Week 5_Technology (Print)

# And eg if y x x 1 13 2 2 3 then marginal products mp

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and E.g. if y x x = 1 1/3 2 2 3 / then

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Marginal Products MP x x 1 1 2 3 2 2 3 1 3 = - / / MP x x 2 1 1/3 2 1/3 2 3 = - and so MP x x x 1 1 1 5 3 2 2 3 2 9 0 = - < - / / MP x x x 2 2 1 1/3 2 4 3 2 9 0 = - < - / . and Both marginal products are diminishing. E.g. if y x x = 1 1/3 2 2 3 / then
Returns-to-Scale Marginal products describe the change in output level as a single input level changes. Returns-to-scale describes how the output level changes as all input levels change in direct proportion ( e.g. all input levels doubled, or halved).

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Returns-to-Scale If, for any input bundle (x 1 ,…,x n ), f kx kx kx kf x x x n n ( , , , ) ( , , , ) 1 2 1 2 = then the technology described by the production function f exhibits constant returns-to-scale. E.g . (k = 2) doubling all input levels doubles the output level.
Returns-to-Scale y = f(x) x’ x Input Level Output Level y’ One input, one output 2x’ 2y’ Constant returns-to-scale

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Returns-to-Scale If, for any input bundle (x 1 ,…,x n ), f kx kx kx kf x x x n n ( , , , ) ( , , , ) 1 2 1 2 < then the technology exhibits diminishing returns-to-scale. E.g . (k = 2) doubling all input levels less than doubles the output level.
Returns-to-Scale y = f(x) x’ x Input Level Output Level f(x’) One input, one output 2x’ f(2x’) 2f(x’) Decreasing returns-to-scale

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Returns-to-Scale If, for any input bundle (x 1 ,…,x n ), f kx kx kx kf x x x n n ( , , , ) ( , , , ) 1 2 1 2 then the technology exhibits increasing returns-to-scale. E.g . (k = 2) doubling all input levels more than doubles the output level.
Returns-to-Scale y = f(x) x’ x Input Level Output Level f(x’) One input, one output 2x’ f(2x’) 2f(x’) Increasing returns-to-scale

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Returns-to-Scale A single technology can ‘locally’ exhibit different returns-to-scale.
Returns-to-Scale y = f(x) x Input Level Output Level One input, one output Decreasing returns-to-scale Increasing returns-to-scale

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Examples of Returns-to-Scale y a x a x a x n n = + + + 1 1 2 2 . The perfect-substitutes production function is Expand all input levels proportionately by k. The output level becomes a kx a kx a kx n n 1 1 2 2 ( ) ( ) ( ) + + +
Examples of Returns-to-Scale y a x a x a x n n = + + + 1 1 2 2 . The perfect-substitutes production function is Expand all input levels proportionately by k. The output level becomes a kx a kx a kx k a x a x a x n n n n 1 1 2 2 1 1 2 2 ( ) ( ) ( ) ( ) + + + = + + +

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Examples of Returns-to-Scale y a x a x a x n n = + + + 1 1 2 2 . The perfect-substitutes production function is Expand all input levels proportionately by k. The output level becomes a kx a kx a kx k a x a x a x ky n n n n 1 1 2 2 1 1 2 2 ( ) ( ) ( ) ( ) . + + + = + + + = The perfect-substitutes production function exhibits constant returns-to-scale.
Examples of Returns-to-Scale y a x a x a x n n = min{ , , , }. 1 1 2 2 The perfect-complements production function is Expand all input levels proportionately by k. The output level becomes min{ ( ), ( ), , ( )} a kx a kx a kx n n 1 1 2 2

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Examples of Returns-to-Scale y a x a x a x n n = min{ , , , }.
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