further_edits

Author: shlff

lectures/supply_demand_heterogeneity.md

@@ -11,12 +11,40 @@ kernelspec:
   name: python3
 ---
 
+(supply_demand_heterogeneity)=
 # Market Equilibrium with Heterogeneity
 
+```{contents} Contents
+:depth: 2
+```
+
+## Overview
+
+In the {doc}`previous lecture
+<supply_demand_multiple_goods>`, we studied competitive equilibria in an economy with many goods.
+
+While the results of the study were informative, we used a strong simplifying assumption: all of the agents in the economy are identical.
+
+In the real world, households, firms and other economic agents differ from one another along many dimensions.
+
+In this lecture, we introduce heterogeneity across consumers by allowing their preferences and endownments to differ.
+
+We will examine competitive equilibrium in this setting.
 
-## An Endowment Economy
+We will also show how a "representative consumer" can be constructed.
 
-Let's study a **pure exchange** economy without production.
+Here are some imports:
+
+```{code-cell} ipython3
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from scipy.linalg import inv
+```
+
+## An simple example
+
+Let's study a simple example of **pure exchange** economy without production.
 
 There are two consumers who differ in their endowment vectors $e_i$ and their bliss-point vectors $b_i$ for $i=1,2$.
 
@@ -80,66 +108,11 @@ $$
 ```{exercise-end}
 ```
 
-## Designing some Python Code
-
-
-Below we shall construct a Python class with the following attributes:
-
- * **Preferences** in the form of
-
-     * an $n \times n$  positive definite matrix $\Pi$
-     * an $n \times 1$ vector of bliss points $b$
-
- * **Endowments** in the form of
-
-     * an $n \times 1$ vector $e$
-     * a scalar "wealth" $W$ with default value $0$
-
- * **Production Costs** pinned down by
-
-     * an $n \times 1$ nonnegative vector $h$
-     * an $n \times n$ positive definite matrix $J$
-
-The class will include a test to make sure that $b  > > \Pi e $ and raise an exception if it is violated
-(at some threshold level we'd have to specify).
-
- * **A Person** in the form of a pair that consists of
-
-    * **Preferences** and **Endowments**
-
- * **A Pure Exchange Economy** will consist of
-
-    * a collection of $m$ **persons**
-
-       * $m=1$ for our single-agent economy
-       * $m=2$ for our illustrations of a pure exchange economy
-
-    * an equilibrium price vector $p$ (normalized somehow)
-    * an equilibrium allocation $c^1, c^2, \ldots, c^m$ -- a collection of $m$ vectors of dimension $n \times 1$
-
- * **A Production Economy** will consist of
-
-    * a single **person** that we'll interpret as a representative consumer
-    * a single set of **production costs**
-    * a multiplier $\mu$ that weights "consumers" versus "producers" in a planner's welfare function, as described above in the main text
-    * an $n \times 1$ vector $p$ of competitive equilibrium prices
-    * an $n \times 1$ vector $c$ of competitive equilibrium quantities
-    * **consumer surplus**
-    * **producer surplus**
-
-Now let's proceed to code.
-
-```{code-cell} ipython3
-# import some packages
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from scipy.linalg import inv
-```
+## Pure exchange economy
 
-### Pure Exchange Economy
+Let's further explore a pure exchange economy with $n$ goods and $m$ people.
 
-Let's first explore a pure exchange economy with $n$ goods and $m$ people.
+### Competitive equilibrium
 
 We'll compute a competitive equilibrium.
 
@@ -163,79 +136,40 @@ c_{i}=\Pi^{-1}b_{i}-(\Pi^{\top}\Pi)^{-1}\mu_{i}p
 $$
 
 
+### Designing some Python code
 
-## Deducing a Representative Consumer
-
-In the class of multiple consumer economies that we are studying here, it turns out that there
-exists a single **representative consumer** whose preferences and endowments can be deduced from lists of preferences and endowments for the separate individual consumers.
-
-Consider a multiple consumer economy with initial distribution of wealth $W_i$ satisfying $\sum_i W_{i}=0$
-
-We allow an initial redistribution of wealth.
 
