Prudence

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Testing for Prudence

August 1, 2008 in Experimental | Permalink

Many economists over time have tried to measure how risk averse (or risk loving) people are. For instance, some risk averse individuals would prefer having $40 for sure compared to playing a game where if the coin lands heads you get $100 and if the coin lands tails you get $0. Risk averse individuals are willing to accept a lower expected value ($40 vs. $50 for the coin flip).

However, another feature of individuals preferences can also influence how individuals evaluate risky situation. This concept is prudence. Let us go back to the coin flip game ($100 heads; $0 tails). Imagine you make $50,000 this year and you are going to get a raise next year so you will earn $60,000. Would you rather play the coin flip game this year or next year? A prudent person would want to take the risk when they are in the better financial situation.

Measuring prudence, however, is not a simple task. A working paper by Deck and Schlesinger tries to estimate prudence preferences in an experimental setting. The experimental question is posed as follows:

  • You will receive $10.50 + (1|-1) if the con lands on Heads or Tails and $9.00 if the coin lands on the Same or Different outcome.

I think the phrasing of the question is unnecessarily complicated, but the question is fairly straight-forward. Everyone gets $10.50. Individuals must choose between Heads or Tails; and Same or Different. To simplify, let us assume that everyone chooses Heads, which means you earn $1 if the coin lands on heads and lose $1 if the coin lands.

Now we must decide between Same or Different. If you choose Same, that means that you get $9 if the first coin toss lands on heads and you also flip the coin a second time to see if you win or lose $1; if the first coin toss lands on tails then you get $0, but do not have to play the win/lose $1 game. If you choose Different, then if the coin lands on heads you get $9, and do not play the second coin toss; if the coin lands on tails you get $0 and do not play the second coin toss.

Individuals who choose Same are prudent because they take the financial risk (win/lose $1) when they are richer ($9 extra). Those who choose Different, are imprudent because they take the financial risk (win/lose $1) when they are poorer ($0 extra)

The authors asked 6 of these prudence of questions. They found that 61% of subjects responded to the questions in a prudent manner, but only 14% of individuals responded to all six questions prudently. A Kolmogorov-Smirnov statistic of 0.2225 indicates that people are making prudent choice more than would be the case if they were choosing randomly.

APPENDIX

The other 6 prudence questions were:

  1. You will receive $30 + (25|-25) if the con lands on Heads or Tails and $25.00 if the coin lands on the Same or Different outcome.
  2. You will receive $12.50 + $9.00 if the coin lands on Heads or Tails and (5|-5) if the coin lands on the Same or Different outcome.
  3. You will receive $12.50 + (5|-5) if the con lands on Heads or Tails and $1.00 if the coin lands on the Same or Different outcome.
  4. You will receive $10.50 + $9.00 if the coin lands on Heads or Tails and (1|-1) if the coin lands on the Same or Different outcome.
  5. You will receive $12.50 + $5.00 if the coin lands on Heads or Tails and (5|-5) if the coin lands on the Same or Different outcome.
  6. You will receive $14.50 + (9|-9) if the con lands on Heads or Tails and $1.00 if the coin lands on the Same or Different outcome.

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Finite Mixture Models

April 1, 2008 in Econometrics | Permalink

Let us assume that there are two types of people: smart people an dumb people. Smart people’s test scores are normally distributed about 80% and dumb people’s tests scores are normally distributed about 40% on their test. If we observe the test score of one person, how do we know if they are smart or dumb? If we see a score of 85%, we are pretty sure they are smart. A dumb person might have had a good day, but this would be a low probability event. Similarly, if we saw a score of 35%, we would be fairly certain that the person is dumb, even though there is a small probability that a smart person may have had a bad day. If we see a score of 62%, however, then it is very difficult to distinguish if the person is smart of dumb. But how can we quantify the probabilities that a person is of a certain type.

One way of doing this is finite mixture models. Jim Hamilton’s Time Series Analysis book has a good explanation of this topic and I will review this material here.

Each type (e.g.: how smart the person is) will be designated as st=1,2,…, or N. Let us assume that there is an observed variable yt (e.g.: the test score) which is distributed according to a N(μs,σj2). What researchers wants to know is that given that we observe yt, what is the probability that the observation is from a person of type st=j.

Let us assume that we know the density of yt is:

  • f(yt|st=j;θ)=(2πσj2)-1/2 * exp{-(yt – μj)/2σj2}

There is also some underlying distribution of types.

  • P(st=j;θ)=λj
  • θ=(μ1,…,μN,σ1,…,σN,λ1,…,λN)

From Bayes Rule, we know that:

  • P(A and B)=P(A|B)*P(B), which implies
  • f(yt,st=j;θ)=λj*(2πσj2)-1/2 * exp{-(yt – μj)/2σj2}

The unconditional density can be found as follows:

  • f(yt;θ)=Σ1 to N p(yt,st=j;θ)
  • f(yt;θ)=λ1*(2πσ12)-1/2 * exp{-(yt – μ1)/2σ12} +…+λN*(2πσN2)-1/2 * exp{-(yt – μN)/2σN2}

Now we can use maximum likelihood estimation techniques to find the θ which will maximize:

  • maxθ L(θ)=Σ1 to Tlog f(yt;θ)
  • s.t.: λ1 + λ2 +…+ λN=1
  • s.t: λj≥0

Once we have the MLE estimated θ, we can figure out what the probability is that observation yt came from a person of type st=j. Using Bayes theory, again, we know that:

  • P(st=j|yt;θ)=f(yt,st=j;θ)/f(yt;θ)=λj*f(yt|st=j;θ)/f(yt;θ)

This value represents the probabilty, given the observed data, that the unobserved type responsible for observation t was in of type j. For example, “…if an observation yt=0,, one could be vertually certain that the observation had come from a N(0,1) distribution rather than a N(4,1) distribution, so that P(st=1|yt;θ) for that date would be near unity. If instead yt were around 2.3, it is equally likely that the observation might have come from either regime so that P(st=1|yt;θ) for such an observation would be close to 0.5.”

