UMAP MODULE 294 - Price Discrimination and Consumer Surplus
MA 314 - Project 1
UMAP 294
John Joseph
Peter Ivancevic
Travis Jegerlehner
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Calculus and Economics
• Calculus methods can be used to compute consumer
surplus and other figures associated with economics.
• Calculus can predict approximate figures for large
situations, but it can not predict exact figures.
• Calculus can be applied to real world economic
problems. However, to be applied correctly to economic
functions and quantities, a full understanding of
economics is required.
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Supply Function
price
• Supply Function:
– The supply function represents the given quantity of a
good that a supplier will produce at a given price.
– Supply is an increasing
function. As price rises
Supply Function
quantity supplied
increases (see
S(q)
graph right).
quantity
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Demand Function
price
• Demand Function
– The demand function represents the given quantity of a
good that is demanded by consumers at a given price.
– Demand is a decreasing
Demand Function
function. As price rises
quantity supplied
decreases (see
graph right).
D(q)
quantity
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Equilibrium Point
• The Supply and Demand Function intersect at the
equilibrium price (p*) and the equilibrium quantity (q*).
• The points can be determined by solving the equation:
D(q) = S(q)
Equilibrium Point (q*, p*)
price
S(q)
(q*, p*)
D(q)
quantity
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Continuity of
D(q) and S(q)
• In reality, the Demand and Supply functions are not
continuous. They are step or discrete functions because
they deal with whole number quantities produced.
Discrete Function
price
price
Step Function
quantity
quantity
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Continuity of
D(q) and S(q)
price
• By assuming that the jumps in the Supply and Demand
functions are small, continuous functions can be used to
approximate the discrete functions. This assumption
allows the tools of
calculus to be used
Continuous Approximaiton
to solve problems
related to supply and
demand.
quantity
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Calculating Consumer Surplus
• To calculate the revenue generated by Demand we can
multiply the price each consumer is willing to pay by the
quantity demanded (discrete Demand function).
Revenue Generated
*Area of each
rectangle
represents how
much revenue is
generated from
the quantity
demanded at the
given price.
price
• These amounts are
represented by the
area of the rectangles
formed by the discrete
demand function.
q1
q2
q3
q4
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Calculating Consumer Surplus
• To calculate the total revenue generated by the firm, we
use the summation represented by the formula below
n

i1
D( qi ) ( qi )
• We can then apply calculus to the continuous Demand
function to use the integral below to approximate the total
revenue.
n


 D( qi ) dq

1
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Calculating Consumer Surplus
• In a competitive market consumers do not pay the price
they are willing to pay for a good (Perfect Price
Discrimination).
• Consumers of a good pay the equilibrium price (p*) for
the good. Therefore, the total revenue generated is p*q*.
• Because some consumers pay less than they were willing
to pay for the good, they experience consumer surplus.
The consumer surplus can be computed using the
following formula:
e
q




1
D( qi )p e dq
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Calculating Consumer Surplus
• On the graph, the consumer surplus (yellow) is the area
located below the Demand function and above the
rectangle that represents the revenue generated (red).
Consumer surplus
S(q)
D(q)
Revenue
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Calculating Producer Surplus
• The supplier also experiences producer surplus. The area
(yellow) above the Supply function and still in the
rectangle representing income is the producer surplus.
Producer Surplus
Supplier surplus
S(q)
Producer Surplus =
q
D(q)
Revenue
e
 e

 p S( qi ) dq

1
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Elasticity of Supply and Demand
• Supply and Demand curve are often approximated as
linear functions.
• The elasticity of demand measures how changes in price
affect changes in quantity demanded. If elasticity is high,
small changes in price have a large effect on the quantity
demanded. If elasticity is small, the opposite is true.
• The elasticity of supply measure how changes in price
affect quantity supplied. If elasticity is high, small
changes in price have a large effect on quantity supplied.
If elasticity is small (just as in the elasticity of demand)
the opposite is true.
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Two Tier Price Discrimination
• Two tier price discrimination occurs when sellers have 2
different prices for products for 2 different consumers.
• If the competitive
equilibrium price p*
is calculated then the
Revenue at higher price
total revenue to seller is
p1*q1 + p*(q*-q1).
• Maximizing the
Revenue at
revenue function
lower price
reveals what price
sellers should charge.
UMAP MODULE 294 - Price Discrimination and Consumer Surplus
Summary
• Using functions to represent Supply and Demand for a
good, we can calculate and equilibrium price and quantity
that meets supply and demand.
• Assuming the discrete functions that represent supply and
demand have small increments, we can use continuous
functions to approximate them and apply Calculus.
• Using Calculus we can calculate consumer surplus,
producer surplus, and total revenue. We can also use
calculus to determine the price that should be charged for
two tier price discrimination to obtain maximum revenue
for the producer.