The instantaneous utility function is crra the

Info icon This preview shows pages 8–16. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 8

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 +
Image of page 9
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 + = +
Image of page 10

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 11
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
Image of page 12

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 +
Image of page 13
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
Image of page 14

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 .
Image of page 15
Image of page 16
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern