Finding a balanced growth path always annoyed me. It felt like we were imposing a condition on the model which may not necessarily be true. However, I feel like this comes down the the fact that what is a very simple idea was very poorly explained.

Definition

What is a balanced growth path?

We are finding out whether a variable in the economy grows at a constant rate.

For example, if we divide this years output Y(t) by last years output Y(t-1) does it equal a constant value? I.E is GDP growth constant over time.

That's all we are doing.

The AK model

Suppose we have the following simple discrete model. I will show in this example that

C(t+1)/C(t) = g where g is a constant value. Thus, consumption follows a balanced growth path.

Y(t) = AK(t)                                                                                         (1)

with the capital evolution path as follows:

K(t+1) = (1-d)K(t) -  I(t)                                                                  (2)

and the resource constraint

Y(t) = C(t) + I(t)                                                                                  (3)

and the Utility function

U = ln[ C(t) ]                                                                                       (4)

We could know set up a lagrangean:

L = ln C(t) + h(t)[K(t+1) - (1-d)K(t) -  AK(t) - C(t)                   (5)

Now derive it wrt to C(t):

dl/dC(t) = 1/C(t) - h(t) = 0                                                            (6)

This implies:

1/C(t) = h(t)                                                                                      (7)

Derive with respect to K(t+1)

dl/dK(t+1) = h(t) - h(t+1)[(1-d) + A] = 0                                 (8)

Which implies

[(1-d) + A] = h(t)/h(t+1)                                                              (9)

Which implies

[(1-d) + A] = 1/c(t) / 1/c(t+1)                                                    (10)

and thus

[(1-d) + A] = c(t+1)/c(t)                                                             (11)

Since d which is depreciation, A the rate of technological change are all assumed to be constant in this model the rate of consumption growth is constant and thus:

g = 1 - d + A

You should be able to solve for the remaining variables Y, I, K.