Merge pull request #169 from QuantEcon/update_sd1

Edits to introductory supply and demand lecture

Author: jstac

lectures/intro_supply_demand.md

@@ -21,12 +21,6 @@ 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.
-
-* **second welfare theorem:** An allocation of goods to consumers that solves a social planning problem can be supported by a competitive equilibrium with an appropriate initial distribution of  wealth.
-
 Key infrastructure concepts that we'll encounter in this lecture are
 
 * inverse demand curves
@@ -37,13 +31,30 @@ 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.
+
+We will provide a version of the [first fundamental welfare theorem](https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics), which was formulated by 
+
+* [Leon Walras](https://en.wikipedia.org/wiki/L%C3%A9on_Walras)
+* [Francis Ysidro Edgeworth](https://en.wikipedia.org/wiki/Francis_Ysidro_Edgeworth)
+* [Vilfredo Pareto](https://en.wikipedia.org/wiki/Vilfredo_Pareto)
+
+Important extensions to the key ideas were obtained by
+
+* [Abba Lerner](https://en.wikipedia.org/wiki/Abba_P._Lerner)
+* [Harold Hotelling](https://en.wikipedia.org/wiki/Harold_Hotelling)
+* [Paul Samuelson](https://en.wikipedia.org/wiki/Paul_Samuelson)
+* [Kenneth Arrow](https://en.wikipedia.org/wiki/Kenneth_Arrow) 
+* [Gerard Debreu](https://en.wikipedia.org/wiki/G%C3%A9rard_Debreu)
+
+
+In our exposition we will use the following imports.
 
 ```{code-cell} ipython3
 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 $p$.
@@ -53,171 +64,564 @@ Quantity $q$ and price  $p$ are  both scalars.
 We assume that inverse demand and supply curves for the good are:
 
 $$
-p = d_0 - d_1 q, \quad d_0, d_1 > 0
+    p = d_0 - d_1 q, \quad d_0, d_1 > 0
 $$
 
 $$
-p = s_0 + s_1 q , \quad s_0, s_1 > 0
+    p = s_0 + s_1 q , \quad s_0, s_1 > 0
 $$
 
 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 $p q$:
+
+Here is a class that stores parameters for our single good market, as well as
+implementing the inverse demand and supply curves.
+
+```{code-cell} ipython3
+class Market:
+
+    def __init__(self, 
+        d_0=1.0,      # demand intercept
+        d_1=0.6,      # 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
+
+```
+
+Let's create an instance.
+
+```{code-cell} ipython3
+market = Market()
+```
+
+
+Here is a plot of these two functions using `market`.
+
+```{code-cell} ipython3
+:tags: [hide-input]
+market = Market()
+
+grid_min, grid_max, grid_size = 0, 1.5, 200
+q_grid = np.linspace(grid_min, grid_max, grid_size)
+supply_curve = market.inverse_supply(q_grid)
+demand_curve = market.inverse_demand(q_grid)
+
+fig, ax = plt.subplots()
+ax.plot(q_grid, supply_curve, label='supply')
+ax.plot(q_grid, demand_curve, label='demand')
+ax.legend(loc='upper center', frameon=False)
+ax.set_ylim(0, 1.2)
+ax.set_xticks((0, 1))
+ax.set_yticks((0, 1))
+ax.set_xlabel('quantity')
+ax.set_ylabel('price')
+plt.show()
+```
+
+
+
+### Consumer Surplus
+
+Let a quantity $q$ be given and let $p := d_0 - d_1 q$ be the
+corresponding price on the inverse demand curve.
+
+We define **consumer surplus** $S_c(q)$ as the area under an inverse demand
+curve minus $p q$:
+
+$$
+    S_c(q) := 
+    \int_0^{q} (d_0 - d_1 x) dx - p q 
+$$
+
+The next figure illustrates
+
+
+```{code-cell} ipython3
+:tags: [hide-input]
+
+q = 1.25
+p = market.inverse_demand(q)
+ps = np.ones_like(q_grid) * p
+
+fig, ax = plt.subplots()
+ax.plot(q_grid, demand_curve, label='demand')
