Homework 6: Nice Demo City, But Will It Scale?
36-402, Advanced Data Analysis
Due at the start of class, 21 February 2011
For data-collection purposes, urban areas of the United States are divided
into several hundred “Metropolitan Statistical Areas” based on patterns of res-
idence and commuting; these cut across the boundaries of legal cities and even
states. In the last decade, the U.S. Bureau of Economic Analysis has begun
to estimate “gross metropolitan products” for these areas — the equivalent of
gross national product, but for each metropolitan area. (See Homework 2 for the
definition of “gross national product”.) Even more recently, it has been claimed
that these gross metropolitan products show a simple quantitative regularity,
called “supra-linear power-law scaling”. If Yis the gross metropolitan product
in dollars, and Nis the number of people in the city, then, the claim goes,
Y≈cNb(1)
where the exponent b > 1 and the scale factor c > 0. This homework will use
the tools built so far to test this hypothesis.
1. (15 points) A metropolitan area’s gross per capita pro duct is y=Y/N.
Show that if Eq. 1 holds, then
log y≈β0+β1log N
How are β0and β1related to cand b?
2. (15 points) The data files gmp 2006.csv and pcgmp 2006.csv on the class
website contain the total gross metropolitan product (Y) in millions of
dollars, and the per capita gross metropolitan product (y) in dollars, for
all metropolitan areas in the US in 2006. Read them in and use them to
calculate the metropolitan populations (N). If it’s done correctly, then
running summary on the population figures should give
Min. 1st Qu. Median Mean 3rd Qu. Max.
54980 135600 231500 680900 530900 18850000
(Your exact results may differ very slightly because of rounding and display
settings.) What is the variance of logy?
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