optimization第四講-在線瀏覽
2024-11-17 05:44本頁(yè)面
【正文】 properties: f ’(x*) = 0 x* = 0 This implies that the following must hold: f ’(x*) ≥ 0 and x*f ’(x*) = 0 Thus: For a maximum: f ’(x*) ≤ 0 and x*f ’(x*) = 0 For a minimum: f ’(x*) ≥ 0 and x*f ’(x*) = 0,12,第十二頁(yè)，共二十九頁(yè)。,Theorem 15.1,If f and g are concave and differentiable, and if Slater’s condition (there exists a point where g 0) is satisfied, then being able to solve this is necessary and sufficient for a solution to the problem! Thus: just find values of lambda and the x’s such that all the conditions are satisfied and you’re done! The book gives a formal proof for this result.,14,第十四頁(yè)，共二十九頁(yè)。,Example,A consumer wants to maximize utility U(x1,x2) = x1189。 subject to budget constraint 2x1 + 3x2 ≤ 10. In this case: f(x1,x2) = x1189。 and g(x1,x2) = 10 – 2x1 – 3x2.,16,第十六頁(yè)，共二十九頁(yè)。,Another example (from last year’s exam!),A consumer maximizes utility given by,Income is 60, both prices are 6. Yet, the consumer cannot carry more than 8 units. Thus, the problem is:,18,第十八頁(yè)，共二十九頁(yè)。,20,第二十頁(yè)，共二十九頁(yè)。,A graph shows what’s going on,Carrying constraint,Budget constraint,The budget constraint is not binding. Note that
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