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Econ407 J. Tessada Spring 2009 Intro Basic Model Tech Progress Growth Rates and Transition Human Capital Conclusions Econ407 Advanced Macroeconomics Lecture 2: Neoclassical Growth Models - The Solow Model Jos´e Tessada University of Maryland - College Park February 5 and 10, 2009 Econ407 J. Tessada Spring 2009 Intro Motivation Basic Model Tech Progress Growth Rates and Transition Human Capital Conclusions Motivation I In the previous lecture we reviewed some basic facts about economic growth. We will now start studying a simple but yet useful and insightful growth model: the Solow-Swan model We will see the role of capital accumulation and technological progress in the process of development The model will deliver important conclusions: Technological progress is the single variable explaining long-run growth It will also help us understand the variables behind differences in income levels Although we will learn a significant amount from this model, more will be needed for a deeper understanding of the process of economic growth
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Econ407 J. Tessada Spring 2009 Intro Basic Model Set-up Steady-state Comparative Statics Cobb-Douglas Golden-rule Tech Progress Growth Rates and Transition Human Capital Conclusions Production Function I Production function: Y ( t ) = F ( K ( t ) , A ( t ) L ( t )) (1) where K ( t ) is the input of capital, L ( t ) is labor input and A ( t ) is labor-augmenting technological progress Also, F ( · ) exhibits constant returns to scale (CRS), i.e., F ( qK , qAL ) = qF ( K , AL ) for all q > 1 Let us assume for now that A ( t ) = 1 t . We will relax this assumption later on. Econ407 J. Tessada Spring 2009 Intro Basic Model Set-up Steady-state Comparative Statics Cobb-Douglas Golden-rule Tech Progress Growth Rates and Transition Human Capital Conclusions Production Function II Define, k = K / L and y = Y / L as capital per worker and output per worker, respectively. (Note: we will maintain this notation throughout our study of economic growth) We can write: 1 L F ( K , L ) = F ( K / L , 1 ) y = f ( k ) . (2) We refer to eq. ( 2 ) as the intensive form of the production function. We assume that f ( 0 ) = 0, f 0 ( k ) > 0, f 00 ( k ) < 0 lim k 0 f 0 ( k ) , lim k f 0 ( k ) 0, where the last conditions are called Inada conditions
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Econ407 J. Tessada Spring 2009 Intro Basic Model Set-up Steady-state Comparative Statics Cobb-Douglas Golden-rule Tech Progress Growth Rates and Transition Human Capital Conclusions Production Function III Notice that we can write: F ( K , L ) = Lf ( k ) Hence our assumptions imply positive but decreasing marginal product of K : F ( K , L ) / K = L f ( = k z}|{ K / L ) K = f 0 ( k ) Econ407 J. Tessada Spring 2009 Intro Basic Model Set-up Steady-state Comparative Statics Cobb-Douglas Golden-rule Tech Progress Growth Rates and Transition Human Capital Conclusions Dynamics of K and L I Labor and total population are equal and grow at a constant rate n 0 L ( t ) = L ( 0 ) e nt ˙ L ( t ) = nL ( t ) , (3) where ˙ X ( t ) = dX ( t ) / dt , and g X ( t ) = ˆ X ( t ) = ˙ X / X is the growth rate of X Notice that d ln [ X ( t )] / dt = ˆ X ( t ) (Chain rule) Given that labor equals population, per capita and per worker units are equivalent (we will use the terms interchangeably for now) What if we remove this assumption? Any ideas?
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