MATH 1325 13 4 Lecture

This paragraph introduces the concept of a continuous income stream, using an oil company as an example. It explains that the income from an oil well can be thought of as a continuous stream due to the constant pumping of oil. The paragraph details how to calculate the total income over a period of time by integrating the rate of change of income, which is given as a function of time. An example problem involving Bailey Oil is presented, where the annual rate of income flow is provided, and the task is to estimate the total income over ten years. The solution involves integration using substitution and emphasizes the importance of considering units, resulting in a total income of two million five hundred ninety-four thousand dollars.

The second paragraph delves into the importance of calculating the present value of a continuous income stream, which is the current worth of future income. This is crucial for making financial decisions such as replacing machinery or evaluating the purchase or sale of assets like oil wells. The paragraph outlines the formula for calculating the present value, which involves integrating the income flow rate multiplied by an exponential decay function. An example is provided using Bailey Oil with a different income flow rate function, and the present value is calculated for a ten-year period with a 10% continuous interest rate, resulting in a present value of 699 thousand dollars. The concept of future value is also introduced, which is the value of an investment at a future date when it has been continuously compounded, and a formula is provided for this calculation.

The third paragraph explores the economic concepts of consumer surplus and producer surplus. Consumer surplus is the difference between what consumers are willing to pay and what they actually pay, representing the benefit they receive from a product being priced lower than their willingness to pay. Producer surplus, on the other hand, is the difference between the price at which a producer is willing to sell a product and the actual selling price. The paragraph explains these concepts with examples and uses supply and demand curves to illustrate equilibrium price, where the quantity supplied equals the quantity demanded. The importance of understanding these surpluses in economics and business decision-making is highlighted.

This paragraph focuses on finding equilibrium points where supply and demand intersect and calculating the associated consumer and producer surpluses. The equilibrium point is identified by setting the supply and demand functions equal to each other and solving for the quantity. Once the equilibrium quantity and price are established, the consumer surplus is calculated by integrating the demand function from zero to the equilibrium quantity and subtracting the total revenue (equilibrium price times quantity). An example is provided where the demand function is given, and the equilibrium price is $20, resulting in an equilibrium quantity of 50 units. The consumer surplus is then calculated to be $3,010.46. The process is also explained for finding the producer surplus, which involves integrating the supply function from zero to the equilibrium quantity and subtracting this value from the total revenue.

The final paragraph presents a problem involving a supply function where the price is a quadratic function of the number of units. Given a supply function and an equilibrium price of $20 per unit, the task is to find the producer surplus. The equilibrium point is determined by setting the supply function equal to the equilibrium price and solving for the quantity. After establishing that the equilibrium quantity is 4 million units, the producer surplus is calculated by subtracting the integral of the supply function from the total revenue at the equilibrium point. The integral of the supply function from 0 to 4 million units is evaluated, and the result is used to find the producer surplus, which is given in million dollars, resulting in a surplus of $50.67 million.