ECON201MathReview

ECON201MathReview - Wellesley College Economics 201...

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Wellesley College Economics 201 Mathematical Concepts and Methods for Economics The major mathematical concept that will interest us in Economics 201 is the optimum of a function. (Optima are either maxima or minima.) Most individual decisions in economics involve either maximization (utility, profits) or minimization (cost, expenditure) and these decisions also often involve constraints that must be taken into consideration (certain amount of income to spend, cost levels). 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 an equilibrium in economics is finding the price and quantity at which supply equals demand. Optimization: Functions of One Variable Start with a function like y = f(x) or A = f(Q). The derivative, which will be denoted as dy/dx or d A /dQ or f N (x), measures the slope of the function at a certain point (i.e. for a particular value of x or Q). When d A /dQ > 0, A is an increasing function of Q; A moves in the same direction as Q. When d A /dQ < 0, A is a decreasing function of Q; A moves in the opposite direction from Q. When d A /dQ = 0, the function has a stationary point (can be a maximum or a minimum or neither.) This last condition is the first order condition for an optimum for a function of one variable. EXAMPLE: A = A (Q) = -2 Q 2 + 24 Q + 5 d A /dQ = -4 Q + 24 so function is increasing when - 4 Q + 24 > 0 or when Q < 6. The function is decreasing for - 4 Q + 24 < 0 or when Q > 6. The function has a stationary point when - 4 Q + 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. The second derivative is denoted: d 2 A / dQ 2 or d/dQ (d A /dQ) or A O (Q). When A O (Q) > 0, then A N (Q) is moving in the same direction as Q (the slope of the function is increasing as Q increases). When A O (Q) < 0, then A N (Q) is moving in the opposite direction from Q (the slope is decreasing as Q increases). When A O (Q) = 0, then A N (Q) is switching from negative to positive (or positive to negative) as you increase Q and there is an inflexion point in A (Q).

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The second order condition for determining whether you have a maximum or a minimum consists of checking the sign of the second derivative of the function at that value for which you found the stationary point. If f O < 0, the stationary point is a maximum. If f O > 0, it is a minimum. (Functions which have maxima look like hills; they are concave functions. Functions which have minima look like valleys; they are convex functions.) If f O = 0, the stationary point is an inflexion point. EXAMPLE:
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ECON201MathReview - Wellesley College Economics 201...

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