+ax.fill_between(q_grid[q_grid <= q], 
+    demand_curve[q_grid<=q], 
+    ps[q_grid <= q], 
+    label='consumer surplus', 
+    color='#EED1CF')
+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.set_ylim(0, 1.2)
+ax.set_xticks((q,))
+ax.set_xticklabels(("$q$",))
+ax.set_yticks((p,))
+ax.set_yticklabels(("$p$",))
+ax.set_xlabel('quantity')
+ax.set_ylabel('price')
+plt.show()
+```
+
+
+Consumer surplus gives a measure of total consumer welfare at quantity $q$.
+
+The idea is that the inverse demand curve $d_0 - d_1 q$ shows willingness to
+pay at a given quantity $q$.
+
+The difference between willingness to pay and the actual price is the surplus.
+
+The value $S_c(q)$ is the "sum" (i.e., integral) of these surpluses when the total
+quantity purchased is $q$ and the purchase price is $p$.
+
+Evaluating the integral in the definition of consumer surplus gives
+
+$$
+    S_c(q) 
+    = d_0 q - \frac{1}{2} d_1 q^2 - p q
+$$
+
+
+
+
+### Producer Surplus
+
+Let a quantity $q$ be given and let $p := s_0 + s_1 q$ be the
+corresponding price on the inverse supply curve.
+
+
+We define **producer surplus** as $p q$ minus the area under an inverse supply curve
 
 $$
-\int_0^q (d_0 - d_1 x) dx - pq = d_0 q -.5 d_1 q^2 - pq
+    S_p(q) 
+    := p q - \int_0^q (s_0 + s_1 x) dx 
 $$
 
-We define **producer surplus** as $p q$ minus   the area under an inverse supply curve:
+The next figure illustrates
+
+```{code-cell} ipython3
+:tags: [hide-input]
+
+q = 0.75
+p = market.inverse_supply(q)
+ps = np.ones_like(q_grid) * p
+
+fig, ax = plt.subplots()
+ax.plot(q_grid, supply_curve, label='supply')
+ax.fill_between(q_grid[q_grid <= q], 
+    supply_curve[q_grid<=q], 
+    ps[q_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.set_ylim(0, 1.2)
+ax.set_xticks((q,))
+ax.set_xticklabels(("$q$",))
+ax.set_yticks((p,))
+ax.set_yticklabels(("$p$",))
+ax.set_xlabel('quantity')
+ax.set_ylabel('price')
+plt.show()
+```
+
+Producer surplus measures total producer welfare at quantity $q$ 
+
+The idea is similar to that of consumer surplus.
+
+The inverse supply curve $s_0 + s_1 q$ shows the price at which producers are
+prepared to sell, given quantity $q$.
+
+The difference between willingness to sell and the actual price is the surplus.
+
+The value $S_p(q)$ is the integral of these surpluses.
+
+Evaluating the integral in the definition of consumer surplus gives
 
 $$
-p q - \int_0^q (s_0 + s_1 x) dx = pq - s_0 q - .5 s_1 q^2
+    S_p(q) = pq - s_0 q -  \frac{1}{2} s_1 q^2
 $$
 
-Sometimes economists measure social welfare by a **welfare criterion** that equals  consumer surplus plus producer surplus
+
+### Social Welfare
+
+Sometimes economists measure social welfare by a **welfare criterion** that
+equals consumer surplus plus producer surplus, assuming that consumers and
+producers pay the same price:
 
 $$
-\int_0^q (d_0 - d_1 x) dx - \int_0^q (s_0 + s_1 x) dx  \equiv \textrm{Welf}
+    W(q)
+    = \int_0^q (d_0 - d_1 x) dx - \int_0^q (s_0 + s_1 x) dx  
 $$
 
-or
+Evaluating the integrals gives
 
 $$
-\textrm{Welf} = (d_0 - s_0) q - .5 (d_1 + s_1) q^2
+    W(q) = (d_0 - s_0) q -  \frac{1}{2} (d_1 + s_1) q^2
 $$
 
-To compute a quantity that  maximizes  welfare criterion $\textrm{Welf}$, we differentiate $\textrm{Welf}$ with respect to   $q$ and then set the derivative to zero.
+Here is a Python function that evaluates this social welfare at a given
+quantity $q$ and a fixed set of parameters.
 
