Lecture 12 - Lecture 12: Production Functions Cost...

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1 Lecture 12: Production Functions Cost Functions 1 Technical Progress z Suppose that the production function is q = A ( t ) f ( k , l ) 2 where A ( t ) represents all influences that go into determining q other than k and l changes in A over time represent technical progress z A is shown as a function of time ( t ) z dA / dt > 0 Technical Progress z Differentiating the production function with respect to time we get 3 dt k df A k f dt dA dt dq ) , ( ) , ( l l + = + + = dt d f dt dk k f k f q A q dt dA dt dq l l l ) , (
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2 Technical Progress z Dividing by q gives us d f dk k f dt dA dt dq l l + + / / / / 4 dt k f dt k f A q l l = ) , ( ) , ( l l l l l l dt d k f f k dt dk k f k k f A dt dA q dt dq / ) , ( / ) , ( / / + + = Technical Progress z For any variable x , [( dx / dt )/ x ] is the proportional growth rate in x denote this by G 5 denote this by x z Then, we can write the equation in terms of growth rates l l l l l G k f f G k f k k f G G k A q + + = ) , ( ) , ( Technical Progress z Since k q e q k k q k f k k f , ) , ( = = l 6 l l G e G e G G q k k q A q , , + + = l l l l l l , ) , ( q e q q k f f = =
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3 Technical Progress in the Cobb- Douglas Function z Suppose that the production function is q = A(t)f(k, l ) =A ( t ) k α l 1- α 7 z If we assume that technical progress occurs at a constant exponential ( θ ) then A ( t ) = Ae θ - t q = Ae θ - t k α l 1- α Technical Progress in the Cobb- Douglas Function z Taking logarithms and differentiating with respect to t gives the growth equation
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Lecture 12 - Lecture 12: Production Functions Cost...

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