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Unformatted text preview: Econ 509, Introduction to Mathematical Economics I Professor Ariell Reshef University of Virginia Lecture notes based on Chiang and Wainwright, Fundamental Methods of Mathematical Economics . 1 Mathematical economics Why describe the world with mathematical models, rather than use verbal theory and logic? After all, this was the state of economics until not too long ago (say, 1950s). 1. Math is a concise, parsimonious language, so we can describe a lot using fewer words. 2. Math contains many tools and theorems that help making general statements . 3. Math forces us to explicitly state all assumptions , and help preventing us from failing to acknowl- edge implicit assumptions. 4. Multi dimensionality is easily described. Math has become a common language for most economists. It facilitates communication between econo- mists. Warning: despite its usefulness, if math is the only language for economists, then we are restricting not only communication among us, but more importantly we are restricting our understanding of the world. Mathematical models make strong assumptions and use theorems to deliver insightful conclusions. But, remember the A-A&C-C&Theorem: & Let C be the set of conclusions that follow from the set of assumptions A. Let A&be a small perturbation of A. There exists such A& that delivers a set of conclusions C& that is disjoint from C. Thus, the insightfullness of C depends critically on the plausibility of A. The plausibility of A depends on empirical validity, which needs to be established, usually using econo- metrics. On the other hand, sometimes theory informs us on how to look at existing data, how to collect new data, and which tools to use in its analysis. Thus, there is a constant discourse between theory and empirics. Neither can be without the other (see the inductivism v deductivism debate). Theory is an abstraction of the world. You focus on the most important relationships that you consider important a priori to understanding some phenomenon. This may yield an economic model. 2 Economic models Some useful notation: 8 for all, 9 exists, 9 ! exists and is unique. If we cross any of these, or pre&x by : or , then it means "not". 1 2.1 Ingredients of mathematical models 1. Equations: De&nitions : & = R & C : Y = C + I + G + E & M : K t +1 = (1 & ) K t + I t Behavioral/Optimization : q d = & p : MC = MR : MC = P Equilibrium : q d = q s 2. Parameters: e.g. , , from above. 3. Variables: exogenous, endogenous. Parameters and functional forms govern the equations, which determine the relationships between variable. Thus, any complete mathematical model can be written as F ( ;Y;X ) = 0 ; where F is a set of functions (say, demand and supply), is a set of parameters (say, elasticities), Y are endogenous variables (price and quantity) and X are exogenous, predetermined variables (income, weather)....
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