-We get
+```{code-cell} ipython3
+def W(q, market):
+    # Unpack
+    d_0, d_1, s_0, s_1 = market.d_0, market.d_1, market.s_0, market.s_1
+    # Compute and return welfare
+    return (d_0 - s_0) * q -  0.5 * (d_1 + s_1) * q**2
+```
+
+The next figure plots welfare as a function of $q$.
+
+
+```{code-cell} ipython3
+:tags: [hide-input]
+q_vals = np.linspace(0, 1.78, 200)
+fig, ax = plt.subplots()
+ax.plot(q_vals, W(q_vals, market), label='welfare')
+ax.legend(frameon=False)
+ax.set_xlabel('quantity')
+plt.show()
+```
+
+Let's now give a social planner the task of maximizing social welfare.
+
+To compute a quantity that  maximizes the welfare criterion, we differentiate
+$W$ with respect to $q$ and then set the derivative to zero.
 
 $$
-\frac{d \textrm{Welf}}{d q} = d_0 - s_0 - (d_1 + s_1) q  = 0
+    \frac{d W(q)}{d q} = d_0 - s_0 - (d_1 + s_1) q  = 0
 $$
 
-which implies
+Solving for $q$ yields
 
 $$
-q = \frac{ d_0 - s_0}{s_1 + d_1}
+    q = \frac{ d_0 - s_0}{s_1 + d_1}
 $$ (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.
 
+
+
 ### Competitive Equilibrium
 
-Instead of equating quantities supplied and demanded, we'll can accomplish the same thing by equating demand price to supply price:
+Instead of equating quantities supplied and demanded, we'll can accomplish the
+same thing by equating demand price to supply price:
 
 $$
-p =  d_0 - d_1 q = s_0 + s_1 q ,
+    p =  d_0 - d_1 q = s_0 + s_1 q 
+$$
+
+If we solve the equation defined by the second equality in the above line for
+$q$, we obtain 
+
 $$
+    q = \frac{ d_0 - s_0}{s_1 + d_1}
+$$ (eq:equilib_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`.
+This is the competitive equilibrium quantity. 
+
+Observe that the equilibrium quantity equals the same $q$ given by equation  {eq}`eq:old1`.
 
 The outcome that the quantity determined by equation {eq}`eq:old1` equates
 supply to demand brings us a **key finding:**
 
-*  a competitive equilibrium quantity maximizes our  welfare criterion
+*  a competitive equilibrium quantity maximizes our welfare criterion
+
+This is a version of the [first fundamental welfare theorem](https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics), 
 
 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
+* 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
 
-### Generalizations
 
-In later lectures,  we'll derive generalizations of the above demand and
-supply curves from other objects.
+## Generalizations
+
+In a {doc}`later lecture <supply_demand_multiple_goods>`,  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.
 
 In addition
 
- * we'll derive  **demand curves** from a consumer problem that maximizes a **utility function** subject to a **budget constraint**.
+ * 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**.
+
+## Exercises
 
- * 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**.
+Suppose now that the inverse demand and supply curves are modified to take the
+form
+
+$$
+    p = i_d(q) := d_0 - d_1 q^{0.6} 
+$$
+
+$$
+    p = i_s(q) := s_0 + s_1 q^{1.2} 
+$$
 
-## Code
+All parameters are positive, as before.
+
+
+
+
+```{exercise}
+:label: isd_ex1
+
+Define a new `Market` class that holds the same parameter values as before by
+changes the `inverse_demand` and `inverse_supply` methods to
+match these new definitions.
+
+Using the class, plot the inverse demand and supply curves $i_d$ and $i_s$
+
+```
+
+
+
+```{solution-start} isd_ex1
+:class: dropdown
+```
 
