Lecture 1: Basics of Math and Economics AGEC 352 Spring 2011 – January 12 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 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?

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

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

‘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 2 nd equation provides no ‘different’ information about my planting problem ◦ Tradeoffs between the two crops ◦ Limits I face in my planting

‘Solution’? B = 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

‘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?

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

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

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

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

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

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)

Optimization The optimization assumption takes care of R for us Says, 1 st 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

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

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

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

Next Week Monday is a holiday Tuesday there will not be a lab session, I will post the first assignment that day Wednesday ◦ Discuss assignment and lab procedures which begin in week 3 ◦ Review some of the stuff from the 2 nd page of the level exam  Calculus, elasticity, more optimization