grow5_solow - 5 5.1 The Solow Growth Model Models and...

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5 The Solow Growth Model 5.1 Models and Assumptions What is a model? A mathematical description of the economy. Why do we need a model? The world is too complex to describe it in every detail. What makes a model successful? When it is simple but e ff ective in de- scribing and predicting how the world works. A model relies on simplifying assumptions. These assumptions drive the conclusions of the model. When analyzing a model it is crucial to spell out the assumptions underlying the model. Realism may not a the property of a good assumption. 67
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5.2 Basic Assumptions of the Solow Model 1. Continuous time. 2. Single good produced with a constant technology. 3. No government or international trade. 4. All factors of production are fully employed. 5. Labor force grows at constant rate n = ˙ L L . 6. Initial values for capital, K 0 and labor, L 0 given. 68
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Production Function Neoclassical (Cobb-Douglas) aggregate production function: Y ( t ) = F [ K ( t ) , L ( t )] = K ( t ) α L ( t ) 1 - α To save on notation write: Y = A K α L 1 - α Constant returns to scale: F ( λ K, λ L ) = λ F ( K, L ) = λ A K α L 1 - α Inputs are essential: F (0 , 0) = F ( K, 0) = F (0 , L ) = 0 69
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Marginal productivities are positive: F K = α AK α - 1 L 1 - α > 0 F L = (1 - α ) AK α L - α > 0 Marginal productivities are decreasing, 2 F K 2 = ( α - 1) α A K α - 2 L 1 - α < 0 2 F L 2 = - α (1 - α ) A K α L - α - 1 < 0 70
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Per Worker Terms Define x = X L as a per worker variable. Then y = Y L = A K α L 1 - α L = A K L a L L 1 - α = A k α Per worker production function has decreasing returns to scale. 71
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Capital Accumulation Capital accumulation equation: ˙ K = sY - δ K Important additional assumptions: 1. Constant saving rate (very specific preferences: no r ) 2. Constant depreciation rate 72
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Dividing by K in the capital accumu equation: ˙ K K = s Y K - δ .
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