Business Calculus

Applications of Integration

5.1 Social Gain

Consider a demand function D(x) and supply function S(x) wherex is the quantity, and the output for both functions is price of theitem.

Demand: as the price of an item increases, the demand for thatitem decreases. Demand is typically a decreasing function.

Supply: as the price of an item increases, the producer of this itemis willing to supply more of the item. Supply is typically anincreasing function.

The point at which supply and demand intersect is called thepoint of equilibrium.

Total Revenue

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

Consumer Surplus can be thought of as the consumer’s satisfactionat 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 anarea of the triangular portion underthe demand function.

Consumer’s Surplus:

q

dxpxD0

)(CS

Consumer Surplus

Producer’s Surplus

Producer’s Surplus can be thought of as the supplier’s satisfactionat 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 theproducer’s surplus.

Producer’s Surplus:

q

dxxSp0

)(PS

Social Gain

The addition of consumer’s surplus and producer’s surplusat 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 GrowthCompound 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:

nt

n

rPA

1

rtPeA

Continuous Money Flow

For investors who have a yearly income P invested throughout theyear (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 investmentof R(t), the future value is:

T

rt dtetRA0

)(

T

rt dtPeA0

Present Value

To determine the amount of a one-time deposit necessary toyield 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 moneyflow with yearly investment of R(t), we would calculate the integral:

This is called accumulated present value.

rtrt AePPeA

T

rt dtetRA0

)(