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.
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)
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)
[(1-d) + A] = h(t)/h(t+1) (9)
[(1-d) + A] = 1/c(t) / 1/c(t+1) (10)
[(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.