7. Introduction to Supply and Demand#

7.1. Overview#

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.

In a subsequent lecture we will investigate settings with many goods.

Key infrastructure concepts that we’ll encounter in this lecture are

inverse demand curves
inverse supply curves
consumer surplus
producer surplus
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 quantity.

(“Affine” means “linear plus a constant” and here 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 subsequent lecture.

We do this in order to simplify the exposition and enable us to use just a few tools from linear algebra, namely, matrix multiplication and matrix inversion.

In our exposition we will use the following imports.

import numpy as np
import matplotlib.pyplot as plt

7.2. Supply and demand#

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.

We assume that inverse demand and supply curves for the good are:

\[ p = d_0 - d_1 q, \quad d_0, d_1 > 0 \]

\[ p = s_0 + s_1 q , \quad s_0, s_1 > 0 \]

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.

Here is a class that stores parameters for our single good market, as well as implementing the inverse demand and supply curves.

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.

market = Market()

Here is a plot of these two functions using market.

_images/b69143db7721008349b41f252d65ff8c42a57d08248ac7eb650929640096d92d.png

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

7.2.1. 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 \(p q\):

(7.1)#\[ S_c(q) := \int_0^{q} (d_0 - d_1 x) dx - p q \]

The next figure illustrates

_images/52ab72d3a0aec9c0ee8e2320d693c9bf6753e72d8f669ab077984957894585ce.png

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 \(q\).

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 (7.1) gives

\[ S_c(q) = d_0 q - \frac{1}{2} d_1 q^2 - p q \]

7.2.2. 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

(7.2)#\[ S_p(q) := p q - \int_0^q (s_0 + s_1 x) dx \]

The next figure illustrates

_images/b8eddca21acd0f2749daecb15d7a188b44be1a527b411e0321f2e58a99b09953.png

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 producer surplus.

The value \(S_p(q)\) is the integral of these surpluses.

Evaluating the integral in the definition of producer surplus (7.2) gives

\[ S_p(q) = pq - s_0 q - \frac{1}{2} s_1 q^2 \]

7.2.4. Competitive equilibrium#

Instead of equating quantities supplied and demanded, we can accomplish the same thing by equating demand price to supply price:

\[ p = d_0 - d_1 q = s_0 + s_1 q \]

If we solve the equation defined by the second equality in the above line for \(q\), we obtain

(7.4)#\[ q = \frac{ d_0 - s_0}{s_1 + d_1} \]

This is the competitive equilibrium quantity.

Observe that the equilibrium quantity equals the same \(q\) given by equation (7.3).

The outcome that the quantity determined by equation (7.3) equates supply to demand brings us a key finding:

a competitive equilibrium quantity maximizes our welfare criterion

This is a version of the first fundamental welfare theorem,

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

7.3. Generalizations#

In a later lecture, 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 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.

7.4. Exercises#

Suppose now that the inverse demand and supply curves are modified to take the form

\[ p = i_d(q) := d_0 - d_1 q^{0.6} \]

\[ p = i_s(q) := s_0 + s_1 q^{1.8} \]

All parameters are positive, as before.

Exercise 7.1

Define a new Market class that holds the same parameter values as before by 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 to Exercise 7.1

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**0.6

    def inverse_supply(self, q):
        return self.s_0 + self.s_1 * q**1.8

Let’s create an instance.

market = Market()

Here is a plot of inverse supply and demand.

_images/7585f016d6530643ea68cf06529f59155fc51a47b2af62f6339ec0574c635c37.png

Exercise 7.2

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:

\[ W(q) = \int_0^q i_d(x) dx - \int_0^q i_s(x) dx \]

Solve the integrals and write a function to compute this quantity numerically at given \(q\).

Plot welfare as a function of \(q\).

Solution to Exercise 7.2

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:

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
    # 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 \(q\).

fig, ax = plt.subplots()
ax.plot(q_vals, W(q_vals, market), label='welfare')
ax.legend(frameon=False)
ax.set_xlabel('quantity')
plt.show()

_images/40b61956be8bd3211ea6901eeefe8f70b260cdddcd4f724a566ee2498d547c6c.png

Exercise 7.3

Due to nonlinearities, the new welfare function is not easy to maximize with pencil and paper.

Maximize it using scipy.optimize.minimize_scalar instead.

Solution to Exercise 7.3

from scipy.optimize import minimize_scalar

def objective(q):
    return -W(q, market)

result = minimize_scalar(objective, bounds=(0, 10))
print(result.message)

Solution found.

maximizing_q = result.x
print(f"{maximizing_q: .5f}")

 0.90564

Exercise 7.4

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 to Exercise 7.4

from scipy.optimize import newton

def excess_demand(q):
    return market.inverse_demand(q) - market.inverse_supply(q)

equilibrium_q = newton(excess_demand, 0.99)
print(f"{equilibrium_q: .5f}")

 0.90564