Merge pull request #64 from QuantEcon/cobweb_2

LECT: Cobweb-Fix citation

Author: jstac

lectures/_static/quant-econ.bib

@@ -2530,4 +2530,34 @@ @article{benhabib_wealth_2019
 	year = {2019},
 	keywords = {Personal Income, Wealth, and Their Distributions, General Aggregative Models: Neoclassical, Macroeconomics: Consumption, Saving, Wealth, Aggregate Factor Income Distribution},
 	pages = {1623--1647},
-}
+}
+
+
+@article{cobweb_model,
+ ISSN = {10711031},
+ URL = {http://www.jstor.org/stable/1236509},
+ abstract = {In recent years, economists have become much interested in recursive models. This interest stems from a growing need for long-term economic projections and for forecasting the probable effects of economic programs and policies. In a dynamic world, past and present conditions help shape future conditions. Perhaps the simplest recursive model is the two-dimensional "cobweb diagram," discussed by Ezekiel in 1938. The present paper attempts to generalize the simple cobweb model somewhat. It considers some effects of price supports. It discusses multidimensional cobwebs to describe simultaneous adjustments in prices and outputs of a number of commodities. And it allows for time trends in the variables.},
+ author = {Frederick V. Waugh},
+ journal = {Journal of Farm Economics},
+ number = {4},
+ pages = {732--750},
+ publisher = {[Oxford University Press, Agricultural & Applied Economics Association]},
+ title = {Cobweb Models},
+ urldate = {2023-02-06},
+ volume = {46},
+ year = {1964}
+}
+
+@article{hog_cycle,
+author = {Harlow, Arthur A.},
+title = {The Hog Cycle and the Cobweb Theorem},
+journal = {American Journal of Agricultural Economics},
+volume = {42},
+number = {4},
+pages = {842-853},
+doi = {https://doi.org/10.2307/1235116},
+url = {https://onlinelibrary.wiley.com/doi/abs/10.2307/1235116},
+eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.2307/1235116},
+abstract = {Abstract A surprisingly regular four year cycle in hogs has become apparent in the past ten years. This regularity presents an unusual opportunity to study the mechanism of the cycle because it suggests the cycle may be inherent within the industry rather than the result of lagged responses to outside influences. The cobweb theorem is often mentioned as a theoretical tool for explaining the hog cycle, although a two year cycle is usually predicted. When the nature of the hog industry is examined, certain factors become apparent which enable the cobweb theorem to serve as a theoretical basis for the present four year cycle.},
+year = {1960}
+}

lectures/cobweb.md

@@ -3,6 +3,8 @@ jupytext:
   text_representation:
     extension: .md
     format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.1
 kernelspec:
   display_name: Python 3 (ipykernel)
   language: python
@@ -13,9 +15,11 @@ kernelspec:
 # The Cobweb Model
 
 
-The cobweb model {cite}:`10.2307/1236509` is a model of prices and quantities in a given market, and how they evolve over time.
+The cobweb model is a model of prices and quantities in a given market, and how they evolve over time.
 
-The model dates back to the 1930s and, while simple, it remains significant
+## Overview 
+
+The cobweb model dates back to the 1930s and, while simple, it remains significant
 because it shows the fundamental importance of *expectations*.
 
 To give some idea of how the model operates, and why expectations matter, imagine the following scenario.
@@ -52,15 +56,46 @@ assumptions regarding the way that produces form expectations.
 
 Our discussion and simulations draw on [high quality lectures](https://comp-econ.org/CEF_2013/downloads/Complex%20Econ%20Systems%20Lecture%20II.pdf) by [Cars Hommes](https://www.uva.nl/en/profile/h/o/c.h.hommes/c.h.hommes.html).
 
-We will use the following imports:
+
++++
+
+We will use the following imports.
 
 ```{code-cell} ipython3
 import numpy as np
 import matplotlib.pyplot as plt
 ```
 
+## History
+
+Early papers on the cobweb cycle include {cite}`cobweb_model` and {cite}`hog_cycle`.
+
+The paper {cite}`hog_cycle` uses the cobweb theorem to explain the prices of hog in the US over 1920--1950
+
+The next plot replicates part of Figure 2 from that paper, which plots the price of hogs at yearly frequency.
+
+Notice the cyclical price dynamics, which match the kind of cyclical soybean price dynamics discussed above.
+
+```{code-cell} ipython3
+hog_prices = [55, 57, 80, 70, 60, 65, 72, 65, 51, 49, 45, 80, 85,
+              78, 80, 68, 52, 65, 83, 78, 60, 62, 80, 87, 81, 70,
+              69, 65, 62, 85, 87, 65, 63, 75, 80, 62]
+years = np.arange(1924, 1960)
+fig, ax = plt.subplots()
+ax.plot(years, hog_prices, '-o', ms=4, label='hog price')
+ax.set_xlabel('year')
+ax.set_ylabel('dollars')
+ax.legend()
+ax.grid()
+plt.show()
+```
+
+
+
 ## The Model
 
