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Lecture 5: Objective Equations

Lecture 5: Objective Equations. AGEC 352 Spring 2011 – January 31 R. Keeney. Linear Programming. Linear All of the functions are linear in the variables Constraint inequalities Objective equation Programming Nothing to do with computer Comes from a term in planning

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Lecture 5: Objective Equations

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  1. Lecture 5: Objective Equations AGEC 352 Spring 2011 – January 31 R. Keeney

  2. Linear Programming • Linear • All of the functions are linear in the variables • Constraint inequalities • Objective equation • Programming • Nothing to do with computer • Comes from a term in planning • First application was in military procurement and distribution

  3. 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) • Combining the inequalities using the most restrictive in any neighborhood • The shape of the feasible space is key in linear programming • Linear segments create kinks or corner points

  4. Level Curves • An equation in 3 variables is difficult to graph in 2 dimensions • In economics, we use the level curve Y Z X

  5. Level Curve Example Y Production Function Y = f(X1 | X2) X1

  6. 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

  7. Level curves in economics • Typically start with Iso- • Prefix meaning equal • Isorevenue • Budget line (isoexpenditure) • Isoquant • Indifference curve (isoutility)

  8. Isorevenue • The assumptions of the PPF (output-output) 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

  9. 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.

  10. Isorevenue graph Qb Intercept = R/Pb R = PaQa + PbQb Qb = R/Pb – (Pa/Pb)Qa Isorevenue PPF Slope = -Pa/Pb Qa

  11. Isorevenue graph Since Pb is fixed, the intercept measures revenue. R2 is optimal because it is the highest intercept that is still feasible given the PPF. Qb Intercepts R1/Pb R2/Pb R3/Pb Slope = -Pa/Pb Qa

  12. Linear programming of the output-output 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

  13. LP PPF model 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 Qa

  14. 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

  15. LP Algebraic Form

  16. Corn and sugar intersection These constraints do not intersect in the 1st quadrant. They do not generate a relevant corner point since we have non-negativity constraints. Only one of these constraints can bind in a solution.

  17. Feasible space graph

  18. Corn and machinery If this is better than producing only PF or only SS, then it is the optimal solution.

  19. Lab on Tuesday • Posted by Noon • Part 1: Use a spreadsheet to graph and solve for a two variable linear program • Save to the same workbook as part 1 from last week (we’ll submit those at midterm) • Discussion questions on blackboard • Part II: Longer assignment with questions due next Monday

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