chp1-5.mw

To the previous section 

1.5  A Rigorous Formulation of Agents' Decision 

 

Having explained the relation between indifference curves and utility functions and the meaning of a unity function we are now in a position to pose the problem solved by the agents. Assume that the price of good 1 is Typesetting:-mrow(Typesetting:-mi( and that of  good 2 is Typesetting:-mrow(Typesetting:-mi( and that the agent has a budget of Typesetting:-mrow(Typesetting:-mi( and a utility function Typesetting:-mrow(Typesetting:-mi(. The reader may have already guessed that the agent needs to solve a maximization problem which is formulated as below: 

 

 

      Typesetting:-mrow(Typesetting:-mi(
s.t.   Typesetting:-mrow(Typesetting:-mi(      
 

     Typesetting:-mrow(Typesetting:-mi( Typesetting:-mrow(Typesetting:-mi( 



Since Typesetting:-mrow(Typesetting:-mi( is a monotone increasing function, the inequality Typesetting:-mrow(Typesetting:-mi(  can be replaced with an equality Typesetting:-mrow(Typesetting:-mi( . Thus, one way of solving this problem is by substituting Typesetting:-mrow(Typesetting:-mi( in terms of Typesetting:-mrow(Typesetting:-mi( and solving an optimization problem with only the non-negativity constraints Typesetting:-mrow(Typesetting:-mi( Typesetting:-mrow(Typesetting:-mi(.
 

 

We will, however, utilize this opportunity to review the optimality conditions as they play a central role in the portfolio choice problem. Furthermore, we will explain or provide a heuristic proof for the optimality conditions. This proof highlights a relation (duality) between two systems of inequalities that is used later in the Arbitrage Pricing Theory (APT). 

 

In order to state the optimality conditions in a generic way we should decide about a canonical form of an optimization problem. We will define a canonical form as a minimization problem with constraints of the type Typesetting:-mrow(Typesetting:-mi(.  That is, a problem of the form
 


Typesetting:-mrow(Typesetting:-mi(

s.t. Typesetting:-mrow(Typesetting:-mi(,  Typesetting:-mrow(Typesetting:-mi(
 


Where Typesetting:-mrow(Typesetting:-mi( is a vector in Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(,  Typesetting:-mrow(Typesetting:-mi(are functions from Typesetting:-mrow(Typesetting:-mi( to Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(, Typesetting:-mrow(Typesetting:-mi(.  
 

 

It is easy to see that there is no loss of generality assuming such a canonical form of an optimization problem. The solution of a maximization problem is the same for a minimization problem in which the negative of the objective function Typesetting:-mrow(Typesetting:-mi(  is minimized. Inequality constraints could be reversed by multiplying the inequality by Typesetting:-mrow(Typesetting:-mo( and an equality can be written as a pair of two inequalities. After we develop the optimality condition for the above problem we will apply it to our maximization of utility problem. 

 

Let us first start with the intuition behind the optimality conditions. If Typesetting:-mrow(Typesetting:-mi( is a minimum point of Typesetting:-mrow(Typesetting:-mi(, subject to the constraint stipulated above, it must be that there is no feasible direction Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo( “along which one can move away” from Typesetting:-mrow(Typesetting:-mi( and reach a point where the value of Typesetting:-mrow(Typesetting:-mi( is smaller than the value of Typesetting:-mrow(Typesetting:-mi( at Typesetting:-mrow(Typesetting:-mi(.  Being a feasible point,  Typesetting:-mrow(Typesetting:-mi( satisfies the constraint, that is Typesetting:-mrow(Typesetting:-mi(,  Typesetting:-mrow(Typesetting:-mi(.  The direction Typesetting:-mrow(Typesetting:-mi( is feasible if there exists an Typesetting:-mrow(Typesetting:-mi(", mathvariant = "normal", fence = "false", separator = "false", stretchy..." align="center" border="0"> such that Typesetting:-mrow(Typesetting:-mi(,  Typesetting:-mrow(Typesetting:-mi(.  Moving away from Typesetting:-mrow(Typesetting:-mi( along the direction Typesetting:-mrow(Typesetting:-mi(, will lead to a point Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi( for some positive Typesetting:-mrow(Typesetting:-mi(.  

