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By Donald A. R. George (auth.)

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If we take a particular linear combination c of a and b a concave function will have the property that f( c) will be greater than (or equal to) to the same linear combination of f(a) and f(b). This is illustrated in Fig. 12. 30 Mathematical Modelling for Economists g I f I f(c)-- I I I --+------------- Xf(a) + (1- X)f(b)--- - - t--- -----I I I I a c c = Xa + ( 1 - X)b Fig. ;; 1) The function f is concave while the function g is not. 12 Let f: S---+ R, where S C Rn is convex. : 0 =:::; A. )b is an element of S.

Dixit (1976) is devoted entirely to optimisation methods and their application in economics. Dixit's treatment is reasonably advanced, though his notation can be rather opaque. Intrilligator Kuhn-Tucker theory 47 (1971) also provides a moderately advanced discussion of optimisation. Baumol (1967) discusses 'satisficing' behaviour of firms, and Simon (1959, 1967) and Cyert and March (1963) interpret it as part of a 'behavioural' theory of the firm. Vanek (1970) is the classic reference on the theory of the labour-managed economy.

It is subject to the constraint that the balance of trade (XqM) must not be negative. ) (i) Set up the problem in Kuhn-Tucker form (ii) Show that the balance of trade is exactly equal to zero (iii) Show that: q ) = 0 a M(1 + _Q£_ 13 (1 + t) 1+t Further reading Utility theory is discussed at length by Green (1976). Baumol (1977) and Chiang (1984) contain discussions of constrained optimisation. Dixit (1976) is devoted entirely to optimisation methods and their application in economics. Dixit's treatment is reasonably advanced, though his notation can be rather opaque.

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