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Lecture10-2010

# Lecture10-2010 - Expected Utility Lecture X Charles B Moss...

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Expected Utility: Lecture X Charles B. Moss September 10, 2010 I. Basic Utility A. A typical economic axiom is that economic agents (consumers, producers, etc.) behave in a way that maximizes their expected utility. The typical formulation is max x 1 ,x 2 U ( x 1 , x 2 ) s . t .p 1 x 1 + p 2 x 2 Y (1) x 1 and x 2 are consumption goods and Y is monetary income. In decision making under risk, we are typically interested in the utility of income U ( Y ). How do these concepts relate? B. The linkage between these two concepts is the indirect utility function which posits optimizing behavior by the economic agent. Specifically, assuming an Cobb-Douglas utility function the gen- eral utility maximization problem can be rewritten as max x 1 ,x 2 x α 1 x β 2 s . t .p 1 x 1 + p 2 x 2 Y (2) Due to the concavity of the utility function, the inequality can be replaced with an equality. The maximization problem can then be reformulated as a Lagrangian L = x α 1 x β 2 + λ ( Y - p 1 x 1 - p 2 x 2 ) (3) The first order conditions are then 1

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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture X Fall 2010 L x 1 = α x α 1 x β 2 x 1 - λ p 1 = 0 L x 2 = β x α 1 x β 2 x 2 - λ p 2 = 0 L ∂λ = Y - p 1 x 1 - p 2 x 2 = 0 (4) Taking the ratio of the first two first order conditions yields x 2 = β α x 1 p 1 p 2 (5)
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Lecture10-2010 - Expected Utility Lecture X Charles B Moss...

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