 

Our next step requires characterization of feasible directions and directions along which the function decreases. To accomplish this task we now define a function Typesetting:-mrow(Typesetting:-mi( on Typesetting:-mrow(Typesetting:-mi( such that 

 

Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo( , for a given Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi( in Typesetting:-mrow(Typesetting:-mi(. 

 

Typesetting:-mrow(Typesetting:-mi( thus equals Typesetting:-mrow(Typesetting:-mi(, and if Typesetting:-mrow(Typesetting:-mi(  is negative it means that moving along the direction Typesetting:-mrow(Typesetting:-mi(,  Typesetting:-mrow(Typesetting:-mi( decreases. Footnote 6  A visualization of direction of decent relative to the gradient and the value of the function along these directions is offered below for the utility function of the former section. The graph below demonstrates the direction Typesetting:-mrow(Typesetting:-mi(for Typesetting:-mrow(Typesetting:-mi( at a point (5,5).   

The directions of the negative of the gradient is drawn as a black line in the Typesetting:-mrow(Typesetting:-mi( axis that points to the origion, with two other directions that are perpendicular to it. All the direction between the two prepndcualr directions are directions of decent. The movement along these directions is also drawn on the graph of the function so one can realize that these are indeed directions of decent.   

 

> NGFradNfun:=plot3d([[5-x,5-x,(5-x)*(5-x)],[5-x,5-x,0],[5-x,5+x,(5-x)*(5+x)],[5+x,5-x,(5+x)*(5-x)],[5-x,5+x,0],[5+x,5-x,0]],x=0..5,y=0..5,axes=box,thickness=4):
 

>
 

> Util:=plot3d(x*y,x=0..10,y=0..10,axes=box,style=patchcontour,contours = 20,filled=true,orientation=[-82,56],title="Gradient, steepest decent, and directions of decent ",transparency=.2):
 

>
 

> plots[display](Util,NGFradNfun);
 

Plot
 

>
 

>
 

 

Similarly if Typesetting:-mrow(Typesetting:-mi( is a point on the boundary of the feasible set, i.e., it satisfies one of the constraints say Typesetting:-mrow(Typesetting:-mi( as equality rather than inequality, and if the function Typesetting:-mrow(Typesetting:-mi( is such that  Typesetting:-mrow(Typesetting:-mi( 

 

Typesetting:-mrow(Typesetting:-mi( is not a feasible direction.  This is because as we move along Typesetting:-mrow(Typesetting:-mi(, Typesetting:-mrow(Typesetting:-mi( increases and since at Typesetting:-mrow(Typesetting:-mi( , Typesetting:-mrow(Typesetting:-mi( there exists an Typesetting:-mrow(Typesetting:-mi(", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric =..." align="center" border="0">such that Typesetting:-mi( violates the constraint since Typesetting:-mrow(Typesetting:-mi( for every ε  in Typesetting:-mrow(Typesetting:-mi(  

 

This is illustrated below where the vector Typesetting:-mrow(Typesetting:-mi( is the red line. Moving along Typesetting:-mrow(Typesetting:-mi( with Typesetting:-mrow(Typesetting:-mi(.5 from the point (2,3) is demonstrated by the green line which leads to the point Typesetting:-mrow(Typesetting:-mi(. Moving along Typesetting:-mrow(Typesetting:-mi( from the point (2,2) with Typesetting:-mrow(Typesetting:-mi( is demonstrated by the blue line which ends at Typesetting:-mrow(Typesetting:-mi(. The feasible set is demonstrated by the black circle and one can visualize that the direction Typesetting:-mrow(Typesetting:-mi( from the point (2,2) is a feasible direction as it is possible to move along it and still stay within the feasible set. On the other hand the direction Typesetting:-mrow(Typesetting:-mi( is not a feasible direction from the point (2,3) since moving along it, even with a very small positiveTypesetting:-mrow(Typesetting:-mi(, will result in a point outside the feasible set.  The feasible set in this illustration is given by the function 

 

Typesetting:-mrow(Typesetting:-mi( . 

