University of California, DavisDepartment of Agricultural and Resource Economics

ARE 252 – Optimization with Economic Applications – Lecture Notes 10Quirino Paris

Risk programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . page 1Freund Treatment of Price Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chance-Constrained Risky Revenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4Empirical Results of Freund Model with Chance Constraint on CARA parameter . . . . . . . . . . . . . . . . 5 Risky Output Prices and Risky Limiting Input Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Empirical results with risky output prices and risky input quantities using Freund’s data . . . . . . . . . . . .9Risky Output Prices and Risky Limiting Input Quantities with Non-Zero Covariance Matrix . . . . . . . 10

Risk ProgrammingRisk programming regards the methodology of dealing with risk in a mathematical programming framework. Risk may involve any component of an economic problem: price, quantity and technicalcoefficients. For lack of time, we will discuss price and quantity risk, one category at a time. In a famous paper, John W. Pratt introduced the notion of risk aversion (Risk aversion in the small and in the large, Econometrica, 1964, pp. 122-136).

Risk aversion can be interpreted as a measure of the curvature of the utility function.

Given a differentiable utility function u(x) , where x represents wealth (money income) the absolute risk aversion is defined as

u′′(x) (second derivative) ≤ 0absolute risk aversion ARA(x) = − = − ≥ 0

u′(x) (first derivative) > 0 The ARA function can be increasing, decreasing or constant as wealth changes. It is known in the literature as the Arrow-Pratt absolute risk aversion function.

Risk Premium Another important notion in risk analysis is the concept of risk premium (RP). Given a risky prospect (of an event), that is, a random variable z! (the symbol ∼ is called tilde), the risk premium is the amount of money that would render a decision maker indifferent between assuming the risk (of theevent) and receiving, for sure, a non-random amount of money. That is, using the symbol E(⋅) to indicate the mathematical expectation,

indifferent Eu(x + z!) = [ z) − RP ] .u x + E( !

The definition of risk premium, RP, is implicit. The purchase of an insurance premium, IP, as a protection against an unfavorable event may be thought to be IP ≤ RP . That is, an economic agent may choose to pay an insurance company for increasing levels of an insurance premium. In this case, theeconomic agent will choose u x + E(z!) − RP When IP > RP , the economic agent will choose [ ] . Eu(x + z!) to deal directly with the risky event. The quantity of money (beside existing wealth, x ) in the squared brackets is called “sure cash” or, more generally, E(z!) − RP ≡ certainty equivalent .

1

Freund Treatment of Price Risk In 1956, (eight years before Pratt’s paper), Rudolf J. Freund presented a paper dealing with price risk asa quadratic programming model (The Introduction of Risk into a Programming Model, Econometrica,1956, 253-263). The empirical contest analyzed farmer behavior in North Carolina.

Freund Assumptions:1. The vector of output market prices is stochastic (a random variable), p! , and it is normally

distributed p! ∼ N[E(p! , Σp )

2. The decision maker’s utility function is the negative exponential function u(r!) = 1− e−φr! , where r! is stochastic revenue and φ > 0 is a constant absolute risk aversion (CARA) parameter. In fact, following Arrow-Pratt definition of the ARA function

u′′(r!) −φ 2e−φr!

ARA = − = − = φ > 0 . u(r!) φe−φr!

3. Economic agents maximize their expected utility: The statistical definition of expected random variable is the integral of the product of the given random variable times its density function.

Given these three assumptions, the expected utility is the following integral (computed in theappendix to these notes):

+∞ ⎛ 1 −[r! − E(r!)]2 ⎞ Eu(r!) = ∫ (1− e−φr! )∗⎜ 2π 2σ r

2 ⎟ ⋅ dr! −∞ ⎝ 2σ r ⎠

1− e−φ[E (r! )−φ 2Var(r! )] =

Since stochastic revenue is r! = p x , expected revenue p)′x and its variance is ! ′ is E(r!) = E( !Var(r!) = x′Σpx . The vector x represents farmer’s production plan and it is not a random variable because farmer chooses it. Therefore, expected utility is

Eu(r!) = 1− e−φ[E(p! )′x−φ 2 x′Σpx]

Maximization of the exponent is equivalent to maximize expected utility. The exponent is φ[E(p! )′x − φ x′Σpx] . Hence, maximization of the exponent requires to max φ and to max 2

[E(p! )′x − φ 2 x′Σpx] . Freund chose to deal only with max [E(p! )′x − φ

2 x′Σpx] . The risk premium under these assumptions is

1− e−φ[ E (r! )−RP ] −φ[E (p! )′x−φ x′Σpx]2Eu(r!) = = 1− eHence, RP = φ

2 x′Σpx . The certainty equivalent (CE) is E(p! )′x − φ x′Σpx . Note that the units of the 2

ARA parameter are 1/$. In fact: xE(p! )′x − φ

2 x′Σp

$ − φ 2 $

2

2

Hence φ = 1$ . Finally, 1 /φ is called the risk tolerance parameter.

