Duality (i)

# Duality (i) - Economics 21: Intermediate Microeconomics...

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Economics 21: Intermediate Microeconomics Topic 2: Consumer Theory – Duality Theory (i) 1 Economics 21: Intermediate Microeconomics Topic 2: Consumer Theory Duality Theory (i) Reference: Varian, p.93 Outline: I. Introduction II. Cobb-Douglass Preferences I. Introduction We explore ‘duality’ in the context of consumer theory. As we have seen, there are three steps to consumer decision-making: (i) Preferences: i.e. what the individual wants to do. (ii) The Budget Constraint: i.e. what the individual is able do. (iii) Constrained Optimisation: i.e. how the individual reconciles (i) and (ii) by endeavouring to reach the highest feasible level of satisfaction. The consumer’s problem can be interpreted as either: (a) Maximising utility subject to an income constraint, yielding the Marshallian demand function; or (b) Minimising expenditure subject to a utility constraint, yielding the Hicksian (or Compensated ) demand function. Duality theory explores the ‘duality’ that exists between these two optimisation problems. It is best explained by exploring the two optimisation problems in detail. We will firstly analyse a general representation of utility and then [Duality Theory (ii)] explore a specific utility function (the Cobb-Douglas function). II. Marshallian Demand Functions The consumer’s problem is: ( ) 2 1 2 1 , , max x x u x x (1) subject to: m x p x p = + 2 2 1 1 (2) The solutions is derived by maximizing the following Lagrange function:

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Economics 21: Intermediate Microeconomics Topic 2: Consumer Theory – Duality Theory (i) 2 max x 1 , x 2 , ! L g = u x 1 , x 2 ( ) + m " p 1 x 1 " p 2 x 2 ( ) (3) The FOCs for maximization are given by: 0 1 1 1 = ! = p u x L x g " # (4) L g x 2 = u x 2 " p 2 = 0 (5) 0 = ! = px m L g "# (6) Dividing (4) by (5) yields the familiar result: 2 1 2 1 p p u u x x = (7) Equation (7) shows that in equilibrium, the consumer’s MRS between the two goods equals the ERS between the two goods. Equation (6) states that the individual must exhaust his entire budget. Thus equations (6) and (7) together imply the tangency between the consumer’s indifference curves and his budget constraint.
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## This note was uploaded on 07/16/2010 for the course ECON 21 taught by Professor Johng.sessions during the Summer '09 term at Dartmouth.

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Duality (i) - Economics 21: Intermediate Microeconomics...

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