+Let's return to our discussion of a hypothetical soy bean market, where price is determined by supply and demand.
+
 We suppose that demand for soy beans is given by
 
 $$
@@ -71,7 +106,7 @@ where $a, b$ are nonnegative constants and $p_t$ is the spot (i.e, current marke
 
 ($D(p_t)$ is the quantity demanded in some fixed unit, such as thousands of tons.)
 
-Supply of soy beans depends on *expected* prices at time $t$, which we denote $p^e_t$.
+Because the crop of soy beans for time $t$ is planted at $t-1$, supply of soy beans at time $t$ depends on *expected* prices at time $t$, which we denote $p^e_t$.
 
 We suppose that supply is nonlinear in expected prices, and takes the form
 
@@ -210,6 +245,7 @@ The function `plot45` defined below helps us draw the 45 degree diagram.
 
 ```{code-cell} ipython3
 :tags: [hide-input]
+
 def plot45(model, pmin, pmax, p0, num_arrows=5):
     """
     Function to plot a 45 degree plot
@@ -352,7 +388,7 @@ The cycle is "stable", in the sense that prices converge to it from most startin
 
 For example,
 
-```{code-cell} ipython3 tags=[]
+```{code-cell} ipython3
 ts_plot_price(m, 10, ts_length=15)
 ```
 
@@ -430,7 +466,6 @@ Let's call the function with prices starting at $p_0 = 5$.
 
 TODO does this fit well in the page, even in the pdf? If not should it be stacked vertically?
 
-
 ```{code-cell} ipython3
 ts_price_plot_adaptive(m, 5, ts_length=30)
 ```
@@ -450,7 +485,7 @@ TODO check / fix exercises
 ```{exercise-start}
 :label: cobweb_ex1
 ```
-Using the default Market model and naive expectations, plot a time series simulation of supply (rather than the price).
+Using the default `Market` class and naive expectations, plot a time series simulation of supply (rather than the price).
 
 Show, in particular, that supply also cycles.
 
@@ -482,10 +517,12 @@ def ts_plot_supply(model, p0, ts_length=10):
             'bo-',
             alpha=0.6,
             lw=2,
-            label=r'$S_t$')
+            label=r'supply')
 
     ax.legend(loc='best', fontsize=10)
     ax.set_xticks(np.arange(ts_length))
+    ax.set_xlabel("time")
+    ax.set_ylabel("quantity")
     plt.show()
 ```
 
@@ -538,30 +575,38 @@ def ts_plot_price_blae(model, p0, p1, alphas, ts_length=15):
     Function to simulate and plot the time series of price
     using backward looking average expectations.
     """
-    fig, ax = plt.subplots()
-    ax.set_xlabel(r'$t$', fontsize=12)
-    ax.set_ylabel(r'$p_t$', fontsize=12)
-    for a in alphas:
+    fig, axes = plt.subplots(len(alphas), 1, figsize=(8, 16))
+
+    for ax, a in zip(axes.flatten(), alphas):
         p = np.empty(ts_length)
-        p[0] = p[0]
-        p[1] = p[1]
+        p[0] = p0
+        p[1] = p1
         for t in range(2, ts_length):
             pe = a*p[t-1] + (1 - a)*p[t-2]
             p[t] = -(model.supply(pe) - model.a) / model.b
         ax.plot(np.arange(ts_length),
                 p,
                 'o-',
                 alpha=0.6,
-                label=r'$\alpha={}$'.format(a),
-                c=np.random.rand(3))
+                label=r'$\alpha={}$'.format(a))
         ax.legend(loc='best', fontsize=10)
+        ax.set_xlabel(r'$t$', fontsize=12)
+        ax.set_ylabel(r'$p_t$', fontsize=12)
     plt.show()
 ```
 
 ```{code-cell} ipython3
 m = Market()
-ts_plot_price_blae(m, 1, 2.5, [0.1, 0.3, 0.5, 0.8], 20)
+ts_plot_price_blae(m, 
+                   p0=5, 
+                   p1=6, 
+                   alphas=[0.1, 0.3, 0.5, 0.8], 
+                   ts_length=20)
 ```
 
 ```{solution-end}
 ```
+
+```{code-cell} ipython3
+
+```