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mathfacts - Some Simple Math Facts A Working with Changes...

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Some Simple Math Facts: A. Working with Changes in Linear Equations: As an example of a linear equation consider the simple consumption function: C = + mpc (Y - T), where the mpc is the marginal propensity to consume and is a constant in the consumption function. We can say that consumption is a linear function of, Y and T because these numbers enter the consumption function additively with no exponents. Many times we will wish to calculate the numerical values of C. For example suppose that Y = 1000, T = 150, = 50 and the mpc = .8. In this case, C(Y = 1000) = 50 + .8(1000 - 150) = 730, where the notation C(Y=1000) is meant to denote the value of consumption at a particular level of income, in contrast to the consumption function itself. Now suppose we wanted to calculate by how much C would change by if Y increased to 1200 and all the other variables remained unchanged. We could of course plug the new values of Y into our original equation; grind out the new value of C, and then subtract the old value from the new value. But watch what happens if we write out this arithmetic: C(Y =1200) = 50 + .8(1200 - 150) = 890 - C(Y = 1000) = - {50 + .8(1000 - 150)} = - 730. C = 0 + .8(1200 - 1000) = 160 where is the symbol that stands for "change." This arithmetic is unnecessarily tedious because 2 out of the 3 terms cancel. We can take advantage of this canceling by the following rule: Math Fact on Changes : In a linear relationship y = mx + b, the change in y (i.e., y) is the change in x times the coefficient on x (i.e., y = m x). In our numerical example of the consumption function the coefficient on Y is .8, and our numerical example follows the general rule since C = 160 = .8(200) = mpc Y. The general result in this case is so
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important I try to emphasize it by telling students that in 99% of the cases the change in consumption is equal to the mpc times the change in disposable income. My 99% rule works for all changes in C due to changes in Y or T, the one percent of the cases where it does not work is when C changes due to a change in ( although the math facts rule still works because the coefficient onis an implicit 1). In this case, C = It is possible to make use of this important tool in more complicated linear equations. For example, we will spend a great deal of time this semester studying factors that cause a change in national saving, S. National Saving, which is simply the sum of public and private saving can be written as: S S p + (T - G) = (Y - T) - C + (T - G); where S P is private saving We can substitute the consumption function for C to write a convenient equation for S: S = - + (1-mpc)(Y - T) + (T - G), or S = - + mps(Y - T) + (T - G), where mps marginal propensity to save = (1-mpc). National saving is a linear function of, Y, T and G because these numbers all enter the equation for S additively with no exponents. Now suppose that Y = 1000, G = 170, T = 150, = 50 and the mpc = .8. From this we can infer that: S(Y = 1000) = -50 + (1-.8)(1000 - 150) + (150-170) = -50 + .2(850) -20 = 100.
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