Calculating Time to Balanced Growth Path

Question

$\textbf{Model:}$ $$\underset{\{c_t,k_t\}}{max}\;\sum_{t=0}^\infty\beta^t\frac{c_t^{1-\gamma}}{1-\gamma}$$ $$s.t.\;c_t=Rk_{t-1}-k_t$$ $$c_t,k_t\geq0$$ 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?

Answer 1

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.

b) On reaching the balanced consumption path:
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.

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A note on reaching the balanced consumption path: This concept usually exists when there is some conflict between balancedness and optimality or when the consumer is boundedly rational. (E.g. Solow model.) Here this does not seem to be the case. However when such conflicts arise there is usually a never-ending (Is this the correct spelling of this word? Please correct if wrong.) convergence towards the balanced path. The distance decreases in every period but never becomes 0.