# The Cobweb Model

Presentation follows Hommes, JEBO 1994. Let $p_t$ denote the observed price of goods and $p_t^e$ the expected price of goods in period $t$. Similarly, let $q_t^d$ denote the quantity demanded of all goods in period $t$ and $q_t^s$ the quantity supplied of all goods in period $t$.

\begin{align} q_t^d =& D(p_t) \tag{1} \\ q_t^s =& S(p_t^e) \tag{2} \\ q_t^d =& q_t^s \tag{3} \\ p_t^e =& p_{t-1}^e + w\big(p_{t-1} - p_{t-1}^e\big) = (1 - w)p_{t-1}^e + w p_{t-1} \tag{4} \end{align}

Equation 1 says that the quantity demanded of goods in period $t$ is some function of the observed price in period $t$. Equation 2, meanwhile, states that the quantity of goods supplied in period $t$ is a function of the expected price in period $t$. Equation 3 is a market clearing equilibrium condition. Finally, equation 4 is an adaptive expectation formation rule that specifies how goods producers form their expectations about the price of goods in period $t$ as a function of past prices.

Combine the equations as follows. Note that equation 3 implies that...

$$D(p_t) = q_t^d = q_t^s = S(p_t^e)$$

...and therefore, assuming the demand function $D$ is invertible, we can write the observed price of goods in period $t$ as...

$$p_t = D^{-1}\big(S(p_t^e)\big). \tag{5}$$

Substituting equation 5 into equation 4 we arrive at the following difference equation

$$p_{t+1}^e = w D^{-1}\big(S(p_t^e)\big) + (1 - w)p_t^e. \tag{7}$$


In [1]:

%matplotlib inline




In [2]:




In [3]:




In [4]:

import functools

import ipywidgets
import matplotlib.pyplot as plt
import numpy as np
from scipy import optimize
import seaborn as sns

import cobweb




In [5]:

def observed_price(D_inverse, S, expected_price, **params):
"""The observed price of goods in a particular period."""
actual_price = D_inverse(S(expected_price, **params), **params)
return actual_price

def adaptive_expectations(D_inverse, S, expected_price, w, **params):
"""An adaptive expectations price forecasting rule."""
actual_price = observed_price(D_inverse, S, expected_price, **params)
price_forecast = w * actual_price + (1 - w) * expected_price
return price_forecast



## Non-linear supply functions

1. ...when prices are low, the quantity supplied increases slowly because of fixed costs of production (think startup costs, etc).
2. ...when prices are high, supply also increases slowly because of capacity constraints.

These considerations motivate our focus on "S-shaped" supply functions...

$$S_{\gamma}(p_t^e) = -tan^{-1}(-\gamma \bar{p}) + tan^{-1}(\gamma (p_t^e - \bar{p})). \tag{10}$$

The parameter $0 < \gamma < \infty$ controls the "steepness" of the supply function.



In [6]:

def quantity_supply(expected_price, gamma, p_bar, **params):
"""The quantity of goods supplied in period t given the epxected price."""
return -np.arctan(-gamma * p_bar) + np.arctan(gamma * (expected_price - p_bar))



### Exploring supply shocks

Interactively change the value of $\gamma$ to see the impact on the shape of the supply function.



In [7]:

ipywidgets.interact?




In [8]:

interactive_quantity_supply_plot = ipywidgets.interact(cobweb.quantity_supply_plot,
S=ipywidgets.fixed(quantity_supply),
gamma=cobweb.gamma_float_slider,
p_bar=cobweb.p_bar_float_slider)






## Special case: Linear demand functions

Suppose the the quantity demanded of goods is a simple, decresing linear function of the observed price.

$$q_t^d = D(p_t) = a - b p_t \implies p_t = D^{-1}(q_t^d) = \frac{a}{b} - \frac{1}{b}q_t^d \tag{11}$$

...where $-\infty < a < \infty$ and $0 < b < \infty$.



