Bellman Equation

\[V(y) = \underset{x}{max} F(x,y) + \beta v(y)\]

Technologies in Neoclassical Growth

Investment: \[k_(t+1) = (1\delta)k_t + s_t\]
Production: \[y = f(k,l)\]
Output: \[c_t + i_t \le y_t\]

ArrowDebreu Equilibrium

TimeZero Trading
\[\sum_{t=0}^{\infty}p_t * c_t \le [\sum_{t=0}^{\infty}p_t * e_t \forall i\]

Sequential Trading Equilibrium

\[c_t + q_t*a_{t+1} \le e_t + a_t\]
\[a_{t+1} \le A_t\]
for each agent i

Natural Borrowing Constraint

\[A_t = \sum_{t=\tau}^{\infty} p_t * e_t\]

ArrowDebreu Market Clearance

\[c_t = e_t\] for each agent i

Sequential Market Clearance

\[c_t = e_t\] for each agent
\[a_{t+1} =0\] across all agents

Neoclassical Firm Optimization

\[\underset{y_t, k_t, n_t}{max} \sum_{t=0}^{\infty} p_t[y_t  r_t*k_t  w_t*l_t\]
s.t. \[y_t \le f(k_t,l_t)\]

Neoclassical Household Optimization

\[\underset{c_t, k_{t+1}}{max} sum_{t=0}^{\infty} beta^t*c^t
s.t:
\[sum_{t=0}^{\infty} p_t[c_t+i_t] \le sum_{t=0}^{\infty} p_t[r_t*k_t + w_t*l_t] + \pi \]
\[k_{t+1} = k_t(1delta_ + i_t\]
\[k_0\] given
\[c_t \ge 0, k_{t+1} \ge 0\]

Neoclassical Market Clearance

\[y_t = c_t + i_t\]
\[n_t^s = n_t^d\]
\[k_t^s = k_t^s\]

\[\Gamma (x)\]

Constraint space of state variable \[(\{(c_t, k_{t+1}) in R^2: c_t +k_{t+1} <= f(k_t)\}\]

\[\Pi (x)\]

Set of feasible plans: \[\{\bar{x}: x_{t+1} \in
\gamma( x_t) \forall t\}\]

Assumptions to map Sequential Problem to Bellman Equation (i.e. the solution to the SP is a solution to the FE)

1. \[\gamma (x)\] is nonempty
2. \[\lim_{t \to \infty} \sum_{t=0}^T beta^t F(x_t, x_{t+1}\] exists for all \[x_0 \in X\] and \[\bar{x} \in \Pi(x_0)\]

Assumption to map Bellman Equation to Sequential Problem

\[lim_{n \to \infty} beta^n v(x_n) =0 \forall x_0 \in X \ and \ x_bar \in Pi(x_0)\]

[\G(x)\]

Policy correspondence: \[\{y \in \gamma (x) \subseteq X: v(x) = F(x,y) + v(y)\}\]

Assumptions to make the Bellman Equation a Fixed Point Problem

1. X is convex; \[\gamma: X \to X\] is compactvalued, continuous, and nonempty
2. \[F:X x X \to R\] is continuous and bounded; \[\beta \in (0,1)\]

Contraction mapping

\[(S,\rho)\] is a metric space and \[T: S\to S\]. T is a contraction mapping with mod \[\beta\] if
\[\rho(Tx, Ty) \le \rho\beta(x,y)\]

Blackwell's Sufficient Conditions

1. Monotonic: \[f(x) \le g(x) \implies Tf(x) \le Tg(x)\]
2. Discounting: There exists \[\beta \in (0,1)\] s.t.
\[[T(f+a)(x)]\le (Tf)(x) + \beta a\]

Properties of v and g

1. V is strictly increasing
2. V is strictly concave
3. G is continuous and singlevalued
4. V is differentiable

Assumptions for V to be strictly increasing

1. For each y, \[F(x,y)\] is strictly increasing in X
2. \[x\le x^\prime\] implies \[\gamma (x) \subseteq \gamma (x^\prime)\]

Assumptions for concavity of V

1. F is strictly concave
2. \[\gamma\] is convex

Neoclassical Growth assumptions on F

1. Continuous
2. \[f(0) = 0\] and there exists \[\bar{x} >0\] s.t. \[f(\bar{x}) = \bar{x}\] (fixed point)
3. Strictly increasing
4. weakly concave
5. Continuously Differentiable

Neoclassical Growth assumptions on u

1. u(0) is finite
2. Continuous
3. Strictly Increasing
4. Strictly concave
5. Continuously Differentiable

Guess and Verify Method

1. Assume functional form.
2. Substitute into FE
3. Take FOCs and find Policy Function.
4. Substitute back into FE for closedform constants.
5. Get closed form Policy Function

VonNeumannMorgenstern Preferences

\[U(c^i) = \sum_{t=0}^{\infty} \sum_{s\inS} \beta^t \pi(s^t) u(c_t^i(s^t))\]

Arrow Securities

\[a_{t+1}^i (s^t, s_{t+1}) \in R\]
The number of units of consumption an agent purchases in a given history contingent on a future realization of a state.

Aggregate State Budget Constraint

\[c_t^i(s^t) + \sum_{s_{t+1}\in S} q_t a^i_{t+1} \le e_t^i(s^t) + a_t^i(s^t)\]

Stochastic natural debt limit

\[a_{t+1}^i(s^t, s_{t+1}) \le A^i(s^t, s_{t+1})\]

Stochastic Sequential Trading Equilibrium

Allocations of consumptions and assets, and arrow prices such that agents optimize subject to budget constraint and debt limit, and markets clear:
\[c^i(s^t) = e^i(s^t)\]
\[a_{t+1}^i(s^t, s_{t+1}) = 0\]

Stochastic Contracting Mapping Assumptions

1. Gamma is nonempty and measurable given some probability space.
2. F is measurable and bounded, and the limit of the sum of its lifetime expectations exists (i.e. there is a finite lifetime utility value)

Big K, Little k Trick

K as aggregate capital stock, k as individual
Price systems as functions of K s.t.
\[V(k,K) = u(R(K)k + w(K)n  k^\prime) + \beta V(k^\prime, K^\prime)\] s.t. \[K^\prime = G(K)\]

Recursive Competitive Equilibrium

Set of pricing functions, law of motion for aggregate capital, household value functions and decision rule, and firm decision rules such that:
1. Given prices and the law of motion, the value and policy function solve the optimization problem.
2. Given prices, firms optimize
3. Markets clear: Capital and labor supply and demand, and consistency between policy function and law of motion.

Rational Expectations

\[g(K,K) = G(K)\]
Consistency Condition

Stochastic NGM

Stochastic parameters are state variables, and choice variables must be defined in terms of them.

Intratemporal Labor Decision

\[u_c(c, l)F_n(k,n) = u_l(c,l)\]

Search Model

\[v_0(w) = max\{v_1(w), b + \beta \int v_o(w^\prime) dF(w^\prime)\}\]
\[v_1(w) = \frac{w}{1\beta}\]

Reservation Wage

\[v_1(w^*) = b + \beta \int v_0(w^\prime)dF(w^\prime)\]
\[w^*  b = \frac{\beta}{1\beta}\int_{w^*}^{\Bar{w}} (w^\prime  w^*) dF(w^\prime)\]

Value functions with Separations

\[v_1(w) = w + \beta(1\delta) v_1(w) + \beta\delta[b+ E[V_0(w)]]\]
\[v_0(w) = max\{v_1(w), b + E[v_0(w)]\}\]
