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Lecture 1: Basics of Math and Economics

Lecture 1: Basics of Math and Economics. AGEC 352 Spring 2012 – January 11 R. Keeney. Basic Algebra. Number of equations Number of unknowns What relationship between these two is required to solve for the unknowns?. Applied Algebra. 20 acres of land 40 hours of labor

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Lecture 1: Basics of Math and Economics

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  1. Lecture 1: Basics of Math and Economics AGEC 352 Spring 2012 – January 11 R. Keeney

  2. Basic Algebra • Number of equations • Number of unknowns • What relationship between these two is required to solve for the unknowns?

  3. Applied Algebra • 20 acres of land • 40 hours of labor • Planting requirements • Corn = 1 acre of land; 2 hours of labor • Soybean = 1 acre of land; 2 hours of labor • What’s an algebraic description of my situation assuming I will use all resources and plant some combination of these two crops?

  4. Applied Algebra • Let C = planted corn • Let B = planted soybeans • C + B = 20 • 2C + 2B = 40 • Can we ‘solve’ this?

  5. ‘Solving’ • C + B = 20 • 2C + 2B = 40 • C = 20 – B (rewrite 1st equation) • Substitute into 2nd equation • 2*(20-B) + 2B = 40 • 40 – 2B + 2B = 40 • 40 + 2B = 2B + 40???

  6. ‘Solving’ • C + B = 20 • 2C + 2B = 40 • We can divide the second equation by 2 without changing the relationship • ½*(2C + 2B = 40) => C + B = 20 • The 2nd equation provides no ‘different’ information about my planting problem • Tradeoffs between the two crops • Limits I face in my planting

  7. ‘Solution’? • 40 + 2B = 2B + 40 • Any value works for B • Once you plug in a choice for B, then you just need to set the value of C to make sure the equation B + C = 20 holds • E.g. : set B = 50 => C = -30 • But, we might not want a value of planted acres that is < 0 so we could change our problem

  8. ‘Solution’? • Choose values of B and C with • 1) B + C = 20 • B >=0, C>=0 • These are called non-negativity conditions • Then B will be some choice on the interval [0,20] • C = 20 – B • What then is your solution to this problem?

  9. Graphical ‘Solution’ Any combination that appears on the line connecting (0,20) and (20,0) is a legitimate ‘solution’.

  10. Need more information, some economics • What would we use to make a choice among the infinite combinations that satisfy the resource (land, labor) equations?

  11. Economic information • Corn Net Returns/acre • $100 • Soybean Net Returns/acre • $50 • First, how do we represent this information mathematically?

  12. Back to algebra • The equation should describe total net returns, so let’s call that R. • Every acre of corn is $100 so that gives • 100*C • Every acre of soybeans is $50 • 50*B • R = 100C + 50B

  13. Collecting our mathematical information we have… • C + B =20 • R = 100C + 50B • That’s two equations for 3 variables • We’re no better off algebraically with the new information

  14. But, we know the solution if… • We are willing to assume that the operator with limited land and labor wants to maximize net returns • Step 1: Compare the net returns between the 2 crops (C > B) • Step 2: Choose to produce as much of that crop as is feasible (C = 20) • Step 3: If resources remain, use those for the other crop (B = 0)

  15. Optimization • The optimization assumption takes care of R for us • Says, 1st find B and C that will make R bigger than any other value it can have • Then, calculate R at the end • C = 20, B =0 • R = 2,000

  16. Simple with two choices • When we have a large number of choices this gets more complicated • Spreadsheet modeling • Formulate the model on paper • Input it correctly into a spreadsheet • Solve • Graphical methods • Algorithm methods • Understand, interpret, and communicate the final solution

  17. In general • Most of the work in this course is in developing the mathematical representation of the problem • Identifying objectives that decision makers use as optimization criteria • Identifying choices available to the decision maker that adjust their objective • Identifying the limits decision makers control in adjusting their objective

  18. Math and Computing • We will get better at writing the equations and building spreadsheets through repetition • In lecture we will often focus on how our decision problems relate to basic economic principles as you might have seen in AGEC 203 or something similar

  19. Next Week • Monday is a holiday • Tuesday there will not be a lab session • Wednesday • Discuss lab procedures which begin in week 3 • Review some of the stuff from the 2nd page of the level exam • Calculus, elasticity, optimization

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