# Math - Mathematical Concepts and Methods for Intermediate...

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Mathematical Concepts and Methods for Intermediate Microeconomics Some of the more basic mathematical concepts you will need to review for this class are the slope and intercept formula for a linear function. How do changes in the formula affect the shape of the line. Is a slope of –2 steeper or flatter than a slope of –5? What is the slope of a horizontal line (zero) and a vertical line (infinity). We will also consider the concept of equilibrium, which involves the solution of a number of equations simultaneously (i.e. finding values for several variables which solve several equations at once). The most obvious example of solving for equilibrium in economics is finding the price and quantity at which supply equals demand. You will need to be comfortable with basic differentiation rules and procedures. We will use derivatives to talk about increasing or decreasing functions and for optimizing (finding the minimum or maximum of a function using the first order and second order conditions). What follows below is a review of important concepts on optimization. Optimization: The major mathematical concept that we will use the most in Intermediate Microeconomic Theory is the optimum of a function. Optima are either maxima or minima. Most individual household of firm decisions in economics involve maximization (of utility, profits) or minimization (of expenditure, cost) and these decisions also often involve constraints that need to be taken into consideration (certain amount of income to spend, cost levels, etc.). Functions with One Variable Start with a function like y=f (x), or Π = f (Q) . The derivative, which will be denoted as dy /dx or d Π/dQ or f′ (x), measures the slope of the function at a certain point (i.e. for a particular value of x or Q ). When dΠ/dQ>0, Π is an increasing function of Q, Π moves in the same direction with Q. When dΠ/dQ<0, Π is a decreasing function of Q, Π moves in opposite direction from Q. When dΠ/dQ=0 , the function has a stationary point (either maximum or minimum). Example: Π = Π (Q)= -2Q 2 + 24Q +5 dΠ/dQ= -4Q +24 , so this function is increasing for -4Q+24>0 or when Q<6. The function is decreasing for -4Q +24 <0 or when Q >6. The function has a stationary point at -4Q +24= 0 or when Q = 6. Stationary Points: Maxima or Minima? How can you tell whether a stationary point is a maximum or a minimum of your function? You need to look at how the derivative is changing at the stationary point. You need to look at the derivative of the derivative, or the second derivative of the function.

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The second derivative is denoted: 2 2 dQ d Π or dQ d dQ d Π or Π″(Q). When Π″(Q) >0 , then Π′(Q) is moving in the same direction as Q (the slope of the function is increasing as Q increases). When
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## This note was uploaded on 08/08/2008 for the course ECON 301 taught by Professor Hansen during the Spring '08 term at University of Wisconsin.

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Math - Mathematical Concepts and Methods for Intermediate...

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