MA 314 - Project 1 UMAP 294

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MA 314 - Project 1UMAP 294
John Joseph Peter Ivancevic Travis Jegerlehner
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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.

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Supply Function

• Supply Function
• The supply function represents the given quantity
  of a good that a supplier will produce at a given
  price.
• Supply is an increasingfunction. As price
  risesquantity supplied increases (see graph
  right).

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Demand Function

• Demand Function
• The demand function represents the given quantity
  of a good that is demanded by consumers at a
  given price.
• Demand is a decreasingfunction. As price
  risesquantity supplied decreases (see graph
  right).

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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)

S(q)
(q, p)
D(q)
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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.

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Continuity of D(q) and S(q)

• 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
  ofcalculus to be usedto solve problemsrelated
  to supply anddemand.

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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).
• These amounts are represented by thearea of the
  rectanglesformed by the discretedemand
  function.

Area of each rectangle represents how much
revenue is generated from the quantity demanded
at the given price.
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Calculating Consumer Surplus

• To calculate the total revenue generated by the
  firm, we use the summation represented by the
  formula below
• We can then apply calculus to the continuous
  Demand function to use the integral below to
  approximate the total revenue.

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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 pq.
• 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

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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
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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
Producer Surplus
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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.

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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 total revenue to seller is
  p1q1 p(q-q1).
• Maximizing the revenue functionreveals what
  pricesellers should charge.

Revenue at higher price
Revenue at lower price
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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.