Formal MDP: Unemployment Branching Portfolio-Consumption¶

Source: packages/dolo/examples/models/doloplus/unemployment_branching_doloplus/

Model¶

An agent enters the period with investable assets \(k \in \mathbb{R}_{++}\). Employment status is then realized exogenously:

with probability \(p_e\), the agent is employed and enters the worker_port stage,
with probability \(1-p_e\), the agent is unemployed and enters the unemployed_port stage.

In the employed branch, the agent chooses a risky portfolio share \(\varsigma \in [0,1]\) before the joint shocks \((\Psi_w, \theta_w)\) are realized. Cash-on-hand is

\[ m = k\bigl(\varsigma \Psi_w + (1-\varsigma)R\bigr) + \theta_w. \]

In the unemployed branch, the agent chooses a risky portfolio share \(\varsigma \in [0,1]\) before a single placeholder return shock \(\Psi_u\) is realized. There is no labour-income shock in this branch, so cash-on-hand is

\[ m = k\bigl(\varsigma \Psi_u + (1-\varsigma)R\bigr). \]

After either branch, the period fans in to a common consumption stage. The agent chooses consumption \(c \in (0,m]\), end-of-period assets are

\[ a = m - c, \]

and the next period is reached via the inter-period twister

\[ \tau : a \mapsto k. \]

That is, the current continuation field a is renamed to the next-period arrival field k; this is a wiring map, not a new economic transition equation.

Preferences are CRRA:

\[ u(c) = \frac{c^{1-\rho}}{1-\rho}, \]

with discount factor \(\beta \in (0,1)\) and risk-aversion parameter \(\rho > 0\).

Perch-Value Bellman Factorization¶

Let \(\mathrm{v}_{\prec}^{+}(k)\) denote the next-period arrival value function. Under the inter-period twister \(\tau : a \mapsto k\), the continuation value entering the current consumption stage is

\[ \mathrm{v}_{\succ}(a) \coloneqq (\tau^{*}\mathrm{v}_{\prec}^{+})(a) = \mathrm{v}_{\prec}^{+}(\tau(a)). \]

This is the canonical pullback rule from the syntax-semantic composition pages: the predecessor stage consumes a continuation value \(\mathrm{v}_{\succ}\), while the successor period provides the arrival value \(\mathrm{v}_{\prec}^{+}\).

Let \(\mathrm{v}_{\mathrm{cons}}\), \(\mathrm{v}_{\mathrm{worker}}\), and \(\mathrm{v}_{\mathrm{unemployed}}\) denote the decision-perch values of the three non-trivial stages. Then:

\[ \mathrm{v}_{\mathrm{cons}}(m) = \max_{0 < c \le m} \left\{ \frac{c^{1-\rho}}{1-\rho} + \beta \mathrm{v}_{\succ}(a) \right\}, \qquad a = m-c. \]

\[ \mathrm{v}_{\mathrm{worker}}(k) = \max_{\varsigma \in [0,1]} \mathbb{E}_{\Psi_w,\theta_w}\left[\mathrm{v}_{\mathrm{cons}}\!\left(k(\varsigma \Psi_w + (1-\varsigma)R) + \theta_w\right)\right]. \]

\[ \mathrm{v}_{\mathrm{unemployed}}(k) = \max_{\varsigma \in [0,1]} \mathbb{E}_{\Psi_u}\left[\mathrm{v}_{\mathrm{cons}}\!\left(k(\varsigma \Psi_u + (1-\varsigma)R)\right)\right]. \]

\[ \mathrm{v}(k) = p_e\, \mathrm{v}_{\mathrm{worker}}(k) + (1-p_e)\, \mathrm{v}_{\mathrm{unemployed}}(k). \]

Stage Decomposition¶

Employment status¶

This is a nature-controlled branching stage with arrival state \(x_{\prec} = \{k\}\), decision state \(x = \{k\}\), and branch-keyed continuation states

\[ x_{\succ,\mathrm{worker}} = \{k_w\}, \qquad x_{\succ,\mathrm{unemployed}} = \{k_u\}. \]

In the syntax-semantic reading, the equality blocks implicitly define

\[ \mathrm{g}_{\prec\sim}^{\mathrm{empl}}(k_{\prec}) = k_{\prec}, \qquad \mathrm{g}_{\sim\succ,\mathrm{worker}}^{\mathrm{empl}}(k) = k, \qquad \mathrm{g}_{\sim\succ,\mathrm{unemployed}}^{\mathrm{empl}}(k) = k. \]

Equivalently, in SYM surface syntax,

\[ k = k[<], \qquad k_w[>] = k, \qquad k_u[>] = k. \]

The branch-specific continuation-value family is \(\left(\mathrm{v}_{\succ,\mathrm{worker}}, \mathrm{v}_{\succ,\mathrm{unemployed}}\right)\), so the backward mover is the probability-weighted aggregation

In YAML surface syntax, this is the branch family declared by values.V[>] = {worker: V_worker, unemployed: V_unemployed}, so the mover reads as V = p_e*V[>][worker] + (1-p_e)*V[>][unemployed].

\[ \mathrm{v}(k) = p_e\, \mathrm{v}_{\succ,\mathrm{worker}}(k) + (1-p_e)\, \mathrm{v}_{\succ,\mathrm{unemployed}}(k), \qquad \mathrm{v}_{\prec}(k) = \mathrm{v}(k). \]

Worker portfolio¶

This stage has arrival state \(\{k_w\}\), control \(\varsigma\), post-decision shocks \((\Psi_w,\theta_w)\), and continuation state \(\{m\}\).

