from compecon import BasisSpline, NLP, qnwnorm, demo
# ## FORMULATION
# In what follows, we are going to solve the problem in term of the log-price, defining $p\equiv\log(P)$. We assume that log-price of the commodity follows a random walk:
# \begin{equation}
# p_{t+1} = p_t + \epsilon_{t+1}
# \end{equation}
#
# where $\epsilon$ is a normal $(\mu, \sigma^2)$ shock. We discretize this distribution by using ```qnwnorm```, assuming that $\mu=0.0001,\quad\sigma=0.008$ and setting up $m=15$ nodes.
# In[3]:
mu, sigma = 0.0001, 0.0080
m = 15
[e, w] = qnwnorm(m, mu, sigma**2)
# We are going to compute the critical exercise price in terms of the time to expiration, up to an horizon of $T=300$ periods. First we allocate memory for the critical prices:
# In[4]:
T = 300
pcrit = np.empty(T + 1)
# The critical exercise price is the price at which the value of exercising the option $K-\exp(p)$ equals the discounted expected value of keeping the option one more period $\delta E_\epsilon V(p + \epsilon)$. To find it, we set it as a nonlinear rootfinding problem by using the ```NLP``` class; here we assume that the option strike price is $K=1$ and that the discount factor is $\delta=0.9998$
# In[5]:
K = 1.0
delta = 0.9998
f = NLP(lambda p: K - np.exp(p) - delta * Value(p))
# ## FORMULATION
# In what follows, we are going to solve the problem in term of the log-price, defining $p\equiv\log(P)$. We assume that log-price of the commodity follows a random walk:
# \begin{equation}
# p_{t+1} = p_t + \epsilon_{t+1}
# \end{equation}
#
# where $\epsilon$ is a normal $(\mu, \sigma^2)$ shock. We discretize this distribution by using ```qnwnorm```, assuming that $\mu=0.0001,\quad\sigma=0.008$ and setting up $m=15$ nodes.
# In[3]:
mu, sigma = 0.0001, 0.0080
m = 15
[e,w] = qnwnorm(m,mu,sigma ** 2)
# We are going to compute the critical exercise price in terms of the time to expiration, up to an horizon of $T=300$ periods. First we allocate memory for the critical prices:
# In[4]:
T = 300
pcrit = np.empty(T + 1)
# The critical exercise price is the price at which the value of exercising the option $K-\exp(p)$ equals the discounted expected value of keeping the option one more period $\delta E_\epsilon V(p + \epsilon)$. To find it, we set it as a nonlinear rootfinding problem by using the ```NLP``` class; here we assume that the option strike price is $K=1$ and that the discount factor is $\delta=0.9998$
# In[5]:
wbar = 100
gamma = 0.40
def transition(w, x, i, j, in_, e):
return wbar + gamma * (w - wbar) + e
# Here, $\epsilon$ is normal $(0,\sigma^2)$ wage shock, where $\sigma=5$. We discretize this distribution with the function ```qnwnorm```.
# In[5]:
sigma = 5
m = 15
e, w = qnwnorm(m, 0, sigma**2)
# ### Approximation Structure
#
# To discretize the continuous state variable, we use a cubic spline basis with $n=150$ nodes between $w_\min=0$ and $w_\max=200$.
# In[6]:
n = 150
wmin = 0
wmax = 200
basis = BasisSpline(n, wmin, wmax, labels=['wage'])
# ## SOLUTION
#
# To represent the model, we create an instance of ```DPmodel```. Here, we assume a discout factor of $\delta=0.95$.
f = np.log(sk)
fx= - sk ** -1
fxx = - sk ** -2
return f, fx, fxx
def transition(s, k, i, j, in_, e):
g = k ** beta
gx = beta * k **(beta - 1)
gxx = (beta - 1) * beta * k ** (beta - 2)
return g, gx, gxx
sigma = 0.1
m=5
e,w = qnwnorm(m, -sigma**2 / 2, sigma**2)
growth_model = DPmodel(basis, reward, transition, bounds,
x=['Investment'],
discount=delta)
# ======== Steady-State
sstar = (beta * delta) ** (beta / (1 - beta)) # steady-state wealth
kstar = beta * delta * sstar # steady-state capital investment
vstar = np.log(sstar - kstar) / (1 - delta) # steady-state value
pstar = 1 / (sstar * (1 - beta * delta)) # steady-state shadow price
b = 1 / (1 - delta * beta)
