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Non-linearity and Modeling

Roman Keeney AGEC 352 12-03-2012. Non-linearity and Modeling. Introduction. In many situations, economic equations are not linear We are usually relying on the fact that a linear equation is a good approximation

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Non-linearity and Modeling

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  1. Roman Keeney AGEC 352 12-03-2012 Non-linearity and Modeling

  2. Introduction • In many situations, economic equations are not linear • We are usually relying on the fact that a linear equation is a good approximation • Even when we assume linearity, sometimes the economics of interest are non-linear • Example: Revenue = Price x Quantity • Quantity = f(Price) • Revenue = Price x f(Price) • dRev/dPrice = Price x df/dPrice + f(Price) • Since this is not a constant, the revenue function when demand depends on price is not constant • Recall our earlier Simon Pies model with the quantity demanded function

  3. Reasons for non-linearity • Non-proportional relationships • Price increases may increase revenue to a point and then decrease it • Depends on demand elasticity at any particular price • Non-additive relationships • E.g. Honey and fruit production • Efficiency of scale • Yield per worker may increase to some point and then decline • Non-linearity of problems results from physical, structural, biological, economic, or logical relationships • Linear models provide good approximations and are MUCH easier to solve

  4. Non-linearity Complications • The degree of non-linearity determines how likely we are to find a solution and have it be the true best choice • Non-linear problems can have local optima • These represent solutions to the problem, but only over a restricted space • Global optima are true best choices, the highest value over the entire feasible set • In LP, any local optima was guaranteed to be a global optima Global Local

  5. Special case of non-linearity: Quadratic Programming • Quadratic programming turns out to be a non-linear problem that is closely related to LP • Quadratic objective equation and linear equality and inequality constraints and non-negativity of variables • The only difference is the functional form (squared terms) of the objective equation • Quadratic function examples • 9X^2 + 4X + 7 • 3X^2 – 4XY + 15Y^2 + 20X - 13Y - 14

  6. QP Example • Min Z = (x – 6)^2 + (y – 8)^2 • s.t. • X <= 7 • Y <= 5 • X + 2Y <=12 • X + Y <= 9 • X, Y >= 0 • Linear constraints, so we could draw them as we always have • Objective equation is quadratic, in fact it is a circle • The 6 and 8 give the coordinates of the center of the circle • Z represents the squared radius of the circle • So, this problem seeks to minimize the squared radius of the circle centered at (6,8) subject to x and y being found in the feasibility set

  7. Example Objective Equation Feasible Space

  8. Spreadsheet modeling • Setup is no different • Need a non-linear formula for the objective • Solver is equipped to solve non-linear problems, just don’t click “assume linear” in the options • Sensitivity • Reduced gradient and Lagrange multiplier replace objective penalties and shadow prices but they are exactly the same • These come from the calculus solution to the problem (Method of Lagrange) • No ranges (allowable increases/decreases)

  9. Why are there differences? • Non-linear functions significantly more complex • Solutions need not occur at corner points of the feasible space • Why is it so useful? • Several models, particularly models involving optimization under risk. • Portfolio model • Minimize the variance of expected returns subject to meeting some minimum expected return

  10. Introduction: Portfolio Model • An individual has 1000 dollars to invest • The 1000 dollars can be allocated a number of ways • Equal split between investments • All in a single investment • Any combination in between • The individual wants to earn high returns • The individual wants low risk

  11. Risk and Returns • Real world investments • Those with high expected returns are those with high risks of losing money • Win big or lose big • Those with low expected returns are those with low risks of losing money • Win small or lose small • Potential losses (downside risk) tend to be larger than upside • Bad outcomes are really bad, Good outcomes are just pretty good

  12. Risks and Returns • Returns are defined by the proportionate gains above the initial investment • Final Amt = (1 + R)* Initial Amt • Risks are defined by the variability (variance) or returns • Given i possible outcomes • Variance is the sum over all i outcomes of • (xi – xmean)^2 • Higher variance means that a given investment produces greater deviations from its average (expected return)

  13. Multiple Objective of Investor • Investors want high returns • Investors want low risk • There are some combined objective equations that look at risk reward tradeoffs but they require knowledge of a decision maker’s risk aversion level • Risk aversion • The concept that people do not like uncertainty about their expected returns/rewards, and in fact will take lower expected returns to avoid some amount of risk/uncertainty when they are making plans or decisions • Absent any knowledge of risk aversion levels we can minimize risk while ensuring a minimum return or maximize return while placing a ceiling on risk

  14. Problem Setup • Two choices • 1) Minimize risk (variance) of the investment strategy • Subject to meeting some minimally acceptable average return for the portfolio • 2) Maximize returns • Subject to not exceeding some maximally acceptable average variance for the portfolio • In practice the second one has become more common • To be a quadratic program, we need to solve option 1 (want the quadratic equation in the objective)

  15. QP Model of Portfolio • Definitions • R1 = returns from investment 1 • Sigma1 = variance of investment 1 • R2 = returns from investment 2 • Sigma2 = variance of investment 2 • Sigma12 = covariance of investments • How much do they vary together? • B = Minimum acceptable return of portfolio • S1 = Maximum share of dollars invested in 1 • S2 = Maximum share of dollars invested in 2

  16. Algebraic Problem • Decision variables X1 and X2 are shares of the total investment • Min Var = X1*X1*sigma1 + X1*X2*sigma12 + X2*X2*sigma2 • Subject to • X1 + X2 = 1 (total investment) • R1*X1 + R2*X2 >= 0.03 (min return) • X1 <= 0.75 (max X1 allocation) • X2 <= 0.90 (max X2 allocation) • Non-negative X1 and X2

  17. Solution • Investment Shares • Inv 1 = 0.36 • Inv 2 = 0.64 • Expected Return • 0.035 • Variance • 0.045 • Sensitivity? • How do investment shares and risk change with changes in minimum expected return

  18. Solve the model for different minimum returns…

  19. How does risk change with investment in X1?

  20. A Different Perspective: Share of X1 and Risk Relative to Mean Return

  21. Results Indicate??? • Risk problems are complex • Investment 1 drives returns up but increases risk • It drives mean returns faster than risk over some range (per last graph) but what is acceptable? • Risk relative to mean is still high for all of these • Only two investments • Adding more choices adds complexity but also adds more ability to mitigate risk • Riskless Assets are often maintained in a portfolio for this reason

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