18. Technology - Solutions

18 Technology- - Chapter 18 NAME Technology Introduction In this chapter you work with production functions relating output of a rm to the inputs

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Chapter 18 NAME Technology Introduction. In this chapter you work with production functions, re- lating output of a Frm 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 indiference 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 indi±erence curve at the bundle ( x 1 ,x 2 ) is the ratio of marginal utilities, MU 1 ( x 1 2 ) /M U 2 ( x 1 2 ). In production theory, the slope of an isoquant at the input combination ( x 1 2 ) is the ratio of the marginal products, MP 1 ( x 1 2 ) /M P 2 ( x 1 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 di±erence between production functions and utility functions. Remember that utility functions were only “unique up to monotonic transformations.” In contrast, two di±erent production func- tions that are monotonic transformations of each other describe di±erent technologies. Example: If the utility function U ( x 1 2 )= x 1 + x 2 represents a person’s preferences, then so would the utility function U ( x 1 2 )=( x 1 + x 2 ) 2 . A person who had the utility function U ( x 1 2 ) would have the same indi±erence curves as a person with the utility function U ( x 1 2 )and would make the same choices from every budget. But suppose that one Frm has the production function f ( x 1 2 x 1 + x 2 , and another has the production function f ( x 1 2 x 1 + x 2 ) 2 . It is true that the two Frms will have the same isoquants, but they certainly do not have the same technology. If both Frms have the input combination ( x 1 2 )=(1 , 1), then the Frst Frm will have an output of 2 and the second Frm 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 2 x 1 / 2 1 x 3 / 4 2 .I fw e 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 .T o c o m p a r e f ( tx 1 2 )t o f ( x 1 2 ), factor out the expressions involving t from the last equation. You get f ( tx 1 2 t 5 / 4 x 1 / 2 1 x 3 / 4 2 = t 5 / 4 f ( x 1 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 2 } .Then f ( tx 1 ,tx 2 { tx 1 2 } =min t { x 1 2 } = t min { x 1 2 } = tf ( x 1 2 ) .
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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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18 Technology- - Chapter 18 NAME Technology Introduction In this chapter you work with production functions relating output of a rm to the inputs

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