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# Adv Macroeconomic Theory ECON 200A

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This 3 page Class Notes was uploaded by Marge Hansen on Monday October 5, 2015. The Class Notes belongs to ECON 200A at California State University - Sacramento taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/218859/econ-200a-california-state-university-sacramento in Economcs at California State University - Sacramento.

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Date Created: 10/05/15

ECON 200A Spring 2008 Growth Rates in the Solow Growth Model Model Assume the following model for the economy Y F K 7 AL n 39 L T A T The growth rate in the labor force n and growth rate in technology 9 are taken as given In pereeffective unit of labor terms7 the production function is a 1 Capital evolves according to km 7ltng6gtE where is the derivative of capital with respect to time1 At steady state7 k 0 N N sfk ng6k Growth Rates Growth rate of capital Capital pereeffective unit of labor Taking the derivative with respect to time 2 RXAL7K1ALLA MD2 27 K KL KA AL ALL ALA K K k wg At steady state0 K K E Em K E n9 So the growth rate of capital is equal to the population growth rate plus the growth rate of technical progress 11 is a continuousitime version of A16 The change in capital A16 is measuring how much capital changes from one period to the next Rather than assuming discrete periods of time period 17 2 etc7 the Solow model uses continuous time This is why the impulse responses we draw are smooth functions rather than ones that jump with each discrete unit of time Growth rate of capital per worker 7 From the expression above g L iAL ALL ALA At steady state0 L A KK K L A DividebyK 5 LELA K 7 L A A K L A 9 Therefore capital periworker grows at a rate equal to the rate of technical progress You can nd the remaining growth rates by take the time derivatives for output consumption wages and the real rental rate You can use the steady state condition since capital per effectivei labor unit is not growing it must be that a 0 andE 0 Growth rate of output per worker 7 We know n this is given We also know that 0 when the economy is at steady state We can take the time derivative of the perieffective worker output and use this information to solve for Take the time derivative of y Take the time derivative 1 7y 7y g Y A Lgt E A2L AL2 1 71 A 71 L AL AL A AL L Use the steady state condition that implies 0 Y L i AL ALA ALL Y A L 7 71 Y L 7 A V L 2 9 Therefore output periworker grows at a rate equal to the rate of technical progress Growth rate of factor payments Real wage Wages are de ned as the marginal product of labor The marginal product of labor is w E Note7 the steady state is de ned in per effective units of laloor7 so the production function is y Y K flt Y AL f Taking the derivative with respect to labor 2 AM 7 ALf a AM 7 Af E Af e Af k A W 7 WW Now7 differentiate with respect to time w A M e f M Aifw 7 MW 7 WW The growth rate of the real wage is therefore fkifnzg flz fMEE w A f 7 f A M 7 g Since 0 at steady state I tl 7 A i E 7 Z 9 Real rental rate The rental rate of capital is equal to 7 E 3 7 6 the factor payment is made net of depreciation Note Y ALfk AL X F BY 1 AL k l k M f lt gt AL f lt gt Therefore7 the rental rate net of depreciation is 7 1 e 6 Taking the time derivative the depreciation rate is constant and dividing through by 7 f 76 r39 7 f r f E e 6 At steady state7 0 0

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