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;
- Capital accumulation over a specified period of time;
- Consumer and producer surplus;
- Lorenz curve and Gini coefficient;
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):
If a marginal revenue function
where integration is carried out over a certain interval of
Marginal cost
The similar relationship exists between the marginal cost
so
Since profit is defined as
we can write the following equation for marginal profit
Capital Accumulation Over a Period
Let
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
The point
The maximum price a consumer is willing and able to pay is defined by the demand curve
Consumer surplus is represented by the area under the demand curve
Consumer surplus
A similar analysis shows that producers also gain if they trade their products at the market equilibrium price. Their gain is called producer surplus
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
The Lorenz Curve is represented by a convex curve. A more convex Lorenz curve implies more inequality in income distribution. The area between the
The Gini coefficient
The Gini coefficient is a relative measure of inequality. It ranges from
Solved Problems
Example 1.
The marginal revenue of a company is given by
Solution.
We find the total revenue function
The constant of integration
So, the total revenue function is given by
Example 2.
The rate of investment is given by
Solution.
Using the integration formula
we have
Example 3.
Assume the rate of investment is given by the function
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
Solution.
First we determine the equilibrium point by equating the demand and supply functions:
The positive solution of the quadratic equation is
The consumer surplus
Similarly we find the producer surplus