 

> V11:=plots[arrow]([<1,1>], width=[0.02, relative], head_length=[0.04, relative], color=red):
 

> V1122:=plots[arrow](<2,2>,<1,1>, width=[0.02, relative], head_length=[0.04, relative], color=blue):
 

> V1123:=plots[arrow](<2,3>,<1.5,1.5>, width=[0.02, relative], head_length=[0.04, relative], color=green):
 

> FeasC:=plot([[2+cos(t),2+sin(t), t=0..2*Pi]], color=[black],thickness=2,scaling=constrained):
 

> plots[display](V11,V1122,V1123,FeasC);
 

Plot_2d
 

>
 

>
 

 

The derivative ofTypesetting:-mrow(Typesetting:-mi( with respect to Typesetting:-mrow(Typesetting:-mi( is given by Typesetting:-mrow(Typesetting:-mi( where ▽ denotes the gradient of Typesetting:-mrow(Typesetting:-mi( (the vector of partial derivatives) and Typesetting:-msubsup(Typesetting:-mi( denotes the transpose of Typesetting:-mrow(Typesetting:-mi( (a row vector). At the point Typesetting:-mrow(Typesetting:-mi(, Typesetting:-mrow(Typesetting:-mi( is positive if  Typesetting:-mrow(Typesetting:-mi( At the point (2,3) we can calculate Typesetting:-mrow(Typesetting:-mi(as below 

> Student:-MultivariateCalculus:-Gradient((x-2)^2+(y-2)^2,[x,y]=[2,3]);
 

Typesetting:-mfenced(Typesetting:-mverbatim(
 

>
 

 

In the graph below the gradient of Typesetting:-mrow(Typesetting:-mi( at the point (2,3) is illustrated by the blue line, the direction Typesetting:-mrow(Typesetting:-mi( from the point Typesetting:-mrow(Typesetting:-mi( by the green line, and the direction  Typesetting:-mrow(Typesetting:-mi( from the point Typesetting:-mrow(Typesetting:-mi( by the red line. 

 

> V0123:=plots[arrow](<2,3>,<0,1>, width=[0.02, relative], head_length=[0.04, relative], color=blue):
 

> V1Neg123:=plots[arrow](<2,3>,<1,-1>, width=[0.02, relative], head_length=[0.04, relative], color=red):
 

> plots[display](V1123,V0123,V1Neg123,FeasC);
 

Plot_2d
 

>
 

 

You may recall that if Typesetting:-mrow(Typesetting:-mi(  than the angle between the vector Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mo( is acute, while if the angle between the vector Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mo( is obtuse Typesetting:-mrow(Typesetting:-mi( . In the case illustrated above by the green line Typesetting:-mrow(Typesetting:-mi( since Typesetting:-mrow(Typesetting:-mi( and hence the angle between the vector Typesetting:-mrow(Typesetting:-mi( and  Typesetting:-mrow(Typesetting:-mo( is acute and the green line is not a feasible direction. Typesetting:-mrow(Typesetting:-mo( 

 

On the other hand the direction Typesetting:-mrow(Typesetting:-mi(  is a feasible direction from the point Typesetting:-mrow(Typesetting:-mi(. It is represented by the red line which points inside the circle. On the circle the value of Typesetting:-mrow(Typesetting:-mi( is 1 and inside the circle it is less than 1.  Hence moving along this direction does not violate the constraint that requires that Typesetting:-mrow(Typesetting:-mi(Feasible directions are therefore those that generate an obtuse angle with the gradients of the binding constraints. A feasible direction Typesetting:-mrow(Typesetting:-mi( at a point Typesetting:-mrow(Typesetting:-mi(, is characterized by the fact that for each binding constraint Typesetting:-mrow(Typesetting:-mi( at Typesetting:-mrow(Typesetting:-mi(, i.e., for each  Typesetting:-mrow(Typesetting:-mi( for which Typesetting:-mrow(Typesetting:-mi(,  the product of  Typesetting:-mrow(Typesetting:-mi( and the gradient of  Typesetting:-mrow(Typesetting:-mi( is negative, that is Typesetting:-mrow(Typesetting:-mi( 

