X1 Level Curve Example Y For any ‘level curve’ below, the X2 value is fixed but it is a variable. Y when X2 = 3 Y when X2 = 2 Y when X2 = 1 X1 Level curves in economics Typically start with Iso◦ Prefix meaning equal Isorevenue Budget line (isoexpenditure) Isoquant Indifference curve (isoutility) Isorevenue The assumptions of the PPF (outputoutput) model ◦ Price taking = decision maker has no impact on prices (exogenous) ◦ Chooses the output mix that maximizes revenue 3 variables: Revenue, quantity of a, quantity of b Isorevenue and Units R = PaQa + PbQb Qb = R/Pb – (Pa/Pb)Qa Qb units = bushels of b R units = dollars Pb units = dollars per bushel of b Pa units = dollars per bushel of a Qa units = bushels of a Cancel out and we see the isorevenue is in units of the Y axis variable. Isorevenue graph Qb Intercept = R/Pb R = PaQa + PbQb Isorevenue Qb = R/Pb – (Pa/Pb)Qa PPF Slope = -Pa/Pb Qa Isorevenue graph Qb Intercepts R1/Pb R2/Pb R3/Pb Since Pb is fixed, the intercept measures revenue. R2 is optimal because it is the highest intercept that is still feasible given the PPF. Slope = -Pa/Pb Qa Linear programming of the outputoutput model Works exactly the same as graphically solving the model with the non-linear PPF There is no tangency result but we are still looking for the isorevenue line with the greatest intercept that is still feasible LP PPF model Qa For given prices, the solution is found by identifying the objective equation level curve that is 1) feasible and 2) has the highest intercept. Qa Implications Linear programming solutions will always occur at a corner point of the feasible space ◦ If the slope of the objective equation is exactly equal to the slope of a boundary constraint, multiple solutions (including 2 corner points) exist We can solve linear programs just by solving for all possible corner points and evaluating the revenue at each one LP Algebraic Form max R 8 PF 6 SS subject to : corn : 5 PF 3SS 600 sugar : 4 PF 2 SS 600 m achinery: PF 2 SS 200 non negativity: PF 0, SS 0 Corn and sugar intersection 5 PF 3SS 600 4 PF 2 SS 600 5 PF 3SS 4 PF 2 SS PF SS 5SS 3SS 600 SS 200 PF 200 These constraints do not intersect in the 1st quadrant. They do not generate a relevant corner point since we have nonnegativity constraints. Only one of these constraints can bind in a solution. Standard Stuff Feasible space graph 350 300 250 200 150 100 50 0 0 30 60 90 120 150 180 210 Premium Finest Corn constraint Mach. constraint Sugar constraint Corn and machinery corn : 5 PF 3SS 600 m achinery: PF 2 SS 200 5 PF 3SS 3PF 6 SS PF 1.5SS 5(1.5SS ) 3SS 600 10.5SS 600 SS 600/ 10.5 PF 900/ 10.5 If this is better than producing only PF or only SS, then it is the optimal solution. Lab for Wednesday Posted after class ◦ Use a spreadsheet to graph and solve for a two variable linear program Discussion questions at end