Business Calculus Applications of Integration

5.1 Social Gain Consider a demand function D(x) and supply function S(x) where x is the quantity, and the output for both functions is price of the item. Demand: as the price of an item increases, the demand for that item decreases. Demand is typically a decreasing function. Supply: as the price of an item increases, the producer of this item is willing to supply more of the item. Supply is typically an increasing function. The point at which supply and demand intersect is called the point of equilibrium.

Total Revenue For the demand and supply curves shown below, the point of equilibrium occurs at approximately $2.75 for 3 items sold. Total revenue would be: price * quantity = the area of the rectangle shown below.

Consumer Surplus Consumer Surplus can be thought of as the consumer’s satisfaction at having spent less for an item than he was willing to spend. A consumer willing to spend $3.50 for one item spends only $2.75 for that item. His satisfaction is in having saved 75 cents. Adding all possibilities results in an area of the triangular portion under the demand function. Consumer’s Surplus:

Producer’s Surplus Producer’s Surplus can be thought of as the supplier’s satisfaction at having sold the first several items for more than the projected price for those items. If the producer sells the first 3 items at $2.75 each, he is selling items one and two at a price higher than his supply curve indicates. This portion of total revenue is the producer’s surplus. Producer’s Surplus:

Social Gain The addition of consumer’s surplus and producer’s surplus at a given price is called social gain. If social gain is calculated for the price at equilibrium, we have the graph below.

5.2 Investment Growth Compound Interest: For a one-time deposit of P dollars, invested at a rate of r % for t years, the future value A of the one-time investments is: Interest compounded n times per year: Interest compounded continuously:

Continuous Money Flow For investors who have a yearly income P invested throughout the year (usually daily or weekly) to be invested at a rate of r % for T years, compounded continuously, the future value A of the continuous money flow is: If the income invested varies over time with a yearly investment of R(t), the future value is:

Present Value To determine the amount of a one-time deposit necessary to yield an amount A from an account compounded continuously, we must solve for P: This P is called present value. If we are looking for the present value A of a continuous money flow with yearly investment of R(t), we would calculate the integral: This is called accumulated present value.