Freund’s primal problem: maxCE = max NTR = E(p! )′x − φ

2 x′Σpx subject to D ≤ S dual variables

Ax ≤ b y (1) with all nonnegative variables. NTR stands for net total revenue. The derivation of Freund’s dual problem follows the familiar procedure

Lagrange function L = E(p! )′x − φ 2 x′Σpx + y′[b − Ax]

Relevant KKT conditions ∂L = E(p! ) −φΣpx − A′y ≤ 0 dual constraints ∂x

x′ ∂L = x′E(p! ) −φx′Σpx − x′A′y = 0 CSC ∂x

Using the CSC to simplify the Lagrange functionL = E(p! )′x − φ x′Σpx + y′[b − Ax]2

= x′A′y +φx′Σp x′Σpx + ′ y′Axx − φ y b −2

= ′ 2 x′Σpy b + φ x

Therefore, Freund’s dual problem is

minTC = TCpp + RP = b y ′ + φ x′Σpx2

subject to MC ≥ MR A′y +φΣpx ≥ E(p! ) (2)

with all nonnegative variables. The dual objective function states that farmer should minimize total cost divided into the portion oftotal cost of the physical plant and the Risk Premium. The dual constraints say that marginal cost isalso divided into two parts: the marginal cost associated with the non-random technology, A′y , and the marginal risk premium, φΣpx .

It is interesting that when price risk is involved, although farmer is still considered a price taker on the output market, the maximization of the expected utility assumption generates the possibility of “expected profit.” In fact, by equating the primal and the dual objective functions, it is possibleto define expected profit as

E(π! ) = NTR − TCpp = ⎡⎣ p)′x − φ 2 x⎤⎦ − b y = φ

2 xE( ! x′Σp ′ x′Σp

It is as if the expected utility hypothesis (under all the assumptions of this model) introduces a fictitious “demand function” of farmer’s outputs in the form of p = E(p! ) −φΣpx . And, furthermore, it is as if farmer behaves as a perfectly discriminating monopolist by defining the Certainty Equivalent as the integral under that fictitious “demand function.” All this discussion has the objective to relate the structure of the risk model to what we have discussed in previouslectures. It would be unsatisfactory to say that the primal economic agent minimizes the total cost

3

of the physical plant and the risk premium because farmer has already paid the risk premium in the primal objective function. Therefore, a dual economic agent who wishes to buy out this farmer operating under price-risky conditions will have to minimize the total cost of the physical plant and the expenditure for reimbursing farmer of the risk premium he had to pay to avoid facing the risky prices directly.

The LCP structure of this problem exhibits the following matrix and vectors ⎡ ⎡⎤A′ −E(p! )

b

⎡⎤ ⎤φΣp xM = ⎢ ⎢⎣

⎥ ⎥⎦ , q = ⎢

⎢⎣ ⎥⎥⎦ , z = ⎢

⎢⎣ ⎥⎥⎦0 y−A

Before solving this Freund’s problem there is to know what to do with the ARA parameter φ . Freund wrote (page 258): “The estimation of the risk aversion constant φ is a purely subjectivetask, and any chosen value is exceedingly difficult to defend.” This seems a strange admission and even more surprising is that Econometrica’s editors accepted the paper. The following section discusses a procedure to estimate the CARA parameter using the available information abutfarmer’s technological and economic environments.

Chance-Constrained Risky RevenueWith some probability, a farmer may survive unfavorable events such as total revenue being less than total cost. Charnes

and Cooper (Charnes, A. and Cooper, W.W. (1959) Chance Constrained Programming, Management Science 6,73-79)

proposed a useful approach to deal with this case. Consider the following probabilistic proposition:

Prob(p x ! ′ ≤ y′Ax) ≤ 1 − β (3)

where the probability that uncertain (risky) total revenue ! ′ y′Ax should be p x be less than or equal to certain total cost

smaller than or equal to 1 − β . Intuitively, for how many years could a farmer survive while operating in the red? As an

example, say once every twenty years. In this case, we could estimate the probability 1 − β = 1 / 20 = 0.05 .