 ```{code-cell} ipython3
-class SingleGoodMarket:
+class Market:
 
     def __init__(self, 
         d_0=1.0,      # demand intercept
-        d_1=0.5,      # demand slope
+        d_1=0.6,      # 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
+        return self.d_0 - self.d_1 * q**0.6
 
     def inverse_supply(self, q):
-        return self.s_0 + self.s_1 * q
+        return self.s_0 + self.s_1 * q**1.8
+
+```
 
-    def equilibrium_quantity(self):
-        return (self.d_0 - self.s_0) / (self.d_1 + self.s_1)
+Let's create an instance.
 
-    def equilibrium_price(self):
-        q = self.equilibrium_quantity()
-        return self.s_0 + self.s_1 * q
+```{code-cell} ipython3
+market = Market()
 ```
 
+
+Here is a plot of inverse supply and demand.
+
 ```{code-cell} ipython3
-def plot_supply_demand(market):
+:tags: [hide-input]
+
+grid_min, grid_max, grid_size = 0, 1.5, 200
+q_grid = np.linspace(grid_min, grid_max, grid_size)
+supply_curve = market.inverse_supply(q_grid)
+demand_curve = market.inverse_demand(q_grid)
+
+fig, ax = plt.subplots()
+ax.plot(q_grid, supply_curve, label='supply')
+ax.plot(q_grid, demand_curve, label='demand')
+ax.legend(loc='upper center', frameon=False)
+ax.set_ylim(0, 1.2)
+ax.set_xticks((0, 1))
+ax.set_yticks((0, 1))
+ax.set_xlabel('quantity')
+ax.set_ylabel('price')
+plt.show()
+```
+
+
+```{solution-end}
+```
+
+
 
+```{exercise}
+:label: isd_ex2
+
+
+As before, consumer surplus at $q$ is the area under the demand curve minus
+price times quantity:
+
+$$
+    S_c(q) = \int_0^{q} i_d(x) dx - p q 
+$$
+
+Here $p$ is set to $i_d(q)$
+
+Producer surplus is price times quantity minus the area under the inverse
+supply curve:
+
+$$
+    S_p(q) 
+    = p q - \int_0^q i_s(x) dx 
+$$
+
+Here $p$ is set to $i_s(q)$.
+
+Social welfare is the sum of consumer and producer surplus under the
+assumption that the price is the same for buyers and sellers:
+
+$$
+    W(q)
+    = \int_0^q i_d(x) dx - \int_0^q i_q(x) dx  
+$$
+
+Solve the integrals and write a function to compute this quantity numerically
+at given $q$. 
+
+Plot welfare as a function of $q$.
+
+```
+
+
+
+```{solution-start} isd_ex2
+:class: dropdown
+```
+
+Solving the integrals gives 
+
+$$
+    W(q) 
+    = d_0 q - \frac{d_1 q^{1.6}}{1.6}
+        - \left( s_0 q + \frac{s_1 q^{2.8}}{2.8} \right)
+$$
+
+Here's a Python function that computes this value:
+
+
+```{code-cell} ipython3
+def W(q, 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)
+    d_0, d_1, s_0, s_1 = market.d_0, market.d_1, market.s_0, market.s_1
+    # Compute and return welfare
+    S_c = d_0 * q - d_1 * q**1.6 / 1.6
+    S_p = s_0 * q + s_1 * q**2.8 / 2.8
+    return S_c - S_p
+```
+
+The next figure plots welfare as a function of $q$.
+
+
+```{code-cell} ipython3
+fig, ax = plt.subplots()
+ax.plot(q_vals, W(q_vals, market), label='welfare')
+ax.legend(frameon=False)
+ax.set_xlabel('quantity')
+plt.show()
+```
+
+```{solution-end}
+```
 
-    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')
+```{exercise}
+:label: isd_ex3
 
-    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')
+Due to nonlinearities, the new welfare function is not easy to maximize with
+pencil and paper.
+
+Maximize it using `scipy.optimize.minimize_scalar` instead.
 
