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May 4, 2023
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Remove numba dependencies #192

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bb6b5d4
Remove unused import
Smit-create May 4, 2023
fd8578a
Remove unwanted comments
Smit-create May 4, 2023
4142cc7
Remove numba njit and use vectorized version
Smit-create May 4, 2023
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75 changes: 37 additions & 38 deletions lectures/commod_price.md
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Original file line number Diff line number Diff line change
Expand Up @@ -13,7 +13,7 @@ For this lecture we need the `yfinance` library.

```{code-cell} ipython3
:tags: [hide-output]
!pip install yfinance
!pip install yfinance
```


Expand Down Expand Up @@ -55,10 +55,10 @@ In general, prices depend on the choices and actions of suppliers, consumers and
speculators.

Our focus will be on the interaction between these parties, viewed through the
lens of the standard competitive storage model
lens of the standard competitive storage model

This model was developed by
{cite}`samuelson1971stochastic`,
{cite}`samuelson1971stochastic`,
{cite}`wright1982economic`, {cite}`scheinkman1983simple`,
{cite}`deaton1992on`, {cite}`deaton1996competitive`, and
{cite}`chambers1996theory`.
Expand All @@ -81,7 +81,7 @@ Supply is exogenous, depending on "harvests".
These days, goods such as basic computer chips and integrated circuits are
often treated as commodities in financial markets, being highly standardized,
and, for these kinds of commodities, the word "harvest" is not
appropriate.
appropriate.

Nonetheless, we maintain it for simplicity.
```
Expand All @@ -97,7 +97,6 @@ We begin with the following imports
```{code-cell} ipython3
import numpy as np
import matplotlib.pyplot as plt
from numba import jit, vectorize
from scipy.interpolate import interp1d
from scipy.optimize import minimize_scalar, brentq
from scipy.stats import beta
Expand All @@ -107,25 +106,25 @@ from scipy.stats import beta
## The Model

Consider a market for a single commodity, whose price is given at $t$ by
$p_t$.
$p_t$.

The harvest of the commodity at time $t$ is $Z_t$.

We assume that the sequence $\{ Z_t \}_{t \geq 1}$ is IID with common
density function $\phi$.
density function $\phi$.

The harvests take values in $S$, a subset of the nonnegative numbers.

Speculators can store the commodity between periods, with $I_t$ units
purchased in the current period yielding $\alpha I_t$ units in the next.

Here $\alpha \in (0,1)$ is a depreciation rate for the commodity.
Here $\alpha \in (0,1)$ is a depreciation rate for the commodity.

For simplicity, the risk free interest rate is taken to be
zero, so expected profit on purchasing $I_t$ units is
zero, so expected profit on purchasing $I_t$ units is

$$
\mathbb{E}_t \, p_{t+1} \cdot \alpha I_t - p_t I_t
\mathbb{E}_t \, p_{t+1} \cdot \alpha I_t - p_t I_t
= (\alpha \mathbb{E}_t \, p_{t+1} - p_t) I_t
$$

Expand Down Expand Up @@ -162,7 +161,7 @@ $$
$$ (eq:pmco)


We also require that the market clears in each period.
We also require that the market clears in each period.

We assume that consumers generate demand quantity $D(p)$ corresponding to
price $p$.
Expand All @@ -173,14 +172,14 @@ Let $P := D^{-1}$ be the inverse demand function.
Regarding quantities,

* supply is the sum of carryover by speculators and the current harvest
* demand is the sum of purchases by consumers and purchases by speculators.
* demand is the sum of purchases by consumers and purchases by speculators.

Mathematically,

* supply $ = X_t = \alpha I_{t-1} + Z_t$
* demand $ = D(p_t) + I_t$
* supply $ = X_t = \alpha I_{t-1} + Z_t$
* demand $ = D(p_t) + I_t$

Thus, the market equilibrium condition is
Thus, the market equilibrium condition is

$$
\alpha I_{t-1} + Z_t = D(p_t) + I_t
Expand All @@ -194,14 +193,14 @@ The initial condition $X_0 \in S$ is treated as given.

### An equilibrium function

Now to find an equilibrium?
Now to find an equilibrium?

Our path of attack will be to seek a system of prices that depend only on the
current state.

In other words, we take a function $p$ on $S$ and set $p_t = p(X_t)$ for every $t$.
In other words, we take a function $p$ on $S$ and set $p_t = p(X_t)$ for every $t$.