To derive a deterministic equivalent of relation (3) it is convenient to standardize the random variable ! ′p x by

x)1/2 :subtracting its expected value E(p! )′x and dividing it by the corresponding standard deviation (x′Σ p

Prob( ! ′ y′Ax) ≤ 1 − βp x ≤

! ′ p y′Ax − E(p! )′x ⎞⎛ p x − E( ! )′xProb

x)1/2 ≤

x)1/2 ⎠⎟ ≤ 1 − β

⎝⎜ (x′Σ p (x′Σ p

(4) ⎛ y′Ax − E(p! )′x ⎞

Prob τ ≤ ⎝⎜ (x′Σ p x)1/2 ⎠⎟

≤ 1 − β .

x)1/2 ≤Prob(E(p! )′x + τ (x′Σ p y′Ax) ≤ 1 − β .

By choosing a value of the standard normal random variable τ , say τ = τ , that corresponds to probability 1 − β , the

deterministic equivalent of relation (4) assumes the specification

x)1/2 ≤E(p! )′x + τ (x′Σ p y′Ax. (5)

4

There remains to establish a relation between the τ parameter and the CARA coefficient φ . This relation is obtained by

subtracting the complementary slackness condition of the dual constraints (2) from relation (5):

x)1/2 ≤E(p! )′x + τ (x′Σ p y′Ax

− [E(p! )′x −φx′Σ p x] = −y′Ax.

The addition of the two relations results in

x)1/2 τ / (x′Σ p + φ ≤ 0. (6)

Recall that φ must be maximized in order to maximize expected utility. Hence, relation (6) is executed as an equation.

Relation (6) (taken as an equation because τ < 0 ) defines the CARA parameter φ simultaneously with the decision

variables x , once the value of τ is selected by the researcher. As an example, if the survival probability is determined to

be 1 − β = 0.05 , the one tail value of the normal random variable is τ = −1.645 . Without the equality sign, the value of

φ could (very likely) be equal to zero because the Freund model does not state any maximization with respect to the φ parameter. The solution of the risky output price problem – a la Freund – is finally achieved by solving the following set

of relations (using the linear complementarity problem (LCP) approach, for example)

dual constraints φΣ p x + A′y ≥ E(p! )

primal constraints Ax ≤ b , x ≥ 0, y ≥ 0

x)1/2 chance constraint τ / (x′Σ p + φ = 0

and the associated complementary slackness conditions. In the chance constraint, the equality sign is selected because to

maximize expected utility the CARA parameter must be maximized too. This programming framework resolves the

dilemma posed by Freund as to the difficulty of “defending any chosen value of the risk aversion constant φ .”

Empirical Results of Freund Model with Chance Constraint on CARA ParameterThe information used by Freund is given in the following GAMS format

Sets j output variables / potatoes, corn, beef, cabbage / i inputs / LandJ-J, Land J-D, CapitalP1, CapitalP2, CapitalP3, LaborP1, LaborP2, LborP2 /;

Expected prices:E( p! j ) = 100 . Freund decided to normalize all expected prices. Availability of inputs: b(i) vector

/ LandJ-J 60 LandJ-D 60 CapitalP1 24 CapitalP2 12 CapitalP3 0 LaborP1 799 LaborP2 867 LaborP3 783 /;

5

Technology matrix: A(I,j)Potatoes corn beef cabbage

LandJ-J 1.199 1.382 2.776 0.0 LandJ-D 0.0 1.382 2.776 0.482 CapitalP1 1.064 0.484 0.038 0.0 CapitalP2 -2.064 0.020 0.107 0.229 CapitalP3 -2.064 -1.504 -1.145 -1.229 LaborP1 5.276 4.836 0.0 0.0 LaborP2 2.158 4.561 0.0 4.198 LaborP3 0.0 4.146 0.0 13.306 ;

Variance- Covariance matrix: Σ ( j, k)Potatoes corn beef cabbage

Potatoes 7304.69 903.89 -688.73 -1862.05 Corn 620.16 -471.14 110.43 Beef 1124.64 750.69 Cabbage 3689.53 ;

Freund chose a value of φ = 0.0008 for the CARA coefficient. We report Freund’s results (with pre-determined CARA coefficient) and two sets of resultsobtained by choosing two values of the normal random variable τ (corresponding to the probability that total revenue be less-than-or-equal to total cost):

τ = −1.645 corresponding to the probability (one tail) 0.05 τ = −2.32 corresponding to the probability (one tail) 0.01

Recall that, for the example of how to estimate the probability that farmer would survive a bad event (total revenue be less than total cost) I chose 1 year out of 20: 1/20 = 0.05.

