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Ebook Mathematics for economics and business (5th edition): Part 2

(BQ) Part 2 book "Mathematics for economics and business" has contents: Further optimization of economic functions, the derivative of the exponential and natural logarithm functions, functions of several variables, partial elasticity and marginal functions, comparative statics,.and other contents. | MFE_C04g.qxd 16/12/2005 11:17 Page 320 Find more at http://www.downloadslide.com section 4.7 Further optimization of economic functions Objectives At the end of this section you should be able to: Show that, at the point of maximum profit, marginal revenue equals marginal cost. Show that, at the point of maximum profit, the slope of the marginal revenue curve is less than that of marginal cost. Maximize profits of a firm with and without price discrimination in different markets. Show that, at the point of maximum average product of labour, average product of labour equals marginal product of labour. The previous section demonstrated how mathematics can be used to optimize particular economic functions. Those examples suggested two important results: (a) If a firm maximizes profit then MR = MC. (b) If a firm maximizes average product of labour then APL = MPL . Although these results were found to hold for all of the examples considered in Section 4.6, it does not necessarily follow that the results are always true. The aim of this section is to prove these assertions without reference to specific functions and hence to demonstrate their generality. Advice You may prefer to skip these proofs at a first reading and just concentrate on the worked example (and Practice Problems 1 and 8) on price discrimination. MFE_C04g.qxd 16/12/2005 11:17 Page 321 Find more at http://www.downloadslide.com 4.7 • Further optimization of economic functions Justification of result (a) turns out to be really quite easy. Profit, π, is defined to be the difference between total revenue, TR, and total cost, TC: that is, π = TR − TC To find the stationary points of π we differentiate with respect to Q and equate to zero: that is, dπ d(TR) d(TC) = − =0 dQ dQ dQ where we have used the difference rule to differentiate the right-hand side. In Section 4.3 we defined MR = d(TR) dQ and MC = d(TC) dQ so the above equation is equivalent to MR − MC = 0 and so MR = MC as required. The stationary .