# The instantaneous utility function is crra the

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The instantaneous utility function is CRRA: The discount factor ρ > n + (1 - θ ) g is needed to ensure U is finite. . )) ( ( ) ( 0 dt t C u H t L e U t - = ρ 0 1 1 ) ( 1 - = - θ θ θ C C u

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ECON 7240 V - 9 Sources of Income Each individual is endowed with a unit of labor that he sells at t for the real wage W ( t ). The household is endowed with capital K (0)/ H at t = 0. At t , the household rents its capital K ( t )/ H at rate r ( t ). The flow of income to the household at t is . ) ( ) ( ) ( ) ( H t K t r t L t W +
ECON 7240 V - 10 Instantaneous Budget Constraint Income can be allocated to consumption or saving. All saving occurs through purchases of capital, so K ( t ) is the stock of saving. The flow of saving is dK ( t )/ dt , which can be positive or negative. ) ( ) ( ) ( ) ( ) ( ) ( ) ( t K t r t L t W dt t dK t C t L + = +

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ECON 7240 V - 11 Compounding of Interest Suppose you have an initial debt D (0) and continue to borrow to pay interest. Let us define The debt has the solution ) ( ) ( ) ( t D t r dt t dD = ). 0 ( ) ( ) ( D t R t D = . ' ) ' ( exp ) ( 0 = t dt t r t R
ECON 7240 V - 12 No-Ponzi-Game Condition To complete the model, we need to specify a terminal condition for K ( t ) at large t . We allow households to borrow, but they cannot roll over their debt forever. The present value of their saving at large t cannot be negative. 0 ) ( ) ( lim t R t K t

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ECON 7240 V - 13 Lifetime Budget Constraint We can rewrite the instantaneous budget constraint as Integrating, we obtain lifetime budget constraint The present value of consumption cannot be more than the present value of income and assets. ). ( ) ( ) ( ) ( ) ( ) ( t L t R t C t W t R t K dt d - = . ) 0 ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 H K dt t R t W H t L dt t R t C H t L +
ECON 7240 V - 14 Household Problem Given K (0), W ( t ), and r ( t ), the household chooses C ( t ) to maximize subject to dt t C u H t L e t - 0 )) ( ( ) ( ρ ) ( ) ( ) ( ) ( ) ( ) ( ) ( t K t r t L t W dt t dK t C t L + = + 0 ) ( ) ( lim 0 t R t K t

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ECON 7240 V - 15 Calculus of Variations Suppose we wish to maximize over the set of all differentiable functions x ( t ) : R R n such that x ( a ) = X a and x ( b ) = X b .
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