Planner

Suppose now, instead, that a central planner is in charge of organising consumption and investment for all individuals. The planner operates by aggregating individual utility, discounting future generations at the rate $\gamma.$
Note: $\gamma$ is the discount rate of future generations, not how young people discount old-age utility.

Her utility considers the utility of all generations, including the initial old generation who owns the initial stock of capital $k_0.$

Planner’s utility

Planner’s utility reads:

$$\sum_{t=-1}^\infty \gamma^t U(c_t, d_{t+1}).$$

Note that, although the planner can decide any allocation, she must respect the resource constraint:

$$f(k_t) + (1-\delta) k_t = c_t + \frac{1}{1+n} d_t + (1+n)k_{t+1}.$$

Assume that $U(c_t, d_{t+1})$ is separable: $U(c_t, d_{t+1}) = u(c_t) + \beta u(d_{t+1}).$ Expanding the utility of the planner, we can reformulate it in more convenient terms:

$$\sum_{t=-1}^\infty \gamma^t \left(u(c_t)+\beta u(d_{t+1}) \right) = $$ $$\gamma^{-1}u(c_{-1}) +\gamma^{-1}\beta u(d_0) + \gamma^0 u(c_0)+ \gamma^0 \beta u(d_1) + \gamma^1 u(c_1) + \gamma^1 \beta u(d_2) + \ldots $$ $$= \sum_{t=0}^\infty \gamma^t \left(u(c_t) + \frac{\beta}{\gamma} u(d_t) \right) + \gamma^{-1}u(c_{-1}).$$ The term $\gamma^{-1}u(c_{-1})$ represents the consumption of the generation born at $t=-1$, but since it is a constant it will not affect the maximisation.

Hence, the planner’s problem is now how to allocate consumption between the young and old that are alive during period $t$.

Maximisation

We use a substitution to obtain the planner’s optimal allocation, namely, the Euler equation;

$$\begin{eqnarray} &\max_{k_{t+1}, c_t, d_t} \sum_{t=0}^\infty \gamma^t \left(u(c_t) + \frac{\beta}{\gamma} u(d_t)\right) \\\ &\mathrm{s.t.}, f(k_t) + (1-\delta)k_{t} = c_t + \frac{1}{1+n}d_t + (1+n)k_{t+1}. \end{eqnarray}$$

$$\max_{c_t, k_{t+1}} \sum_{t=0}^\infty \gamma^t \left( u(c_t) + \frac{\beta}{\gamma}u\Bigg(\underbrace{\left[1+n\right]\left[f(k_t)+(1-\delta)k_t -(1+n)k_{t+1} - c_t\right]\Bigg)}_{d_t}\right).$$

Taking derivatives and equating them to zero yields:

$$\gamma^t u^\prime(c_t) = \gamma^t \frac{\beta}{\gamma} u^\prime(d_t)(1+n)$$ $$\gamma^t\frac{\beta}{\gamma}u^\prime (d_t)(1+n)(1+n)=\gamma^{t+1}\frac{\beta}{\gamma}(1+n)u^\prime (d_{t+1})\left(f^\prime (k_{t+1}) + 1 - \delta \right).$$

Combining both equations we get the Euler equation:

$$u^\prime (c_t) = \beta u^\prime (d_{t+1})\left(f^\prime (k_{t+1}) + 1 -\delta \right).$$

The planner’s Euler equation coincides with the decentralised one, where we had $R_{t+1} = f^\prime(k_{t+1}) + 1 -\delta.$

However, the planner also allocates consumption between the young and the old at time $t$. $$\gamma^t u^\prime(c_t) = \gamma^t \frac{\beta}{\gamma} u^\prime(d_t)(1+n)$$ In the centralised equilibrium, individuals do not arbitrage between young and old consumption at time $t.$ We missed this equation because individuals are short-sighted, and only derive utility while alive. Hence, they have no interest in trading off utility with the young generation once they are old.

Steady state and modified golden rule

Using the fact that $$\gamma^t\frac{\beta}{\gamma}u^\prime (d_t)(1+n)(1+n)=\gamma^{t+1}\frac{\beta}{\gamma}(1+n)u^\prime (d_{t+1})\left(f^\prime (k_{t+1}) + 1 - \delta \right)$$ we can easily characterise the steady state knowing that $d_t = d_{t+1} = \bar{d}, k_t = k_{t+1} = \bar{k}.$

$$f^\prime (\bar{k}) = \frac{1+n}{\gamma}+1-\delta.$$

This equation provides us with the modified golden rule: the level of capital that maximises the planner’s utility. Clearly, if $\gamma=1$, the planner attributes the same weight to all generations and we recover the golden rule: $f^\prime(k) = n + \delta.$ As we discussed before, it is quite unlikely the decentralised equilibrium converges towards the golden rule (modified or not).