# How do we solve for initial consumption in the Ramsey model?

How do we solve for the initial consumption level in the Ramsey model? Assume we have the Euler equation $$\frac {\dot c_t}{c_t}=r_t-\rho$$ Then $$c_t=c_0e^{R_t-\rho t}$$ Putting this into the lifetime budget constraint: $$\int_0^\infty e^{-R_t}(w_t-c_t)dt=k_0$$ gives $$\int_0^\infty e^{-R_t}w_t dt-\frac {c_0} {\rho}=-k_0$$

Therefore $$c_0=\rho\left(\int_0^\infty e^{-R_t}w_t dt+k_0\right)$$

Where we know that $w_t=f(k_t)-k_t r_t$ and that $r_t=f'(k_t)$, and $R_t=\int_0^t r_\tau d\tau$.

However, the problem is that the initial amount of consumption will influence the path of $k_t$, because $\dot k_t=f(k)-c_t$, and therefore of $w_t$ and $r_t$ as well. Therefore it seems to me that solving for $c_0$ is almost impossible here.

So let's say we would have a computer that could help us solve this, how would we do it? e.g. how does Dynare/matlab do it?

• what is $r_t$ and $R_t$? How are they related? Commented Dec 8, 2017 at 11:50
• @Dmitry, sorry forgot that one. Edited. Commented Dec 8, 2017 at 17:55

An algorithm to solve for equilibrium in a steady state (so that $\dot k_t=0$) is as follows:
1. Guess $k^*$, the steady state level of capital;
2. Compute steady state prices, $r^*$ and $w^*$;
3. Compute steady state consumption, $c^*$;
4. Check the market clearing condition $f(k^*)-c^*=0$;
5. Adjust your guess for $k^*$ and repeat until $f(k^*)-c^*$ is close enough to zero.
Out of steady state, the procedure is similar. You should first obtain the analogous formulas in discrete time and you have to start from some $k_0$ which is set exogenously. Then, compute $k^*$ following the algorithm above and guess a path for $\{k_t\}_{t=0}^\infty$ which converges to $k^*$ after $T$ periods. Compute prices for every period, consumption for every period, then check the market clearing at every period and use it to adjust the guess for $\{k_t\}_{t=0}^\infty$. Once this converges, increase $T$ and repeat until the path stops changing.