<p>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}
&amp; \max_{h_i} h^{\alpha_i}l_i^{1-\alpha_i} \\
&amp; \text{s.t.} \quad h_i + l_i = \bar{T} \\
&amp; 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
$$
h_1 = h_2 = \min_i \alpha_i \cdot \overline{T}.
$$</p>

CC BY-SA 3.0
2016-03-06T23:18:51.227Z
2016-03-06T23:49:24.357Z
2016-03-06T23:49:24.357Z
Giskard
<p>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:</p>

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

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

<p>(Note that typical Inada conditions and non-negativity constraints apply.)</p>

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

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

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

CC BY-SA 3.0
2016-03-06T22:57:02.997Z
Do Inada conditions apply?
ChinG
CC BY-SA 3.0
2016-03-06T23:01:00.530Z
Yes, I'll add a note on that.
Kitsune Cavalry
<utility><public-economics><dynamic-programming>
Joint Dynamic Programming: Group Activity
Kitsune Cavalry
# Joint Dynamic Programming: Group Activity

(Asked by: username_0 on 2016-03-06)

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 $\lambda \in (0,1)$

**Tags:** utility, public-economics, dynamic-programming

> **Comments:**
> **username_1**: Do Inada conditions apply?  
> **username_0**: Yes, I'll add a note on that.  

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## 1 answers

### Answer by username_2 on 2016-03-06. Score: 1 | Accepted answer

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
$$
h\_1 = h\_2 = \min\_i \alpha\_i \cdot \overline{T}.
$$


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