Fix supply demand

lectures/intro_supply_demand.md

@@ -57,9 +57,9 @@ 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 $p$.
+We study a market for a single good in which buyers and sellers exchange a quantity $q$ for a price $p$.
 
-Quantity $q$ and price  $p$ are  both scalars.
+Quantity $q$ and price $p$ are  both scalars.
 
 We assume that inverse demand and supply curves for the good are:
 
@@ -177,7 +177,7 @@ 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
+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.
@@ -312,16 +312,15 @@ $$
     q = \frac{ d_0 - s_0}{s_1 + d_1}
 $$ (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.
+Let's remember the quantity $q$ given by equation {eq}`eq:old1` that a social planner would choose to maximize consumer surplus 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 that equates supply to demand.
 
 
 
 ### Competitive Equilibrium
 
-Instead of equating quantities supplied and demanded, we'll can accomplish the
+Instead of equating quantities supplied and demanded, we can accomplish the
 same thing by equating demand price to supply price:
 
 $$
@@ -347,29 +346,24 @@ supply to demand brings us a **key finding:**
 
 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
+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
 
 ## Generalizations
 
-In a {doc}`later lecture <supply_demand_multiple_goods>`,  we'll derive
+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.
+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**.
+   taker and maximizes his profits minus total costs that are described by a **cost function**.
 
 ## Exercises
 
@@ -387,21 +381,18 @@ $$
 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
+changing 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
 ```
@@ -432,7 +423,6 @@ Let's create an instance.
 market = Market()
 ```
 
-
 Here is a plot of inverse supply and demand.
 
 ```{code-cell} ipython3
@@ -455,16 +445,13 @@ 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:
 
@@ -500,7 +487,6 @@ Plot welfare as a function of $q$.
 ```
 
 
-
 ```{solution-start} isd_ex2
 :class: dropdown
 ```
@@ -515,7 +501,6 @@ $$
 
 Here's a Python function that computes this value:
 
-
 ```{code-cell} ipython3
 def W(q, market):
     # Unpack
@@ -528,7 +513,6 @@ def W(q, market):
 
 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')
@@ -541,11 +525,9 @@ plt.show()
 ```
 
 
-
 ```{exercise}
 :label: isd_ex3
 
-
 Due to nonlinearities, the new welfare function is not easy to maximize with
 pencil and paper.
 
@@ -554,7 +536,6 @@ Maximize it using `scipy.optimize.minimize_scalar` instead.
 ```
 
 
-
 ```{solution-start} isd_ex3
 :class: dropdown
 ```
@@ -569,8 +550,6 @@ result = minimize_scalar(objective, bounds=(0, 10))
 print(result.message)
 ```
 
-
-
 ```{code-cell} ipython3
 maximizing_q = result.x
 print(f"{maximizing_q: .5f}")
@@ -580,7 +559,6 @@ print(f"{maximizing_q: .5f}")
 ```
 
 
-
 ```{exercise}
 :label: isd_ex4
 
@@ -605,12 +583,10 @@ price, in line with the first fundamental welfare theorem.
 ```
 
 
-
 ```{solution-start} isd_ex3
 :class: dropdown
 ```
 
-
 ```{code-cell} ipython3
 from scipy.optimize import newton
 
@@ -621,7 +597,5 @@ equilibrium_q = newton(excess_demand, 0.1)
 print(f"{equilibrium_q: .5f}")
 ```
 
-
 ```{solution-end}
 ```
-