Joint Dynamic Programming: Group Activity

Question

Here we have two agents who can spend their time doing some group activity ($h$) or staying at home ($l$). Each agent $i$ is trying to maximize their respective dynamic programming problem:

\begin{align} & \max \sum^\infty_{t=0} \beta^t h^{\alpha_i}l_i^{1-\alpha_i} \\ & \text{s.t.} \quad h_i + l_i = \bar{T} \\ & h = \min\left\{h_i, h_{-i}\right\} \end{align}

• Is this problem well defined? That is, is there a solution/does the question make sense economically?
• Is there an equilibrium where $h_i = h_{-i}$, and if so, when? Only when $\alpha_i = \alpha_{-i}$?

(Note that typical Inada conditions and non-negativity constraints apply.)

Edit: Okay, to make this an actual dynamic programming question (derp), I offer a modified version of this question as well for your perusal:

\begin{align} & \max \sum^\infty_{t=0} \beta^t h_t^{\alpha_i}l_i^{1-\alpha_i}k_t\\ & \text{s.t.} \quad h_{i,t} + l_i = \bar{T} \\ & h_t = \min\left\{h_{i,t}, h_{-i, t}\right\} \\ & k_t = (1-\lambda + \frac{h_t}{\bar{T}})k_{t-1} \cdot \end{align}

Where $k$ is some "capital" stock in the group activity that makes people enjoy the activity more, the more it is done. Obviously $\lambda \in (0,1)$

Answer 1

If I understand correctly the temporal dimension is merely a distraction as agents face a separate optimization problem every round with no intertemporal budget constraint. Hence \begin{align} & \max_{h_i} h^{\alpha_i}l_i^{1-\alpha_i} \\ & \text{s.t.} \quad h_i + l_i = \bar{T} \\ & h = \min\left\{h_i, h_{-i}\right\} \end{align} A continuous best response function exists for each agent: $$ BR_i(h_{-i}) = \min\left(\alpha_i \cdot \bar{T}; h_{-i} \right) $$ The equilibrium is $$ h_1 = h_2 = \min_i \alpha_i \cdot \overline{T}. $$