1. ECO 6120: The Ramsey-Cass-Koopmans model
1-Assumptions
Endogenous saving rate (intertemporal optimization)
Same production function as in Solow
No uncertainty
Continuous time, infinite horizon (the options are overlapping-generations models, Blanchard’s probability of dying)
We follow B and S in notation
n is the growth rate of the size (members)
of the representative household
The problem: Max U(t) under constraint

2. The representative household utility function is:
c(t) ≡ C(t)/L(t)
u() is the instantaneous utility function
is a concave function with u' >0 and u'' <0
ρ > 0 is the discount rate (explanation)
u() are then weighted by the growth rate of the family n
and by the discount rate for the present
ρ > n condition for convergence of lifetime utility (to solve
relatively easily the optimization problem)

3. Decentralized (versus central planner) problem
The state variable is a(t) the assets hold by the representative family
Alternatively, in a central planner problem, the state variable
is simply dk/dt as in Solow (do it yourself)
a could be negative (borrowing) but the present value
of the household’s asset holding cannot, at the limit, be negative
This is the no-Ponzi-game-condition:
Will come back to this later with the transversality condition (and for λ)

4. The economic problem:
Households
Max: 2.1 s/c 2.2 et 2.3, a(0)
Complex problem, we use the technique developed by
Pontryagin (Maximum principle)(1950, 1962)
Firms
Simple problem, the solution is:

5. Let λ be the shadow price associated with the state variable a
It is the present value of the implicit price of income da/dt
And define φ as the discount factor:
We form the present value Hamiltonian J:

6. A special case: log utility
Lets assume first that u(c) = log (c)
And lets abstract from the transversally condition
Focus the simplest possible way on Euler equation

7. c is the control variable
The first order conditions for the maximization of U given 2.2 are:

8. We take the log of equation (1)
And since
We derive both sides with respect to time
We equalize (1’) with (2)

9. The two n drop and we get the Euler equation:
In an optimal path, discussion r > ρ, r < ρ

10. Lets come back to the general case
c is the control variable
The first order conditions for the maximization of U given 2.2 are:
And the transversality condition

11. The transversality condition
Implies that the no-Ponzi-game condition is respected with equality
Implies that the intertemporal budget constraint is respected with equality
For the point of view of the course, more important to concentrate on (1) and (2) in order to get a tractable formulation of the Euler equation

12. Lets derive equation (1) with respect to time in order
to get another expression for dλ/dt:
Since we have taken the derivative of a product
And
We equalize (1') with (2)
We use again (1) to eliminate λ:

13. We divide both side by φu', the n drop, we get:
The big question is: how interpret the right-hand side term? First:
Then:

14. And:
θ is the the coefficient of relative risk aversion
1/θ is the elasticity of substitution for consumption
between two point of time
Small θ implies that household accept easily c to vary over time
A well-known instantaneous utility function is the
Constant-relative-risk-aversion (CRRA):
Demonstrate than in this case:

15. Finally, our tractable solution to the maximization problem is
The Euler equation or, the Keynes/Ramsey rule:

16. Alternatively, the state variable might be define as k and the
equation of motion as:
In this case, demonstrate that the Euler equation is:
The interest rate r is replace by f'(k)-δ

17. The Euler equation in discreet time (material borrow from Aghion and Howitt (2009)
Assume no technological progress and population growth, the household is maximizing a weighted sum of utilities:
The capital accumulation constraint is:
The Langrangian equation is:

18. The first order conditions are:

19. (1.1) and (1.2) imply