IH1_SL - The Growth Model In Discrete Time Prof. Lutz...

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The Growth Model In Discrete Time Prof. Lutz Hendricks September 8, 2009 L. Hendricks () Growth Model September 8, 2009 1 / 78
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The standard growth model The neoclassical growth model, aka the standard growth model, is the most important model in macro. It underlies entire branches of the literature (parts of growth theory and business cycle theory, for example). Here, we study this model in discrete time. The main issues of this section are: Tools: Dynamic programming The neoclassical growth model L. Hendricks () Growth Model September 8, 2009 2 / 78
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Model structure There are many versions of the growth model. This is a basic version. 1 Households are identical and live forever. 2 Firms produce a single good using capital and labor. 3 All agents are price takers . 4 L. Hendricks () Growth Model September 8, 2009 3 / 78
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How to justify this? Reduced form of an OLG model with altruism . Stochastic deaths (perpetual youth models). L. Hendricks () Growth Model September 8, 2009 4 / 78
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The representative household Demographics All households are identical. This is stronger than needed (see notes on aggregation later on). We can think of a single, price-taking household. The measure of households is 1. Therefore, per capita and aggregate variables are the same. L. Hendricks () Growth Model September 8, 2009 5 / 78
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The representative household Preferences The household values discounted utility from consumption: t = 0 β t u ( c t ) , 0 < β < 1 (1) Utility is time separable (for tractability). Discounting is exponential (to avoid time consistency problems). Time consistency means: If f c t g t = 0 solves the problem with start date 0, then f c t g t = τ solves the problem with start date τ . The household does not want to change past plans. L. Hendricks () Growth Model September 8, 2009 6 / 78
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Technology The household enters the world with k 0 units of "the good." Resource constraint: k t + 1 = f ( k t ) c t (2) We assume Inada conditions for f . Capital cannot be negative: k t ± 0 . L. Hendricks () Growth Model September 8, 2009 7 / 78
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Planning Problem The planner maximizes discounted utility of the representative household t = 0 β t u ( c t ) , 0 < β < 1 Constraints: k t + 1 = f ( k t ) c t k t + 1 ± 0 k 0 given L. Hendricks () Growth Model September 8, 2009 8 / 78
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Lagrangian Γ = t = 0 β t u ( c t ) + t = 0 λ t [ f ( k t ) c t k t + 1 ] FOCs for an interior solution: β t u 0 ( c t ) = λ t λ t + 1 f 0 ( k t + 1 ) = λ t L. Hendricks () Growth Model September 8, 2009 9 / 78
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Euler equation β u 0 ( c t + 1 ) f 0 ( k t + 1 ) = u 0 ( c t ) (3) This is exactly the same Euler equation we saw many times before.
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This note was uploaded on 10/29/2009 for the course ECON 720 at UNC.

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IH1_SL - The Growth Model In Discrete Time Prof. Lutz...

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