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Chapter3_BasicModel_II_012009

# Chapter3_BasicModel_II_012009 - The Basic Model II Juan...

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The Basic Model II Juan Rubio-Ram°rez Duke University and Federal Reserve Bank of Atlanta January 20, 2009 Juan Rubio-Ram°rez (DUKE) Basic Model January 20, 2009 1 / 15

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Firms All ±rms are identical. Normalize the number of ±rms to 1 . Competitive behavior. Hence, we can work with a representative °rm that stands in for the entire production sector in the economy. Juan Rubio-Ram°rez (DUKE) Basic Model January 20, 2009 2 / 15
Technology Neoclassical Cobb-Douglas production function: y t = f ( k t , l t ) = k α t ( A t l t ) 1 ° α where α 2 ( 0 , 1 ) . A t is the current level of technology. Labor augmenting form. Juan Rubio-Ram°rez (DUKE) Basic Model January 20, 2009 3 / 15

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Characteristics 1 Constant returns to scale ( λ k t ) α ( A t λ l t ) 1 ° α = λ k α t ( A t l t ) 1 ° α = λ y t for λ > 0 2 Both inputs are essential. If either the capital k t or labor l t input is zero, output is also equal to zero. 3 Marginal productivities are positive f k = α k α ° 1 t ( A t l t ) 1 ° α > 0 f l = ( 1 ° α ) A t k α t ( A t l t ) ° α > 0 but decreasing. 4 Inada conditions: lim k t ! 0 f k = lim l t ! 0 f l = lim k t ! f k = lim l t ! f l = 0 Juan Rubio-Ram°rez (DUKE) Basic Model January 20, 2009 4 / 15
Interpretation α is the elasticity of output with respect to the capital input. Take logs of the production function to obtain: log y t = log A t + α log k t + ( 1 ° α ) log l t and thus d log ( y t ) d log ( k t ) = α . Similarly, the elasticity of output with respect to labor is 1 ° α because: d log ( y t ) d log ( l t ) = 1 ° α . In addition, when the ±rm uses k t machines in period t , a fraction δ of them wear down. This process is called depreciation.

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