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  * NB: Sicnce $\succeq$ is a 'preference relation', it must satisfy the axioms of ''completeness'' and ''transitivity''.   * NB: Sice $\succeq$ is a 'preference relation', it must satisfy the axioms of ''completeness'' and ''transitivity''.
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   * Suppose $x_1,x_2 \in X$ and $x_1\succeq x_2$. Now if $x_3\succeq x_1$ then by transitivity: $x_3\succeq x_1 \succeq x_2 \Rightarrow x_3 \succeq x_2$ which implies that $\succeq \subset \succeq$.
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   * We can use the same argument as above.
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   * Suppose $x' \ni \succ(x)$ and $x' \ni \sim(x)$ then by completeness, $x' \preceq x$ but this means that $x' \ni \succeq(x)$. However, if either one of the preceeding conditions are satisfied, then by 1. and 2. above, $x' \in \succeq(x)$, and since these are all the preference possibilities, we conclude that $\succ \, \cup \, \sim \, = \, \succeq$.
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   * Suppose there does exist some $x' \in \succ(x)\, \cap\, \sim(x)$ then this would imply that both $x'\succ x$ and $x' \sim x$ which by transitivity would imply that $x' \succ x \sim x'$, that is, $x' \succ x'$ which is a contradiction.

Unit

ECC5650

Topics

Consumer Preference Theory, Utility Functions

Preference Relations

  1. [JR Ex 1.2] Let 

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    be a preference relation, prove the following:

    • NB: Sice 

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      is a 'preference relation', it must satisfy the axioms of completeness and transitivity. 

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  2. [JR Ex 1.6] Cite a credible example where the preferences of an "ordinary consumer" would be unlikely to satisfy the axiom of convexity.

Utility Functions

  1. [JR Ex 1.12] Suppose 

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    and 
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    are both utility functions.

    1. If both u and v are homogeneous of degree r. Prove that the function 

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      is also homogeneous of degree r.

    2. If both u and v are quasiconcave, then 

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      is also quasiconcave.
  2. [JR Ex 1.17] Suppose that preferences are convex but not strictly so. Give a clear and convincing argument that a solution to the consumer's problem still exists, but that it need not be unique. Illustrate your argument with a two-good example.

Indirect Utility Functions Consider the Indirect Utility Function version of the Utility Maximization Problem: 

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such that 

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. Prove the following two propositions:

  1. Prop I v(p,y) is homogeneous of degree 0 in (p,y).

  2. Prop II v(p,y) is increasing in y. (Hint: use the Envelope Theorem)

MonashU/ECC5650MicroTheory/ProblemSet02 (last edited 2009-04-06 05:30:36 by Supervisor2012)