Most of the above content came is from:

  • James D. Hamilton (1994) Time Series Analysis, Princeton University Press, Princeton, NJ; pp. 685-689.

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Risk Preferences and Technology Adoption in China

December 18, 2007 in Economics - General | Permalink

Development economists have long sought the answers as to why new innovations do or do not get implemented in developing countries. Giliches (1957) found that hybrid corn adoption has an S-shaped function over time. Other studies have found that an individual’s social network is the primary determinant of technology adoption. If your friends try out a new technology and it works, you will be more likely to hear about this advance if you have a large social network. Other economists blame credit constraints for the slow adoption of many farming technologies. A large up-front cost of some fertilizer or new seeds may be prohibitive, even if there is a high payback rate in terms of crop yield. Finally, it is possible that the “new and improved” technology may not be better. A pesticide developed in the West may work on American or European pests but may prove impotent against different other farm threats in other countries.

A paper by Elaine Liu, however, argues that there is another driving force which may explain technology adoption: risk preferences. To test this, Liu surveys Bt cotton adoption of farmers in the Henan, Shandong, Hebei and Anhui provinces in China. Bt cotton is slightly more expensive that regular cotton seeds, but farmers who use these new seeds spray 82% less pesticides than with the original seeds.

To test for risk aversion, Liu employs a Holt and Laury (2002) methodology but uses prospect theory to fit parameters of the individual’s utility functions. This allows individuals to be loss averse and also to use nonlinear probability weighting.

After controlling for various covariates, Liu finds the following results.

  • Individuals with higher levels of risk aversion adopt Bt cotton later.
  • Individuals with higher levels of risk aversion continue using higher levels of pesticide even though less pesticide is needed when Bt cotton is used compared to traditional cotton seeds.
  • Farmers with more education were not found to adopt Bt cotton earlier, but once they did begin using the Bt seeds, they wisely used less pesticide.

Although Liu does not mention this, prudence may play a factor in these technology adoption decisions. Taking a sure loss from the higher price of Bt cotton may not outweigh the gain from decreasing the probability of crop loss.

By 2006, the adoption of Bt cotton was nearly 100% and it seems that Chinese farmers are reaping the rewards of this new technology. Once the farmers understood better the benefits of the new seed, risk decreased and farmers were more likely to adopt the new Bt cotton technology.

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Survey Measures of Risk Aversion and Prudence

November 1, 2007 in Current Events, Econometrics, Experimental | Permalink

Can we estimate risk aversion and prudence using a survey question for the general public? This is what a paper by Eisenhauer and Ventura attempts to do.

Methods

In the 1995 Survey of Italian Households’ Income and Wealth, one question asked:

You are offered the opportunity of acquiring a security permitting you, with the same probabilities, either to gain 10 million lire [5165€] or to lose all the capital invested. What is the most you are prepared to pay for this security?

Assuming, the respondents answer honestly and precisely (which is a big assumption to make), the authors can create and individual’s utility function:

  • U(w)=0.5U(w-z)+0.5*U(w-z+10)

The variable w represents initial wealth and z is the amount individual would pay for a security. Using a Taylor expansion, we can create an estimate of absolute risk aversion.

  • 2U(w)=U(w)-zU’(w)+0.5z2U”(w) + (10-z)U’(w) + .5(10-z)2U”(w), or
  • [(50-10z+z2)/(10-2z)]*U”(w)=-U’(w)
  • A(w)=[(10-2z)/(50-10z+z2)]
  • R(w)=A(w)*w

The term A(w) represents the Arrow-Pratt measure of absolute risk aversion while R(w) is equal to relative risk aversion. If we differentiate the second equation above with respect to initial income, w, we can calculate a measure of prudence (-U”’/U”).

  • η(w)=A(w) + {(10-z)-1 + [2z/(100+z2)]}*∂z/∂w
  • Ï?(w)=w*η(w)

The term η(w) measures absolute prudence while Ï?(w) measures relative prudence.

Results

Since the authors have information regarding each individual’s initial earnings and various sociodemographic factors, they can analyze which type of people are risk averse.

  • Relative risk aversion is between 7.18 and 8.59.
  • Relative prudence is between 7.32 and 8.65.
  • The most risk averse groups are those in poor health and those with only an elementary school education.
  • The least risk averse are the college educated and those with health insurance.
  • Those with risk assets such as stocks or loans are less risk averse.
  • The authors claim that generally R(w)<Ï?(w)<R(w)+1 and risk aversion and prudence are highly correlated.

Healthcare Economist critique

Finding that people are risk averse and prudent is unsurprising, but the levels of risk aversion and prudence are very high compared to other studies. While having a vast array of sociodemographic information is important, simply eliciting a willingness to pay for a risky gamble is likely not a precise estimate of risk aversion. Likely, most people will respond to the question categorically (5 million lire, 4.5 million lire, 4 million lire, etc.). Further, finding that people with health insurance are less risk averse is counter-intuitive. One explanation is that having health insurance may be a proxy for wealth. Thus people with heath insurance in general could be more risk averse, but since this group of people is also richer (and more affluent people are generally less risk averse) we could have opposing effects.

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