Rule of 72

The Rule of 72 is a helpful approximation used to determine the time frame required for an investment to double. It’s a nice trick to remember when a quick calculation is necessary. Simply divide 72 by the prevailing interest rate (expressed as 100*r) and the solution is the resulting time necessary for the investment to double. The derivation is simple for compound and continuous compound problems. The future value of an asset with an interest
rate compounded per annum for t years is given by,

To determine the time period at which the future value of the asset is doubled, we simply solve the previous equation for t while setting the future value to twice that of the present value.

A first order Taylor series approximation for ln(1+r) is simply r, and
easily simplifies our result,

Given that the ln(2)=0.693, the rule can be applied as follows%


72 or 70 are commonly used for the convenience of division, and the approximation is relatively accurate. For cases involving continuous compounding the derivation is even more straight forward.




Because interest is compounded continuously a Taylor series approximation isn’t necessary, and the rule applies similarly,


As an example, assume you want to put $1,000 in an account accruing an annual percentage rate of 8%. According to our rule, it would take around 9 years to double the investment to $2,000. The actual calculation is 8.66 years.

NCAA

My mom and my wife have each filled out NCAA brackets for the first time this year. They seem to be getting the feel for it already; agonizing over match ups and a growing distaste for teams. My wife struggled with any first round losses and I’ve tried to convince her to be patient. For perspective, let’s review a few characteristics of the tournament. First and foremost, there are a possible unique bracket outcomes for this tournament, ignoring the one game playoff,

For a perfect bracket, an individual needs to select the single bracket out of the total possible set. Of course, certain brackets are more likely to occur than others, and using information we can eliminate unlikely brackets and highlight brackets that are most likely. This should reduce the current odds of a correct bracket, , significantly.

Consider a crude probability model where the probability a team wins is strictly dependent on seed. One such specification is

where represents the prescribed seed for a team. For example, the probability a four seed beats the thirteenth seed is about 0.76 using this formulation. Assuming a team with the higher seed wins each game we can thus calculate the likelihood of such a bracket.

The odds of such a bracket occurring are Still not a likely prospect but it is significantly smaller then the original computation. I imagine there is even a more likely bracket that would reduce the odds even more. A more efficient probability model could find such an answer. Nevertheless, the likelihood of a perfect bracket is tremendously small.

Rational Addiction

The Becker-Murphy Rational Addiction framework incorporates two important means for describing addictive behavior: reinforcement and tolerance. The current period utility function,

states that well being is determined by consumption of an addictive good, , the capital accumulation of the addictive good, , and a non-addictive good, Reinforcement is imposed on the model if and is consistent in agents with myopic preferences. More clearly stated, reinforcement is observed in a good whose past consumption increases the marginal utility of current consumption. Tolerance requires and describes how past
consumption of an addictive good reduces current utility. In other words, a greater quantity of the addictive substance is required to receive the original euphoric effect.

Presented in this manner, steady state solutions are found and analyzed for the dynamic optimization problem. The model can describe a variety of circumstances, and the existence of unstable steady states illustrate the difference between highly addicted people who continue consuming harmful substances and those who eventually quit entirely. Two other important implications of the model are: First, individuals who heavily discount the future are more likely to become addicted, and second, reinforcement causes addictive goods across time periods to be complements.

Because robust data on illegal drug prices doesn’t exist, the implications for the model are purely hypothetical. However, the model has been empirically tested on cigarette, alcohol, and gambling consumption. From these results a permanent reduction in the price of drugs is expected to increase consumption in the short run. It is likely that addiction would increase in the long run as well. Furthermore, addiction among lower income groups and the young are likely to be affected the most.

The results are certainly not complete and a more thorough study is necessary to evaluate the impact of such a policy. Nevertheless, legalizing drugs like heroin, cocaine, and marijuana are likely to increase
consumption, particularly in low income and younger groups.

Economics of Drugs

Earlier, I discussed comments made by Gary Becker that addressed the issue
of legalizing illicit drugs. While I was unable to locate any information
regarding the effects on inner-city youth, I did find an interesting
discussion detailing the impact legalization would have on consumption.
Certainly prices would fall, but the corresponding change in quantity
demanded is the question to be answered. If this were an introductory
economics class we would analyze the change in policy through a series of
graphs. The following figure, for example, illustrates the current policy,
where legal barriers increase the risk of bringing goods to the market. The
demand curve is relatively inelastic, showing that changes in price result
in relatively unresponsive changes in the quantity demanded.

Legalizing drugs would cause a shift in the supply curve, S1 to S2; as a
result, a new market equilibrium point is determined. This new equilibrium
represents a solution corresponding to a lower market price and higher
market quantity. A simplistic representation and one that provides a
generally intuitive example. However, this static realization fails to
address the question of how much the actual quantity will change. Is it
enough to know that consumption will increase, or is it more important to
know the magnitude of such a change. One issue is the usage of a linear
demand curve. Naturally, it is generally assumed that demand functions for
addictive goods are inelastic. However, the computed elasticity for a linear
demand curve is not constant, and in fact encompasses the entire permissible
range of elasticity, . The problem can be easily addressed by
substituting a constant elasticity demand curve in place of the linear
specification. But, consumption of an addictive good has dynamic
implications as well. This is why the Becker-Murphy Rational Addiction model
is so promising. Next time we’ll discuss the implications of the Rational
Addiction model and consumption estimates for the change in policy.

arg(max)

I must be in a mood today because I find myself picking on helpless newspaper bloggers. I’m more frustrated in the quality of arguments people carelessly throw out on the web. From a perspective, the cost of posting columns is so low that it makes sense. But when you’re backed by a newspaper I expect more thoughtful arguments. Not the typical, “in my experience,” or, ” I knew a family when…” A single case does not make a convincing argument. Here is a comment I posted on one such website discussing a possible cigarette tax in Iowa.

