19. Profit Maximisation - Solutions

19. Profit Maximisation - Solutions - Chapter 19 NAME Prot...

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Chapter 19 NAME Proft Maximization Introduction. A frm in a competitive industry cannot charge more than the market price For its output. IF it also must compete For its inputs, then it has to pay the market price For inputs as well. Suppose that a proft- maximizing competitive frm can vary the amount oF only one Factor and that the marginal product oF this Factor decreases as its quantity increases. Then the frm will maximize its profts by hiring enough oF the variable Factor so that the value oF its marginal product is equal to the wage. Even iF a frm uses several Factors, only some oF them may be variable in the short run. Example: A frm has the production Function f ( x 1 ,x 2 )= x 1 / 2 1 x 1 / 2 2 . Sup- pose that this frm is using 16 units oF Factor 2 and is unable to vary this quantity in the short run. In the short run, the only thing that is leFt For the frm to choose is the amount oF Factor 1. Let the price oF the frm’s output be p , and let the price it pays per unit oF Factor 1 be w 1 .W e want to fnd the amount oF x 1 that the frm will use and the amount oF output it will produce. Since the amount oF Factor 2 used in the short run must be 16, we have output equal to f ( x 1 , 16) = 4 x 1 / 2 1 .T h em a r g i n a l product oF x 1 is calculated by taking the derivative oF output with respect to x 1 . This marginal product is equal to 2 x 1 / 2 1 . Setting the value oF the marginal product oF Factor 1 equal to its wage, we have p 2 x 1 / 2 1 = w 1 . Now we can solve this For x 1 . We fnd x 1 =(2 p/w 1 ) 2 . Plugging this into the production Function, we see that the frm will choose to produce 4 x 1 / 2 1 =8 p/w 1 units oF output. In the long run, a frm is able to vary all oF its inputs. Consider the case oF a competitive frm that uses two inputs. Then iF the frm is maximizing its profts, it must be that the value oF the marginal product oF each oF the two Factors is equal to its wage. This gives two equations in the two unknown Factor quantities. IF there are decreasing returns to scale, these two equations are enough to determine the two Factor quantities. IF there are constant returns to scale, it turns out that these two equations are only sufficient to determine the ratio in which the Factors are used. In the problems on the weak axiom oF proft maximization, you are asked to determine whether the observed behavior oF frms is consistent with proft-maximizing behavior. To do this you will need to plot some oF the frm’s isoproft lines. An isoproft line relates all oF the input-output combinations that yield the same amount oF proft For some given input and output prices. To get the equation For an isoproft line, just write down an equation For the frm’s profts at the given input and output prices. Then solve it For the amount oF output produced as a Function oF the amount oF the input chosen. Graphically, you know that a frm’s behavior is consistent with proft maximization iF its input-output choice in each period lies below the isoproft lines oF the other periods.
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244 PROFIT MAXIMIZATION (Ch. 19) 19.1 (0)
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This note was uploaded on 07/16/2010 for the course ECON 21 taught by Professor Johng.sessions during the Summer '09 term at Dartmouth.

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19. Profit Maximisation - Solutions - Chapter 19 NAME Prot...

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