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# newsolow - New Notes on the Solow Growth Model Roberto...

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New Notes on the Solow Growth Model Roberto Chang September 2009 1 The Model The f rstingredientofadynamicmodelisthedescriptionofthetimehorizon. In the original Solow model, time is continuous and the horizon is in f nite. Without loss of generality assume that time is indexed by in [0 ) At each point in time, there is only one f nal good that is produced via the aggregate production function : = (   ) Here  and denote output, labor productivity, capital, and labor, and are functions of time (i.e. we should really write ( ) and so on, but we omit time arguments when not needed). The product  is called e f ective labor. The function is has neoclassical properties: is continuously di f erentiable, strictly increasing, strictly concave has constant returns to scale A key example is the Cobb Douglas function: (   )= (  ) 1 0  1 Exercise: Show that the Cobb Douglas function enjoys the neoclassical prop- erties listed above. Once you assume the existence of the aggregate production function, it is clear that the growth of output can be due to growth of   or  Solow assumed that and grow at exogenous rates: ˙ =   =  ˙ =  These are revised but still rough notes for Econ 504. 1

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At time =0  and are given by history. Then one can describe the value of and at each point in time: ( )= (0)  ( )= (0)  So it remains to describe the motion of capital. Assume (0) is given by history. The accumulation of capital must follow: ˙ =  where denotes investment and the depreciation rate of capital. As usual, investment is assumed to equal savings. The key behavioral equa- tion of the Solow model is that savings equal a constant fraction of output, so: =  This completes the description of the model. 2 The Solution A solution is a description of the values of    and at each point in time. Now, from the capital accumulation equation: ˙ =   =  (  )  =  (  1)  =  [  (  1)  ] Here we have de f ned =  De f ne ( )= (  1) the intensive pro- duction function. Now we can rewrite the previous equation as: ³ ˙  ´ =  ( )  But =  implies ˙  = ˙  ( + ) Comb inethetwotoobta in : ˙ =  ( ) ( + + )
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newsolow - New Notes on the Solow Growth Model Roberto...

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