### Abstract

The perceived and actual value of the United States one cent coin ("penny" hereafter) are compared. The comparison determines whether the energy and time expended result in a net gain or net loss to a test subject encountering a penny which has left circulation and settled in a lower energy state. Obverse/reverse influence on perceived value is examined, and a standard for evaluating the caloric cost of the retrieval is established. High variability of contributing factors led to the creation of guidelines for establishing the expected profit of the sidewalk-to-pocket transaction.

### Introduction

The United States of America has circulated a small-denomination coin called the "penny" since 1793. The vast majority of pennies are worth one one-hundredth of a dollar, or one "cent". In 2003-2005, the United States Mint produced approximately 1.1 trillion pennies, so the occurrence of anomalous pennies, while driving up the value of the individual anomalous penny, is insignificant when calculating the average value of a penny. This study examines any legal tender penny, and assumes all pennies are worth \$0.01.

A penny's immediate value, for reasons discussed above, is one hundredth of a U.S. Dollar. However, half of the pennies encountered (on average) will have their obverse exposed and their reverse concealed -- this is colloquially known as being in a "heads up" state because the silhouette engraving of Abraham Lincoln is visible. These pennies are anecdotally connected to varying amounts of monetary gain, and are said to exert an anomalous influence on local statistical rates of success in situations where probability would otherwise be against the subject. Further research will need to be conducted to determine if this affects the value of the penny. For purposes of this paper, we will only be examining pennies which are "tails up", or "heads up" and assumed to be luck-neutral or better. Since the aim of this study is to incentivize penny retrieval only when it is already profitable, we assume that all pennies picked up have an intrinsic value of at least one one-hundredth of a U.S. Dollar.

The energy expended in bending down to pick up a penny will be calculated and converted to a monetary cost. The time taken to pick up a penny is measured and converted to a monetary cost. These costs are summed and compared to the expected financial gain of the action.

### Materials and Methods

A group of ten pennies were dropped on a thin-pile carpeted floor. The pennies were dated as follows: 1968, 1969, 1970, 1979, 1980, 1982, 1998, 2000, 2001, 2001. The time required to pick up the pennies from a standing position was recorded. Each penny required no more than three seconds to pick up.

A model for the energy expended picking up the pennies was established. Isaac Newton's laws of motion give a widely-accepted formula for the work (energy) required to move a mass up a given distance:

E = mgh

where "m" is the mass of the object in kilograms, g is the acceleration of gravity (vector quantity, assumed to point straight down), and "h" is the distance (parallel to gravity) that the object must be moved. The mass of a penny has varied historically, but is dwarfed by the fluctuations in mass of the test subject, who must bend over to pick up the penny, and who must raise himself back up to the standing position. Statistical and census data were consulted to determine the average height and mass of a human; the reader is welcome to substitute his or her mass and height. The subject is otherwise assumed to be 1.8m tall and 80kg in mass. Bending from the waist requires moving half of the human body (cf. Da Vinci's "Vitruvian Man"), and assuming a nearly-equal mass distribution, this requires moving half of the body's mass from .75h to a height of .50h. Thus in the above formula, we have:

E = 40kg x 9.8m/s2 x 0.45m = 176 J or .00421 calories

It remains now only to convert both the caloric cost and time cost into monetary units. First the issue of calories is addressed. Calories can be purchased for wildly varying amounts, based on the perceived pleasure one gains from ingesting the calories. Maine lobster and a McDonald's hamburger were compared to arrive at rational maximum and minimum caloric costs. A 3.5oz serving of Maine Lobster tail contains 98 calories and can be purchased for \$16.95, resulting in a cost-per-calorie of \$0.173; a 3.7oz McDonald's hamburger (with standard toppings) contains 260 calories and costs \$0.59, resulting in a cost-per-calorie of \$.0023/calorie. If the test subject has eaten lobster recently, then his or her calories are worth almost seventeen cents each, and so the financial cost of bending down is \$0.00729, or 73 hundredths of a cent, yielding an expected gain of 27 hundredths of a cent. At the low end, the subject has even more incentive to pick up the penny; the cost is only \$9.69X10-5, resulting in almost pure profit from picking up the penny.

Now the issue of the time cost of money (where applicable) is addressed. The median income of a male full-time year-round worker is \$40,688 (2003 figures - the reader is welcome to insert his or her annual income instead). Assuming 50 weeks of work and approximately 40 hours of work per week, we arrive at an average hourly wage of \$20.34. The three seconds required to pick up a penny -- assuming that one would otherwise be compensated for one's time -- cost the working test subject \$0.01695, or 1.7 cents. A test subject walking back to work from lunch should therefore not bend down to pick up the penny unless he or she is paid \$11.88/hr or less.

