L16Optimisation

L16Optimisation - Introduction to Microeconomics Lecture 16...

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Unformatted text preview: 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 The general approach: mathematical analysis of optimisation. Examples with functional forms: Consumer choice Cost minimisation Oligopoly Examples without functional forms: Profit maximisation Perfect and imperfect competition Optimisation Economics views agents (people, firms) as optimisers • Consumers maximise utility • Firms 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 ) 1.Differentiate it : f’ ( x ) 1.Set f’ ( x )=0 (1 st order condition) and check the sign of f’’ ( x ) (2 nd order condition) 4. Work out what f’ ( 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: If we substitute the budget constraint in for the problem becomes: 1 2 1 1 2 2 ln ln u a x b x p x p x m = + + = 1 1 1 1 2 max ln ln x m p x u a x b p - = + 2 x We need to find the maximum point of this function So differentiate it with respect to [the 2 nd order condition is fine] 1 1 1 2 2 1 1 1 1 1 2 ln ln m p x u a x b p p p du a b dx x m p x p - = + =-- 1 x Optimisation – E.g. Consumers Set the derivative equal to zero to identify the turning point: Remain calm and solve for This is the Marshallian demand function for 2 1 1 1 1 1 2 p p du a b dx x m p x p =- =- 1 x 1 1 a m x a b p = + 1 x Optimisation – E.g. Consumers To get the demand for we use the budget constraint (spending on each good sums to total income) Stay calm and solve for This is the Marshallian demand function for...
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L16Optimisation - Introduction to Microeconomics Lecture 16...

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