In [9]:

def quantity_demand(observed_price, a, b):
"""The quantity demand of goods in period t given the price."""
quantity = a - b * observed_price
return quantity

def inverse_demand(quantity_demand, a, b, **params):
"""The price of goods in period t given the quantity demanded."""
price = (a / b) - (1 / b) * quantity_demand
return price



### Exploring demand shocks

Interactively change the values of $a$ and $b$ to get a feel for how they impact demand. Shocks to $a$ shift the entire demand curve; shocks to $b$ change the slope of the demand curve (higher $b$ implies greater sensitivity to price; lower $b$ implies less sensitivity to price).



In [10]:

interactive_quantity_demand_plot = ipywidgets.interact(cobweb.quantity_demand_plot,
D=ipywidgets.fixed(quantity_demand),
a=cobweb.a_float_slider,
b=cobweb.b_float_slider)






## Supply and demand

Market clearing equilibrium price, $p^*$, satisfies...

$$D(p_t) = S(p_t^e).$$

Really this is also an equilibrium in beliefs because we also require that $p_t = p_t^e$!



In [13]:

interactive_supply_demand_plot = ipywidgets.interact(cobweb.supply_demand_plot,
D=ipywidgets.fixed(quantity_demand),
S=ipywidgets.fixed(quantity_supply),
a=cobweb.a_float_slider,
b=cobweb.b_float_slider,
gamma=cobweb.gamma_float_slider,
p_bar=cobweb.p_bar_float_slider)






## Analyzing dynamics of the model via simulation...

Model has no closed form solution (i.e., we can not solve for a function that describes $p_t^e$ as a function of time and model parameters). BUT, we can simulate equation 7 above to better understand the dynamics of the model...

We can simulate our model and plot time series for different parameter values. Questions for discussion...

1. Can you find a two-cycle? What does this mean?
2. Can you find higher cycles? Perhaps a four-cycle? Maybe even a three-cycle?
3. Do simulations with similar initial conditions converge or diverge over time?

Can we relate these things to other SFI MOOCS on non-linear dynamics and chaos? Surely yes!



In [30]:

interactive_time_series_plot = ipywidgets.interact(cobweb.time_series_plot,
F=ipywidgets.fixed(model),
X0=cobweb.initial_expected_price_slider,
T=cobweb.T_int_slider,
a=cobweb.a_float_slider,
b=cobweb.b_float_slider,
w=cobweb.w_float_slider,
gamma=cobweb.gamma_float_slider,
p_bar=cobweb.p_bar_float_slider)






## Forecast errors

How do we measure forecast error? What does the distribution of forecast errors look like for different parameters? Could an agent learn to avoid chaos? Specifically, suppose an agent learned to tune the value of $w$ in order to minimize its mean forecast error. Would this eliminate chaotic dynamics?



In [32]:

interactive_forecast_error_plot = ipywidgets.interact(cobweb.forecast_error_plot,
D_inverse=ipywidgets.fixed(inverse_demand),
S=ipywidgets.fixed(quantity_supply),
F=ipywidgets.fixed(model),
X0=cobweb.initial_expected_price_slider,
T=cobweb.T_int_slider,
a=cobweb.a_float_slider,
b=cobweb.b_float_slider,
w=cobweb.w_float_slider,
gamma=cobweb.gamma_float_slider,
p_bar=cobweb.p_bar_float_slider)




---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
/Users/drpugh/Teaching/sfi-complexity-mooc/notebooks/cobweb.py in forecast_error_plot(D_inverse, S, F, X0, T, **params)
132     """Plot forecast errors."""
133     trajectory = _simulate(X0, F, T, **params)
--> 134     errors = _forecast_error(D_inverse, S, trajectory , **params)
135
136     fig, ax = plt.subplots(1, 2)

/Users/drpugh/Teaching/sfi-complexity-mooc/notebooks/cobweb.py in _forecast_error(D_inverse, S, expected_price, **params)
56 def _forecast_error(D_inverse, S, expected_price, **params):
57     """Difference between observed price and expected price."""
---> 58     return observed_price(D_inverse, S, expected_price, **params) - expected_price
59
60

NameError: name 'observed_price' is not defined



## Other things of possible interest?

Impulse response functions? Compare constrast model predictions for rational expectations, naive expectations, adaptive expectations. Depending on what Cars might have in mind, we could also add other expectation formation rules from his more recent work and have students analyze those...



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