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Sunday, February 18, 2007

Holiday Cards and Zipf's Law

Friday's Fort Worth Star Telegram ran an Associated Press story concerning Hallmark and its attempts to beef up card sales during the non-Christmas period, i.e. all the other months of the year, by creating so-called niche cards for some of life's most difficult aspects (repeated here at the Detroit Free Press).

What I found interesting was the data on card sales for the various holidays:

Christmas - 2 billion
Valentine's Day - 188 million
Mother's Day - 150 million
Father's Day - 103 million
Easter - 80 million
Halloween - 29 million
Thanksgiving - 28 million
St. Patrick's Day - 10 million

I never receive a St. Patrick's Day card - nor do about twenty-nine in thirty people in the United States in a given year. Nevertheless, the data are interesting. The Christmas card is a long-standing tradition, observed by individuals sending cards to family and friends and companies who send cards to their employees and clients. As we can see from the dramatic drop-off in cards sent during Easter (see below), it would seem that the vast majority of Christmas cards are sent out of a sense of "doing business" or "obligation" rather than because of heart-felt sentiment.

The number of cards on Mother's Day is interesting. Mom's get about 30% more cards (as a group) than Dads. I don't think it is too hard to understand why that is the case.

Those concerned about the religiosity of Christmas can relax a bit. If Christmas is important to Christians, so too must be Easter. However, the number of cards sent on Easter equal only 4% of the cards sent during Christmas. Does this imply that only 4% of the cards sent during Christmas are done so by those celebrating the religious holiday rather than the secular holiday? This is an interesting question - someone should write a paper...


I took the data (all eight observations) and threw them into Stata. Here's a bar graph of the reported numbers of cards sent for each holiday:

Image will be uploaded 2/19/2007

A common empirical finding is that sizes (whether they be city sizes, blog hits, or other rank-orderings) tend to follow a Pareto distribution where the rank of individual i is a function of it's size:

ri = Asi-a

where r = rank, A is a scale parameter, s = size, and a = shape parameter. The famous Zipf's Law suggests that a = 1. When a>1 the sizes are a bit more evenly distributed and when a<1 the sizes are bit less evenly distributed.

I estimated the rank-size relationship (with a whopping 8 observations) and obtained the following:
. reg lnrank lncards,r

Regression with robust standard errors Number of obs = 8
F( 1, 6) = 195.72
Prob > F = 0.0000
R-squared = 0.9369
Root MSE = .19077

------------------------------------------------------------------------------
| Robust
lnrank | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lncards | -.4254901 .0304136 -13.99 0.000 -.4999094 -.3510708
_cons | 3.233197 .1564306 20.67 0.000 2.850426 3.615969

The hypothesis that the parameter estimate on the natural log of the number of cards sold for a given holiday is equal to one in absolute value is strongly rejected (t= 13.99, p = 0.000). The upshot is that the distribution of card sales is disproportionately weighted towards Christmas. To this extent, Hallmark is completely justified in its attempt to build customer awareness and increase card sales during the other parts of the year.

One use of econometrics/statistics is to confirm what others have already intuited from their situation. I doubt if the good folks at Hallmark had any econometrician/statistician look at their card sales and suggest that the rank-size rule was violated. Rather, the higher-ups at Hallmark likely figured out that sales during Christmas were really strong and the sales in August were not. Therefore, Hallmark decides to introduce new gimmicks to the card industry and, at least from the limited information we can glean from their set-piece in the Associated Press, they seem to be justified in what they are doing.

Whether the new gimmicks will actually work or not is another issue.

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