STOCHASTIC

SOME EFFECTS OF TAXES ON RISK-TAKING 295
and therefore the sign of (1.15) is determined by
bc-ad = -±
2 (i_r2)-.
Thus if the asset with the higher yield is riskier (e.g. ml>tn2 and eri>o-2), an increase in
taxes would produce higher risk-taking.
2. CHANCE—CONSTRAINED PROGRAMMING 1
Work on the extension of the linear programming formulation to allow for randomness
^n the problems studied has been going on for over a decade. Many papers have been
published on the subject, but among these contributions one can distinguish three main
approaches, namely:
(a) stochastic linear programming,
(6) linear programming under uncertainty, and
(c) chance-constrained programming.
Method (a) was first developed by Tintner [19]. It enables the decision-maker to
affect the probability distribution of the optimum by allocating the scarce resources to
various activities prior to the realization of the random variables.
Method (ft) partitions the decision problem into two or more stages. First a decision
is made, then the random variables are observed and finally it is necessary to undertake
corrective action to restore the proper form of those constraints that are violated due to the
randomness in the problem.
The authors working in this field have been concerned with a decision-maker who
minimizes the expected value of the functional. It is assumed that there always exist
second stage variables that restore the problem to its proper form. Furthermore, it is
assumed that the costs associated with these second stage variables can be specified.
The class of problems dealt with in chance-constrained programming can be described
by the following formulation 2
max/(cx) ...(2.1)
subject to
P(Ax£b)^a, ...(2.2)
where / is a concave function of x,
A is a random m x n matrix,
6 is a random column vector with m elements,
c is a random row vector with n elements,
x is a column vector with n elements,
and P means probability and a is a column vector with m elements
0 g a, S 1.
Instead of seeking a maximum of (2.1) over all x subject to (2.2), we often require
a maximum over only such vectors generated by some decision rule. In general, such
a rule will involve a set of x values that depend upon the random variables A, b and c.
That is,
x = D(A, b, c), ...(2.3)
1 For a more complete discussion of mathematical programming under risk see Naslund [14].
2 For a description of this class of problems, see Charnes and Cooper [3],
EFFECTS OF TAXES ON RISK TAKING