# The Solow Model

# Models of Optimal Growth

# Finite periods Model - planner's problem

$$\max_{\{c_t,k_{t+1}\}_{t=0}^T} \sum_{t=0}^T \beta^t u(c_t) \\ \text{s.t. } c_t+k_{t+1} \leq F(k_t,l) + (1-\delta)k_t,\ t=0,\dots,T \\ c_t \geq 0,\ k_{t+1} \geq 0,\ k_0 \text{ is given}.$$

# Infinite periods Model - planner's problem

$$\max_{\{c_t,k_{t+1}\}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u(c_t) \\ \text{s.t. } c_t+k_{t+1} \leq F(k_t,l) + (1-\delta)k_t,\ t=0,1,2\dots \\ c_t \geq 0,\ k_{t+1} \geq 0,\ k_0 \text{ is given}, \\ \lim_{t\rightarrow\infty} \beta^t u'(c_t)f'(k_t)k_t = 0 \text{ (TVC)}.$$

# Recursive Analysis

$$V(k_t) = \max_{\{k_{s+1}\}_{s=t}^{\infty}} \sum_{s=t}^{\infty} \beta^{s-1} F(k_s,k_{s+1}) \\ \text{s.t. } k_{s+1} \in \Gamma(k_s),\ \forall s \geq t$$

The problem is equavalent with:

$$V(k_t) = \max_{k_{t+1} \in \Gamma(k_t)} \left\{F(k_t,k_{t+1}) + \max_{k_{s+1} \in \Gamma(k_s),\ k_{s+1}\in \Gamma(k_s),\ \forall s \geq t} \sum_{s=t+1}^{\infty} \beta^{s-t} F(k_{s+1},k_{s+2})\right\} \\ \Leftrightarrow V(k_t) = \max_{k_{t+1} \in \Gamma(k_t)} \left\{F(k_t,k_{t+1}) + \beta V(k_{t+1})\right\} \\ \Leftrightarrow V(k) = \max_{k' \in \Gamma(k)} \left\{F(k,k') + \beta V(k')\right\} \text{ (Bellman Equation)}\\$$

and

$$k'^{*} = g(k) = \argmax_{k'\in \Gamma(k)} \left\{F(k,k') + \beta V(k')\right\}$$

# Competitive Equilibrium Models

# Endowment Economy Model

# Date-0 Trade Model

Definiton: A date-0 competitive equilibrium is a set of quantities $\{c_t^*\}_{t=0}^{\infty}$ and prices $\{p_t\}_{t=0}^{\infty}$ such that

• $\{c_t^*\}_{t=0}^{\infty}$ solves the household's problem:

$$\max_{\{c_t\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^t u(c_t) \\ \text{s.t. } \sum_{t=0}^{\infty}p_t c_t \leq \sum_{t=0}^{\infty} p_t w_t,\ c_t \geq 0, \forall t$$

• In equilibrium: $c_t^* = w_t,\ \forall t$.

# Sequential Trade Model

Definition: A sequential competitive equilibrium is a set of sequence $\{c_t^*,a_{t+1}^*\}_{t=0}^{\infty}$ and rates $\{R_t\}_{t=0}^{\infty}$ such that

• $\{c_t^*,a_{t+1}^*\}_{t=0}^{\infty}$ solves the household's problem:

$$\max_{\{c_t,a_{t+1}\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^t u(c_t) \\ \text{s.t. } c_t+a_{t+1} \leq a_t R_t + w_t,\ c_t \geq 0,\ \forall t,\ a_0 = 0\text{ is given,}\\ \lim_{t \rightarrow 0} \left(\prod_{s=0}^t R_{s+1}\right)^{-1} a_{t+1} = 0$$

• In equilibrium: $c_t^* = w_t,\ a_t^* = 0,\ \forall t$.

# The Neoclassical Growth Model

# Date-0 Trade Model

Definiton: A date-0 competitive equilibrium is a set of quantities $\{c_t^*,n_t^*,k_{t+1}^*\}_{t=0}^{\infty}$ and prices $\{p_t,r_t,w_t\}_{t=0}^{\infty}$ such that

• $\{c_t^*,n_t^*,k_{t+1}^*\}_{t=0}^{\infty}$ solves the household's problem:

$$\max_{\{c_t,n_t,k_{t+1}\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^t u(c_t) \\ \text{s.t. } \sum_{t=0}^{\infty}p_t (c_t + k_{t+1}) \leq \sum_{t=0}^{\infty} p_t (r_t k_t + (1-\delta) k_t + w_t n_t),\ c_t \geq 0, \forall t,\ k_0\text{ is given}$$

• $\{n_t^*,k_t^*\}_{t=0}^{\infty}$ solves the firm's problem:

$$\max_{\{n_t,k_t\}_{t=0}^{\infty}} p_t F(k_t,n_t) - p_t r_t k_t - p_t w_t n_t$$

• Market clear: $c_t^* + k_{t+1}^*= F(k_t^*,n_t^*) + (1-\delta) k_t^*,\ \forall t$.