Results Freund Prob = 0.01 Prob = 0.05 CARA parameter 0.0008 0.0007249 0.000477 Production plan Potatoes 10.29 16.46 22.14 Corn 26.76 12.94 0.0 Beef 2.68 5.86 11.62 Cabbage 32.35 53.60 57.55 Input prices LandJ-D 34.74 31.03 31.35 CapitalP1 93.98 84.42 83.53 LaborP3 6.12 1.64 3.08

Certainty equivalent 5383.09 5174.40 6299.00 Standard deviation of revenue 2135.51 3200.46 3443.30 Risk Premium 4742.54 3712.54 2832.12

6

Note that when the CARA coefficient φ is estimated by chance constraint programming, the results“approximate” Freund solution only when the Prob( p x ! ′ ≤ y′Ax) = 1− β = 0.01 , practically zero. Ofcourse these results depend crucially on the technology matrix and the variance-covariance matrix.

Risky Output Prices and Risky Limiting Input QuantitiesIn a farm environment risk affects heavily the limiting input availability. Rain determines thepossibility of labor to enter the fields. Machinery breaks down at unexpected time. Surface and underground water for irrigation depends on the previous rainy season (or seasons). Therefore, wecan think that it might be possible to estimate a variance–covariance matrix of limiting inputquantities that we indicate as Σ s , where s! is the stochastic vector of input supply.

Assumptions about the stochastic market prices and limiting input quantities. We assume normality for both output prices and input quantities:

p! ∼ N[E( ! ] , s ∼ N[E(! ] , p, ! 0p), Σp ! s),Σ s C ov( ! s) = [ ] = null matrix.

The technology is not stochastic and is the usual matrix A. The utility function is the negative exponential function u(π! ) = 1− exp [−φπ! ] already discussed.Farmer maximizes the expected utility of profit subject to the non-stochastic technology relations.The production plan, x , and the vector of input shadow prices, y , are not stochastic because they are the decision variables of farmer.

Under these assumptions:π! ! ′ s y = p x − !′ profit is random, x and y are not E(π! ) = E(p! )′x − E(s!)′y Var(π! ) = x′Σpx + y′Σ sy C ov( ! s) = [ ] p, ! 0

Eu(π! ) = 1− exp ⎡⎣−φ {E(π! ) − φ 2 Var(π! )}⎤⎦

⎡ ⎧ ⎫⎤ = 1− exp ⎢−φ ⎪⎨E(p! )′x − E(s!)′y − φ

2 (x′Σpx + y′Σ sy)⎪⎬⎥ ⎢ "$$#$$% ⎥⎪ ⎪⎣ ⎩ Risk Premium ⎭⎦

Notice that we do not maximize the expected utility directly. In this case, the maximization of expected utility takes a roundabout path. Attention !!! Furthermore, notice that knowledge of Symmetric Quadratic Programming is essential.

Primal max ENTR = E(p! )′x − φ 2 (x′Σpx + y′Σ sy)

subject to Ax ≤ E(s!) +φΣ sy

with all nonnegative variables. ENTR stands for expected net total revenue. The term E(s!)′y does not enter the primal objective function because it would wash out of theLagrange function (precisely as for the price taker and the monopolist).

7

The supply of limiting inputs, under risky quantities, takes the form of supply = E(s!) +φΣ sy . That is, the wise farmer would procure an amount of limiting inputs that is equal to the expected quantity plus an additional amount determined by φΣ sy . Presumably, he knows his risk aversion. With thehelp of the Agricultural Extension Service he may know also the variance-covariance matrix Σ s . Finally, the term φΣ sy can be either positive or negative (depending on the sign and magnitude ofthe covariances). That is, farmer will procure more or less risky inputs than its expected quantities.

The dual of this primal model.

