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class3 - Consumption and Savings Introduction to Dynamic...

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Consumption and Savings Introduction to Dynamic Programming Xavier Ragot Macroeconomics October 2010 Class 3 The Saving Choice
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One of the most important macroeconomic choice is the households°choice between consumption and savings. Class 3 The Saving Choice
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One of the most important macroeconomic choice is the households°choice between consumption and savings. Consumption is the biggest component of demand (between 60% and 70% of GDP). Households save to consume later, what allows to increase the capital stock : households savings is lent to ±rms and the State thanks to ±nancial market (not present in IS/LM). Class 3 The Saving Choice
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One of the most important macroeconomic choice is the households°choice between consumption and savings. Consumption is the biggest component of demand (between 60% and 70% of GDP). Households save to consume later, what allows to increase the capital stock : households savings is lent to ±rms and the State thanks to ±nancial market (not present in IS/LM). In IS/LM saving rate s exogenous. What are the determinants of consumption and savings : Income wealth, Interest Rates ? Class 3 The Saving Choice
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Personal Saving rate in the US 0,0 2,0 4,0 6,0 8,0 10,0 12,0 14,0 16,0 1954-10-03 1968-06-11 1982-02-18 1995-10-28 2009-07-06 Class 3 The Saving Choice
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In this class : 1 Saving choice with in±nite horizon 2 Bellman equations 3 The random Walk Hypothesis of consumption 4 Asset Pricing 5 The Equity Premium Puzzle Class 3 The Saving Choice
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A two-period economy Representative household : Instantaneous utility function u ( . ) . u 0 > 0 , u 00 < 0. Von Neuman-Morgenstern framework. The subjective rate of time of time prefenrece is ρ . The discount factor is β = 1 1 + ρ The household receives an income Y t at period t and Y t + 1 at period t + 1 . The real interest rate is r between t and t + 1. Households maximize u ( c t ) + β u ( c t + 1 ) Class 3 The Saving Choice
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A two-period economy Representative household : Instantaneous utility function u ( . ) . u 0 > 0 , u 00 < 0. Von Neuman-Morgenstern framework. The subjective rate of time of time prefenrece is ρ . The discount factor is β = 1 1 + ρ The household receives an income Y t at period t and Y t + 1 at period t + 1 . The real interest rate is r between t and t + 1. Households maximize u ( c t ) + β u ( c t + 1 ) subject to period t and t + 1 budget constraint c t + A t + 1 = Y t Y t + 1 + ( 1 + r ) A t + 1 = c t + 1 where A t + 1 ( <> 0 ) is the ±nancial savings between period t and t + 1 ( Be careful about the notations !). Class 3 The Saving Choice
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Substituting for A t + 1 in the two previous equations yield c t + 1 1 + r c t + 1 = Y t + 1 1 + r Y t + 1 Class 3 The Saving Choice
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Substituting for A t + 1 in the two previous equations yield c t + 1 1 + r c t + 1 = Y t + 1 1 + r Y t + 1 The program of the households is max f c t , c t + 1 g u ( c t ) + β u ( c t + 1 ) c t + 1 1 + r c t + 1 = Y t + 1 1 + r Y t + 1 Class 3 The Saving Choice
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Important Result : The ±rst important result is that consumption depends on W t = Y t + 1 1 + r Y t + 1 .
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