# Sequential Trade Model

Definiton: A sequential competitive equilibrium is a sequence $\{c_t^*,n_t^*,k_{t+1}^*,R_t,w_t\}_{t=0}^{\infty}$ such that

• $\{c_t^*,n_t^*,k_{t+1}^*\}_{t=0}^{\infty}$ solves the household's problem:

$$\max_{\{c_t,n_t,k_{t+1}\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^t u(c_t) \\ \text{s.t. } c_t+k_{t+1} \leq R_t k_t + w_t n_t,\ c_t \geq 0,\ \forall t,\ k_0 \text{ is given,}\\ \lim_{t \rightarrow 0} \left(\prod_{t=0}^\infty R_{t+1}\right)^{-1} k_{t+1} = 0$$

• $\{n_t^*,k_t^*\}_{t=0}^{\infty}$ solves the firm's problem:

$$\max_{\{n_t,k_t\}_{t=0}^{\infty}} F(k_t,n_t) - r_t k_t - w_t n_t$$

• Market clear: $c_t^* + k_{t+1}^*= F(k_t^*,n_t^*) + (1-\delta) k_t^*,\ \forall t$.

# N-Household Model

# Date-0 Trade Model

Definiton: A date-0 competitive equilibrium of the N-agent economy is a set of quantities $\{c_t^*,n_t^*,k_{t+1}^*,\{c_t^{i*},n_t^{i*},k_{t+1}^{i*}\}_{i=1}^N\}_{t=0}^{\infty}$ and prices $\{p_t,r_t,w_t\}_{t=0}^{\infty}$ such that

• $\{c_t^{i*},n_t^{i*},k_{t+1}^{i*}\}_{t=0}^{\infty}$ solves the household $i$'s problem for each $i = 1,\dots,N$:

$$\max_{\{c_t^{i},n_t^{i},k_{t+1}^{i}\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^t_{i} u(c_t^{i}) \\ \text{s.t. } \sum_{t=0}^{\infty}p_t (c_t^{i} + k_{t+1}^{i}) \leq \sum_{t=0}^{\infty} p_t (r_t k_t^{i} + (1-\delta) k_t^{i} + w_t n_t^{i}),\ c_t^{i} \geq 0 \text{ and } 0 \leq n_t^{i} \leq l_t^{i}, \forall t,\ k_0^{i}\text{ is given}$$

• $\{n_t^*,k_t^*\}_{t=0}^{\infty}$ solves the firm's problem:

$$\max_{\{n_t,k_t\}_{t=0}^{\infty}} p_t F(k_t,n_t) - p_t r_t k_t - p_t w_t n_t$$

• Market clear:
  • Aggregation: $n_t^* = \sum_{t=1}^N n_t^{i*}$, $k_t^* = \sum_{t=1}^N k_t^{i*}$, $c_t^* = \sum_{t=1}^N c_t^{i*}$;
  • $c_t^* + k_{t+1}^*= F(k_t^*,n_t^*) + (1-\delta) k_t^*,\ \forall t$.

# Government Debt and Tax Model

# The Neoclassical Recursive Model

Definition: A recursive competitive equlibrium is a set of functions: quantities $G(\bar{k}),g(k,\bar{k})$, value $V(k,\bar{k})$, prices $R(\bar{k}),w(\bar{k})$ such that

• $V(k,\bar{k})$ solves the household's problem, $k' = g(k,\bar{k})$ is the individual asscociated policy function:

$$V(k,\bar{k}) = \max_{c,k'}\left\{ u(c) + \beta V(k',\bar{k'})\right\} \\ \text{s.t. } c + k' = R(\bar{k})k + w(\bar{k}) l\\ \bar{k'} = G(\bar{k})$$

• Prices are competitive determined:
  • $R(\bar{k}) = F_1(\bar{k},l) + 1 - \delta$;
  • $w(\bar{k}) = F_2(\bar{k},l)$.
• Individual decisions are consistent with aggregator:
  • $G(\bar{k}) = g(\bar{k},\bar{k}),\ \forall \bar{k}$.