Its transition equalities define the implicit maps

\[ \mathrm{g}_{\prec\sim}^{\mathrm{worker}}(k_w) = k_w, \]

\[ \mathrm{g}_{\sim\succ}^{\mathrm{worker}}(k_d,\varsigma,\Psi_w,\theta_w) = k_d(\varsigma \Psi_w + (1-\varsigma)R) + \theta_w. \]

Equivalently, the SYM equations are \(k_d = k_w\) and \(m = k_d(\varsigma \Psi_w + (1-\varsigma)R) + \theta_w\).

Because the shocks are realized after the portfolio decision, the expectation sits inside cntn_to_dcsn_mover, not in dcsn_to_arvl_mover. The backward recursion is

\[ \mathrm{v}_{\mathrm{worker}}(k_d) = \max_{\varsigma \in [0,1]} \mathbb{E}_{\Psi_w,\theta_w}\left[ \mathrm{v}_{\succ}\!\left(\mathrm{g}_{\sim\succ}^{\mathrm{worker}}(k_d,\varsigma,\Psi_w,\theta_w)\right) \right], \qquad \mathrm{v}_{\prec,\mathrm{worker}}(k_w) = \mathrm{v}_{\mathrm{worker}}(k_w). \]

Unemployed portfolio¶

This stage has arrival state \(\{k_u\}\), control \(\varsigma\), shock \(\Psi_u\), and continuation state \(\{m\}\).

Its transition equalities define

\[ \mathrm{g}_{\prec\sim}^{\mathrm{unemployed}}(k_u) = k_u, \]

\[ \mathrm{g}_{\sim\succ}^{\mathrm{unemployed}}(k_d,\varsigma,\Psi_u) = k_d(\varsigma \Psi_u + (1-\varsigma)R). \]

Equivalently, the SYM equations are \(k_d = k_u\) and \(m = k_d(\varsigma \Psi_u + (1-\varsigma)R)\).

Its backward recursion is

\[ \mathrm{v}_{\mathrm{unemployed}}(k_d) = \max_{\varsigma \in [0,1]} \mathbb{E}_{\Psi_u}\left[ \mathrm{v}_{\succ}\!\left(\mathrm{g}_{\sim\succ}^{\mathrm{unemployed}}(k_d,\varsigma,\Psi_u)\right) \right], \qquad \mathrm{v}_{\prec,\mathrm{unemployed}}(k_u) = \mathrm{v}_{\mathrm{unemployed}}(k_u). \]

Consumption¶

This stage has arrival state \(\{m\}\), control \(c\), and continuation state \(\{a\}\).

Its transition equalities define

\[ \mathrm{g}_{\prec\sim}^{\mathrm{cons}}(m) = m, \qquad \mathrm{g}_{\sim\succ}^{\mathrm{cons}}(m_d,c) = m_d - c. \]

Equivalently, the SYM equations are \(m_d = m\) and \(a = m_d - c\).

Its backward mover is

\[ \mathrm{v}_{\mathrm{cons}}(m_d) = \max_{0 < c \le m_d} \left\{ \frac{c^{1-\rho}}{1-\rho} + \beta \mathrm{v}_{\succ}(a) \right\}, \qquad \mathrm{v}_{\prec,\mathrm{cons}}(m) = \mathrm{v}_{\mathrm{cons}}(m). \]

The EGM representation uses the marginal continuation value \(\partial_a \mathrm{v}_{\succ}(a)\) and the inverse map associated with \(\mathrm{g}_{\sim\succ}^{\mathrm{cons}}\):

\[ c(a) = \left(\beta\,\partial_a \mathrm{v}_{\succ}(a)\right)^{-1/\rho}, \]

and

\[ \left(\mathrm{g}_{\sim\succ}^{\mathrm{cons}}\right)^{-1}(a,c) = a + c. \]

Composition and Wiring¶

The period wiring is

employment_status -> worker_port      -> cons_stage
employment_status -> unemployed_port  -> cons_stage

The branching stage fans out via the disjoint continuation fields k_w and k_u. The two portfolio stages then fan in to the shared arrival field m of cons_stage. In the syntax-semantic composition rules, this is an identity composition on a common arrival space: both upstream stages expose the same continuation field m, so no intra-period connector is needed.

The inter-period twister is the standard rename

{a: k}

so the next period starts again from investable assets k. This twister induces the pullback \(\mathrm{v}_{\succ} = \tau^{*}\mathrm{v}_{\prec}^{+}\) used in the Bellman factorization above.

Calibration¶

Symbol	Value	Description
\(\beta\)	0.96	discount factor
\(\rho\)	2.0	CRRA risk aversion
\(R\)	1.02	gross risk-free return
\(p_e\)	0.95	employment probability
\(\mu_{\Psi_w}\)	0.04	worker risky-return log-mean
\(\sigma_{\Psi_w}\)	0.15	worker risky-return log-std
\(\mu_{\theta_w}\)	0.0	worker labour-income log-mean
\(\sigma_{\theta_w}\)	0.10	worker labour-income log-std
\(\rho_{\zeta_w}\)	0.0	worker shock-correlation placeholder
\(\mu_{\Psi_u}\)	0.04	unemployed risky-return log-mean
\(\sigma_{\Psi_u}\)	0.15	unemployed risky-return log-std