562_lecture1_intro_models-differential_calculus

562_lecture1_intro_models-differential_calculus - SI 562...

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1 Page 1 SI 562 Lecture 1 1 Differential Calculus and Optimization Professor Yan Chen Fall 2009 AGENDA THIS WEEK Syllabus and Course Policies Introduction to Microeconomics Optimization 2 Basic concepts: profit, revenue, cost Differential Calculus and Optimization NEXT WEEK Consumer Theory Market Demand SYLLABUS & COURSE POLICIES Course Materials Lectures, Class Notes, Text Course Requirements Homework assignments 3 Final exam Grading Policy: median B+
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2 Page 2 Economics Economics: Optimal allocation of scarce resources Example: How many books should I purchase? Key: How much of other good things am I willing to give 4 up? Budget constraint implies tradeoffs. If the price of books rises, should I spend same smaller or larger total amount? Electronic books are developed. How much should I switch my reading from print-on-paper to electronic? Optimizing Behavior Optimizing behavior: solving problems by doing the best possible given scarcity and other constraints Best possible: optimization We will assume our goal or objective can be summarized (approximately) by a single valued function V(x) where 5 (approximately) by a single valued function, V(x) , where V = performance, value of outcome; x = choice variables, e.g., number of labor hours, etc. Subject to scarcity, other constraints I want to maximize my satisfaction in life, or utility, by purchasing food, clothing, books and CDs. I like more of each, but don’t have infinite budget, so I have to max V(x) s.t. p x = Y. Managerial Economics Optimal allocation problems facing organizational decision makers Focus on for profit firms Most jobs Most relevant for policy making 6 Most directly relevant for NFPs E.g., cost minimization, incentive contracts, purchasing from for-profits When not, learning how to solve for a for-profit shows you the method to solve for a NFP Problem solving: set prices? Set output?
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3 Page 3 Differential Calculus 7 Differentiation and Optimization Differential Calculus Y = f(x) Assume: f(x) is continuous and “smooth” Note:(1) Most real decisions are discrete (2) Many decisions are “continuous enough” (3) Calculus: easy Integer programming: hard 8 What is a derivative? or Intuitively: slope (“rate of change”) of a curve in the immediate vicinity of a point dx df(x) (x) f y x f(x) = ax + b then the derivative of f(x), f (x) = a a dx df(x) = y B D Y=f(x) x A C y 1 x 1 x 0 y 0 Δ Y Δ X What is Slope at C: Slope CA too flat (low) = Δ y y y 0 1 = 9 •Slope CA too flat (low) = •Slope CB better •Slope CD just right Derivative of f(x) is the limit of the ratio Δ Y/ Δ X as Δ X approaches zero Δ X x x 0 1 Δ x Δ y lim dx dy (x) f 0 Δ X =
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4 Page 4 Q Δ Q) (Q C(Q) Δ Q) C(Q lim dQ dC(Q) Q 2 20 C(Q) : Example 0 Δ Q 2 + + = + = Δ Q 2Q 20 ) Δ Q 2Q Δ Q 2(Q 20 lim Δ Q ) 2Q (20 Δ Q) 2(Q 20 lim 2 2 2 o Δ Q 2 2 o Δ Q + + + = + + + = 10 4Q Q 2 4Q lim Δ Q Q 2 4Q Δ Q lim 0 Δ Q 2 0 Δ Q = Δ + = Δ + = 0 Note: As we saw with numerical approximations, dc/dQ increases in Q.
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562_lecture1_intro_models-differential_calculus - SI 562...

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