-    plt.show()
+```
+
+
+
+```{solution-start} isd_ex3
+:class: dropdown
 ```
 
 ```{code-cell} ipython3
-market = SingleGoodMarket()
+from scipy.optimize import minimize_scalar
+
+def objective(q):
+    return -W(q, market)
+
+result = minimize_scalar(objective, bounds=(0, 10))
+print(result.message)
 ```
 
+
+
 ```{code-cell} ipython3
-plot_supply_demand(market)
+maximizing_q = result.x
+print(f"{maximizing_q: .5f}")
+```
+
+```{solution-end}
+```
+
+
+
+```{exercise}
+:label: isd_ex4
+
+Now compute the equilibrium quantity by finding the price that equates supply
+and demand.
+
+You can do this numerically by finding the root of the excess demand function
+
+$$
+    e_d(q) := i_d(q) - i_s(q) 
+$$
+
+You can use `scipy.optimize.newton` to compute the root.
+
+Initialize `newton` with a starting guess somewhere close to 1.0.
+
+(Similar initial conditions will give the same result.)
+
+You should find that the equilibrium price agrees with the welfare maximizing
+price, in line with the first fundamental welfare theorem.
+
+```
+
+
+
+```{solution-start} isd_ex3
+:class: dropdown
+```
+
+
+```{code-cell} ipython3
+from scipy.optimize import newton
+
+def excess_demand(q):
+    return market.inverse_demand(q) - market.inverse_supply(q)
+
+equilibrium_q = newton(excess_demand, 0.1)
+print(f"{equilibrium_q: .5f}")
+```
+
+
+```{solution-end}
 ```
 

lectures/supply_demand_multiple_goods.md

@@ -2,7 +2,16 @@
 
 ## Overview
 
-We study a setting with $n$ goods and $n$ corresponding prices.
+In a {doc}`previous lecture <intro_supply_demand>` we studied supply, demand
+and welfare in a market with just one good.
+
+In this lecture, we study a setting with $n$ goods and $n$ corresponding prices.
+
+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.
+
+* **second welfare theorem:** An allocation of goods to consumers that solves a social planning problem can be supported by a competitive equilibrium with an appropriate initial distribution of  wealth.
 
 
 
@@ -30,30 +39,35 @@ $$
 
 ## From utility function to demand curve
 
-Let
+Our study of consumers will use the following primitives
 
 * $\Pi$ be an $m \times n$ matrix,
-* $c$ be an $n \times 1$ vector of consumptions of various goods,
 * $b$ be an $m \times 1$ vector of bliss points,
 * $e$ be an $n \times 1$ vector of endowments, and
-* $p$ be an $n \times 1$ vector of prices
 
-We assume that $\Pi$ has linearly independent columns, which implies that $\Pi^\top \Pi$ is a positive definite matrix.
 
-* it follows that $\Pi^\top \Pi$ has an inverse.
+We will analyze endogenous objects $c$ and $p$, where
+
+* $c$ is an $n \times 1$ vector of consumptions of various goods,
+* $p$ is an $n \times 1$ vector of prices
+
 
 The matrix $\Pi$ describes a consumer's willingness to substitute one good for every other good.
 
-We shall see below that $(\Pi^T \Pi)^{-1}$ is a matrix of slopes of (compensated) demand curves for $c$ with respect to a vector of prices:
+We assume that $\Pi$ has linearly independent columns, which implies that $\Pi^\top \Pi$ is a positive definite matrix.
+
+* it follows that $\Pi^\top \Pi$ has an inverse.
+
+We shall see below that $(\Pi^\top \Pi)^{-1}$ is a matrix of slopes of (compensated) demand curves for $c$ with respect to a vector of prices:
 
 $$
-    \frac{\partial c } {\partial p} = (\Pi^T \Pi)^{-1}
+    \frac{\partial c } {\partial p} = (\Pi^\top \Pi)^{-1}
 $$
 
 A consumer faces $p$ as a price taker and chooses $c$ to maximize the utility function
 
 $$
