Mathematical Techniques in Economics
... x = -4 or x – 4 = 0 x = 4 KS 3.2, 3.3, 3.4 Q.4 Present Values KS 4.1, 4.2, 4.3 Q.5 Continuous Growth KS 2.6, 2.7, 4.4 Q.6 Differentiation Use the seven basic rules (KS 5.2) KS 5.1, 5.2 Q.7 Locating Stationary Points Use product rule(KS 5.2): Break down equation into the separate, simpler functions then apply KS 5.2 point 1. eg, y = x2(2x + 1)3 the two simpler functions are x2 and (2x + 1)3. Label on u and one v, doesn’t matter which. du/dx = 2x and dv/dx = 6(2x + 1)2 dy/dx = u(dv/dx) + v(du/dx) Remember to use the derivative (!), then factorise and set to 0. KS 5.1, 5.2, 5.3, 6.1 Q. 8 Deriving and Plotting MC and/or MR MC is a derivative of TC (as f(q)) MR is a derivative of TR (as f(q)) For a demand function, first invert the demand function to give AR then multiply by Q to give TR then derive! KS 5.4, 6.4, 6.5, 2.2 Additional Notes for KS 4 Geometric Progression A sequence of numbers with the following structure: a, ab, ab2, ab3 etc. where the ratio of each term and the next is a constant value. xi = abi-1 A firm decides to increase output at a constant rate from current level 50, 000 (‘a’ below) to 60, 000 (‘b’ below) over the next 5 years (‘t’ below). The rate of increase is r%: a(1 +(r/100))t = b 50, 000(1 + (r/100))5 this is the equation so 50, 000(1 + (r/100))5 = 60, 000 divide by 50, 000, (1 + (r/100))5 = 1.2 1 + (r/100) = (1.2)1/5 = 1.037 Þ 3.7% Future Earnings If you are to receive a sum of money in the future, you can borrow against that sum at a rate of interest/price of money. xt = y(1/1+r)t R = rate of interest Y = value of future earnings T = numbe...