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Unformatted text preview: Microeconomics I  Lecture #5, March 17, 2009 5 Profit maximization, Supply We already described the technological possibilities now we analyze how the firm chooses the amount to produce so as to maximize its profits. Profits are defined as revenues minus cost. We assume that the firm faces fixed prices (is on a competitive market). Example: A firm is on a competitive market, i.e. takes price of the output as given. Production function is given by f ( x 1 ,x 2 ) = x 1 / 4 1 x 1 / 4 2 , prices of inputs are w 1 = 4, w 2 = 4 and price of output is p = 1. Costminimization approach: Consists of two stages: First, we find minimum cost for producing any given level of output y . Second, we find optimal value of output y . First stage: find minimum cost for arbitrary level of output y : min { x 1 ,x 2 } w 1 x 1 + w 2 x 2 min { x 1 ,x 2 } 4 x 1 + 4 x 2 such that x 1 / 4 1 x 1 / 4 2 = y x 2 = y 4 x 1 min x 1 4 x 1 + 4 y 4 x 1 FOC: 4 4 y 4 x 2 1 = 0 x 1 = y 2 and x 2 = y 2 So in this example, our cost function is: c ( y ) = 4 x 1 + 4 x 2 = 4 y 2 + 4 y 2 = 8 y 2 Second stage: find optimal level of output y : max y py c ( y ) max y y 8 y 2 FOC: 1 16 y = 0 y = 1 16 x 1 = x 2 = y 2 = 1 256 Profitmaximization approach: We maximize profit (revenues minus costs) of the firm. max { x 1 ,x 2 } py w 1 x 1 w 2 x 2 max { x 1 ,x 2 } 1 x 1 / 4 1 x 1 / 4 2 4 x 1 4 x 2 FOC [ x 1 ] : x 2 4( x 1 x 2 ) 3 / 4 4 = 0 FOC [ x 2 ] : x 1 4( x 1 x 2 ) 3 / 4 4 = 0 20 Solving these two equations with two unknowns gives: x 1 = x 2 = 1 256 Profit maximization Cost minimization. If a firm is maximizing profits and if it chooses to supply some output y , then it must be minimizing the cost of producing y . If this were not so, then there would be some cheaper way of producing y units of output, which would mean that the firm was not maximizing profits in the first place. This simple observation turns out to be quite useful in examining firm behavior. 5.1 Profit maximization in shortrun: In shortrun the amount of at least one inputs is fixed. In longrun all inputs can be changed. max x 1 pf ( x 1 , x 2 ) w 1 x 1 w 2 x 2 where: p price of output f ( x 1 , x 2 )  production function x 1 , x 2 inputs, x 2 is in shortrun fixed at the level x 2 w 1 ,w 2 prices of inputs x 1 ,x 2 For profit maximizing quantity the first order condition has to hold: p f ( x 1 , x 2 ) x 1 = w 1 or pMP 1 = w 1 In other words, the value of the marginal product of a factor should equal its price. In order to understand this rule, think about the decision to employ a little more of factor 1. As you add a little more of it, x 1 , you produce y = MP 1 x 1 more output that is worth pMP 1 x 1 ....
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 Spring '11
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 Microeconomics

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