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mathGraphical1-07-draft

mathGraphical1-07-draft - ARE211 Fall 2007 LECTURE#7 THU...

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Preliminary draft only: please check for final version ARE211, Fall 2007 LECTURE #7: THU, SEP 20, 2007 PRINT DATE: AUGUST 21, 2007 (GRAPHICAL1) Contents 2. Graphical Overview of Calculus and Optimization Theory 1 2.1. Necessary and Sufficient Conditions 1 2.2. Unconstrained Maximization: One Variable. 6 2.3. Convex sets, Concave and Convex Functions. 10 2.4. Strictly Concave and Convex Functions and Strict (Locally Unique) Maxima 13 2. Graphical Overview of Calculus and Optimization Theory This section will take about five-six lectures. The idea is partly to provide a purely graphical roadmap of much of the territory we are going to cover later, partly to introduce a number of hard conceptual issues (e.g., necessary and sufficient conditions) in a low-tech environment. Doesn’t mean that the material is necessarily easy. Just that I’m going to concentrate entirely on the pictures to the exclusion of (almost) all symbols. Later on, we’ll come back and redo all these concepts. Some duplication, but it will be useful. 2.1. Necessary and Sufficient Conditions Everybody takes a while to get the hang of these relationships. They will pop up throughout our calculus overview, so we’ll try to deal with them in an abstract context. 1
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2 LECTURE #7: THU, SEP 20, 2007 PRINT DATE: AUGUST 21, 2007 (GRAPHICAL1) Consider the following groups of people (1) Economists (2) World renowned experts in the economics of spinach production (3) Really clever people who know about agriculture. (4) Agricultural economists What can we say about these groups of people in terms of necessary and sufficient conditions? In particular, we’re interested in whether or not membership in one or other of the above groups is a necessary and/or sufficient condition for a person to be an ag economist. Mathematics is a bit more precise than this example, but it’s good to think about imprecise things. Think: does belonging to one group imply anything about belonging to another? (1) If you are an agric economist, then you pretty much have to be an economist (2) If you are an expert in the economics of spinach production, you pretty much have to be an agricultural economist (3) What about in the other direction? Obviously, there are a lot of economists who aren’t agricultural economists. And a lot of ag economists who know nothing about spinach. (4) What about cleverness? Do you have to be really clever to be an ag economist? So the answers are: (1) A necessary condition for a person to be an ag economist is that she’s an economist (2) A sufficient condition for a person to be an ag economist is that she’s a world renowned expert on the economics of spinach production. (3) Cleverness and knowledge of spinach production are neither necessary nor sufficient. The best way to think about necessary and sufficient conditions is in terms of set containment. In general notation:
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ARE211, Fall 2007 3 W = clever people Y = Economists X = Ag economists Z = World renowned spinachists W’ = clever with 7 AJAE’s on spinach Figure 1. Necessary and Sufficient Conditions for an Ag Economist.
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