-We have the following objects
-
-
-- The demand curve:
-  
-$$ 
-c_{i}=\Pi^{-1}b_{i}-(\Pi^{\top}\Pi)^{-1}\mu_{i}p 
-$$
-
-- The marginal utility of wealth:
-  
-$$ 
-\mu_{i}=\frac{-W_{i}+p^{\top}\left(\Pi^{-1}b_{i}-e_{i}\right)}{p^{\top}(\Pi^{\top}\Pi)^{-1}p}
-$$
-
-- Market clearing:
-  
-$$ 
-\sum c_{i}=\sum e_{i}
-$$
-
-Denote aggregate consumption $\sum_i c_{i}=c$ and $\sum_i \mu_i = \mu$.
-
-Market clearing requires
+Below we shall construct a Python class with the following attributes:
 
-$$ 
-\Pi^{-1}\left(\sum_{i}b_{i}\right)-(\Pi^{\top}\Pi)^{-1}p\left(\sum_{i}\mu_{i}\right)=\sum_{i}e_{i}
-$$
-which, after a few steps, leads to
+ * **Preferences** in the form of
 
-$$
-p=\mu^{-1}\left(\Pi^{\top}b-\Pi^{\top}\Pi e\right)
-$$
+     * an $n \times n$  positive definite matrix $\Pi$
+     * an $n \times 1$ vector of bliss points $b$
 
-where
+ * **Endowments** in the form of
 
-$$ 
-\mu = \sum_i\mu_{i}=\frac{0 + p^{\top}\left(\Pi^{-1}b-e\right)}{p^{\top}(\Pi^{\top}\Pi)^{-1}p}.
-$$
+     * an $n \times 1$ vector $e$
+     * a scalar "wealth" $W$ with default value $0$
 
-Now consider the representative consumer economy specified above.
 
-Denote the marginal utility of wealth of the representative consumer by $\tilde{\mu}$.
+The class will include a test to make sure that $b  > > \Pi e $ and raise an exception if it is violated
+(at some threshold level we'd have to specify).
 
-The demand function is
+ * **A Person** in the form of a pair that consists of
 
-$$
-c=\Pi^{-1}b-(\Pi^{\top}\Pi)^{-1}\tilde{\mu} p
-$$
+    * **Preferences** and **Endowments**
 
-Substituting this into the budget constraint gives
+ * **A Pure Exchange Economy** will consist of
 
-$$
-\tilde{\mu}=\frac{p^{\top}\left(\Pi^{-1}b-e\right)}{p^{\top}(\Pi^{\top}\Pi)^{-1}p}
-$$
+    * a collection of $m$ **persons**
 
-In an equilibrium $c=e$, so
+       * $m=1$ for our single-agent economy
+       * $m=2$ for our illustrations of a pure exchange economy
 
-$$
-p=\tilde{\mu}^{-1}(\Pi^{\top}b-\Pi^{\top}\Pi e)
-$$
+    * an equilibrium price vector $p$ (normalized somehow)
+    * an equilibrium allocation $c^1, c^2, \ldots, c^m$ -- a collection of $m$ vectors of dimension $n \times 1$
 
-Thus, we have  verified that, up to choice of a numeraire in which to express absolute prices,  the price vector in our representative consumer economy is the same as that in an underlying  economy with multiple consumers.
+Now let's proceed to code.
 
 ```{code-cell} ipython3
 class ExchangeEconomy:
@@ -244,7 +178,7 @@ class ExchangeEconomy:
                  bs, 
                  es, 
                  Ws=None, 
-                 thres=4):
+                 thres=1.5):
         """
         Set up the environment for an exchange economy
 
@@ -253,12 +187,13 @@ class ExchangeEconomy:
             bs (list): all consumers' bliss points
             es (list): all consumers' endowments
             Ws (list): all consumers' wealth
+            thres (float): a threshold set to test b >> Pi e violated
         """
         n, m = Pi.shape[0], len(bs)
 
         # check non-satiation
         for b, e in zip(bs, es):
-            if np.min(b / np.max(Pi @ e)) <= 1.5:
+            if np.min(b / np.max(Pi @ e)) <= thres:
                 raise Exception('set bliss points further away')
 
         if Ws == None:
@@ -305,11 +240,17 @@ class ExchangeEconomy:
         return p, c_s, mu_s
 ```
 
-### Example: Two-Person Economy **without** Production
-  * Study how competitive equilibrium $p, c^1, c^2$ respond to  different
+## Implementation
+
+Next we use the class ``ExchangeEconomy`` defined above to study 
+
+* a two-person economy without production,
+* a dynamic economy, and
+* an economy with risk and arrow securities.
 
