# Social Planner vs Representative Household

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.

• Can you write down what these problems are meant to be? Commented May 12 at 17:51
• @MichaelGreinecker I edited. Commented May 12 at 18:15
• Typically the social planner's problem includes welfare weights such that your utility function would be: $$\sum_{i=1}^n \lambda_i \sum_{t=0}^\infty \beta^t U(c_t^i, 1-l_t^i)$$ Though I think this would be isomorphic to maximizing each of their individual lifetime utilities(?) Commented May 12 at 21:43
• @Brennan yes but under some conditions I think Commented May 12 at 21:53
• Symmetric weights $\lambda_i = \lambda_j$ for $i\neq j$, surely. Not sure what else is required. Also, having riskless bonds isn't necessarily the same as having access to capital. For example you ensure capital is non-negative but are agnostic about the lower bound of assets, which could be negative implying the individual can borrow. I imagine this is just a mistake though, $R_t$ can just be the interest less the depreciation. Commented May 13 at 4:50