If the point is a minimum point, it must be that there exists no feasible direction along which the objective function decreases. Footnote 7  Since if one existed there would be a feasible point say Typesetting:-mrow(Typesetting:-mi(=Typesetting:-mrow(Typesetting:-mi( for some γ>0  such that Typesetting:-mrow(Typesetting:-mi(, but that means that Typesetting:-mrow(Typesetting:-mi( could not be the minimum point. Thus, if Typesetting:-mrow(Typesetting:-mi( is a minimum point it must be that 

 

Typesetting:-mrow(Typesetting:-mi(for every Typesetting:-mrow(Typesetting:-mi( such that Typesetting:-mrow(Typesetting:-mi( then Typesetting:-mrow(Typesetting:-mi(,  

i.e., the function Typesetting:-mrow(Typesetting:-mi(  increases along each feasible direction Typesetting:-mrow(Typesetting:-mi( from Typesetting:-mrow(Typesetting:-mi(.  

 

This condition can also be stated as:   

 

Every Typesetting:-mrow(Typesetting:-mi( such that Typesetting:-mrow(Typesetting:-mi(  for  Typesetting:-mrow(Typesetting:-mi( such that Typesetting:-mrow(Typesetting:-mi(,  must satisfy Typesetting:-mrow(Typesetting:-mi(.  

Put into words, every feasible direction from point Typesetting:-mrow(Typesetting:-mi(, must be a direction along which Typesetting:-mrow(Typesetting:-mi( does not decrease. This condition can also be geometrically visualized.  

 

Assume that there are two binding constraints at a point Typesetting:-mrow(Typesetting:-mi(, hence we have vectors representing the gradient of Typesetting:-mrow(Typesetting:-mi(, and Typesetting:-mrow(Typesetting:-mi(.  A feasible direction must make an acute angle with each of these vectors. We start by plotting a vector representing Typesetting:-mrow(Typesetting:-mo(and the vectors that are making an acute angle with it.  Assume that Typesetting:-mrow(Typesetting:-mo( is the vector Typesetting:-mrow(Typesetting:-mi(. Two vectors that make a right angle with it are given by Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(, hence vectors that are making an acute angle with Typesetting:-mrow(Typesetting:-mi( are between these two vectors. (Note the vectors that make a right angle with Typesetting:-mrow(Typesetting:-mi( must have a product of zero with it, e.g., Typesetting:-mrow(Typesetting:-mi( . This is illustrated below: 

 

> Lg1:=plots[textplot]([1,2,`-grad(g1)`],align=ABOVE):
 

> Neggradg1:=plots[arrow]({<1,2>,<-2,1>,<2,-1>}, width=[0.02, relative], head_length=[0.04, relative], color=red):
 

> plots[display](Lg1,Neggradg1);
 

Plot_2d
 

>
 

 

The vectors that are making an acute angle withTypesetting:-mrow(Typesetting:-mi( are those that are between the two vectors making a right angle with it. These are the vectors with an end point above the red line (that passes via the origin).   

 

Assume that Typesetting:-mrow(Typesetting:-mi( and let us now graph Typesetting:-mrow(Typesetting:-mi( and all the vectors making an acute angle with it. This can still be displayed in the figure below. 

 

> Neggradg2:=plots[arrow]({<2,0>,<0,-2>,<0,2>}, width=[0.02, relative], head_length=[0.04, relative],color=blue):
 

> Lg2:=plots[textplot]([1.6,0.1,`-grad(g2)`],align=ABOVE):
 

> plots[display](Lg2,Neggradg2);
 

Plot_2d
 

>
 

 

Let us now visualize the vectors that are making acute angles with bothTypesetting:-mrow(Typesetting:-mo(andTypesetting:-mrow(Typesetting:-mo(This will be the intersection of the two sets of vectors described in the above two graphs. That is the vectors with an end point above the red line that passes via the origin, and to the left of the blue line that passes through the origin.  

