> **Remark:** More generally there is a class of density functions

> that possesses this feature, i.e.

>

>

> $$

> \exists g: \mathbb{R}_+ \mapsto \mathbb{R}_+ \ \ \text{ and } \ \ c \geq 0,

> \ \ \text{s.t. the density } \ \ f \ \ \text{of} \ \ Z \ \

> \text{ has the form } \quad f(z) = c g(z\cdot z)

> $$

>

\exists g: \mathbb{R}_+ \mapsto \mathbb{R}_+ \ \ \text{ and } \ \ c \geq 0,

\ \ \text{s.t. the density } \ \ f \ \ \text{of} \ \ Z \ \

\text{ has the form } \quad f(z) = c g(z\cdot z)

$$

>

> This property is called **spherical symmetry** (see p 81. in Leamer

> (1978) {cite}`leamer1978specification`).

1. choose an initial $(w_0, \theta_0) \in S$;

1. generate a sustainable outcome recursively by iterating on {eq}`chang501`, which we repeat here for convenience:

> $$

> \begin{aligned}

> \hat h_t & = h(w_t,\theta_t) \\

> m_t & = m(h_t, w_t,\theta_t) \\

> x_t & = x(h_t, w_t,\theta_t) \\

> w_{t+1} & = \chi(h_t, w_t,\theta_t) \\

> \theta_{t+1} & = \Psi(h_t, w_t,\theta_t)

> \end{aligned}

> $$

>

>

$$

\begin{aligned}

\hat h_t & = h(w_t,\theta_t) \\

m_t & = m(h_t, w_t,\theta_t) \\

x_t & = x(h_t, w_t,\theta_t) \\

w_{t+1} & = \chi(h_t, w_t,\theta_t) \\

\theta_{t+1} & = \Psi(h_t, w_t,\theta_t)

\end{aligned}

$$

## Calculating the Set of Sustainable Promise-Value Pairs

Choose $\{c_t, s_t, i_t, h_t, k_t, g_t\}_{t=0}^\infty$ to maximize

> $$

> - \frac{1}{2}\mathbb{E}\sum_{t=0}^\infty \beta^t [(s_t-b_t)\cdot(s_t - b_t) + g_t \cdot g_t]

> $$

>

>

$$

-\frac{1}{2}\mathbb{E}\sum_{t=0}^\infty \beta^t [(s_t-b_t)\cdot(s_t - b_t) + g_t \cdot g_t]

$$

subject to the linear constraints