-    -.5 (\Pi c -b) ^\top (\Pi c -b )
+    - \frac{1}{2} (\Pi c -b) ^\top (\Pi c -b )
 $$ (eq:old0)
 
 subject to the budget constraint
@@ -62,25 +76,28 @@ $$
     p^\top (c -e ) = 0
 $$ (eq:old2)
 
-We shall specify examples in which  $\Pi$ and $b$ are such that it typically happens that
+We shall specify examples in which  $\Pi$ and $b$ are such that 
 
 $$
-    \Pi c < < b
+    \Pi c \ll b
 $$ (eq:bversusc)
 
-so that utility function {eq}`eq:old2` tells us that the consumer has much less  of each good than he wants.
+This means that the consumer has much less of each good than he wants.
+
+The deviation in {eq}`eq:bversusc` will ultimately  assure us that competitive equilibrium prices  are  positive.
 
-Condition {eq}`eq:bversusc` will ultimately  assure us that competitive equilibrium prices  are  positive.
 
 ### Demand Curve Implied  by Constrained Utility Maximization
 
 For now, we assume that the budget constraint is {eq}`eq:old2`.
 
 So we'll be deriving what is known as  a **Marshallian** demand curve.
 
+Our aim is to maximize [](eq:old0) subject to [](eq:old2).
+
 Form a Lagrangian
 
-$$ L = -.5 (\Pi c -b) ^\top (\Pi c -b ) + \mu [p^\top (e-c)] $$
+$$ L = - \frac{1}{2} (\Pi c -b)^\top (\Pi c -b ) + \mu [p^\top (e-c)] $$
 
 where $\mu$ is a Lagrange multiplier that is often called a **marginal utility of wealth**.
 
@@ -107,8 +124,13 @@ $$ (eq:old4)
 
 Equation {eq}`eq:old4` tells how marginal utility of wealth depends on  the endowment vector  $e$ and the price vector  $p$.
 
-**Remark:** Equation {eq}`eq:old4` is a consequence of imposing that $p^\top (c - e) = 0$.  We could instead take $\mu$ as a parameter and use {eq}`eq:old3` and the budget constraint {eq}`eq:old2p` to solve for $W.$ Which way we proceed determines whether we are constructing a **Marshallian** or **Hicksian** demand curve.
+```{note}
+Equation {eq}`eq:old4` is a consequence of imposing that $p^\top (c - e) = 0$.  
+
+We could instead take $\mu$ as a parameter and use {eq}`eq:old3` and the budget constraint {eq}`eq:old2p` to solve for wealth. 
 
+Which way we proceed determines whether we are constructing a **Marshallian** or **Hicksian** demand curve.
+```
 
 ## Endowment economy
 
@@ -134,52 +156,63 @@ This amounts to choosing a common  unit (or numeraire) in which prices of all go
 
 We'll set $\mu=1$.
 
-**Exercise:** Verify that setting $\mu=1$ in {eq}`eq:old3` implies that   formula {eq}`eq:old4` is satisfied.
+```{exercise}
+:label: sdm_ex1
 
-**Exercise:** Verify that setting  $\mu=2$ in {eq}`eq:old3` also implies that formula {eq}`eq:old4` is satisfied.
+Verify that setting $\mu=1$ in {eq}`eq:old3` implies that formula {eq}`eq:old4` is satisfied.
+
+```
+
+```{exercise}
+:label: sdm_ex2
+
+Verify that setting  $\mu=2$ in {eq}`eq:old3` also implies that formula
+{eq}`eq:old4` is satisfied.
+
+```
 
 
 ## Digression: Marshallian and Hicksian Demand Curves
 
-**Remark:** Sometimes we'll use budget constraint {eq}`eq:old2` in situations in which a consumers's endowment vector $e$ is his **only** source of income. Other times we'll instead assume that the consumer has another source of income (positive or negative) and write his budget constraint as
+Sometimes we'll use budget constraint {eq}`eq:old2` in situations in which a consumers's endowment vector $e$ is his **only** source of income.
+
+Other times we'll instead assume that the consumer has another source of income (positive or negative) and write his budget constraint as
 
 $$
-p ^\top (c -e ) = W
+p ^\top (c -e ) = w
