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Unformatted text preview: Introduction to Microeconomics Lecture 3 Maths: Algebra Simon Cowan Outline Solving equations Linear and quadratic equations Solving simultaneous equations Linear demand and supply equations Indices and logarithms Solving equations: an example Lets suppose that a firm is the only supplier of operating systems and the demand for operating systems is given by q = A  Bp where q is the number of units bought, p is the price and A and B are positive but unspecified numbers. This holds whenever 0 p A / B . (Why?) If the quantity bought is 10 , what must be the price? This asks us to find p* in 10 = A  Bp*. Take Bp* to the lefthand side and subtract 10 from both sides to give Bp* = A  10 . Thus p * = ( A  10)/ B is the solution. We need to check that this satisfies the constraint on the price. (i) It is certainly true that p * < A / B because p * = A / B  10/ B . (ii) For p * we require that A 10 . A quadratic equation (i) Now the firm is threatened with a competition case for abusing its dominant position. Suppose that there is a maximum level of profit that the competition authority will allow, denoted by M . Assume this is nonnegative (otherwise the firm goes bankrupt). What are the feasible prices that the firm can set to ensure that profits equal M ? Suppose the firms costs consist entirely of Research and Development (R&D) costs which are fixed and thus independent of output. Denote the fixed cost by F > 0. Quadratic equation (ii) The firms profit is Revenue minus Cost = Price x Quantity  Cost = p ( A  Bp )  F The feasible prices must satisfy p ( A  Bp )  F = M Rearranging yields an equation which is quadratic in p : Bp 2  Ap + F + M = 0. Quadratic equation (iii)...
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 Spring '10
 Vines
 Economics, Microeconomics

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