# Calculating growth rate of capital when not in steady state

Given a Cobb-Douglas production function, the annual population growth rate, savings rate, alpha, annual depreciation rate, and annual technological progress rate, how would one calculate the growth rate of capital?

We're assumed to not be in steady state so we would need a growth rate in order to calculate how long it takes to reach the true steady state.

Perhaps I misunderstood something but it seems to me you can calculate the growth rate by using all the parameters to calculate $K_{t+1}$. Then get the growth rate by taking $\frac{K_{t+1}}{K_t}$.

You will never reach the true steady state, as outlined in my answer here.

• Could you point me towards an equation where I could calculate K(t+1)? That's what i've been looking for but I couldn't find it. – Anthony Oct 16 '16 at 16:51
• I'm mostly confused about turning it into a function of time. – Anthony Oct 16 '16 at 16:54
• You cannot calculate $K_{t+1}$ as pure function of time, just a function of some earlier $K_s$. Given $K_t$ and alpha you can calculate $Y_t$, right? From this and the savings rate you have $I_t$, and so on. I'm sorry but I don't see where your problem lies because you seem to understand the underlying model. – Giskard Oct 16 '16 at 18:05

In the steady state by definition the growth rate of capital is equal to the growth rate of technology (gA) plus the growth rate of labor force (gN).

In the steady state, investment (s*F(K/AN)) is equal to depreciation of capital (d), growth rate of technology, and growth rate of labor force so that capital per effective worker (K/AN) is held constant.

Since you're asking the overall level of capital it will be increasing at the same growth rate of technology plus the growth rate of labor force.

In steady state: s*F(K/AN) = ((gA + gN + d)K)/AN.

The depreciation (d) will offset everything but the growth of labor and technology.