 $$ (eq:old2p)
 
-where $W$ is measured in "dollars" (or some other **numeraire**) and component $p_i$ of the price vector is measured in dollars per unit of good $i$.
+where $w$ is measured in "dollars" (or some other **numeraire**) and component $p_i$ of the price vector is measured in dollars per unit of good $i$.
 
-Whether the consumer's budget constraint is  {eq}`eq:old2` or {eq}`eq:old2p` and whether we take $W$ as a free parameter or instead as an endogenous variable   will  affect the consumer's marginal utility of wealth.
+Whether the consumer's budget constraint is  {eq}`eq:old2` or {eq}`eq:old2p` and whether we take $w$ as a free parameter or instead as an endogenous variable   will  affect the consumer's marginal utility of wealth.
 
 Consequently, how we set $\mu$  determines whether we are constructing
 
 * a **Marshallian** demand curve, as when we use {eq}`eq:old2` and solve for $\mu$ using equation {eq}`eq:old4` below, or
-* a **Hicksian** demand curve, as when we  treat $\mu$ as a fixed parameter and solve for $W$ from {eq}`eq:old2p`.
+* a **Hicksian** demand curve, as when we  treat $\mu$ as a fixed parameter and solve for $w$ from {eq}`eq:old2p`.
 
 Marshallian and Hicksian demand curves contemplate different mental experiments:
 
-* For a Marshallian demand curve, hypothetical changes in a price vector  have  both **substitution** and **income** effects
+For a Marshallian demand curve, hypothetical changes in a price vector  have  both **substitution** and **income** effects
 
-  * income effects are consequences of  changes in $p^\top e$ associated with the change in the price vector
+* income effects are consequences of  changes in $p^\top e$ associated with the change in the price vector
 
-* For a Hicksian demand curve, hypothetical price vector  changes  have only **substitution**  effects
+For a Hicksian demand curve, hypothetical price vector  changes  have only **substitution**  effects
 
-  * changes in the price vector leave the $p^\top e + W$ unaltered because we freeze $\mu$ and solve for $W$
+* changes in the price vector leave the $p^\top e + w$ unaltered because we freeze $\mu$ and solve for $w$
 
-Sometimes a Hicksian demand curve is called a **compensated** demand curve in order to emphasize that, to disarm the income (or wealth) effect associated with a price change, the consumer's wealth $W$ is adjusted.
+Sometimes a Hicksian demand curve is called a **compensated** demand curve in order to emphasize that, to disarm the income (or wealth) effect associated with a price change, the consumer's wealth $w$ is adjusted.
 
 We'll discuss these distinct demand curves more  below.
 
 
 
-## Dynamics and Risk as Special Cases of Pure Exchange Economy
+## Dynamics and Risk as Special Cases
 
 Special cases of our $n$-good pure exchange  model can be created to represent
 
-* dynamics
-  - by putting different dates on different commodities
-* risk
-  - by interpreting delivery  of goods as being contingent on states of the world whose realizations are described by a **known probability distribution**
+* **dynamics** --- by putting different dates on different commodities
+* **risk** --- by interpreting delivery  of goods as being contingent on states of the world whose realizations are described by a *known probability distribution*
 
 Let's illustrate how.
 
@@ -188,7 +221,7 @@ Let's illustrate how.
 Suppose that we want to represent a utility function
 
 $$
-  -.5 [(c_1 - b_1)^2 + \beta (c_2 - b_2)^2]
