# 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

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

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).

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