 

 

> plots[display](Lg1,Neggradg1,Lg2,Neggradg2);
 

Plot_2d
 

>
 

 

Recall that if Typesetting:-mrow(Typesetting:-mi( is a minimum point, the angle between each feasibleTypesetting:-mrow(Typesetting:-mi( andTypesetting:-mrow(Typesetting:-mi(must also be acute, that is Typesetting:-mrow(Typesetting:-mi( for every Typesetting:-mrow(Typesetting:-mi( which is a feasible direction.  Observing the set of feasible directions in the graph above, the reader should realize that forTypesetting:-mrow(Typesetting:-mi( to make an acute angle with each feasible Typesetting:-mrow(Typesetting:-mi(, Typesetting:-mrow(Typesetting:-mi( must be between Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(An example of a vector that satisfies this condition is illustrated below. It is labeled as grad(f) and it is the green vector in the graph below: 

 

> gradf:=plots[arrow](<3,2>, width=[0.02, relative], head_length=[0.04, relative],color=green):
 

> Lf:=plots[textplot]([2.7,2,`grad(f)`],align=ABOVE):
 

> plots[display](Lg1,Neggradg1,Lg2,Neggradg2,gradf,Lf);
 

Plot_2d
 

>
 

 

However algebraically for  Typesetting:-mrow(Typesetting:-mo( to be between Typesetting:-mrow(Typesetting:-mo( and Typesetting:-mrow(Typesetting:-mo( it means that Typesetting:-mrow(Typesetting:-mo( is a positive linear combination of  Typesetting:-mrow(Typesetting:-mo( and Typesetting:-mrow(Typesetting:-mo(i.e., Typesetting:-mrow(Typesetting:-mo(=Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mi( for some positive Typesetting:-mrow(Typesetting:-mi(and  Typesetting:-mrow(Typesetting:-mi(In general therefore, this condition is satisfied if  Typesetting:-mrow(Typesetting:-mo(  is a positive linear combination of Typesetting:-mrow(Typesetting:-mo(, for Typesetting:-mrow(Typesetting:-mi( such that Typesetting:-mrow(Typesetting:-mi(. That is Typesetting:-mrow(Typesetting:-mo( where the summation is over  Typesetting:-mrow(Typesetting:-mi( such that Typesetting:-mrow(Typesetting:-mi( and the Typesetting:-mrow(Typesetting:-mi( are all nonnegative. Footnote 8  

 

We are now ready to stipulate the optimality conditions for the canonical minimization problem. It is common to stipulate the optimality conditions in terms of a function called Lagrangian, associated with the canonical optimization problem. The Lagrangian of our canonical minimization problem is defined to be a  function of  Typesetting:-mrow(Typesetting:-mi(, a vector in Typesetting:-mrow(Typesetting:-mi( (Typesetting:-mrow(Typesetting:-mi( is the number of constraints) and Typesetting:-mrow(Typesetting:-mi(. 



Typesetting:-mrow(Typesetting:-mi(
 

The optimality conditions are stated in terms of the partial derivative of the Lagrangian Typesetting:-mrow(Typesetting:-mi( below:
 


Typesetting:-mrow(Typesetting:-mi(  i=1,...,n
Typesetting:-mrow(Typesetting:-mi(   Typesetting:-mrow(Typesetting:-mi(  j=1,...,m
Typesetting:-mrow(Typesetting:-mi( j=1,...,m
 

The reader is asked to verify that the first condition above states that Typesetting:-mrow(Typesetting:-mo(, the third condition ensures that if the Typesetting:-mrow(Typesetting:-mi( constraint is not binding Typesetting:-msub(Typesetting:-mi(=0 and hence the sum in Typesetting:-mrow(Typesetting:-mo( is only over the binding constraints. Finally the second constraint simply guarantees that Typesetting:-mrow(Typesetting:-mi( is a feasible point and satisfies the constraints.   