+  - \frac{1}{2} [(c_1 - b_1)^2 + \beta (c_2 - b_2)^2]
 $$
 
 where $\beta \in (0,1)$ is a discount factor, $c_1$ is consumption at time $1$ and $c_2$ is consumption at time 2.
@@ -223,7 +256,13 @@ The right side is the **discounted present value** of the consumer's endowment.
 
 The relative price  $\frac{p_1}{p_2}$ has units of time $2$ goods per unit of time $1$ goods.
 
-Consequently, $(1+r) = R \equiv \frac{p_1}{p_2}$ is the  **gross interest rate** and $r$ is the **net interest rate**.
+Consequently, 
+
+$$
+    (1+r) := R := \frac{p_1}{p_2}
+$$ 
+
+is the  **gross interest rate** and $r$ is the **net interest rate**.
 
 ### Risk and state-contingent claims
 
@@ -242,7 +281,7 @@ As an example, our consumer confronts **risk** meaning in particular that
 Before the outcome is realized, the the consumer's **expected utility** is
 
 $$
--.5 [\lambda (c_1 - b_1)^2 + (1-\lambda)(c_2 - b_2)^2]
+- \frac{1}{2} [\lambda (c_1 - b_1)^2 + (1-\lambda)(c_2 - b_2)^2]
 $$
 
 where
@@ -312,7 +351,7 @@ to maximize total revenue minus total costs.
 The firm's total  revenue equals $p^\top q$ and its total cost equals $C(q)$  where $C(q)$ is a total cost function
 
 $$
-C(q) = h ^\top q + .5 q^\top J q
+C(q) = h ^\top q +  \frac{1}{2} q^\top J q
 $$
 
 
@@ -336,7 +375,7 @@ $$
 where
 
 $$
-H = .5 (J + J')
+H =  \frac{1}{2} (J + J')
 $$
 
 An $n \times 1$ vector of marginal revenues for the price-taking firm is $\frac{\partial p^\top q}
@@ -410,15 +449,15 @@ Thus, instead of being a price-taker, a monopolist sets prices to maximize profi
 So the monopolist's total profits as a function of its  output $q$ is
 
 $$
-[\mu^{-1} \Pi^\top (b - \Pi q)]^\top  q - h^\top q - .5 q^\top J q
+[\mu^{-1} \Pi^\top (b - \Pi q)]^\top  q - h^\top q -  \frac{1}{2} q^\top J q
 $$ (eq:monopprof)
 
 After finding
 first-order necessary conditions for maximizing monopoly profits with respect to $q$
 and solving them for $q$, we find that the monopolist sets
 
 $$
-q = (H + 2 \mu^{-1} \Pi^T \Pi)^{-1} (\mu^{-1} \Pi^\top b - h)
+q = (H + 2 \mu^{-1} \Pi^\top \Pi)^{-1} (\mu^{-1} \Pi^\top b - h)
 $$ (eq:qmonop)
 
 We'll soon  see that a monopolist sets a **lower output** $q$ than does either a
@@ -428,7 +467,13 @@ We'll soon  see that a monopolist sets a **lower output** $q$ than does either a
  * a competitive equilibrium
 
 
-**Exercise:** Please  verify the monopolist's supply curve {eq}`eq:qmonop`.
+
+```{exercise}
+:label: sdm_ex3
+
+Please  verify the monopolist's supply curve {eq}`eq:qmonop`.
+
+```
 
 
 
@@ -438,19 +483,20 @@ We'll soon  see that a monopolist sets a **lower output** $q$ than does either a
 Our welfare maximization problem -- also sometimes called a  social planning  problem  -- is to choose $c$ to maximize
 
 $$
--.5 \mu^{-1}(\Pi c -b) ^\top (\Pi c -b )
+    - \frac{1}{2} \mu^{-1}(\Pi c -b) ^\top (\Pi c -b )
 $$
 
 minus the area under the inverse supply curve, namely,
 
 $$
-h c + .5 c^\top J c  .
+    h c +  \frac{1}{2} c^\top J c  
 $$
 
 So the welfare criterion is
 
 $$
--.5 \mu^{-1}(\Pi c -b) ^\top (\Pi c -b ) -h c - .5 c^\top J c
+    - \frac{1}{2} \mu^{-1}(\Pi c -b)^\top (\Pi c -b ) -h c 
+        -  \frac{1}{2} c^\top J c
 $$
 
 In this formulation, $\mu$ is a parameter that describes how the planner weights interests of outside suppliers and our representative consumer.
@@ -472,8 +518,4 @@ We can deduce a competitive equilibrium price vector from either
   * the inverse demand curve, or
 
   * the inverse supply curve
-
-<!-- #endregion -->
-
-<!-- #region -->
-
+