Roman KeeneyAGEC 352

12-03-2012

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

FeasibleSpace

ObjectiveEquation

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 0.035

Variance 0.045

Sensitivity? How do investment shares and risk change

with changes in minimum expected return

0.04

0.05

0.05

0.06

0.06

0.07

0.03 0.04 0.04 0.05 0.05

Min

imum

Va

riance (R

isk)

Expected Return

Mean-Variance Frontier

0.0400

0.0450

0.0500

0.0550

0.0600

0.0650

0.0700

0.3 0.4 0.5 0.6 0.7 0.8

To

tal R

isk o

f P

ortf

olio

(Va

riance o

f R

etu

rns)

Share of Investment in X1

1.1600

1.1800

1.2000

1.2200

1.2400

1.2600

1.2800

1.3000

1.3200

1.3400

1.3600

0.35 0.45 0.55 0.65 0.75

Re

lati

ve R

isk

(Va

riance/M

ean)

Share of Investment in X1(Higher Variance 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