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Lecture16 Optimisation

Lecture16 Optimisation - Introduction to Microeconomics...

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11/23/2007 1 Introduction to Microeconomics Lecture 16 Unconstrained Optimisation Ian Crawford Microeconomics is SIMPLE There are only two main guiding principles: Optimisation and Equilibrium So far we have been examining the behavioural implications of optimising behaviour for: - consumers - individual firms - industries Mathematics helps. Outline z The general approach: mathematical analysis of optimisation. z Examples with functional forms: z Consumer choice z Cost minimisation z Oligopoly z Examples without functional forms: z Profit maximisation z Perfect and imperfect competition Optimisation Economics views agents (people, firms) as optimisers • Consumers maximise utility •F irms maximise profits and/or minimise costs We know that we can represent consumers’ preferences and firms’ technology mathematically. Calculus then provides a powerful analytical technique for drawing out the implications of optimising behaviour. Optimisation The general approach is 1. Write down the objective function : f ( x ) 2. Differentiate it : f’ ( x ) 3. Set ( x )=0 (1 st order condition) and check the sign of f’’ ( x ) (2 nd order condition) 4. Work out what ( x )=0 implies about economic behaviour. 1 st order conditions tell you what has to hold if f ( x ) is being optimised. Optimisation – E.g. Consumers The consumer wants to maximise utility subject to being on the budget constraint: 12 ln ln uaxbx = + If we substitute the budget constraint in for the problem becomes: 11 2 2 p xp xm + = 1 1 2 max ln ln x mp x uaxb p ⎛⎞ =+ ⎜⎟ ⎝⎠ 2 x
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11/23/2007 2 We need to find the maximum point of this function 11 1 2 ln ln mp x uaxb p ⎛⎞ =+ ⎜⎟ ⎝⎠ Optimisation – E.g. Consumers So differentiate it with respect to [the 2 nd order condition is fine] 21 1 1 2 pp du a b dx x m p x p =− 1 x Set the derivative equal to zero to identify the turning point: 1 1 2 0 du a b dx x m p x p = −= Optimisation – E.g. Consumers Remain calm and solve for This is the Marshallian demand function for 1 x 1 1 am x abp = + 1 x To get the demand for we use the budget constraint (spending on each good sums to total income) 2 2 px m += 2 x Optimisation – E.g. Consumers Stay calm and solve for
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Lecture16 Optimisation - Introduction to Microeconomics...

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