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QABE_Notes21

# QABE_Notes21 - QABE Lecture 21 Multi-variable Optimisation...

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Unformatted text preview: QABE Lecture 21 Multi-variable Optimisation School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Unconstrained optimisation 2 2.1 First order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Second order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Saddle points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Constrained optimisation 6 3.1 Dealing with the constraint I: substitution . . . . . . . . . . . . . . . . . . 6 3.2 Dealing with the constraint II: the Lagrange multiplier . . . . . . . . . . . 7 3.3 First order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Second Order Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Introduction We now come to our final lecture in calculus. We met last time the partial derivative as a way of finding the rate of change of a function of more than one variable. Further, we could then use the partial derivative to find the total derivative of a function and so, get at how infinitesimal changes in one of the inputs would change the output. This time, we use this knowledge, to go one step further and use the idea of multi-variable calculus to perform optimization. In some ways, this will look very familiar to our optimization problems under normal single-variable calculus, but in others, we’ll need extra tools. In the first case, the first- and second- order conditions that we apply will look very similar, but in the second case – where we have to deal with constraints on where we are allowed to look for maxima or minima – things will get quite a bit more complicated! In particular, we have to somehow perform our normal optimisation, but at the same time, ensure that we don’t get off the constraint line. This sounds diﬃcult, but in fact, a device called the Lagrange multiplier will come in very handy. It should be pointed out that some of the steps in between in the later cases are not shown, and are not required in this course – we’ll just take the conditions ‘as is’. For those interested however, any undergraduate text in calculus will be a good place to look, most probably under the title of ‘Jacobian’ or ‘Hessian’ matrices. For the rest of us, we just need to be able to apply these nice results to find out when (and if) we really are at the top (or bottom) of the hill (or depression). 1 ECON 1202/ECON 2291: QABE c School of Economics, UNSW Agenda 1. Extrema in two-variable functions; 2. First and second order criteria to find them; 3. Dealing with constraints; 4. Applying the technique....
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QABE_Notes21 - QABE Lecture 21 Multi-variable Optimisation...

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