The Comparative Statistics of Self-Protection

By Sweeney, George H.; Beard, T. Randolph | Journal of Risk and Insurance, June 1992 | Go to article overview

The Comparative Statistics of Self-Protection


Sweeney, George H., Beard, T. Randolph, Journal of Risk and Insurance


The expenditure of resources to modify the probabilities of suffering losses is called "self-protection."(1) As noted by Briys and Schlesinger (1990), expenditures on self-protection do not merely trade income in one state of the world for income in another as market insurance contracts do. Rather, self-protection reduces income in all states, shifting the support of the wealth distribution to the left. Hence, increases in self-protection spending do not in general lead to less risky income prospects in the sense of stochastic dominance.(2) Consequently, the willingness of agents to engage in self-protection fails to parallel agents' willingness to buy market insurance in several important respects. Both risk-loving and risk-averse agents may buy self-protection, and more risk-averse agents will not generally spend more on self-protection than less risk-averse agents.(3) This characteristic of self-protection makes its analysis considerably more difficult than that of insurance. In addition, analysis of self-protection choice must account for the fact that any agent's optimal level of self-protection spending depends on an exogenous, technical" relationship between spending and loss probability. This relationship summarizes the agent's opportunities to reduce loss probabilities and is a critical component in determining optimal self-protection choice.

The purpose of this article is to present a set of comparative static results for the simplest and most widely-used model of self-protection choice. Interest focuses on the effects of changes in initial wealth and potential losses on optimal self-protection expenditures.(4) The goal is to seek results of a general character which do not depend on the particular form of the technical relationship between spending and loss probability, nor on restrictive characterizations of agents' von Neumann - Morgenstern utility functions for wealth. Further, the main findings will be stated in terms of the characteristics of agents' attitudes towards risk in a relatively straight-forward fashion, and a set of results useful for evaluating economic problems that involve self-protection will be derived.

Optimal Self-Protection Choice

Consider the simplest model of self-protection choice in which an agent faces the potential of losing a fixed sum of money L. Letting [y.sub.o] be the agent's initial wealth, s her spending on self-protection, p(s) the probability of suffering the loss L given self-protection expenditure s > 0, and u(.), the agent's von-Neumann Morgenstern utility function for money, write the agent's expected utility given self-protection spending level s as v(s,L,yo), where

v(s,L,[y.sup.o]) = p(s)u([y.sup.o]-s-L) + [1 - p(s)]u([y.sub.o]-s). (1)

Letting y = [y.sub.o] - s, and assuming the required differentiability of p(-) and u(-), the optimal level of self-protection spending s* is given by the firstorder condition

-p'(s*)[u(y) - u(y-L)l - [1 - p(s*)]u'(y) - p(s*)u'(y-L) [less than or equal to] 0, (2)

with equality holding if s* > 0. The second-order necessary condition for an interior solution requires

-p"(s*)[u(y) - u(y-L)] + 2p'(s*)[u'(y) - u'(y - L)]

+ [1 - p(s*)]u"(y) + p(s*)u"(y-L) < 0. (3)

The first-order condition given by (2) has an immediate conventional interpretation in terms of costs and benefits: the (expected) marginal benefit of self-protection spending (-p'(s*)[u(y) - u(y-L)]) should be equal to the (expected) marginal "cost" of such spending ([1-p(s*)]u'(y) + p(s*)u'(y-L)) at the optimal choice s*.

Comparative Static Results for Changes in Initial Wealth

The initial goal is evaluating the effect of changes in the agent's initial wealth [y.sub.o] on optimal self-protection spending s*. If the purchase of selfprotection is identified with the purchase of an economic good such as a burglar alarm, security guard service, or smoke detector, then the issue of how changes in (initial) wealth [y. …

The rest of this article is only available to active members of Questia

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

The Comparative Statistics of Self-Protection
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    Buy instant access to save your work.

    Already a member? Log in now.

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.