A person's "free time" has no expected gain -- and in fact, according to the Bureau of Labor Statistics, there is often an expected cost. A single-person consumer unit expends, on average, \$27,042 annually (2002 figures). We assumed earlier that a person works 2,000 hours in a year. This leaves 6,766 hours of time, throughout which one is spending an average of \$3.99/hr, or \$0.0011/second. During one's free time, then, the profit from picking up a penny actually defers some of the costs of living (for example, the cost of eating that Maine Lobster), to the tune of \$0.0133 for each penny picked up.

### Results

Expected Gain from Picking Up A Penny
```PROFITS
=======
monetary value:   \$0.0100
time value    :   \$0.0033 (assumes penny discovered in free time)
SUBTOTAL:   \$0.0133

LOSSES
======
caloric value :  -\$0.00729 (maximum - assumes Lobster dinner)
lost wages    :  -\$0.01695 (assumes penny discovered during work)
SUBTOTAL:  -\$0.02424

EXPECTED GAIN
=============
AT WORK       : -\$0.01424
AT LEISURE    : +\$0.00601

A general function for one's expected gain from a penny is presented below:
Profit = \$0.01 + (Annual Expenditures)/(1200 * Leisure Hours)
- (Cost of last Meal)*(.0029155 x Mass x Height)/(Calories in last meal)
- (Annual Income)/(1200 * Work Hours)
```
NOTE: The reader is advised to perform this calculation during leisure hours, where, if it takes one minute, can result in a leisure-time savings of \$0.067, with proportional gains with longer calculations.

### Conclusion

The United States one cent coin offers a strong picking-up incentive to the intelligent leisure-seeker, who can defer his normal cost-rate-of-leisure by stooping to bring a penny up from the sidewalk. Workers who would otherwise be paid for their time lose a quantifiable but minimal amount of money in the effort, and are encouraged to leave the penny until they are no longer being paid for their time. The time required to calculate an individual's expected profit function can offset losses from accidentally retrieving two pennies, provided it is calculated during leisure hours. The effects of obverse/reverse mechanics on localized probability have not been measured but may have significant influence on the total expected gain from retrieval of a penny.

### Works Cited

1. Dogood, Silence. Poor Richard's Almanack, Letter to the Editor: "A Penny Saved"; 1733, Franklin Press, Philadelphia.
2. Bureau of Labor Statistics, "Consumer Expenditures in 2002": February 2004, Government Printing Office, Washington D.C.
3. United States Census, "Median Income in 2003": June 2004, Government Printing Office, Washington D.C.
4. Traditional Rhyme - "See a Penny Pick It Up"
5. United States Mint, "Coin Production Figures" 2003-2005, Washington D.C.
6. Adams, Cecil, The Straight Dope article, "Is it worth it to pick up a penny?", July 1983

Not only do American pennies have value as money that can be exchanged for goods and services, but they also have the intrinsic value of the metals contained within them. In most cases, actual value of the metal within the coin is far below the face value of that coin, however this is not the case of pennies minted before 1982.

Pennies today weigh 2.5 grams and are made up of 97.5% zinc and 2.5% copper and this is the way it's been since 1982. Before that time, pennies weighed 3.11g and contained a mixture of 95% copper and 5% zinc. A sharp rise in copper prices in the early 1980s caused the US Mint to change to today's ratios which makes pennies less expensive to manufacture.

Which begs one to ask the question: Has the intrinsic value of the copper in pre-1982 pennies exceeded the face value of the currency, and if so, how does that impact if the penny is worth picking up?

Utilizing the values above, we know that old pennies contain 2.95g of copper. Copper is currently trading at a price of \$3,365 per tonne1. Since there are 1,000,000 grams in a tonne, we use simple algebra where x equals the number of pennies we need:

2.95x = 1000000

Gives us that we will need to extract the copper from about 338,983 pennies in order to get 1 tonne of copper.

The face value of the pennies we have is equal to \$3,389.83, slightly more than the \$3,365 we would recieve for the copper on the market. The face value of \$.01 per individual penny is currently more than the intrinsic value of \$.0099 per individual penny. However, we must realize that the price of copper only needs to rise 1% in order for the value of the copper in a pre-1982 penny to be worth more than the penny itself, and thereby changing the "value" of penny-pinching that Jurph shows above.

Thus, when one is asking the question if a penny worth picking up, one should not only take into account the cost of their food and their wages, but also the date the penny was minted and the current price of copper.

1 London Metal Exchange - May 6, 2005