# Two Agents Endowment Model

Definition: A recursive competitive equlibrium of the two agents endowment economy is a set of functions: quantities $G(A_1), g_1(a_1,A_1), g_2(a_2,A_1)$, value $V_1(a_1,A_1), V_2(a_2,A_1)$, prices $q(A_1)$ such that

• $V_i(a_i,A_1)$ solves the type $i$ household's problem, $a_i' = g_i(a_i,A_1)$ is the individual asscociated policy function, $i=1,2$:

$$V_i(a_i,A_1) = \max_{c_i,a_i'} \left\{u_i(c_i)+\beta_i V_i(a_i',A_1')\right\} \\ \text{s.t. } c_i + a_i' q(A_1) = a_i + w_i \\ a_i' \geq a,\ A_1' = G(A_1)$$

• Consistency: $g_1(A_1,A_1) = G_(A_1)$, $g_2(A_1,A_1) = -G_(A_1)$

# Uncertainty and the Neoclassical Growth Model

# Planner's Problem

$$\max_{\{c_t(z^t),k_{t+1}(z^t)\}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t \sum_{z^t \in Z^{t+1}} \pi(z^t) u(c_t^t) \\ \text{s.t. } c_t(z^t)+k_{t+1}(z^t) \leq z_t F(k_t(z^{t-1}),1) + (1-\delta)k_t(z^{t-1}),\ t=0,1,2\dots \\ c_t(z^t) \geq 0,\ k_{t+1}(z^{t}) \geq 0,\ k_0 \text{ is given}$$

# Recursive Formulation

Assumption: $\{z_t\}_{t=0}^\infty$ is a first order Markov process, i.e. $\text{Prob}\{(z_{t+1},z^t)|z^t\} = \text{Prob}\{(z_{t+1},z^t)|z_t\}$.

$$V(k,z) = \max_{k'} \left\{u(z f(k) - k' + (1-\delta)k) + \beta \sum_{z'\in Z} \pi(z'|z) V(k',z')\right\}$$

and the associated policy function $k' = g(k,z)$.

# Competitive Equilibrium under Uncertainty

# Date-0 Trade Model

Definition: Arrow-Debreu date-0 trading competitive equilibrium is a sequence $\{c_t(z^t),k_{t+1},l_t(z^t),p_t(z^t),r_t(z^t),w_t(z^t)\}_{t=0}^\infty$ such that:

• $\{c_t(z^t),k_{t+1}(z^t),l_t(z^t)\}_{t=0}^\infty$ solves household's problem:

$$\max_{\{c_t(z^t),k_{t+1}(z^t),l_t(z^t)\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \sum_{z^t\in Z^{t+1}} \beta^t \pi(z^t) u(c_t(z^t),1-l_t(z^t)) \\ \text{s.t. } \sum_{t=0}^{\infty} \sum_{z^t\in Z^{t+1}} p_t(z^t) (c_t(z^t) + k_{t+1}(z^t)) \leq \sum_{t=0}^{\infty} \sum_{z^t\in Z^{t+1}} p_t(z^t) (r_t(z^t) k_t(z^t) + (1-\delta) k_t(z^{t-1}) + w_t(z^t) l_t(z^t))\\ c_t(z^t) \geq 0,\ k_{t+1}(z^{t}) \geq 0, \forall t,\ k_0\text{ is given}$$

• $\{l_t(z^t),k_t(z^{t-1})\}_{t=0}^{\infty}$ solves the firm's problem:

$$\max_{\{l_t(z^t),k_t(z^t)\}_{t=0}^{\infty}} z^t p_t(z^t) F(k_t(z^t),l_t(z^t)) - p_t(z^t) r_t(z^t) k_t(z^{t-1}) - p_t(z^t) w_t(z^t) n_t(z^t)$$

• Market clear: $c_t(z^t) + k_{t+1}(z^{t})= z_t F(k_t(z^{t-1}),l_t(z^t)) + (1-\delta) k_t(z^{t-1}),\ \forall t,\ \forall z^t$.
• No-arbitrage condition: $p_t(z^t) = \sum_{z^t\in Z^{t+1}} p_{t+1}(z_{t+1},z^t)(r_t(z_{t+1},z^t)+1-\delta)$.

# Sequential Trade Model

# General Equilibrium under Uncertainty

Assumptions:

• Random shock:
  • $z \in \{z_1,z_2,\dots,z_n\}$;
  • $\pi_j = \text{Prob}(z = z_j)$;
  • The expected value of $z$: $\bar{z} = \sum_{j=1}^n \pi_j z_j,\ i=1,2$.
• Preference:
  • $U_i = u_i(c_0^i) + \beta \sum_{j=1}^n \pi_j u_i(c_j^i),\ i=1,2$,
    • $u_1(x) = x$, $u_2(x)$ is strictly concave and $\lim_{x\rightarrow 0}u_2'(x) = \infty$.
      • That is, Agent 1 is risk neutral, Agent 2 is risk averse.
• Endowments:
  • $w_0$ consumption goods in period 0;
  • $1$ unit of labor in period 1.
• Technology (Production):
  • $y_j = z_j k^\alpha (\frac{n}{2})^{1-\alpha},\ n=2$.
  • $\Rightarrow y_j = z_j k^\alpha$

# Case 1: Two-type Agent, Two-period Setting

# Case 2: Two-type Agent, Multi-period Setting