Calculus

Applications of Integrals

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Applications of Integrals in Economics

The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics:

Marginal and Total Revenue, Cost, and Profit

Marginal revenue (MR) is the additional revenue gained by producing one more unit of a product or service.

It can also be described as the change in total revenue (TR) divided by the change in number of units sold (Q):

MR=dTRdQ.

If a marginal revenue function MR(Q) is known, the total revenue can be obtained by integrating the marginal revenue function:

TR(Q)=MR(Q)dQ,

where integration is carried out over a certain interval of Q.

Marginal cost (MC) denotes the additional cost of producing one extra unit of output.

The similar relationship exists between the marginal cost MC and the total cost TC:

MC=dTCdQ,

so

TC(Q)=MC(Q)dQ.

Since profit is defined as

TP=TRTC,

we can write the following equation for marginal profit (MP):

MP=MRMC,ordTPdQ=dTRdQdTCdQ.

Capital Accumulation Over a Period

Let be the rate of investment. The total capital accumulation during the time interval can be estimated by the formula

Consumer and Producer Surplus

The demand function or demand curve shows the relationship between the price of a certain product or service and the quantity demanded over a period of time.

The supply function or supply curve shows the quantity of a product or service that producers will supply over a period of time at any given price.

Both these price-quantity relationships are usually considered as functions of quantity

Generally, the demand function is decreasing, because consumers are likely to buy more of a product at lower prices. Unlike the law of demand, the supply function is increasing, because producers are willing to deliver a greater quantity of a product at higher prices.

The point where the demand and supply curves intersect is called the market equilibrium point.

The point of equilibrium is the intersection of demand and supply curves.
Figure 1.

The maximum price a consumer is willing and able to pay is defined by the demand curve For quantities it is greater than the equilibrium price in the market. Consumers gain by buying at the equilibrium price rather than at a higher price. This net gain is called consumer surplus.

Consumer surplus is represented by the area under the demand curve and above the horizontal line at the level of the market price.

Consumer surplus and producer surplus.
Figure 2.

Consumer surplus is thus defined by the integration formula

A similar analysis shows that producers also gain if they trade their products at the market equilibrium price. Their gain is called producer surplus and is given by the equation

Lorenz Curve and Gini Coefficient

The Lorenz curve is a graphical representation of income or wealth distribution among a population.

The horizontal axis on a Lorenz curve typically shows the portion or percentage of total population, and the vertical axis shows the portion of total income or wealth. For instance, if a Lorenz curve has a point with coordinates this means that the first of population (ranked by income in increasing order) earned of total income.

Lorenz curve and Gini coefficient
Figure 3.

The Lorenz Curve is represented by a convex curve. A more convex Lorenz curve implies more inequality in income distribution. The area between the degree line (the line of equality) and the Lorenz curve can be used as a measure of inequality.

The Gini coefficient is defined as the area between the line of equality and the Lorenz curve, divided by the total area under the line of equality:

The Gini coefficient is a relative measure of inequality. It ranges from (or ) to (or ), with representing perfect equality in a population and representing perfect inequality.

Solved Problems

Example 1.

The marginal revenue of a company is given by where is amount of units sold for a period. Find the total revenue function if at it is equal to

Solution.

We find the total revenue function by integrating the marginal revenue function

The constant of integration can be determined using the initial condition Hence,

So, the total revenue function is given by

Example 2.

The rate of investment is given by Calculate the capital growth between the and the years.

Solution.

Using the integration formula

we have

Example 3.

Assume the rate of investment is given by the function Compute the total capital accumulation between the and the years.

Solution.

To calculate the capital accumulation, we use the formula

Integrating by parts, we have

Hence

Example 4.

For a certain product, the demand function is and the supply function is Compute the consumer and producer surplus.

Solution.

First we determine the equilibrium point by equating the demand and supply functions:

The positive solution of the quadratic equation is The market equilibrium price is

The consumer surplus is given by

Similarly we find the producer surplus

See more problems on Page 2.

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