The Rebelo AK Growth Model

Rebelo (1991) examines a model in which a social planner maximizes the discounted sum of utility

in an economy with an $AK$ production function:

subject to

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 $\CRRA$ one can show that optimal behavior

requires

Note that this equation comes about because the marginal product of

capital in this model is always $A = \rProd$, because $\fFunc^{\prime}(K) = A$. 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 $\gamma = \dot{K}/K$,

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 $A$ 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 $AK$ 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.