By Mike Mesterton-Gibbons
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Extra info for A Concrete Approach to Mathematical Modelling (Wiley-Interscience Paperback Series)
33), for the specific growt h rat e of an economy. But what good is it ? 33) to obtain an explicit expression for the growth rat e of output? 32) is a single relation among three functions, namely, K, L, and Q. It is because ther e are fewer relations than functions that we are unable to obtain an explicit expression for any of them. Mor e generally, thi s situation would arise if we had η relations among η + m functions. I f the word "closed" were used t o mean explicit, then such a model would not be closed, because explicit expressions could not be deduced for all n + m functions.
9e ’ . S. population in millions, 1790-1850. 3 Data for x(i) are taken from Historical Statistics of the United States, Colonial Times to 1970, Bicenten› nial Edition, Part I , p. S. , 1975). S. 2 for x(2). Can the same model predict population magnitude in later years? 57) are compared wit h observed values for the years 1 8 5 0 - 1 9 7 0. 57) soon fail s to describe population growth wit h acceptable accuracy. 24 for the years 1 8 6 0 - 1 8 8 0, suggesting the improved model \ 3 1 i ? 24(/-7) ( ( > 6 5 But although thi s is correct to tw o significant figures for the years 1 8 6 01880, and even to one significant figure for 1850 and 1890, it seriously overestimates the magnitude of the population in the present century.
What do we mean by goods per year at tim e f? We mean that if the production rat e could somehow be held constant for a whole year beginning at tim e t, then the amount of goods produced in that year would be QXt). Provided, as we have assumed, 14 Chapte r 1 Growt h and Decay . 26) QXr)dr, j th e error term vanishing in the limi t of integration. Because "instantaneous annual output" is rather a mouthful , we’ll refer t o Q simply as output. I n producing thi s output, worker s need buildings, raw materials, ma› chinery, tools, and a variety of other aids.
A Concrete Approach to Mathematical Modelling (Wiley-Interscience Paperback Series) by Mike Mesterton-Gibbons