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solutions_Chapters%2019%20and%2020

# solutions_Chapters%2019%20and%2020 - Chapter 19 NAME Prot...

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Chapter 19 NAME Profit Maximization Introduction. A firm 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 profit- maximizing competitive firm can vary the amount of only one factor and that the marginal product of this factor decreases as its quantity increases. Then the firm will maximize its profits by hiring enough of the variable factor so that the value of its marginal product is equal to the wage. Even if a firm uses several factors, only some of them may be variable in the short run. Example: A firm has the production function f ( x 1 , x 2 ) = x 1 / 2 1 x 1 / 2 2 . Sup- pose that this firm 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 firm to choose is the amount of factor 1. Let the price of the firm’s output be p , and let the price it pays per unit of factor 1 be w 1 . We want to find the amount of x 1 that the firm 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 . The marginal 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 find x 1 = (2 p/w 1 ) 2 . Plugging this into the production function, we see that the firm will choose to produce 4 x 1 / 2 1 = 8 p/w 1 units of output. In the long run, a firm is able to vary all of its inputs. Consider the case of a competitive firm that uses two inputs. Then if the firm is maximizing its profits, 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 profit maximization, you are asked to determine whether the observed behavior of firms is consistent with profit-maximizing behavior. To do this you will need to plot some of the firm’s isoprofit lines. An isoprofit line relates all of the input-output combinations that yield the same amount of profit for some given input and output prices. To get the equation for an isoprofit line, just write down an equation for the firm’s profits 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 firm’s behavior is consistent with profit maximization if its input-output choice in each period lies below the isoprofit lines of the other periods.

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