# 18s - 18 2010 W W Norton Company Inc Technology...

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© 2010 W. W. Norton & Company, Inc. 18 Technology

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© 2010 W. W. Norton & Company, Inc. 2 Technologies A technology is a process by which inputs are converted to an output. E.g. labor, a computer, a projector, electricity, and software are being combined to produce this lecture.
© 2010 W. W. Norton & Company, Inc. 3 Technologies Usually several technologies will produce the same product -- a blackboard and chalk can be used instead of a computer and a projector. Which technology is “best”? How do we compare technologies?

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© 2010 W. W. Norton & Company, Inc. 4 Input Bundles x i denotes the amount used of input i; i.e. the level of input i. An input bundle is a vector of the input levels; (x 1 , x 2 , … , x n ). E.g. (x 1 , x 2 , x 3 ) = (6, 0, 9 × 3).
© 2010 W. W. Norton & Company, Inc. 5 Production Functions y denotes the output level. The technology‟s production function states the maximum amount of output possible from an input bundle. y f x x n ( , , ) 1

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© 2010 W. W. Norton & Company, Inc. 6 Production Functions y = f(x) is the production function. x‟ x Input Level Output Level y‟ y‟ = f(x‟) is the maximal output level obtainable from x‟ input units. One input, one output
© 2010 W. W. Norton & Company, Inc. 7 Technology Sets A production plan is an input bundle and an output level; (x 1 , … , x n , y). A production plan is feasible if The collection of all feasible production plans is the technology set. y f x x n ( , , ) 1

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© 2010 W. W. Norton & Company, Inc. 8 Technology Sets y = f(x) is the production function. x‟ x Input Level Output Level y‟ y” y‟ = f(x‟) is the maximal output level obtainable from x‟ input units. One input, one output y” = f(x‟) is an output level that is feasible from x‟ input units.
© 2010 W. W. Norton & Company, Inc. 9 Technology Sets The technology set is T x x y y f x x and x x n n n {( , , , ) | ( , , ) , , }. 1 1 1 0 0

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© 2010 W. W. Norton & Company, Inc. 10 Technology Sets x‟ x Input Level Output Level y‟ One input, one output y” The technology set Technically inefficient plans Technically efficient plans
© 2010 W. W. Norton & Company, Inc. 11 Technologies with Multiple Inputs What does a technology look like when there is more than one input? The two input case: Input levels are x 1 and x 2 . Output level is y. Suppose the production function is 1/3 1/3 1 2 1 2 ( , ) 2 . y f x x x x 

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© 2010 W. W. Norton & Company, Inc. 12 Technologies with Multiple Inputs E.g. the maximal output level possible from the input bundle (x 1 , x 2 ) = (1, 8) is And the maximal output level possible from (x 1 ,x 2 ) = (8,8) is 1/3 1/3 1/3 12 2 2 1 8 2 1 2 4. y x x       2 2 8 8 2 2 2 8. y x x      
© 2010 W. W. Norton & Company, Inc. 13 Technologies with Multiple Inputs Output, y x 1 x 2 (8,1) (8,8)

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© 2010 W. W. Norton & Company, Inc.
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18s - 18 2010 W W Norton Company Inc Technology...

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