Roman Keeney AGEC

 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

 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

 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 Local Global

 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

 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

Feasible Space Objective Equation

 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)

 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

 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

 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

 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)

 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

 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)

 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

 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

 Investment Shares  Inv 1 = 0.36  Inv 2 = 0.64  Expected Return   Variance   Sensitivity?  How do investment shares and risk change with changes in minimum expected return

 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