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Merge pull request #278 from QuantEcon/review_supply_demand
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Review supply demand
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@jstac
jstac authored Aug 2, 2023
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76 changes: 37 additions & 39 deletions lectures/intro_supply_demand.md
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Original file line number Diff line number Diff line change
Expand Up @@ -3,6 +3,8 @@ jupytext:
text_representation:
extension: .md
format_name: myst
format_version: 0.13
jupytext_version: 1.11.5
kernelspec:
display_name: Python 3 (ipykernel)
language: python
Expand All @@ -11,7 +13,7 @@ kernelspec:

# Introduction to Supply and Demand

## Outline
## Overview

This lecture is about some models of equilibrium prices and quantities, one of
the main topics of elementary microeconomics.
Expand All @@ -31,9 +33,9 @@ Key infrastructure concepts that we'll encounter in this lecture are
* social welfare as the sum of consumer and producer surpluses
* relationship between equilibrium quantity and social welfare optimum

Throughout the lectures, we'll assume that inverse demand and supply curves are **affine** functions of output.
Throughout the lectures, we'll assume that inverse demand and supply curves are **affine** functions of quantity.

("Affine" means "linear plus a constant".)
("Affine" means "linear plus a constant" and [here](https://math.stackexchange.com/questions/275310/what-is-the-difference-between-linear-and-affine-function) is a nice discussion about it.)

We'll also assume affine inverse supply and demand functions when we study models with multiple consumption goods in our {doc}`subsequent lecture <supply_demand_multiple_goods>`.

Expand All @@ -46,7 +48,6 @@ 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 $p$.
Expand All @@ -73,11 +74,11 @@ 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
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
Expand All @@ -87,22 +88,19 @@ class Market:
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
Expand All @@ -122,7 +120,7 @@ ax.set_ylabel('price')
plt.show()
```

In the above graph, an **equilibrium** price, quantity pair occurs at the intersection of the supply and demand curves.
In the above graph, an **equilibrium** price-quantity pair occurs at the intersection of the supply and demand curves.

### Consumer surplus

Expand All @@ -135,7 +133,7 @@ curve minus $p q$:
$$
S_c(q) :=
\int_0^{q} (d_0 - d_1 x) dx - p q
$$
$$ (eq:cstm_spls)
The next figure illustrates
Expand All @@ -149,11 +147,11 @@ 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.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)
Expand All @@ -168,18 +166,17 @@ ax.set_ylabel('price')
plt.show()
```

Consumer surplus provides a measure of total consumer welfare at quantity $q$.
The idea is that the inverse demand curve $d_0 - d_1 q$ shows a consumer's willingness to
pay for an additional increment of the good at a given quantity $q$.
pay for an additional increment of the good at a given quantity $q$.
The difference between willingness to pay and the actual price is consumer 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
Evaluating the integral in the definition of consumer surplus {eq}`eq:cstm_spls` gives
$$
S_c(q)
Expand All @@ -200,7 +197,7 @@ We define **producer surplus** as $p q$ minus the area under an inverse supply c
$$
S_p(q)
:= p q - \int_0^q (s_0 + s_1 x) dx
$$
$$ (eq:pdcr_spls)
The next figure illustrates
Expand All @@ -213,11 +210,11 @@ 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.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)
Expand All @@ -243,7 +240,7 @@ The difference between willingness to sell and the actual price is producer surp
The value $S_p(q)$ is the integral of these surpluses.
Evaluating the integral in the definition of consumer surplus gives
Evaluating the integral in the definition of producer surplus {eq}`eq:pdcr_spls` gives
$$
S_p(q) = pq - s_0 q - \frac{1}{2} s_1 q^2
Expand Down Expand Up @@ -275,14 +272,15 @@ 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
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')
Expand Down Expand Up @@ -394,11 +392,11 @@ Using the class, plot the inverse demand and supply curves $i_d$ and $i_s$
```{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
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
Expand All @@ -408,7 +406,6 @@ class Market:
def inverse_supply(self, q):
return self.s_0 + self.s_1 * q**1.8
```
Let's create an instance.
Expand Down Expand Up @@ -498,7 +495,8 @@ Here's a Python function that computes this value:
```{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
d_0, d_1 = market.d_0, market.d_1
s_0, s_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
Expand Down Expand Up @@ -577,7 +575,7 @@ price, in line with the first fundamental welfare theorem.
```
```{solution-start} isd_ex3
```{solution-start} isd_ex4
:class: dropdown
```
Expand Down

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