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Suppose there are two goods, X and Y. A consumer will choose a
vector (x,y) in R^2 that is feasible (non-negative consumption
levels) and which she can afford, given income I and prices Px and
Py. Assume I, Px, and Py are all positive. So the domain for her
utility maximization problem is the set D of all vectors (x,y)
which is an element of R^2 satisfying x>=0, y>=0, and
PxX+PyY<=I.

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Question: Suppose there are two goods, X and Y. A consumer will choose a vector (x,y) in R^2 that is feasib…
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Show this set D of vectors in R^2 is compact, i.e., both bounded
and closed.

there are 3 hints:

1) All linear functions are continuous

2) The intersection of closed sets are closed

3) If g: X subset R^n projected on to R is continuous on closed
X, then for any c in R, both of the sets x element of X such that
g(x) >=c and x element of X such that g(x) <=c are
closed.

there is another hint saying if that doesnt work, use the
definition: X is closed if and only if its complement (in R^2) is
open.

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