Social Planner vs Representative Household

Question

Is there any difference between solving the social planner's problem or the representative household's in the Ramsey-Cass-Koopmans model?

The social planner chooses a plan $\left\{c_t, l_t, k_{t+1}\right\}_{t=0}^{\infty}$ so as to maximize utility subject to the resource constraint of the economy, taking initial $k_0$ as given: $$ \begin{gathered} \max \mathcal{U}_0=\sum_{t=0}^{\infty} \beta^t U\left(c_t, 1-l_t\right) \\ c_t+k_{t+1} \leq(1-\delta) k_t+F\left(k_t, l_t\right), \quad \forall t \geq 0 \\ c_t \geq 0, \quad l_t \in[0,1], \quad k_{t+1} \geq 0 ., \quad \forall t \geq 0 \\ k_0>0 \text { given } \end{gathered} $$ This is called the social planner's problem.

Given a price sequence $\left\{R_t, w_t\right\}_{t=0}^{\infty}$, household $j$ chooses a plan $\left\{c_t^j, l_t^j, k_{t+1}^j\right\}_{t=0}^{\infty}$ so as to maximize lifetime utility subject to its budget constraints $$ \begin{gathered} \max \mathcal{U}_0^j=\sum_{t=0}^{\infty} \beta^t U\left(c_t^j, 1-l_t^j\right) \\ \text { s.t. } c_t^j+a_{t+1}^j \leq\left(1+R_t\right) a_t^j+w_t l_t^j \\ c_t^j \geq 0, \quad l_t^j \in[0,1], \quad a_{t+1}^j \geq \underline{a}_{t+1} \end{gathered} $$ This is called the household's problem.

Answer 1

If we assume that there is a single household (there are some subtleties in how one defines a representative household), then the Pareto optimality is the same as solving the planner's problem.

The first welfare theorem gives conditions under which every competitive equilibrium is Pareto optimal, and the second welfare theorem gives conditions under which every Pareto optimum is a competitive equilibrium after a suitable redistribution of endowments. With one agent, there is nothing to redistribute.

There are two caveats in how the welfare theorems differ from the usual models with finitely many commodities known from microeconomics courses. To make the proof of the first welfare theorem go through, you have to be able to rewrite the recursive problem as a sequence problem. In particular, prices have to be such that total values are finite. For the second welfare theorem, one needs some more technical assumptions to ensure that the infinite-dimensional version of the separating hyperplane theorem, the geometric Hahn-Banach theorem, applies. For suitable assumptions and a proof, you can take a look at Chapter 5 of Acemoglu's "Introduction to Modern Economic Growth."