# where $\epsilon \sim N(0,\sigma^2)$.
# In[7]:
def transition(p, x, i, j, in_, e):
return pbar + gamma * (p - pbar) + e
# The continuous shock must be discretized. Here we use Gauss-Legendre quadrature to obtain nodes and weights defining a discrete distribution that matches the first 10 moments of the Normal distribution (this is achieved with $m=5$ nodes and weights).
# In[8]:
m = 5
e, w = qnwnorm(m,0,sigma ** 2)
# On the other hand, the age of the asset is deterministic: if it is replaced now it will be new ($a=0$) next period; otherwise its age will be $a+1$ next period if current age is $a$.
# In[9]:
h = np.zeros((2, A),int)
h[0, :-1] = np.arange(1, A)
# ## Model Structure
# In[10]:
pbar = 1.0
gamma = 0.7
def transition(p, x, d, j, in_, e):
return pbar + gamma * (p - pbar) + e
# In the transition function $\epsilon_t$ is an i.i.d. normal(0, $σ^2$), with $\sigma=1$. We discretize this distribution by using a discrete distribution, matching the first 10 moments of the normal distribution.
# In[4]:
m = 5 # number of profit shocks
sigma = 1.0
[e,w] = qnwnorm(m,0,sigma **2)
# The collocation method calls for the analyst to select $n$ basis functions $\varphi_j$ and $n$ collocation nodes $(\pi_i,d_i)$, and form the value function approximant $V(\pi,d) \approx \sum_{j=1}^{n} c_j\varphi_j(\pi,d)$ whose coefficients $c_j$ solve the collocation equation
#
# \begin{equation}
# \sum_{j=1}^{n} c_j\varphi_j(\pi_i,d_i) = \max_{x\in\{0,1\}}\left\{\pi_i x − K_1(1−d_i)x − K_0 d_i(1−x) + \delta\sum_{k=1}^{m}\sum_{j=1}^{n}w_k c_j \varphi_j(\hat{\pi}_{ik},x)\right\}
# \end{equation}
#
# where $\hat\pi_{ik}=g(\pi_i,\epsilon_k)$ and where $\epsilon_k$ and $w_k$ represent quadrature nodes and weights for
# the normal shock.
#
# For the approximation, we use a cubic spline basis with $n=250$ nodes between $p_{\min}=-20$ and $p_\max=20$.
# In[5]:
pbar = 1.0
gamma = 0.7
def transition(p, x, d, j, in_, e):
return pbar + gamma * (p - pbar) + e
# In the transition function $\epsilon_t$ is an i.i.d. normal(0, $σ^2$), with $\sigma=1$. We discretize this distribution by using a discrete distribution, matching the first 10 moments of the normal distribution.
# In[4]:
m = 5 # number of profit shocks
sigma = 1.0
[e, w] = qnwnorm(m, 0, sigma**2)
# The collocation method calls for the analyst to select $n$ basis functions $\varphi_j$ and $n$ collocation nodes $(\pi_i,d_i)$, and form the value function approximant $V(\pi,d) \approx \sum_{j=1}^{n} c_j\varphi_j(\pi,d)$ whose coefficients $c_j$ solve the collocation equation
#
# \begin{equation}
# \sum_{j=1}^{n} c_j\varphi_j(\pi_i,d_i) = \max_{x\in\{0,1\}}\left\{\pi_i x − K_1(1−d_i)x − K_0 d_i(1−x) + \delta\sum_{k=1}^{m}\sum_{j=1}^{n}w_k c_j \varphi_j(\hat{\pi}_{ik},x)\right\}
# \end{equation}
#
# where $\hat\pi_{ik}=g(\pi_i,\epsilon_k)$ and where $\epsilon_k$ and $w_k$ represent quadrature nodes and weights for
# the normal shock.
#
# For the approximation, we use a cubic spline basis with $n=250$ nodes between $p_{\min}=-20$ and $p_\max=20$.
# In[5]:
n = 250