6 Lecture The Ramsey Model
Pareto Efficiency
An allocation of goods across a group of people is called Pareto efficient if there is no way to reallocate the goods to make someone better off without making someone worse off. This is equivalent to the allocation maximizing some Pareto social welfare function.
A linear combination of the utility functions of every member of the group.
ECON 7240 VI - 2
A Representative Agent
Pareto efficiency concept simplifies if there is a representative agent.
There is only one agent. Pareto social welfare function must be the utility function of this agent.
In Cass-Koopmans model, a technologically feasible allocation that maximizes the utility function is Pareto optimal.
Pareto optimality does not involve markets.
ECON 7240 VI - 3
The Ramsey Problem
Given K(0), the social planner chooses C(t) and K(t) to maximize
subject to
0
e t L(t )u (C (t ))dt
dK (t ) L(t )C (t ) + = F ( K (t ), A(t ) L(t )) K (t ). dt
ECON 7240 VI - 4
The Ramsey Hamiltonian
The Hamiltonian for the Ramsey problem is
H=e
t
+ (t )[ F ( K (t ), A(t ) L(t )) K (t ) L(t )C (t )].
L(t )u (C (t ))
We have not imposed enough conditions to get a unique solution. The transversality condition:
ECON 7240
lim (t ) K (t ) = 0.
t
VI - 5
Ramsey Equations of Motion
H t = e L(t )u' (C (t )) (t ) L(t ) = 0 C (t ) H dK (t ) = F ( K (t ), A(t ) L(t )) K (t ) L(t )C (t ) = (t ) dt H d (t ) = (t )[ FK ( K (t ), A(t ) L(t )) ] = K (t ) dt
ECON 7240 VI - 6
Simplifying the Ramsey Model
Let us define
r (t ) = FK ( K (t ), A(t ) L(t )) .
This is not a market-determined interest rate; just the net marginal product of capital.
The C and equations to
d simplify (t ) = r (t ) (t ) dt
ECON 7240
u ' (C (t )) = e t (t ).
Same as in the C-K model.
VI - 7
Dynamics of Consumption
With CRRA utility, we get the same C-K Euler equation:
d ln C (t ) r (t ) = . dt
The intuition is nearly the same.
If social planner defers consumption C(t) from t to t + t, the additional capital produces r(t) C(t) t more goods in the interim.
ECON 7240 VI - 8
Income-Product Identity
Let us define
F ( K (t ), A(t ) L(t )) W (t ) = = A(t ) FL ( K (t ), A(t ) L(t )) L
With constant returns to scale production,
F ( K (t ), A(t ) L(t )) = (r (t ) + ) K (t ) + W (t ) L(t ).
The budget constraint becomes
ECON 7240
dK (t ) = W (t ) L(t ) + r (t ) K (t ) L(t )C (t ). dt
VI - 9
Transversality Condition
The solution to the (t) equation is
(0) (t ) = . R (t )
Thus the transversality condition is
(0) K (t ) lim = 0. t R(t )
This is just the no-Ponzi condition.
ECON 7240 VI - 10
Equivalence of Models
Laws of motion and boundary conditions for Ramsey model are identical to C-K model. The competitive equilibrium in the CassKoopmans model is Pareto efficient. This is an example of the Welfare Theorems.
1. Absent frictions, a competitive equilibrium is Pareto efficient. 2. With more caveats, a Pareto efficient allocation can be supported as a market equilibrium. VI - 11 ECON 7240
Summary
1. Pareto efficient allocations can be found by solving a social planners problem. 2. A social planner will choose the same allocation for consumption in the CassKoopmans model as a rational consumer trading in markets. 3. We can often solve for competitive equilibria by solving easier social planners problem.
ECON 7240 VI - 12