Week 5_Technology (Print)

# But mp x x 1 1 13 2 2 3 2 3 diminishes as x 1

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But MP x x 1 1 1/3 2 2 3 2 3 = - / diminishes as x 1 increases

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Returns-to-Scale y x x x x a a = = 1 2 3 2 2 3 1 2 1 2 / / a a 1 2 4 3 1 + = so this technology exhibits increasing returns-to-scale. But MP x x 1 1 1/3 2 2 3 2 3 = - / diminishes as x 1 increases and MP x x 2 1 2 3 2 1/3 2 3 = - / diminishes as x 2 increases.
Returns-to-Scale So a technology can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Why?

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Returns-to-Scale A marginal product is the rate-of-change of output as one input level increases, holding all other input levels fixed. Marginal product diminishes because the other input levels are fixed, so the increasing input’s units have each less and less of other inputs to work with.
Returns-to-Scale When all input levels are increased proportionately, there need be no diminution of marginal products since each input will always have the same amount of other inputs to work with. Input productivities need not fall and so returns-to-scale can be constant or increasing.

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Technical Rate-of-Substitution At what rate can a firm substitute one input for another without changing its output level?
Technical Rate-of-Substitution x 2 x 1 y ≡ 100 x 2 ' x 1 '

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Technical Rate-of-Substitution x 2 x 1 y ≡ 100 The slope is the rate at which input 2 must be given up as input 1’s level is increased so as not to change the output level. The slope of an isoquant is its technical rate- of-substitution . x 2 ' x 1 '
Technical Rate-of-Substitution How is a technical rate-of-substitution computed?

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Technical Rate-of-Substitution How is a technical rate-of-substitution computed? The production function is A small change (dx 1 , dx 2 ) in the input bundle causes a change to the output level of y f x x = ( , ). 1 2 dy y x dx y x dx = + 1 1 2 2 .
Technical Rate-of-Substitution dy y x dx y x dx = + 1 1 2 2 . But dy = 0 since there is to be no change to the output level, so the changes dx 1 and dx 2 to the input levels must satisfy 0 1 1 2 2 = + y x dx y x dx .

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Technical Rate-of-Substitution 0 1 1 2 2 = + y x dx y x dx rearranges to y x dx y x dx 2 2 1 1 = - so dx dx y x y x 2 1 1 2 = - / / .
Technical Rate-of-Substitution dx dx y x y x 2 1 1 2 = - / / is the rate at which input 2 must be given up as input 1 increases so as to keep the output level constant. It is the slope of the isoquant.

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Technical Rate-of-Substitution; A Cobb-Douglas Example y f x x x x a b = = ( , ) 1 2 1 2 so y x ax x a b 1 1 1 2 = - y x bx x a b 2 1 2 1 = - . and The technical rate-of-substitution is dx dx y x y x ax x bx x ax bx a b a b 2 1 1 2 1 1 2 1 2 1 2 1 = - = - = - - - / / .
x 2 x 1 Technical Rate-of-Substitution; A Cobb-Douglas Example TRS ax bx x x x x = - = - = - 2 1 2 1 2 1 1 3 2 3 2 ( / ) ( / ) y x x a and b = = = 1 1/3 2 2 3 1 3 2 3 / ;

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x 2 x 1 Technical Rate-of-Substitution; A Cobb-Douglas Example TRS ax bx x x x x = - = - = - 2 1 2 1 2 1 1 3 2 3 2 ( / ) ( / ) y x x a and b = = = 1 1/3 2 2 3 1 3 2 3 / ; 8 4 TRS x x = - = - × = - 2 1 2 8 2 4 1
x 2 x 1 Technical Rate-of-Substitution; A Cobb-Douglas Example TRS ax bx x x x x = - = - = - 2 1 2 1 2 1 1 3 2 3 2 ( / ) ( / ) y x x a and b = = = 1 1/3 2 2 3 1 3 2 3 / ; 6 12 TRS x x = - = - × = - 2 1 2 6 2 12 1 4

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Assumptions on Technologies Common assumptions on technology are monotonic, and convex.
Assumptions on Technologies - Monotonicity Monotonicity: More of any input generates more output.

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