solutions_Chapter 18

solutions_Chapter 18 - Chapter 18 NAME Technology...

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Chapter 18 NAME Technology Introduction. In this chapter you work with production functions, re- lating output of a firm to the inputs it uses. This theory will look familiar to you, because it closely parallels the theory of utility functions. In utility theory, an indifference curve is a locus of commodity bundles, all of which give a consumer the same utility. In production theory, an isoquant is a lo- cus of input combinations, all of which give the same output. In consumer theory, you found that the slope of an indifference curve at the bundle ( x 1 , x 2 ) is the ratio of marginal utilities, MU 1 ( x 1 , x 2 ) /MU 2 ( x 1 , x 2 ). In production theory, the slope of an isoquant at the input combination ( x 1 , x 2 ) is the ratio of the marginal products, MP 1 ( x 1 , x 2 ) /MP 2 ( x 1 , x 2 ). Most of the functions that we gave as examples of utility functions can also be used as examples of production functions. There is one important difference between production functions and utility functions. Remember that utility functions were only “unique up to monotonic transformations.” In contrast, two different production func- tions that are monotonic transformations of each other describe different technologies. Example: If the utility function U ( x 1 , x 2 ) = x 1 + x 2 represents a person’s preferences, then so would the utility function U ( x 1 , x 2 ) = ( x 1 + x 2 ) 2 . A person who had the utility function U ( x 1 , x 2 ) would have the same indifference curves as a person with the utility function U ( x 1 , x 2 ) and would make the same choices from every budget. But suppose that one firm has the production function f ( x 1 , x 2 ) = x 1 + x 2 , and another has the production function f ( x 1 , x 2 ) = ( x 1 + x 2 ) 2 . It is true that the two firms will have the same isoquants, but they certainly do not have the same technology. If both firms have the input combination ( x 1 , x 2 ) = (1 , 1), then the first firm will have an output of 2 and the second firm will have an output of 4. Now we investigate “returns to scale.” Here we are concerned with the change in output if the amount of every input is multiplied by a number t > 1. If multiplying inputs by t multiplies output by more than t , then there are increasing returns to scale. If output is multiplied by exactly t , there are constant returns to scale. If output is multiplied by less than t , then there are decreasing returns to scale. Example: Consider the production function f ( x 1 , x 2 ) = x 1 / 2 1 x 3 / 4 2 . If we multiply the amount of each input by t , then output will be f ( tx 1 , tx 2 ) = ( tx 1 ) 1 / 2 ( tx 2 ) 3 / 4 . To compare f ( tx 1 , tx 2 ) to f ( x 1 , x 2 ), factor out the expressions involving t from the last equation. You get f ( tx 1 , tx 2 ) = t 5 / 4 x 1 / 2 1 x 3 / 4 2 = t 5 / 4 f ( x 1 , x 2 ). Therefore when you multiply the amounts of all inputs by t , you multiply the amount of output by t 5 / 4 . This means there are increasing returns to scale.
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232 TECHNOLOGY (Ch. 18) Example: Let the production function be f ( x 1 , x 2 ) = min { x 1 , x 2 } . Then f ( tx 1 , tx 2 ) = min { tx 1 , tx 2 } = min t { x 1 , x 2 } = t min { x 1 , x 2 } = tf ( x 1 , x 2 ) .
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