By Donald A. R. George (auth.)
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Additional info for Mathematical Modelling for Economists
Increasing the number of men but keeping the number of shovels constant has no effect on output. Decreasing the number of men, with shovels constant, will decrease output. There can therefore be no meaning to the expression 'marginal product of labour' in this case. Exercise Suggest a (non-differentiable) function which could be used to model the above relationship between holes, men and shovels. Partial derivatives can often themselves be differentiated to give second, third and nth partial derivatives.
3 Let f, g1 , • • • , gk> be concave and differentiable. - as domain. e. assume there exists x E R"+- such that gi (x) > 0 for j = 1, ... , k). t- i i j = = = 1, ... , n 1, ... , n 1, ... 3 are sometimes referred to as the Kuhn-Tucker conditions. Condition (b) requires that either a state variable X; be zero or the derivative of the Lagrangian with respect to X; be zero. 2) is sometimes called the complementary slackness condition. It requires that, at an optimum either a constraint is binding or its multiplier is zero (or possibly both).
There is indeed a similarity but we do not pursue the KuhnTucker approach here. Instead we state three theorems relating primal and dual programmes. The first gives a sufficient condition for optima of primal and dual, in terms of their values. 4) respectively. If: b'x* = c'y* then x* andy* are optima for primal and dual respectively. The next theorem is a partial converse of this result. 4) both have feasible vectors then optimal vectors x* and y* exist for primal and dual respectively and: b'x* = c'y* The third duality theorem deals with the relationship between dual variables and the binding or slackness of the corresponding constraints.
Mathematical Modelling for Economists by Donald A. R. George (auth.)