Varian_Chapter18_Technology

Varian_Chapter18_Technology - Chapter Eighteen Technology...

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Chapter Eighteen Technology
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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.
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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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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).
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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 xx n = (, , ) 1 L
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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
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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 . yf x x n (, , ) 1 L
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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.
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Technology Sets The technology set is Tx x y y f x x a n d xx nn n = ≥≥ {( , , , ) | (, , ) ,, } . 11 1 00 LL K
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Technology Sets x’ x Input Level Output Level y’ One input, one output y” The technology set
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Technology Sets x’ x Input Level Output Level y’ One input, one output y” The technology set Technically inefficient plans Technically efficient plans
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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 yf x x x x == (, ) . 12 1 1/3 2 2
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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 yx x == × × = × × = 22 1 8 2 1 2 4 1 1/3 2 . x × × = × × = 8 8 2 2 2 8 1 2 .
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Technologies with Multiple Inputs Output, y x 1 x 2 (8,1) (8,8)
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Technologies with Multiple Inputs ± The y output unit isoquant is the set of all input bundles that yield at most the same output level y.
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Isoquants with Two Variable Inputs y ≡ 8 y ≡ 4 x 1 x 2
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Isoquants with Two Variable Inputs ± Isoquants can be graphed by adding an output level axis and displaying each isoquant at the height of the isoquant’s output level.
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Isoquants with Two Variable Inputs Output, y x 1 x 2 y ≡ 8 y ≡ 4
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Isoquants with Two Variable Inputs ± More isoquants tell us more about the technology.
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Isoquants with Two Variable Inputs y ≡ 8 y ≡ 4 x 1 x 2 y ≡ 6 y ≡ 2
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Isoquants with Two Variable Inputs Output, y x 1 x 2 y ≡ 8 y ≡ 4 y ≡ 6 y ≡ 2
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Varian_Chapter18_Technology - Chapter Eighteen Technology...

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