optimization第二講-資料下載頁(yè)
2024-11-17 05:44本頁(yè)面
【正文】 ses we run into, we’ll have convex and concave functions. Note also that a utility function that is strict quasiconcave if and only if it yields indifference curves that are strictly convex.,30,第三十頁(yè)，共三十九頁(yè)。,Theorem 13.7,In a constrained maximization problem,If f is quasiconcave, all g’s are quasiconvex, then any locally optimal solution to the problem is also globally optimal. Thus, if these conditions are satisfied, solving the Lagrange yields the global optimum!,31,第三十一頁(yè)，共三十九頁(yè)。,Theorem 13.8: Uniqueness,In a constrained maximization problem,where f and all the g’s are increasing, then if f is strictly quasiconcave and the g’s are convex, or f is quasiconcave and the g’s are strict convex, then a locally optimal solution is unique and also globally optimal. Example: the consumer problem! Utility function is increasing and strictly quasiconcave, Budget constraint is increasing and convex. The theorem says that solving the FOCs yields a unique and global optimum.,32,第三十二頁(yè)，共三十九頁(yè)。,Another example,A firm produces some output using the following production function,This is a CES production function (constant elasticity of supply). It’s general form is:,33,第三十三頁(yè)，共三十九頁(yè)。,Another example,A firm produces some output using the following production function,Prices of inputs:,Question: What are the minimal costs of producing 1 unit of output?,34,第三十四頁(yè)，共三十九頁(yè)。,Of course, second order conditions also have to be checked.,35,第三十五頁(yè)，共三十九頁(yè)。,Example,A representative student spends 60 hours per week studying. She takes two subjects. Her objective: allocate time between the two subjects such that the average grade is maximized. Subjects differ with respect to their production function.,Objective function:,36,第三十六頁(yè)，共三十九頁(yè)。,Thus,We have that f is concave and g is convex. So this is a maximum. Spending one hour more studying leads to an increase in grade average of 1.5.,37,第三十七頁(yè)，共三十九頁(yè)。,This week’s exercises,pg. 615: 1, 3, 5, 7. pg. 622: 1.,38,第三十八頁(yè)，共三十九頁(yè)。,內(nèi)容總結(jié),Optimization Lecture 2。Let’s maximize this:。This week’s exercises。pg. 622: 1.。38,第三十九頁(yè)，共三十
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