4. Define the set inclusion partial ordering on the powersets of a set X. (a) Show...

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4. Define the set inclusion partial ordering on the powersets of a set X. (a) Show that budget correspondence B(p,m) = {x 0|p1x1 + p2x2 m} is monotone increasing in m under set inclusion for fixed p >> 0. (b) let the rational preferences of a consumer be represented by a utility function u(c) that is continuous on C = R. Consider the following Marshallian demand problem: for p >> 0, m > 0, define the value function v*(p,m) in v (p,m) max u(c) cB(p.m) where under our assumptions v*(p,m) is well-defined. First assume preferences are "strictly more is better". Show that under this assumption, v*(p,m) must be strictly increasing in m for each fixed p >> 0. (c) Now, say preferences are not even monotone (let alone strictly monotone as in part (c)). Show that v* (p,m) is "non-decreasing" in m, but not necessarily strictly increasing in m. 4. Define the set inclusion partial ordering on the powersets of a set X. (a) Show that budget correspondence B(p,m) = {x 0|p1x1 + p2x2 m} is monotone increasing in m under set inclusion for fixed p >> 0. (b) let the rational preferences of a consumer be represented by a utility function u(c) that is continuous on C = R. Consider the following Marshallian demand problem: for p >> 0, m > 0, define the value function v*(p,m) in v (p,m) max u(c) cB(p.m) where under our assumptions v*(p,m) is well-defined. First assume preferences are "strictly more is better". Show that under this assumption, v*(p,m) must be strictly increasing in m for each fixed p >> 0. (c) Now, say preferences are not even monotone (let alone strictly monotone as in part (c)). Show that v* (p,m) is "non-decreasing" in m, but not necessarily strictly increasing in m.