misc

Author: jstac

lectures/intro_supply_demand.md

@@ -1,12 +1,17 @@
 # Introduction to Supply and Demand
 
-This lecture is about some  linear models of equilibrium prices and
-quantities, one of the main topics of elementary microeconomics.
-
-Our approach is first to offer a  scalar version with one good and one price.
++++
 
 ## Outline
 
+This lecture is about some models of equilibrium prices and quantities, one of
+the main topics of elementary microeconomics.
+
+Throughout the lecture, we focus on models with one good and one price.
+
+({doc}`Later <supply_demand_multiple_goods>` we will investigate settings with
+many goods.)
+
 We shall describe two classic welfare theorems:
 
 * **first welfare theorem:** for a  given a distribution of wealth among consumers,  a competitive  equilibrium  allocation of goods solves a  social planning problem.
@@ -23,6 +28,13 @@ Key infrastructure concepts that we'll encounter in this lecture are
 * social welfare as a sum of consumer and producer surpluses
 * competitive equilibrium
 
+We will use the following imports.
+
+```python
+import numpy as np
+import matplotlib.pyplot as plt
+```
+
 ## Supply and Demand
 
 We study a market for a single good in which buyers and sellers exchange a  quantity $q$ for a price $p$.
@@ -41,7 +53,7 @@ $$
 
 We call them inverse demand and supply curves because price is on the left side of the equation rather than on the right side as it would be in a direct demand or supply function.
 
-
+### Surpluses and Welfare
 
 We define **consumer surplus** as the  area under an inverse demand curve minus $p q$:
 
@@ -83,15 +95,20 @@ $$ (eq:old1)
 
 Let's remember the quantity $q$ given by equation {eq}`eq:old1` that a social planner would choose to maximize consumer plus producer surplus.
 
-We'll compare it to the quantity that emerges in a competitive  equilibrium equilibrium that equates
-supply to demand.
+We'll compare it to the quantity that emerges in a competitive  equilibrium
+equilibrium that equates supply to demand.
+
++++
+
+### Competitive Equilibrium
 
 Instead of equating quantities supplied and demanded, we'll can accomplish the same thing by equating demand price to supply price:
 
 $$
 p =  d_0 - d_1 q = s_0 + s_1 q ,
 $$
 
++++
 
 It we solve the equation defined by the second equality in the above line for $q$, we obtain the
 competitive equilibrium quantity; it equals the same $q$ given by equation  {eq}`eq:old1`.
@@ -105,15 +122,102 @@ It also brings  a useful  **competitive equilibrium computation strategy:**
 
 * after solving the welfare problem for an optimal  quantity, we can read a competitive equilibrium price from  either supply price or demand price at the  competitive equilibrium quantity
 
-Soon we'll derive generalizations of the above demand and supply
-curves from other objects.
+### Generalizations
+
+In later lectures,  we'll derive generalizations of the above demand and
+supply curves from other objects.
 
-Our generalizations will extend the preceding analysis of a market for a single good to the analysis
-of $n$ simultaneous markets in $n$ goods.
+Our generalizations will extend the preceding analysis of a market for a
+single good to the analysis of $n$ simultaneous markets in $n$ goods.
 
 In addition
 
  * we'll derive  **demand curves** from a consumer problem that maximizes a **utility function** subject to a **budget constraint**.
 
  * we'll derive  **supply curves** from the problem of a producer who is price taker and maximizes his profits minus total costs that are described by a  **cost function**.
-<!-- #endregion -->
+
++++
+
+## Code
+
++++
+
+```python
+class SingleGoodMarket:
+
+    def __init__(self, 
+        d_0=1.0,      # demand intercept
+        d_1=0.5,      # demand slope
+        s_0=0.1,      # supply intercept
+        s_1=0.4):     # supply slope
+
+        self.d_0, self.d_1 = d_0, d_1
+        self.s_0, self.s_1 = s_0, s_1
+
+    def inverse_demand(self, q):
+        return self.d_0 - self.d_1 * q
+
+    def inverse_supply(self, q):
+        return self.s_0 + self.s_1 * q
+
+    def equilibrium_quantity(self):
+        return (self.d_0 - self.s_0) / (self.d_1 + self.s_1)
+
+    def equilibrium_price(self):
+        q = self.equilibrium_quantity()
+        return self.s_0 + self.s_1 * q
+```
+
+```python
+def plot_supply_demand(market):
+
+    # Unpack
+    d_0, d_1 = market.d_0, market.d_1
+    s_0, s_1 = market.s_0, market.s_1
+    q = market.equilibrium_quantity()
+    p = market.equilibrium_price()
+    grid_size = 200
+    x_grid = np.linspace(0, 2 * q, grid_size)
+    ps = np.ones_like(x_grid) * p
+    supply_curve = market.inverse_supply(x_grid)
+    demand_curve = market.inverse_demand(x_grid)
+
+    fig, ax = plt.subplots()
+
+    ax.plot(x_grid, supply_curve, label='Supply', color='#020060')
+    ax.plot(x_grid, demand_curve, label='Demand', color='#600001')
+
+    ax.fill_between(x_grid[x_grid <= q], 
+        demand_curve[x_grid<=q], 
+        ps[x_grid <= q], 
+        label='Consumer surplus', 
+        color='#EED1CF')
+    ax.fill_between(x_grid[x_grid <= q], 
+        supply_curve[x_grid <= q], 
+        ps[x_grid <= q], 
+        label='Producer surplus', 
+        color='#E6E6F5')
+
+    ax.vlines(q, 0, p, linestyle="dashed", color='black', alpha=0.7)
+    ax.hlines(p, 0, q, linestyle="dashed", color='black', alpha=0.7)
+
+    ax.legend(loc='upper center', frameon=False)
+    ax.margins(x=0, y=0)
+    ax.set_ylim(0)
+    ax.set_xlabel('Quantity')
+    ax.set_ylabel('Price')
+
+    plt.show()
+```
+
+```python
+market = SingleGoodMarket()
+```
+
+```python
+plot_supply_demand(market)
+```
+
+```python
+
+```