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Douglas A. Ruby Macroeconomic Principles Macroeconomic Theory Microeconomic Theory |
Linear Equations A linear equation is defined by two parameters: the intercept term (or constant) b and the slope m: y = mx + bThe intercept defines the position of the line in the coordinate system and the slope provides direction. For example, a Demand curve is often modeled as a straight line. The equation might be: P = 40 - 8QIn the above diagram enter '40' for the vertical intercept and '-8' for the slope value and then press the "Plot Equation" button. The points of the line in the positive quadrant could be representative of this demand relationship. This line is downward sloping indicating that as the 'Y' value (Price) decreases, the 'X' value (Quantity demanded) increases. Note that the line passes through the vertical axis at a numerical value of 40. A Supply curve may be defined as: P = 10 + 4QJust enter '10' for the vertical intercept and '4' for the slope and press "Plot Equation". You will notice that in this example, the line is sloping upwards indicating that as the 'Y' value (Price) increases, the 'X' value (Quantity supplied) also increases. Quadratic Equations A quadratic equation is defined by three parameters: a -- the "squared" term, b -- the "linear" term, and c -- the "constant". A common example in economics would be a Utility function: U = 24Q - 3Q^{2}In this case, the constant term is equal to zero -- consistent with the notion that if the quantity of a particular good consumed is zero, Utility is also zero. Enter the value of '-3' for the a term and '24' for the b term and press "Plot Equation". Note that when the a term is negative, then the curve opens downward. Also since the constant term is equal to zero, the curve crosses the vertical axis at the origin. A Marginal Cost equation might provide a good example of a curve that opens up. Given the following equation: MC = 2Q^{2} - 8Q + 20Enter '2' for the a term, '-8' for the b term and '20' for the constant c term and press "Plot equation". Note that the curve crosses the vertical axis at a value of '20' and is "u-shaped" upwards. Third-degree Polynomials A Third-degree polynomial is defined by four parameters: a -- the "cubic" term, b -- the "squared" term, c -- the "linear" term and d -- the "constant". In this exercise, we will set the constant equal to zero thus forcing the equation through the origin. A common example in economics would be a Short-run Production Function that exhibits both increasing marginal productivity and ultimately diminishing marginal productivity of the variable input (i.e., labor): Q = -0.20L^{3} + 2L^{2} + 8LIn this case, the constant term is equal to zero -- consistent with the notion that if the quantity of labor is zero, Output is also zero (labor is an essential factor of production). Enter the value of '-0.20' for the a term, '2' for the b term, 8 for the c term and press "Plot Equation". |