Prices and quantities then follow
Prices and quantities then follow

$$
p_t = p(X_t), \quad I_t = X_t - D(p_t), \quad X_{t+1} = \alpha I_t + Z_{t+1}
Expand All @@ -218,7 +217,7 @@ To this end, suppose that there exists a function $p^*$ on $S$
satisfying

$$
p^*(x) = \max
p^*(x) = \max
\left\{
\alpha \int_0^\infty p^*(\alpha I(x) + z) \phi(z)dz, P(x)
\right\}
Expand All @@ -233,14 +232,14 @@ $$
$$ (eq:einvf)

It turns out that such a $p^*$ will suffice, in the sense that [](eq:arbi)
and [](eq:pmco) hold for the corresponding system [](eq:eosy).
and [](eq:pmco) hold for the corresponding system [](eq:eosy).

To see this, observe first that
To see this, observe first that

$$
\mathbb{E}_t \, p_{t+1}
= \mathbb{E}_t \, p^*(X_{t+1})
= \mathbb{E}_t \, p^*(\alpha I(X_t) + Z_{t+1})
= \mathbb{E}_t \, p^*(X_{t+1})
= \mathbb{E}_t \, p^*(\alpha I(X_t) + Z_{t+1})
= \int_0^\infty p^*(\alpha I(X_t) + z) \phi(z)dz
$$

Expand All @@ -249,20 +248,20 @@ Thus [](eq:arbi) requires that
$$
\alpha \int_0^\infty p^*(\alpha I(X_t) + z) \phi(z)dz \leq p^*(X_t)
$$

This inequality is immediate from [](eq:dopf).

Second, regarding [](eq:pmco), suppose that
This inequality is immediate from [](eq:dopf).

Second, regarding [](eq:pmco), suppose that

$$
\alpha \int_0^\infty p^*(\alpha I(X_t) + z) \phi(z)dz < p^*(X_t)
$$

Then by [](eq:dopf) we have $p^*(X_t) = P(X_t)$

But then $D(p^*(X_t)) = X_t$ and $I_t = I(X_t) = 0$.
But then $D(p^*(X_t)) = X_t$ and $I_t = I(X_t) = 0$.

As a consequence, both [](eq:arbi) and [](eq:pmco) hold.
As a consequence, both [](eq:arbi) and [](eq:pmco) hold.

We have found an equilibrium.

Expand Down Expand Up @@ -293,16 +292,16 @@ To implement our update step, it is helpful if we put [](eq:dopf) and
This leads us to the update rule

$$
p_{k+1}(x) = \max
p_{k+1}(x) = \max
\left\{
\alpha \int_0^\infty p_k(\alpha ( x - D(p_{k+1}(x))) + z) \phi(z)dz, P(x)
\right\}
$$ (eq:dopf2)

In other words, we take $p_k$ as given and, at each $x$, solve for $q$ in
In other words, we take $p_k$ as given and, at each $x$, solve for $q$ in

$$
q = \max
q = \max
\left\{
\alpha \int_0^\infty p_k(\alpha ( x - D(q)) + z) \phi(z)dz, P(x)
\right\}
Expand All @@ -313,7 +312,7 @@ points $x_1, \ldots, x_n$.

Then we get the corresponding values $q_1, \ldots, q_n$.

Then we compute $p_{k+1}$ as the linear interpolation of
Then we compute $p_{k+1}$ as the linear interpolation of
the values $q_1, \ldots, q_n$ over the grid $x_1, \ldots, x_n$.

Then we repeat, seeking convergence.
Expand All @@ -330,7 +329,7 @@ The integral in [](eq:dopf3) is computed via Monte Carlo.


```{code-cell} ipython3
α, a, c = 0.8, 1.0, 2.0
α, a, c = 0.8, 1.0, 2.0
beta_a, beta_b = 5, 5
mc_draw_size = 250
gridsize = 150
Expand All @@ -339,7 +338,7 @@ grid = np.linspace(a, grid_max, gridsize)

beta_dist = beta(5, 5)
Z = a + beta_dist.rvs(mc_draw_size) * c # Shock observations
D = P = lambda x: 1.0 / x
D = P = lambda x: 1.0 / x
tol = 1e-4


Expand All @@ -348,8 +347,8 @@ def T(p_array):
new_p = np.empty_like(p_array)

# Interpolate to obtain p as a function.
p = interp1d(grid,
p_array,
p = interp1d(grid,
p_array,
fill_value=(p_array[0], p_array[-1]),
bounds_error=False)

Expand Down Expand Up @@ -388,8 +387,8 @@ prices.

```{code-cell} ipython3
# Turn the price array into a price function
p_star = interp1d(grid,
price,
p_star = interp1d(grid,
price,
fill_value=(price[0], price[-1]),
bounds_error=False)

Expand Down
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