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||'''Unit'''|| ECC5650 ||
||'''Topics'''|| Consumer Preference Theory, Utility Functions ||

'''Preference Relations'''
 1. [JR Ex 1.2] Let $\succeq$ be a preference relation, prove the following:
  * NB: Sice $\succeq$ is a 'preference relation', it must satisfy the axioms of ''completeness'' and ''transitivity''.
  1. $\succeq \, \subset \, \succeq$
  2. $\sim \, \subset \, \succeq$
  3. $\succ \, \cup \, \sim \, = \, \succeq$
  4. $\succ \, \cap \, \sim \, = \, \emptyset$
 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 $u(x_1,x_2)$ and $v(x_1,x_2)$ are both utility functions.
  1. If both ''u'' and ''v'' are homogeneous of degree ''r''. Prove that the function $$s(x_1,x_2) = u(x_1,x_2) + v(x_1,x_2)$$ is also homogeneous of degree ''r''.
  2. If both ''u'' and ''v'' are quasiconcave, then $$m(x_1,x_2) \equiv \min \left\{u(x_1,x_2),v(x_1,x_2)\right\}$$ 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: $$v(p,y) = \max u(x)$$ such that $$ p\cdot x = y$$. 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)''