Chap. 17 Optimization with constrained conditions...
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1 Cha. 7 Otimization with constrained conditions Under thecondition k ma f min f Bdet constraint... nn I I Constraint of a rodctionfnction q f... f... q n n
2 lower contor set Uer Contor set
3 3 Constraint set constraint line } { k Continosly differentiable Partial derivatives are ositive
4 4 Analysis by the imlicit fnction theorem there eists By theimlicit fnction theorem h k h h h at differentiate Totally d d k d d
5 Gradient vector of the constraint crve k k k qasi - conve 5
6 Gradient vector of the objective fnction qasi - concave 6
7 The tanent condition of the otimm soltion * * * constraint k * 7
8 Method of aranean mltilier ma [min ] sbject to k : strictly qasi -concave fnction d is weakly monotone increasin d at k 8
9 9 arane mltilier and arane fnction aranian k mltilier : arane arane fnction aranian The st order conditions
10 First order conditions and tanency conditions k k k
11 Second order conditions Conditions for maimm: is strictly qasi-concave * * constraint : k k qasi - conve
12 Second order conditions Conditions for minimm: is strictly qasi-conve Constraint: k k qasi-concave * * k
13 Second order conditions Conditions for maimm: is strictly qasi-concave Constraint: k k qasi-conve * * k 3
14 Second order conditions Conditions for maimm: is strictly qasi-concave * Constraint: k k qasi-concave * k 4
15 Second order conditions for maimization : strictly qasi-concave and : qasi-conve : strictly qasi-concave and : qasi-cancave 5
16 6 Second order conditions for maimization
17 Second order conditions for maimization 3 4 : qasi-concave and : strictly qasi-conve : qasi-concave and : strictly qasi-concave 7
18 8 Second order conditions for maimization
19 9 Second order conditions for maimization The sloe of s contor crve increases faster than that of
20 Smmary of the otimization conditions ] [min ma f f sb to f First order conditions
21 Smmary of Second order conditions for maimization f: strictly qasi-concave : linear f: strictly qasi-concave > : qasi-conve f: strictly qasi-concave < : qasiconcave f: qasi-concave > : strictly qasi-conve f: qasi-concave < : strictly qasiconcave
22 Smmary of Second order conditions for minimization f: strictly qasi-conve : linear f: strictly qasi-conve > : qasi-concave f: strictly qasi-conve < : qasi-conve f: qasi-conve > : strictly qasi-concave f: qasi-conve < : strictly qasi-conve
23 The maimm vale fnction and arane mltilier ma sbject to [min k ] varies accordin to the arameter k k k k 3
24 The maimm vale fnction and arane mltilier d d dk d d d d d d d dk d k dk 4
25 5 Utility-maimization roblem of a consmer sbject to I qasi-concave linear I I ma
26 6 The tility-maimization roblem of the consmer I I I I is sbstitted to the bdet constraint and solve
27 7 Utility-maimization roblem of a consmer I di d I I I
28 Utility-maimization roblem of a consmer s. t. I qasi-concave ma I conve set linear qasi-conve fnction 8
29 9 I I The arane's mltilier law Make arane fnction Since is a qasi-concave fnction and the bdet constraint is linear the second order conditions of maimization are met. Partially differentiate w.r.t. three variables
30 Cost minimization roblem of a rodcer conve set C K rk w linear qasi-conve min C K K s. t. F K Q Q F K strictly qasi-concave 3
31 The arane's mltilier law Make arane fnction K rk w [ Q F K ] K r F K K wf Q K F K Since the objective fnction is linear and F is strictly qasi-concave the second order conditions of minimization are met. 3
λ. It is usually positive; if it is zero then the constraint is not binding.
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