The power-law distribution of agricultural land size

ABSTRACT

Power-law distributions explain a variety of natural and man-made processes spanning various disciplines including economics and finance. This paper demonstrates that the distribution of agricultural land size in the United States is best described by a power-law distribution. Maximum likelihood estimation is carried out using county-level data of over 3000 observations gathered at five-year intervals by the USDA Census of Agriculture. Our analysis indicates that U.S. agricultural land size is heavy-tailed, that variance estimates generally do not converge, and that the top 5% of agricultural counties account for about 25% of agricultural land between 1997 and 2012. The goodness of fit of power-law distribution is evaluated using likelihood ratio tests and regression-based diagnostics. The power-law distribution of farm size has important implications for the design of more efficient regional and national agricultural policies as counties close to the mean account for little of the cumulative distribution of total agricultural land.

KEYWORDS:

• Farm size
• goodness-of-fit testing
• heavy-tailedness
• Pareto distribution
• power law

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 Previous research to understand this decline of agricultural land size in the U.S. has focused on factors of farm survival and exit [35], effects of governmental policies and subsidies on farms [2,23,24], and the individual farm [40].

2 Gibrat’s law of proportionate growth posits that the growth rate of a stochastic process does not depend on its size, but is proportionate to it [19]. Further, Gibrat [19] showed that the law of proportionate growth can generate the lognormal distribution for the size of the process. Later, Gabaix [16] demonstrated that the proportionate growth process can also give rise to power law behavior at the upper tail of the process.

3 Examples include firm size [4,28,39], city size [13,16,21,26,27], frequency of words [22,50], income and wealth [9,25,32,38,41,48], consumption [42,43], carbon dioxide emissions [3], and natural gas and oil production [5], among others. See Gabaix [17] for a review.

4 The log-log plot is constructed by taking the logarithm of the rank of $x$ in the data and the logarithm of $x$, and then plotting log rank of $x$ against log $x$. Note that the counter- (complimentary-) CDF (also known as survival function) for power law distribution is $\text{Prob}\left(X>x\right)=k/x^{\gamma }$, where $k$ is a constant. Now, taking the log of both sides of the counter-CDF of power law produces a linear relationship between log counter-cumulative probability and log data (i.e., $\ln x$), with the power-law parameter $\gamma$ being the slope of the line.

5 To identify fraction $L$ of the total land held by the top fraction $P$ of counties, use $L=P^{\left(\alpha -2\right)/\left(\alpha -1\right)}$, which is derived from the complementary-CDF of power law distribution.

6 According to the raw data, the top 5% of counties accounted for 25.75% of all agricultural land in 1997, 25.73% in 2002, 25.52% in 2007, and 25.42% in 2012.

Additional information

Funding

This work was supported by Utah Agricultural Experiment Station [grant number 1306].