The most pathetic argument to be used in favor of a cigarette tax is one that generalizes a single, familiar experience to justify a policy that impacts a certainly diverse set. While I technically agree that a cigarette tax, among other policies, is appropriate; you neglect to mention a single point that lends support for the tax. First, people are allowed to make consumption decisions without those decisions being termed wasteful. Your definition of wasteful is neither complete or absolute. And if, as you claim, people are augmenting food income with food stamps to free up income for cigarettes, your argument should be in favor of eliminating food aid programs. Second, you haven’t reported the numbers for the population who smoke and don’t have health insurance. According to studies, health care costs shared by the public will increase, yet these studies fail to reflect the increase in coverage required for longevity effects. Everyone at a stage in their life will get sick. If it’s not smoking related then there will be another factor (obesity possibly). These effects need to be recognized if an honest debate on the issue can occur.
It is careless to approach your columns in this fashion. It serves little purpose in furthering the argument and only amounts to stratifying your audience. If I were to write a column concerning stay-at-home mothers and their self-importance I would be promoting a careless opinion. It is certainly not true and I have no evidence to support the claim. What is necessary, however, is a subjective understanding of the solution. Present facts and step back from your ideas for a second. Even questioning yourself can be an important exercise. But carelessly expressing your opinion with little aforethought does little to strengthen your ideas.

Patent Reform

In a story from wired news, patent reform is again in the congressional spotlight. The patent system in the U.S., for years, has failed to promote ingenuity and technological advances. The losers are the true innovators, while lawyers and patent trolls take advantage of the system. Japan in particular, with less strict patent laws, benefits greatly from relatively unconstrained innovative processes. One likely case of abusing the patent system: method of swinging on a swing. Apparently, licenses are available for those interested.

First Order Conditions

It has come to my attention that having a blog entitled “First Order Conditions” will direct some math searches this way. Not surprisingly, a common search theme that has directed users to the site is the first order conditions for a concave function. I might as well post these now and have them available in the archives. Enjoy!

A general maximization problem is

where V(⋅) is the objective function to be maximized with respect to the choice variables x_{i},i=1,..,n. The first order necessary conditions for a maximum are

The first order conditions are necessary in the sense that the values for the choice variables must satisfy the above equation for the solution to maximize the objective function. However, this is not a sufficient condition for a maximum and the solution could characterize a minimum. Second order sufficient conditions are needed to sufficiently characterize the solution. Although, if V(⋅) is a concave function then the first order conditions are also sufficient for a maximum.

I’m Hooked on Oriental Drugs

I’ve yet to link to more then a couple sites, but one of those sites I enjoy immensely. That is The Becker-Posner Blog. Becker is a Nobel Prize winning economist and someone with whom I rarely disagree. His arguments are thoughtful and sophisticated. He doesn’t find data and statistics that support his point. Rather, he uses classical theory and a broad econometric background to form his positions, something that I greatly appreciate. If you haven’t read his post concerning raising the minimum wage, I strongly recommend it.

Anyway, a recent comment in a post about growing global wealth inequality grabbed my attention. While discussing ways to improve the distribution of wealth domestically, Becker subtly includes the legalization of illegal drugs as a way to discourage inner city youth from dropping out of school. By removing an attractive substitute, i.e. selling illicit drugs, inner city youths are more willing to invest in education, and positive returns to schooling are key in reducing wealth inequality. What strikes me as interesting is the almost casual inclusion of the drug legalization argument. On a theoretical level, legalizing illegal substances is the most efficient solution to the high costs incurred in the illicit market. However, I have a hard time convincing myself practically that this policy is correct. Deep down, I want to believe that legalizing banned substances will be welfare improving, but some part of me hesitates to make the conclusion. Perhaps a literature review and an econometric study can sway me.

Akerlof’s Lemons

Efficient markets rely on complete information to succeed, while information asymmetries can create a market collapse. As was the case, one year ago, when my wife and I bought our first used car. Sixty days and one catastrophic break down later, I was reviewing Akerlof’s “Market for Lemons,” a work for which he received the Nobel prize in 2001. Lemons described a market where buyers and sellers had different information about the quality of used cars. The result was a complete market break down, a situation where no trade would take place.

When buyers are unable to discern the true quality of a good, they are unwilling to pay a premium for goods that actually are of quality. Producers are now unwilling to supply quality goods to the market, which creates a market of only poor goods. More specifically, buyers are willing to pay a price equal to the average quality of the goods. Any good whose quality is above average will not be supplied to the market. The market now contains only goods with average or below quality. While buyers cannot directly observe quality, they are aware that the departure of quality goods will lower the average quality of the remaining set. Buyers realize that average quality in the market will always lie below any given price, resulting in no trades taking place.

Do I believe the pre-owned car industry is a lemons market? No. However, I do believe there are many markets where information is not complete and uncertainty exists regarding quality. Consequently, these markets suffer and good products are driven out by lemons. So, during this 4th quarter shopping binge, look for products offering definite guarantees and full disclosure of information. And be wary of late night infomercials promising quality wares, because the market may be slowly collapsing.

Akerlof, George A. “The Market for ‘Lemons’: Quality Uncertainty and The Market Mechanism.” The Quarterly Journal of Economics, 84, no. 3 (1970), 488-500.