Choosing among risky alternatives



Choosing among risky alternatives

using stochastic dominance

When is one risky outcome preferable to another?

Stochastic dominance tries to resolve choice under weakest possible assumptions.

Generally, SD assumes an expected utility maximizer then adds further assumptions relative to preference for wealth and risk aversion.

Basic Assumptions

#1 - individuals are expected utility maximizers.

#2 - two alternatives are to be compared and these are mutually exclusive, i.e., one or the other must be chosen not a convex combination of both.

#3 - the stochastic dominance analysis is developed based on population probability distributions.

Choosing among risky alternatives

using stochastic dominance

Background

Expected utility of wealth maximization.

Assume x is the level of wealth while f(x) and g(x) give the probability of each level of wealth for alternatives f and g. The difference in the expected utility between the prospects as follows.

[pic]

and this equation can be rewritten as:

[pic]

If f is preferred to g then the sign of the above equation would be positive. Conversely, if g is preferred to f, the sign of the above equation be negative.

Choosing among risky alternatives

using stochastic dominance

Integration by Parts

Classical calculus technique

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where a and b are functions of x.

First Degree Stochastic Dominance

Apply the integration by parts formula to

a = u(x)

b = (F(x) - G(x))

where [pic]

[pic]

in turn the differential terms are:

[pic]

Choosing among risky alternatives

using stochastic dominance

First Degree Stochastic Dominance

Under this substitution the integration of

[pic]

equals

[pic]

In the left part when F(x) and G(x) are evaluated at

x equals -∞ both equal zero

x equals +∞ both equal one

and plus infinity where they equal one

so the left part equals zero.

Now let us look at the right part which is:

[pic]

If the overall sign is positive then f dominates g.

We restrict the sign by adding assumptions.

1.) nonsatiation more is preferred to less or u'(x) > 0 for all x.

2) F(x) ( G(x) for all x

Under these conditions then implies f dominates g

Choosing among risky alternatives

using stochastic dominance

First Degree Stochastic Dominance

Given 1) two probability distributions f and g,

2) the assumption that the decision maker has positive marginal utility of wealth u’(x)>0 for all x

Then when for all x the cumulative probability under f is less than or equal to cumulative probability under g with strict inequality for some x. we say distribution f dominates distribution g by first degree stochastic dominance (FSD)

Not revolutionary. Mean of f is greater than that for g and for every level of probability you make at least as much money under f as you do under g.

Not very discriminating practically.What one has to observe is that one crop variety always has to consistently out perform the other. This may not be the case.

Choosing among risky alternatives

using stochastic dominance

First Degree Stochastic Dominance -- Geometry

[pic]

Note here that for all x the area under f(x) is less than the are under g(x) so f dominates g. Why? Requires for all x the cdf for f is always to the right of cdf for g or that for every x the cumulative probability of that level of wealth or higher is greater under f than g.

Notice in Figure 1 that for a value of x equal to 7 that there is no meaningful area under the f (x) distribution but there is under the g(x) distribution so one can make that much or less under g but will always make more under f.

Note, for any point x where there is an area under both distributions the area underneath the g distribution is greater than it is for the f distribution.

Choosing among risky alternatives

using stochastic dominance

Second Degree Stochastic Dominance

The FSD stochastic dominance development while theoretically elegant is not terribly useful so lets go further

Above FSD derivation says Expected utility of f minus g can be expressed as

[pic]

Applying integration by parts

[pic]

so that:

[pic]

where F2 and G2 are the second integral of the cdfs

[pic]

[pic]

Under these circumstances if we plug in our integration by parts formula we get the equation.

[pic]

Choosing among risky alternatives

using stochastic dominance

Second Degree Stochastic Dominance

The formula above has two parts. Let us address the right hand part of it first.

[pic]

Right part contains the second derivative of the utility function times the difference in the integrals of the cdf with a positive sign in front of it.

[pic]

To guarantee that f dominates g the sign of this whole term must be positive. Second degree stochastic dominance adds two assumptions

1) the second derivative of the utility function with respect to x is negative everywhere

2) F2(x) is less than or equal to G2(x) for all x with strict inequality for some x.

Must also sign the left hand part. First, add the assumption on nonsatiation U’(X)>0.

This term then multiplies by F2(x) - G2(x) which is zero at x equals minus infinity and under assumption 2 above is non-positive at plus infinity.

Choosing among risky alternatives

using stochastic dominance

Second Degree Stochastic Dominance

Given f and g if an individual has

1) positive marginal utility u'>0;

2) diminishing marginal utility of income u" ................
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