Response to “Comment on Metallic Mineral Resources in the Twenty-First Century: I. Historical Extraction Trends and Expected Demand” by D. A. Singer and W. D. Menzie

Abstract

Singer and Menzie (2010, p. 24) state: ‘‘The pattern of increasing per-capita consumption with increasing income...defines a growth curve, which may be modeled by a logistic function...’’. Menzie et al. (2005, p. 46) write: ‘‘This pattern defines a growth curve that may be modeled by a logistic function...’’, and then state that they used this function to estimate future levels of copper consumption for the twenty most populous countries, for the year 2020. In their comment, Singer and Menzie (2015) suggest that I may have misinterpreted these statements as implying that they have said that future metal consumption can be modeled by a logistic function of time. There is no misinterpretation. The function that they present (e.g., Singer and Menzie 2010, Eq. 2.1) is a function of time. The variable P (population) in the authors equation is a function of time, and the same is true of the variable i in their equation, that stands for percapita GDP. These facts make consumption, C, a function of time too. Whether time is an explicit variable in the function, or is introduced as a hidden variable that controls the behavior of another variable (or variables), that is (are) explicitly shown in the function, the goal is the same: one is attempting to predict per-capita consumption as a function of time. The authors use the word future, they comment on metal consumption in the year 2020, and they plot Cu consumption as a function of time (Singer and Menzie 2010, Fig. 2.7; Menzie et al. 2005, Figs. 2–7). Singer and Menzie (2010) and Menzie et al. (2005) appear to have used GDP and population estimates generated by the UN (see Singer and Menzie 2009) in order to estimate 2020 copper consumption with a logistic function. Those UN estimates must have been generated by functions of time. Moreover, Singer and Menzie (2010, p. 25) discuss growth of world copper consumption in terms of annual growth rate, i.e., of the first derivative of consumption relative to time. One can only differentiate a variable relative to another variable that it is a function of. Singer and Menzie (2015) also state that ‘‘...direct use of time as a predictor of metal consumption doesn t make sense...’’. I prefer to base scientific arguments on data and rigorous mathematical arguments, rather than on ‘‘sense’’. None of the references given by Singer and Menzie (2015), nor any other publications that I am aware of, proves that worldwide metal consumption of any metal follows a logistic function, nor a Kuznets curve, nor, for that matter, any other specific functional relationship, regardless of whether time appears in the function explicitly or as a hidden variable. Conclusions are derived either from philosophical considDepartment of Geology, University of Georgia, Athens, GA 30602, USA. To whom correspondence should be addressed; e-mail: alpatino@uga.edu