Since utility increases in consumption assume

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Since utility increases in consumption, assume lifetime budget constraint holds with equality. ) 0 ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 K dt t R t C t L dt t R t L t W + = ) 0 ( ' ) ' ( ) 1 ( 1 exp ) 0 ( 0 0 0 K dt nt t dt t r L C t + + - - = ρ θ θ
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ECON 7240 V - 24 Euler Equation Consumption evolves according to Consumption increases when the return on saving is greater than the discount rate. The elasticity of intertemporal substitution is θ - 1 . . ) ( ) ( ln θ ρ - = t r dt t C d
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ECON 7240 V - 25 A Thought Experiment Consider the following experiment. For simplicity let H = 1. 1. The household decreases its consumption by - L ( t ) C ( t ) 2. Household saves an additional L ( t ) C ( t ). 3. Return on this saving is consumed at t + t . ) ( ) ( ) ) ( exp( ) ( ) ( t C t L t t r t t C t t L = + +
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ECON 7240 V - 26 Optimal Consumption and Saving The loss of utility at t is The gain in utility at t + t is Marginal rate of substitution equals relative price if ). ( ) ( ) ) exp(( t C t C t n - - - θ ρ ). 1 ( ) ( )) )( exp(( + + + - - t C t t C t t n θ ρ ( 29 t t r t C t t C t - = + - - ) ( exp ) ( ) ( ) exp( θ ρ
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ECON 7240 V - 27 Intuition about Euler Equation We can rewrite the marginal condition as In the limit of small t , this becomes θ ρ t t r t dt t dC t C - + + ) ) ( ( 1 ) ( ) ( 1 1 - = + θ ρ t t r t C t t C ) ) ( ( exp ) ( ) ( θ ρ - = ) ( ) ( ln t r dt t C d
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ECON 7240 V - 28 Competitive Equilibrium The factor prices are endogenous. In a competitive equilibrium, the path of K ( t ) has to give r ( t ) = r ( K ( t )) and W ( t ) = W ( K ( t )) that give C ( t ) consistent with K ( t ). + - = ) 0 ( ' ) ' ( ) ' ( ) ' ( ) ' ( ) ( ) ( 0 K dt t L t R t C t W t R t K t
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ECON 7240 V - 29 Dynamics of K Combining the equations for C and K , we get This is a second-order differential equation. We have boundary conditions specifying K (0) and lim t K ( t )/ R ( t ) . This can be solved with a shooting algorithm.
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