I've got a model of endogenous growth due to spillovers.
$\textbf{Model:}$ $$K_t=\frac{1}{n}\sum_{t=1}^nk_t$$ In this model, $k_t$ is chosen by agents, and $K_t=\bar{k}_t$ (the average of all $k_t$).
Now, agents want to dynamically maximize utility (under certain constraints) and they have CRRA (constant relative risk aversion) utility, so the maximization looks like: $$\sum_{t=0}^\infty\beta^t\bigg(\frac{c_t^{1-\gamma}}{1-\gamma}\bigg)$$ $$s.t.\;Y_t=k_t^\alpha(E_tL)^{1-\alpha}$$ $$c_t+i_t=Y_t$$ $$k_{t+1}=(1-\delta)k_t+i_t$$ $$c_t,i_t\geq0$$
$E_tL$ is effective labor and the rest of the variables are typical (I can give their definitions if requested).
One last addition to the model is that there are two equilibrium constraints: $$E_t=\frac{K_t}{L}$$ $$k_t=K_t$$ $\textbf{Solution:}$
Using an Euler equation approach, two terms in the objective: $$...\frac{\beta^t[k_t^\alpha K_t^{1-\alpha}+(1-\delta)k_t-k_{t+1}]^{1-\gamma}}{1-\gamma}+\frac{\beta^{t+1}[k_{t+1}^\alpha K_{t+1}^{1-\alpha}+(1-\delta)k_{t+1}-k_{t+2}]^{1-\gamma}}{1-\gamma}...$$ FOC:
w.r.t $k_{t+1}$ and substituting in consumption:
$$\beta^tc_t^{-\gamma}=\beta^{t+1}c_{t+1}^{-\gamma}[\alpha k_{t+1}^{\alpha-1}K_{t+1}^{1-\alpha}+1-\delta]$$ Substituting in the equilibrium constraints: $$c_t^{-\gamma}=\beta c_{t+1}^{-\gamma}[\alpha+1-\delta]$$ $$\implies \frac{c_{t+1}}{c_t}=[\beta(\alpha+1-\delta)]^{\frac{1}{\gamma}}$$ This implies that consumption grows at a constant rate which depends on preference parameters.
The next thing I want to prove that we have a balanced growth path. By this I mean that all variables grow at the same constant rate. $$\frac{k_{t+1}}{k_t}=\frac{c_{t+1}}{c_t}?$$ I have started to answer the question, but I have gotten stuck. Here is what I have so far: $$k_{t+1}=k_t^\alpha K_t^{1-\alpha}+(1-\delta)k_t-c_t$$ In equilibrium $K_t=k_t$: $$k_{t+1}=k_t+(1-\delta)k_t-c_t$$ $$\implies \frac{k_{t+1}}{k_t}=1+(1-\delta)-\frac{c_t}{k_t}$$ If we have a constant rate of capital growth, suppose: $$\frac{k_{t+1}}{k_t}<\frac{c_{t+1}}{c_t}$$ If this is true: $$\underset{t\rightarrow \infty}{lim}\;\frac{k_{t+1}}{k_t}=\underset{t\rightarrow \infty}{lim}\;1+(1-\delta)-\frac{c_t}{k_t}=-\infty$$ This means that the growth rate will continue decreasing. Now there are two ways this can happen. The first way is if $\underset{t\rightarrow \infty}{lim}\;k_t=-\infty$, which would clearly show that it couldn't be the case because capital cannot be negative. The second way which this could happen is if $\underset{t\rightarrow \infty}{lim}\;k_t=D$ where $D$ is some positive horizontal asymptote. If it did approach some asymptote, there is no reason why it could not be the case that consumption grows at a faster rate than capital. This is where I am stuck. How can I show that it cannot converge to some asymptote, and that $\underset{t\rightarrow \infty}{lim}\;k_t=-\infty$? Also, if there is an easier way to show that this model exhibits balanced growth paths, what is it? Any help would be greatly appreciated!
