In 1976 Basil Rennie introduced in his James Cook Mathematical Notes (JCMN) a question from the first examination in Pure Mathematics at Adelaide University (1876). In the following year my contribution saw the question as a 'prehistory' of linear programming and Giffen's paradox which would be of interest to High School students.
'Prehistory' of LP and Giffen's paradox
Extract from JCMN, #4, June 1976
It is a little hard for Adelaide to compete with the Cook bicentenary but a hundred years ago lectures started at the University of Adelaide, and you might like to try question VI of the first Pure Mathematics I exam set by the first Professor of Mathematics, Horace Lamb, in 1876.
Question VI. A man sets apart 28 pounds a year to be spent in drink, and considers that he requires in the year a quantity of alcohol amounting to 24 (reputed) quarts. He prefers claret to ale, but claret costs 40 shillings a dozen, ale only 12 shillings a dozen. The percentage of alcohol in the claret being 10, and in the ale 6, how much does he buy of each? If the price of ale rises, will he drink more ale, or less, than before.
Extract from JCMN, #8, February 1977
Question VI (from JCMN 4).
The first problem set by Adelaide's first professor of Mathematics can lay some claim to another pair of 'firsts'. It predates by 65 years the first problems in linear programming - F.L. Hitchcock's transportation problem and J. Cornfield's diet problem; and it predates by 19 years Alfred Marshall's recognition of the Giffen paradox.
A senior secondary student of today would solve the problem graphically by considering the problem in the following form: find the quantities, x and y, of claret and ale that are to be bought at the given prices if x is to he maximised subject to the given budget constraint and the 'nutritiion' constraint. The effect of an increase in the price of ale on the optimal value of y then becomes apparent.
However, the solution may be exposed more readily if we lapse into a literary approach that would not have been out of place in the nineteenth century.
We can image oure bibulous friend filling his basket in the following way. Let him first buy enough of ale alone to satisfy his craving. Then the remainder of his purse can be used to finance the return of some ale in exchange for its alcoholic equaivalent in claret.
Now suppose that the price of ale rises (but not so much that his craving cannot be satisfied by ale alone). Then the value of the contents of the basket will increase. Hence, some claret must be 'liquidated' by returning it in exchange for its alcoholic equivalent in ale. We thus have an example of Giffen's paradox - a situation where an increase in the price of a commodity causes more of it to be consumed.
© 1998 D.R. Watson