Lagrange function L = E(p! )′x − φ (x′Σpx + y′Σ sy) + y′(E(s!) +φΣ sy − Ax)2

All KKT conditions ∂L = E(p! ) −φΣpx − A′y ≤ 0 dual constraints ∂x

x′ ∂L = x′E(p! ) −φx′Σpx − x′A′y = 0 Dual CSC ∂x

∂L = −φΣ sy + E(s!) + 2φΣ sy − Ax ≥ 0∂y

= E(s!) +φΣ sy − Ax ≥ 0 primal constraints Using the dual CSC to simplify the Lagrange function

L = E(p! )′x − φ 2 (x′Σpx + y′Σ sy) + y′(E(s!) +φΣ sy − Ax)

= φx′Σpx + x′Ay − φ 2 x′Σpx − φ

2 y′Σ sy + y′E(s!) +φy′Σ sy − y′Ax

= E(s!)′y + φ y′Σ sy + φ 2 x′Σpx2"#$ "%%#%%$

TC of expected Total Risk Premium inputs

Therefore, the dual problem isDual minTC = E(s!)′y + φ y′Σ sy + φ

2 x′Σpx2"#$ "%%#%%$TC of expected Total Risk Premium inputs

subject to A′y +φΣpx ≥ E(p% )&!#"#$

ExpectedTotal Marginal Cost MarginalRevenue

with all nonnegative variables.The LCP structure of this risky problem is

⎡ ⎡⎤A′ −E(p! ) E(s!)

⎡⎤ ⎥ ⎥⎦ , z = ⎢

⎢⎣

φΣp ⎤xM = ⎢ ⎢⎣

⎥ ⎥⎦, q = ⎢

⎢⎣ ⎥⎥⎦−A yφΣ s

To show that the solution of this LC problem (dual pair of SQP problem) solves the original max Eu(π! ) objective function we equate the primal objective function to the dual objective function

8

E(p! )′x − φ (x′Σpx + y′Σ sy) = E(s!)′y + φ y′Σ sy + φ x′Σpx2 2 2

and rearrange terms to show that φ y + φCE = E(p! )′x − E(s!)′y − φ

2 (x′Σpx + y′Σ sy) = 2 y′Σ s 2 x′Σpx which corresponds exactly to the Certainty Equivalent of the expected utility.

Empirical results with risky output prices and risky input quantity using Freund’s dataIn order to implement this more general model of risk programming I generated (guessed) a variance-covariance matrix of the input quantities. The empirical results are as follows: For easy comparison, I reprinted the results without the covariance matrix Σ s and added a column to the right of the table for the results with the matrix Σ s .

Without Σ s With Σ s

Results Freund Prob = 0.01 Prob = 0.05 Prob = 0.05 CARA parameter 0.0008 0.0007249 0.000477 0.0004693 Production plan Potatoes 10.29 16.46 22.14 21.61 Corn 26.76 12.94 0.0 13.32 Beef 2.68 5.86 11.62 5.64 Cabbage 32.35 53.60 57.55 53.77 Input prices LandJ-J 0.0 0.0 0.0 8.07 LandJ-D 34.74 31.03 31.35 24.56 CapitalP1 93.98 84.42 83.53 68.34 LaborP3 6.12 1.64 3.08 3.26

Certainty equivalent 5383.09 5174.40 6299.00 -2698.45 Standard deviation of revenue 2135.51 3200.46 3443.30 3443.31 Risk Premium 4742.54 3712.54 2832.12 2832.12

The solution with risky inputs is close to the solution without risky inputs and probability = 0.01 that corresponds to a normal random variable equal to -2.32. The certainty equivalent of the solution with risky inputs is negative. This result may be due to the selection (guess) of an unrealistic variance-covariance matrix Σ s .

Scaling of data series is very importantIn nonlinear programming (actually in every computation), scaling of the data series is very important. To illustrate the consequences of scaling we report below the empirical results of using Freund’s data (prices and quantities) scaled by a factor of 10 (the price variance-covariance matrix is scaled by a factor of 100 because a variance is measured in the square of the original units). Freund’s choice of the CARA parameter is now φ = 0.0008 ∗10 = 0.008 since the units of this coefficient are φ = 1 / $ . The results are as follows:

9

Without Σ s

Results Freund Prob = 0.05 CARA parameter 0.008 0.0477 Production plan Potatoes 2.21 2.21 Corn 0.0 0.0 Beef 1.16 1.16 Cabbage 5.75 5.75 Input prices LandJ-J 0.0 0.0 LandJ-D 3.36 3.13 CapitalP1 9.04 8.35 LaborP3 0.51 0.31

Certainty equivalent 86.57 62.99 Standard deviation of revenue 34.43 34.43 Risk Premium 4.74 28.32

With Σ s

Prob = 0.05 0.0305

4.27 0.64 0.0 7.60

4.99 0.0 0.47 0.51

3.85 53.97 44.39

Scaling (choice of units) of original data series affects the results in nonlinear systems of relations in mathematical programming problems and in econometric models.

