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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