February 14, 2012, Christopher D. Carroll RebeloAK
February 14, 2012, Christopher D. Carroll RebeloAK
Growth Model Rebelo (1991) examines a social planner maximizing the discounted sum of
utility in an economy with an 
production function:
 | (1) |
subject to
 | (2) |
where we have assumed zero population growth and zero depreciation to make the analysis less cluttered.
This problem can be solved with the same Hamiltonian apparatus we used to
solve the Ramsey/Cass-Koopmans model. In particular, with CRRA
utility with risk aversion 
one can show that optimal behavior requires
Note that this equation comes about because the marginal product of capital
in this model is always 
, because 
. Note further that
according to this equation, the growth rate of consumption is always the same;
unlike the Cass-Koopmans model with a normal production function, this model
has no transitional dynamics.
We can also use (2) to obtain an expression for the steady-state growth rate:
If the model has a steady-state growth rate of 
, this equation
implies that
This is a model with a constant saving rate, because
Thus, the steady-state growth rate in a Rebelo economy is directly proportional to the saving rate.
A further requirement for the Rebelo model to have a well-defined solution is that
Recalling that 
is effectively the real interest rate in this model, this
equation can be interpreted as the ‘impatience’ condition that we imposed in the
infinite horizon perfect foresight consumption model. In fact, the Rebelo 
model is essentially just a way of reinterpreting the perfect foresight infinite
horizon consumption problem as a model for economic growth. The principal
distinction is that we usually use the perfect foresight infinite horizon
model to analyze circumstances where the agent has both labor and
capital income, whereas the Rebelo model rules out labor income by
assumption.