Risky Output Prices and Risky Limiting Input Quantities with Non-Zero Covariance Matrix This case is a more realistic specification than the previous one because it is likely that when considering two series of random variables, tied by economic and technological environments, their covariance is non-zero. We assume:

p! ∼ N[E(p! ), Σp ] , s! ∼ N[E(s!),Σ s C ov(p! ,s!) = Σp,s [ ] .] , ≠ 0The technology is not stochastic and is the usual matrix A. The utility function is the negative exponential function u(π! ) = 1− exp[−φπ! ] already discussed.Farmer maximizes the expected utility of profit subject to the non-stochastic technology relations.The production plan, x , and the vector of input shadow prices, y , are not stochastic because they are the decision variables of farmer.

Under these assumptions:π! = p x − !′ profit is random, x and y are not! ′ s y E(π! ) = E(p! )′x − E(s!)′y Var(π! ) = x′Σpx + y′Σ sy − 2x′Σp,sy

Eu(π! ) = 1− exp ⎡⎣−φ {E(π! ) − φ 2 Var(π! )}⎤⎦

⎡ ⎧ ⎫⎤⎪ ⎪ = 1− exp ⎢−φ ⎨E(p! )′x − E(s!)′y − φ 2 (x′Σpx + y′Σ sy − 2x′Σp,sy)⎬⎥ ⎢ %"$$$$$#$$$$$ ⎥⎪ ⎪⎣ ⎩ Risk Premium ⎭⎦

10

This problem is another example of an economic problem for which neither a primal nor a dualobjective function exists to be used as an optimization goal. This impossibility is not due to the problem of integration. The expected utility is well defined. It is due to the fact that the two covariance terms ( 2x′Σp,sy ) are not quadratic forms; they are bilinear forms. Said in equivalent way, it is not possible to specify a dual pair of optimization problems to solve this more general risky problem.

A brute way to show algebraically this impossibility is to guess the introduction of the covariance term ( x′Σp,sy ) into the primal objective function and show that the alleged KKT conditions are inconsistent. Try it to convince yourself.

The Economic Equilibrium structure comes to the rescue, once again.

Let us begin with MC ≥ MR and D ≤ S as we know them from the previous case without covariance matrix: Let us suppose that the technology matrix A is of dimensions (m × n),m < n . Then

⎢⎣ ⎡y

⎤⎡ ⎥ ⎥ ⎥⎦

x⎢ ⎢ ⎢⎣

⎤ ⎥⎦

⎢⎣

(m × n)(n ×1) ≤ (m ×1) + (m × m)(m ×1) (m ×1) ≤ (m ×1) + (m ×1)

The covariance terms are

⎡y

⎤ ⎥⎦

⎤⎡ ⎥ ⎥ ⎥⎦

E(p! )

⎢⎣

(n ×1)

⎡y

E(p! )

⎤ ⎥ ⎥⎦

⎡ ⎢⎣

⎢ ⎢ ⎢⎣

≥

≥(n ×1) + (n ×1)

≥⎤ ⎥⎦

⎡ ⎢ ⎢⎣

φΣpx + A′y

⎤⎡⎤⎡⎤⎡

+φ⎤ ⎥⎦

E(s!)⎡ ⎢⎣

⎥ ⎥ ⎥⎦

A′⎢ ⎢ ⎢⎣

+

E(s!) +

⎥ ⎥ ⎥⎦

x

≤

⎤⎡

Ax

≤⎥ ⎥ ⎥⎦

x⎢ ⎢ ⎢⎣

⎤⎦A⎡

⎣

⎢ ⎢ ⎢⎣

⎥ ⎥ ⎥⎦

Σp

⎤ ⎥⎦

⎢ ⎢ ⎢⎣

⎥ ⎥ ⎥⎦

φ

⎤⎡

Σp,s ⎢ ⎢ ⎢⎣(n × m)(m ×1) (m × n)(n ×1)