In our utility maximization problem,  Typesetting:-mrow(Typesetting:-mi(  is Typesetting:-mrow(Typesetting:-mi( , (Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi( are Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(), the non-negativity constraints on Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi( can be written in terms of the Typesetting:-mrow(Typesetting:-mi(, and we also have one inequality constraint that can be written as an equality constraint  Typesetting:-mrow(Typesetting:-mi(because of the monotonic of Typesetting:-mrow(Typesetting:-mi(. The equality constraint can be written as two inequality constraints.
 

 

The reader is asked to show that this will alter the optimal conditions only by allowing the Typesetting:-mrow(Typesetting:-mi( associated with this constraint to be positive or negative instead of the nonnegative constraint that is imposed on  λ that is associated with an inequality constraint of the type less than or equal. Our problem is a maximization problem.  We can transform the maximization of Typesetting:-mrow(Typesetting:-mi( to a minimization of Typesetting:-mrow(Typesetting:-mo(.  Therefore we have
 


Typesetting:-mrow(Typesetting:-mi(
s.t.  Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
 

Writing the Lagrangian of this problem and the optimality conditions where, Typesetting:-mrow(Typesetting:-mi(and Typesetting:-mrow(Typesetting:-mi(  are associated with 

the non-negativity constraints on Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi( respectively, yields:  



Typesetting:-mrow(Typesetting:-mi(+Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mi(cTypesetting:-mrow(Typesetting:-mi(

Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(

Typesetting:-mrow(Typesetting:-mi(=Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(
 

 

Typesetting:-mrow(Typesetting:-mi(=Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo( 

Typesetting:-mrow(Typesetting:-mi(=Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo( 

Typesetting:-mrow(Typesetting:-mi(cTypesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo( 

Typesetting:-mrow(Typesetting:-mi(cTypesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo( 

Typesetting:-mrow(Typesetting:-mi(
 

If we assume that the non-negativity constraints on Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi( are not binding, that is that the optimal solution is an interior point of the positive orthant we can try to satisfy the equations:
 


Typesetting:-mrow(Typesetting:-mi(

Typesetting:-mrow(Typesetting:-mi(
 

which implies that

 

Typesetting:-mrow(Typesetting:-mi(
 

or that
 


             Typesetting:-mrow(Typesetting:-mi(
 


Note that Typesetting:-mrow(Typesetting:-mi( is the slope of the budget line andTypesetting:-mrow(Typesetting:-mo(Typesetting:-mrow(Typesetting:-mi( is the slope of the indifference curve. Since Typesetting:-mrow(Typesetting:-mi( we have the second equation:
 

        Typesetting:-mrow(Typesetting:-mi(
 

Solving these two equations will thus produce the same solution as produced by the graphical procedure.  

 

 

We conclude this chapter with an example. Consider the optimization problem 

 

Typesetting:-mrow(Typesetting:-mi(
s.t.   Typesetting:-mrow(Typesetting:-mi(      
 

     Typesetting:-mrow(Typesetting:-mi( Typesetting:-mrow(Typesetting:-mi( 

 


Where  Typesetting:-mrow(Typesetting:-mi(, Typesetting:-mrow(Typesetting:-mi(, Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(.  The KT optimality conditions of this problem are:
 

 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

This system of equalities and inequalities can be solved by Maple as demonstrated below.  

 

> solve({c1+c2/(1.14)=80,-diff(c2+1.07*c1,c1)+lambda+eta||1=0,\
-diff(c2+1.07*c1,c2)+lambda*(1/1.14)+eta||2=0,\
c1>=0,c2>=0,eta||1<=0,eta||2<=0,eta||1*c1=0,eta||2*c2=0},\
{c1,c2,lambda,eta||1,eta||2});
 

{c1 = 0., eta2 = 0., eta1 = -0.6999999994e-1, c2 = 91.20000000, lambda = 1.140000000}
 

>
 

 

One of the exercises at the end of the chapter asks the reader to solve this problem using the graphing approach (which can also be done using our procedure BudnIndif(c1+c2/1.14, c2+1.07c2)) and to explain and compare the results of the graphical approach with that of the formal approach using KT conditions.  

 

 

Footnote 

 

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