RandD_RQ

RandD_RQ - Review Problems: Innovation and Growth Prof....

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Review Problems: Innovation and Growth Prof. Lutz Hendricks. August 7, 2009 1 Technology Adoption [Based on Romer (2001) question 3.12] The world consists of 2 regions: i = N;S . Output is produced according to Y i = K i ( A i [1 a Li ] L i ) 1 Capital is accumulated according to _ K i = s i Y i New technologies are developed in the North: _ A N = B a LN L N A N : The South learns from the North: _ A S = LS L S ( A N A S ) as long as A N > A S . Otherwise _ A S = 0 . Labor supplies ( L i ) are constant over time. 1. Determine the long-run growth rate of Y N . Z = A S =A N . Derive the law of motion for Z . Is Z stable? If so, which value does it converge to? What is the long-run growth rate of Y S ? 3. Assume a LN = a LS and s N = s S . Find the balanced growth output ratio Y N =Y S . 1.1 Answer: Technology adoption 1. The growth rates obey g ( Y i ) = ±g ( K i ) + (1 ± ) ( g ( A i ) + n ) (1) g ( K i ) = s i Y i =K i (2) g ( A N ) = B a LN L N (3) g ( A S ) = LS L S ( A N =A S 1) (4) 1
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In the long-run g ( Y i ) = g ( K i ) and therefore g ( Y i ) = g ( A i ) + n (5) 2. The law of motion for Z is g ( Z ) = g ( A S ) g ( A N ) = LS L S (1 =Z 1) B a LN L N This has a unique, stable steady state. Its value is 1 =Z = 1 + B a LN L N LS L S This implies that larger adopting countries have higher productivity ( Z ) - the scale e/ect in action. 3. Equals s i implies equal K=Y . Write the production function as Y i =L i = ( K i =Y i ) (1 ± ) A i (1 a Li ) (6) Then the ratio of output per worker is y S =y N = Z 1 a LS 1 a LN (7) so that larger adopting countries are richer. 2 Delayed Adoption [Based on Romer (2001) question 3.13] The world consists of two regions: i = N;S . The North is described by Y N = A N (1 a LN ) L N _ A N = a L L N A N The South uses technologies developed in the North with a lag of ± : A S ( t ) = A N ( t ± ) Y S ( t ) = A S ( t ) L S Labor inputs are constant over time. If g ( Y ) = 0 : 03 , how old must technologies be in the South to yield a 10-fold output gap? Assume a L is close to 0 and n = 0 : 01 . 2
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2.1 Answer: Delayed adoption From the production functions y N ( t ) y S ( t ) = (1 a LN ) A N ( t ) A N ( t ) A N ( t ) =A N ( t ) which measures how far behind the South is in terms of technology. A grows at rate g ( A ) = g ( Y ) n so that A N ( t ) =A N ( t ) = e
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RandD_RQ - Review Problems: Innovation and Growth Prof....

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