# Calculating Time to Balanced Growth Path

$\textbf{Model:}$ $$\underset{\{c_t,k_t\}}{max}\;\sum_{t=0}^\infty\beta^t\frac{c_t^{1-\gamma}}{1-\gamma}$$ $$s.t.\;c_t=Rk_{t-1}-k_t$$ $$c_t,k_t\geq0$$ At time $t$, $c_t$ is consumption and $k_{t-1}$ is the capital used in production. $0<\beta<1,\;\gamma>0,\;\gamma\neq1$

$\textbf{(a)}$ Compute a balanced growth path in which consumption and capital grow at constant rates.

Solving this using the Euler equation, we get $$\frac{c_{t+1}}{c_t}=\big(\beta R\big)^{\frac{1}{\gamma}}$$ We know that capital must grow at the same rate as consumption in a balanced growth path, so $$\frac{k_{t+1}}{k_t}=\big(\beta R\big)^{\frac{1}{\gamma}}$$ This means that: $$c_t=\bigg(\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1\bigg)k_t$$ $\textbf{(b)}$ What restrictions are necessary for both capital and consumption to grow at a positive rate on the balanced growth path?

For this question, it seems like all we need is $R>1$

Now, my question is the following: How long will it take for consumption and capital to reach the balanced growth path? In general, how would one calculate time to balanced growth path? Or is it more based on economic intuition?

$\textbf{Edit:}$ According to another student, the professor said this will never reach a balanced growth path. However, the professor never stated why. Can someone give me a reasoning as to why this would never reach a balanced growth path?

• @KitsuneCavalry We need to invert the growth rate because we are going backwards in time rather than forwards in time Jun 20 '16 at 21:04
• I missed the minus and plus signs. I understand now. I'll delete my comment (out of sheer """"humiliation"""" at the hands of arithmetic) Jun 20 '16 at 21:08
• Isn't it strange to talk about growth without a production sector? This problem is similar to a cake-eating problem. You start with a capital $k_0$ and choose how much to consume each period, given that capital depreciates if $R<1$. Consumption and capital then tend to 0 in the long run. Jun 21 '16 at 10:24
• @DornerA Indeed the question seems ill-defined. Where do we start and what behavioral rule do we follow? I am guessing this is the Solow model but it is not stated, nor are any properties of the production function. If we start on the balanced growth path and stay on it, there is no problem. If we start there but eat everything on day one of course we will never reach it again. So more information is needed. Jun 21 '16 at 18:03
• @denesp the only other information we are given is about the values of the parameters. I will include it in the question, but it doesn't seem like it will give you the information you need. I really wish I could give more, but I am just trying to figure out the question/general intuition of how to figure out time to balanced growth path/steady state. Again, as I stated in the little bounty box, I do not need an answer to this question exactly. I just want to know if there is a cookbook or intuitive approach to solving for time to balanced growth path or steady state. Jun 22 '16 at 17:17

a) Your calculations are correct, but in order for consumption to be positive, so for $$c_t=\bigg(\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1\bigg)k_t > 0,$$ you will need to additional conditions. The first is the obvious $k_t > 0$. If there is nothing to gain interest on, there will be no growth and no consumption. The second is the more nuanced $$\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1 > 0.$$ This is actually a necessary condition for the existence of an optimal solution. The inequality may be reformulated as $$\beta \cdot R^{1 - \gamma} < 1.$$ If this does not hold, then given any consumption path the consumer would gain by pushing off all consumption one period further. Since there is no infinitely distant timeperiod, no optimum exists.
Perhaps there is a trick with the indeces. $c_0$ seems ill defined unless there is a $k_{-1}$. If consumption only starts at $t = 1$, the above conditions are satisfied and the consumer is rational then $$\forall t: \ c_t=\bigg(\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1\bigg)k_t$$ defines the optimal consumption path as I assume $k_0$ is given and $$\forall t: \ k_t = k_0 \cdot (\beta R)^{\frac{t}{\gamma}}.$$ Hence $$\forall t: \ c_t=\bigg(\frac{R}{(\beta R)^{\frac{1}{\gamma}}}-1\bigg) \cdot k_0 \cdot (\beta R)^{\frac{t}{\gamma}}.$$ It is straightforward to check that this path is feasible (if the conditions set out in a) are met) and balanced.