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ECON203 Microeconomic Analysis 2010 Lecture 2: Optimisation and demand (Perloff chs. 3&4) 1 Lecture 2: Optimisation and demand
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ECON203 Microeconomic Analysis 2010 Lecture 2: Optimisation and demand (Perloff chs. 3&4) 2 Utility functions and indifference curves ) q , q ( u 2 1 1 u 0 u 1 q 2 q 1 2 1 lu ) q ( f q = 0 2 1 lu ) q ( f q = A utility function with two goods needs a three dimensional representation. The indifference curves are the level sets (contours) of the utility function. They are shown in two dimensions. They show the combinations of q 1 and q 2 that yield a fixed level of utility.
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ECON203 Microeconomic Analysis 2010 Lecture 2: Optimisation and demand (Perloff chs. 3&4) 3 Optimisation: Choice as utility maximization The process of choice can be represented as maximizing a utility function subject to the budget constraint * 1 q * q 2 2 1 21 U U MRS slope = = 21 21 MRT MRS choice optimal = = 2 1 21 p p MRT slope = = 1 q 2 q
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ECON203 Microeconomic Analysis 2010 Lecture 2: Optimisation and demand (Perloff chs. 3&4) 4 Optimisation: using tangency rule The condition for an optimum using the tangency rule is 2 1 21 q q p p MRS MU MU 2 1 = = for any specific differentiable utility function this rule can be used to find the optimum first obtain the expression for the 21 MRS and set it equal to the price ratio then substitute this expression into the budget constraint.
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ECON203 Microeconomic Analysis 2010 Lecture 2: Optimisation and demand (Perloff chs. 3&4) 5 An example of the tangency rule For example if the utility function is Cobb-Douglas, say 3 2 2 3 1 1 q q u = then: Suppose the prices are p 1 = 1 and p 2 = 2 Substitution of the prices into the tangency condition defines the equation of the ICC 1 2 1 2 q q 2 1 q 2 q = = The optimal choice is the intersection of this line q 1 =q 2 with the budget constraint. If income is 300 then the budget constraint is 300 q 2 q 1 2 1 = + So 300 q 2 q 1 1 1 = + and q 1 = q 2 = 100 21 2 1 1 2 3 1 2 3 1 1 3 2 2 3 2 1 21 MRT p p q 2 q q q 3 2 q q 3 1 MRS = = = = - -
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ECON203 Microeconomic Analysis 2010 Lecture 2: Optimisation and demand (Perloff chs. 3&4) 6 Graphical depiction of the solution 1 q 2 q 2 1 q q = 100 u = 100 100 150 300
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ECON203 Microeconomic Analysis 2010 Lecture 2: Optimisation and demand (Perloff chs. 3&4) 7 Optimisation: Maximising utility subject to the budget constraint Lagrange multiplier method General method to solve constrained optimisation problems: Example: maximize a Cobb-Douglas utility function subject to a budget constraint (Perloff pp 83-84) Y q p q p . t . s q q u max 2 2 1 1 3 2 2 3 1 1 = + = Set up the function ( 29 Y q p q p q q L 2 2 1 1 3 2 2 3 1 1 - + - = λ Partially differentiate this with respect to the two choice variables q 1 and q 2 and with respect to the new variable This new variable has the interpretation of the value to the objective function (utility) of a u problem is called the marginal utility of income.
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ECON203 Microeconomic Analysis 2010 Lecture 2: Optimisation and demand (Perloff chs. 3&4) 8 Partial differentiation gives three first order conditions.
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