(n ×1) (m ×1)

Therefore, conformability requires that the first covariance term go with the dual constraints and the second covariance term go with the primal constraints:

(n × n)(n ×1) + (n × m)(m ×1) ≥ (n ×1)

φΣ sy

Σ s

Σ s,pΣ s,px =Σp,s y =

11

φΣpx + A′y ≥ E(p! ) +φΣp,sy Ax +φΣ s,px ≤ E(s!) +φΣ sy

Note that the assignment of the covariance ters was done by following the rules of conformability and nothing else.

The solution of this problem can be performed by LCP. Restate the above relations with the samedirection of inequalities:

φΣpx + (A′ −φΣp,s )y − E(p! ) ≥ 0 −(A +φΣ s,p )x +φΣ sy + E(s!) ≥ 0

The matrix M and vectors q and z are defined as

⎡ (A′ −φΣp,s ) ⎡⎤ ⎥ ⎥⎦, q = ⎢

⎢⎣

−E(p! ) E(s!)

⎡⎤ ⎥ ⎥⎦ , z = ⎢

⎢⎣

φΣp ⎤xM = ⎢ ⎢⎣

⎥⎥⎦−(A +φΣ ,s,p ) φΣ s y

The matrix M is interesting. First of all it is not anti-symmetric as all our previous M matrices; for example

⎡ D A′ ⎤M = ⎢

⎣ ⎥⎦

.−A E

The off-diagonal matrices of the risky M matrix, namely (A′ −φΣp,s ) and −(A +φΣ ,s,p ) are obviously not the negative transpose of each other.

In previous economic problem we interpreted the A matrix as the technology matrix used by the economic agent. But now that input quantities are stochastic, their randomness is transmitted to outputs via the covariance matrix. Therefore, we can interpret the matrix A as the nominal technology and the matrix (A +φΣ ,s,p ) as the effective technology. Similarly, then, (A′ −φΣp,s ) may be interpreted as the effective marginal cost technology under risky inputs.

To show that this LCP solves the expected utility problem originally formulated on page 9, consider the complementary slackness conditions of the dual and primal constraints:

φx′Σpx + x′A′y −φx′Σp,sy − x′E(p! ) = 0 −y′Ax −φy′Σ s,px +φy′Σ sy + y′E(s!) = 0

Adding up the two CSC and rearranging terms we can write

E(p! )′x − E(s!)′y − φ ⎡⎣x′Σpx + y′Σ sy − 2x′Σp,sy⎤⎦ = φ ⎡⎣x′Σpx + y′Σ sy − 2x′Σp,sy⎤⎦2 2"$$$$$$$$#$$$$$$$$% "$$$$$# %$$$$$ Certainty Equivalent or Exponent of the Expected Utility Function Risk Premium

12

APPENDIX

Exercises 95

10.9 Given the utility function u = 1 - e-</Jr, where ¢ > 0 is a constant risk aversion coefficient and f is stochastic revenue distributed as f ,...., N[E(f), Var(f)], (N = norma l density), ( a) Find an explicit expression for the expected utility (hint: complete

the square). (b) Show that th e maximization of expected utilit y is equivalent to

the maximization of [E(f) - !Var( f)].

Solution

(a) Let µr = E(f).and O"; = Var(f). The desired integrat ion can be carried out by completing the square in the exponent of the definition of expected utility , where utility is a normal random variable:

1+00 1 [{ ,,. 2- -2 2 - ,,. 2 2" 2 ,,.2( 2)2 ,,.2( 2)2} /2 2] E[u(f)] =1- e - 2.,,0'rr -r -µr+2rµr+2.,,µrO'r- .,,µrO'r+.,, O'r -.,, (J'r (J'r df -oo O"r-./2ir ·

extract the constant t erm s e- 2<PJ.Lr0';/20'; and e<P2 (0'; )2 / 20'; from the in-tegral sign

and collect the square on th e remaining terms

(2)

since the int egra l in the first row of (2) corr esponds to the density func-tion of a normal r andom variable with mean (µr - <pO";) and varianc e

2 (Tr·

(b) Since expected utili ty is a monoto nic nonlinear function of µr and O";, its m aximization can be achieved by concentratin g on the expon ent and maximizing (µr - ¥0";) 13