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Lecture 5

Lecture 5.pdf

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Leslie Shiell

ECO 3145 Mathematical Economics I Lecture 5 Static Optimization Equality ConstraintsChiang ch 12SummaryI Structure of the problem II Necessary conditions for an optimum the method of Lagrange III Sufficient conditions for a local optimum IV Sufficient conditions for a global optimum V Comparative Statics VI Envelope theoremI Structure of the Problem1 structure of the problem xxf objective functionn 1cxxg constraintswhere c is some scalar constant n 1 the problem an economic agent seeks to choose a combination of the xs that maximizes the value of f subject to the condition that the constraint is also satisfiedvocabularythe xs are choice variablesc is a parameter a constant value that is outside the control of the economic agent who chooses the xs 2 example utility maximization consumer seeks to maximize utility subject to a budget constraint xx n goodsn 1 xxf utility functionn 1n budget constraintwhere p denotes the price of good i and M is the M x pii i1 iconsumers incomechoice variables the xs s p parameters theand M i31455doc Shiell 1II Necessary Conditions for an Optimum the Method of Lagrange1 substitution method in very simple problems one can use the substitution method of simultaneous equations to turn a constrained optimisation into an unconstrained optimisationexample a discriminating monopolist Simon et Blume p64950 there are two distinct markets represented 2 1 i PQ let represent the quantity and sale price in market i i i QG Pthe inverse demand function in market i is given by i i i the monopolist has only one factory therefore the cost of production depends upon Q Q Q Q Q Ctotal outputand the cost function is represented as2 12 1 the monopolists profit function is Q Q C Q P Q PP P QQ2 12 2 1 1 2 1 2 1 the goal of the monopolist is to maximize its profit given the constraint of the inverse demand function by substituting the inverse demand function into the profit function the constrained maximization problem is transformed into an unconstrained problem Q QC QQG QQGQQ 2 12 2 2 1 1 12 1 however most problems are too complicated for this approachtherefore another approach is needed2 failure of unconstrained necessary conditions letrepresent an optimumxx xn1 recall from Lecture 1 the firstorder necessary conditions for an unconstrained optimum the partial derivatives of the objective must all equal zero ief o r a l l i 1 0xx fn1 ithis condition may also be satisfied at a constrained optimum but in general it is not for two reasonspoints which satisfy 1 may not satisfy the constraint iec xxgn1 a particular objective function may be such that there does not exist any point which 1 satisfies 1 therefore 1 cannot be used to find a constrained optimum 1 For example it is standard in consumer theory to assume that more is always better ieregardless of the 0 fichoice of the xs31455doc Shiell 2
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