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Unformatted text preview: ue of the independent variable. 14 Y Curve Y* Tangent line 0 X* X For example, if the straight line Yline = f(X) and the curve Ycurve = g(X) are tangent to each other at the point of tangency (X*, Y*), then we have... Y*line = Y*curve = Y* f(X*) = g(X*) So, at each point on a curve, the slope of the curve at that point is defined as the slope of the tangent line at the point of tangency. Why do we care so much about slopes and tangencies in economics? We care because they are very closely related to various marginal measures that we find interesting and useful in economic analysis. 15 Curves / Functions Utility function Production function Total cost Total revenue Total profit Indifference curve Isoquant Differentiation Rules Slopes / Tangents Marginal utility Marginal product Marginal cost Marginal revenue Marginal profit Marginal rate of Substitution Marginal rate of Technical Substitution To calculate the slope of a curve we take the derivative of the function with respect to the independent variable. For example, Curves / Functions Y=a Y = aX + b Y = aX2 + bX + c Y = aX3 + bX2 + cX + d Y = Xn Slopes / Tangents Y / X = 0 (Constant) Y / X = a (Linear) Y / X = 2aX + b (Quadratic) Y / X = 3aX2 + 2bX + c (Cubic) Y / X = nXn-1 (Exponential) In addition, we can calculate the slope of a combination of two curves, f(X) and g(X). Curves / Functions Y = f(X) + g(X) Y = f(X) - g(X) Y = f(X) g(X) Y = f(X) / g(X) Slopes / Tangents Y / X = [f / X] + [g / X] Y / X = [f / X] - [g / X] Y / X = [f / X] g(X) + [g / X] f(X) Y / X = [f / X] g(X) - [g / X] f(X) [g(X)]2 (Addition) (Subtraction) (Multiplication) (Division) 16 Let's use each of these rules for the example of f(X) = 2X3 and g(X) = 7X2. Y = f(X) + g(X) Y / X = [f / X] + [g / X] 6X2 + 14X Y = f(X) - g(X) Y / X = [f / X] - [g / X] 6X2 - 14X Y = f(X) g(X) Y / X = [f / X] g(X) + [g / X] f(X) (6X2)( 7X2) + (14X)( 2X3) 42X4 + 28X4 70X4 Y / X = [f / X] g(X) - [g / X] f(X) [g(X)]2 (6X2)( 7X2) - (14X)( 2X3) ( 7X2)2 42X4 - 28X4 49X4 14X4 49X4 Envelopes A family of curves is a collection of curves which are related to each other by a common variable (parameter). A typical example is a family of short run cost curves, CSR each of which is associated with a plant siz...
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- Spring '10