-     * $b^i$'s
-     * $e^i$'s
+### Two-person economy without production
+
+Here we tudy how competitive equilibrium $p, c^1, c^2$ respond to different $b^i$ and $e^i$, $i \in \{1, 2\}.
 
 ```{code-cell} ipython3
 Pi = np.array([[1, 0],
@@ -384,7 +325,7 @@ print('Competitive equilibrium price vector:', p)
 print('Competitive equilibrium allocation:', c_s)
 ```
 
-### A **Dynamic Economy**
+### A dynamic economy
 
 Now let's use the tricks described above to study a dynamic economy, one with two periods.
 
@@ -405,7 +346,7 @@ print('Competitive equilibrium price vector:', p)
 print('Competitive equilibrium allocation:', c_s)
 ```
 
-### Example: **Arrow Securities**
+### Risk economy with Arrow securities
 
 We use the tricks described above to interpret  $c_1, c_2$ as "Arrow securities" that are state-contingent claims to consumption goods.
 
@@ -427,3 +368,76 @@ p, c_s, mu_s = EE_AS.competitive_equilibrium()
 print('Competitive equilibrium price vector:', p)
 print('Competitive equilibrium allocation:', c_s)
 ```
+
+## Deducing a Representative Consumer
+
+In the class of multiple consumer economies that we are studying here, it turns out that there
+exists a single **representative consumer** whose preferences and endowments can be deduced from lists of preferences and endowments for the separate individual consumers.
+
+Consider a multiple consumer economy with initial distribution of wealth $W_i$ satisfying $\sum_i W_{i}=0$
+
+We allow an initial redistribution of wealth.
+
+We have the following objects
+
+
+- The demand curve:
+  
+$$ 
+c_{i}=\Pi^{-1}b_{i}-(\Pi^{\top}\Pi)^{-1}\mu_{i}p 
+$$
+
+- The marginal utility of wealth:
+  
+$$ 
+\mu_{i}=\frac{-W_{i}+p^{\top}\left(\Pi^{-1}b_{i}-e_{i}\right)}{p^{\top}(\Pi^{\top}\Pi)^{-1}p}
+$$
+
+- Market clearing:
+  
+$$ 
+\sum c_{i}=\sum e_{i}
+$$
+
+Denote aggregate consumption $\sum_i c_{i}=c$ and $\sum_i \mu_i = \mu$.
+
+Market clearing requires
+
+$$ 
+\Pi^{-1}\left(\sum_{i}b_{i}\right)-(\Pi^{\top}\Pi)^{-1}p\left(\sum_{i}\mu_{i}\right)=\sum_{i}e_{i}
+$$
+which, after a few steps, leads to
+
+$$
+p=\mu^{-1}\left(\Pi^{\top}b-\Pi^{\top}\Pi e\right)
+$$
+
+where
+
+$$ 
+\mu = \sum_i\mu_{i}=\frac{0 + p^{\top}\left(\Pi^{-1}b-e\right)}{p^{\top}(\Pi^{\top}\Pi)^{-1}p}.
+$$
+
+Now consider the representative consumer economy specified above.
+
+Denote the marginal utility of wealth of the representative consumer by $\tilde{\mu}$.
+
+The demand function is
+
+$$
+c=\Pi^{-1}b-(\Pi^{\top}\Pi)^{-1}\tilde{\mu} p
+$$
+
+Substituting this into the budget constraint gives
+
+$$
+\tilde{\mu}=\frac{p^{\top}\left(\Pi^{-1}b-e\right)}{p^{\top}(\Pi^{\top}\Pi)^{-1}p}
+$$
+
+In an equilibrium $c=e$, so
+
+$$
+p=\tilde{\mu}^{-1}(\Pi^{\top}b-\Pi^{\top}\Pi e)
+$$
+
+Thus, we have  verified that, up to choice of a numeraire in which to express absolute prices,  the price vector in our representative consumer economy is the same as that in an underlying  economy with multiple consumers.