Topic10 - Theorem of the Maximum Economists are often...

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1 Theorem of the Maximum Economists are often interested in characterizing the effect of some environmental parameter on the decision made by a maximizing agent. Consider the following problem, where we allow both the maximand and the feasible set to depend upon a vector of environmental parameters α . () {(,) | } x FM a x f x x X αα =∈ . Let the set of maximizing values be * X , that is, * () a r g | } x XM a x f x x X . We seek to characterize the properties of the set of maximizers * X if the maximand (, ) f x is continuous and the correspondence X is also continuous (that is, the correspondence X is both upper and lower hemi-continuous). Definition: Upper hemi-continuous correspondence The set valued mapping X is upper hemi-continuous at 0 if for any open neighborhood, V of 0 X there exists a δ neighborhood of 0 , 0 (, ) N , such that XV ,for all 0 ) N αδ Definition: Lower hemi-continuous correspondence The set valued mapping X is lower hemi-continuous at 0 if for any open set V that intersects 0 X , there exists a neighborhood of 0 , 0 ) N , such that X intersects V for all 0 ) N A mapping is continuous if it is both upper and lower hemi-continuous. We first consider the case when there is a unique maximize * x . Consider the diagram below.
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2 Consider the sequence 0 (1 ) {} t t t αα =− . As depicted, this sequence does not converge due to the discontinuity of * () x α at 0 . However the subsequence * {( ) } s x where 2 st = converges to 00 x and the subsequence * } s x where 21 = + converges to 0 x . Also, since X is continuous (and hence upper hemi-continuous) , there exists some sequence ˆ s x where s s xX such that *0 ˆ s xx . We will use these observations in the proof below. Proposition C.4-1: Theorem of the maximum I Define F = { ( , ) | 0, ( ) , } nm x Max f x x x X ≥∈ Α \\ where f is continuous. If (i) for each there is a unique * () a r g {(,) | 0 , , } x xM a x f x x x X = Α . and (ii) X is a compact-valued correspondence that is continuous at 0 , then * x is continuous at 0 . 0 * x x 0 x 00 x X
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3 Proof : Since () X α is compact, * x exists. If * x is discontinuous at 0 , there exists some 0 ε > and a sequence 0 {} t , such that ** 0 () () t xx αε > . Since X is bounded, the sequence * {( ) } t x has a convergent sub-sequence. That is, for some sub-sequence s x , *0 * 0 } ( ) s x →≠ . Since x is maximal, it follows that *0 0 00 (( ) , ) (, ) fx αα > Since X is continuous, there exists a sequence ˆ s x where ˆ s s xX such that ˆ s .
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Topic10 - Theorem of the Maximum Economists are often...

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