Week7 Revised Handout (6pp)

Week7 Revised Handout (6pp) - 4/18/2011 Technology...

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4/18/2011 1 1 Technology Chapter 18 Technology Production functions Isoquants Marginal Product Returns to scale Technical rate of substitution Long run vs Short run Technology A technology decribes how inputs are converted to some output Lecture is being produced by using computer, projector, software, labour, etc… Production function: The maximum amount of output (y) one can produce using inputs x 1 , x 2 , x 3 ,…, x n Production Function 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 Production function with more more than one input Labour and Capital used for manufacturing production Labour and land for agricultural production How are different factors/inputs combined to produce output? Isoquant: Combination of different inputs that produce the same level of output Fixed-proportion technologies Y = Digging holes, X 1 = labourers, X 2 = shovels Y = Min{ X 1 , X 2 } X 1 = 1, X 2 = 1; Y = Min{ 1, 1 } = 1 X 1 = 2, X 2 = 1; Y = Min{ 2, 1 } = 1 X 1 = 1, X 2 = 2; Y = Min{ 1, 2 } = 1 These combinations all produce the same output: one unit. These inputs have be used in fixed proportions. Adding more of one without having more the other does not help.
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4/18/2011 2 7 p 1 x 1 * Fixed p 2 and y. p 1 = p 1 y/p 2 x 2 x 1 isoquants Fixed-proportion isoquants - proportion technologies Perfect substitutes technologies Y = Grading multiple choice exams, X 1 = labour, X 2 = machines Y = X 1 + X 2 X 1 = 1, X 2 = 1; Y = 2 X 1 = 0, X 2 = 2; Y = 2 X 1 = 2, X 2 = 0; Y = 2 These combinations all produce the same output: two units. 9 p 1 x 1 * Fixed p 2 and y. y/p 2 x 2 x 1 isoquants Perfect substitutes isoquants - proportion technologies 10 Cobb-Douglas technologies Let a = ½ , b = ½ and A = 1. Find input combinations such that y = 2. X 1 = 1, X 2 = 4; Y = 2 X 1 = 4, X 2 = 1; Y = 2 X 1 = 2, X 2 = 2; Y = 2 Note that if X 1 = 0 or X 2 = 0 then y = 0. Thus both inputs are essential. . 2 1 b a x Ax y 5 . 0 2 5 . 0 1 2 x x 11 p 1 x 1 * Fixed p 2 and y. y/p 2 isoquants Substitutes isoquants -proportion technologies x 2 x 1 y  x 2 ' x 1 ' 12 Marginal Product (MP) The MP of any input i is the rate of change of the output level as the level of input i changes. Production function: y = f(L,K) MP of Labour (MPL) is: MP of Capital (MPK) is: Properties (that typically hold): MP of L depends on K, and similarly MP of K depends on L Diminishing MP: MP of input i declines as the use of input i increases (i.e., its quantity increases) L K L f L y ) , ( K K L f K y ) , (
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4/18/2011 3 13 Computing MP MPL is Note: For Cobb-douglas technology MPL increases as K increases MPL diminishes as L increases: 3 / 2 3 / 1 ) , ( K L K L f y 3 / 2 3 / 2 3 1 ) , ( K L L K L f L y 0 9
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Week7 Revised Handout (6pp) - 4/18/2011 Technology...

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