MATHEMATICS II

In the first two chapters, we have introduced the notions of indefinite and definite integrals both of which have widespread applications in economics. Depending on the type of problem we are dealing with, we either employ the indefinite or the definite integral. In marginal analysis, we know that the quantities are differentiated to obtain the rate of their changes. Reversing this process, we may obtain the total functions, i.e. using the integral we may acquire the totality of the functions such as cost, revenue, profit and so on. Integration may therefore be used to calculate the total income during a given period from a continuous income stream, the consumer and producer surplus, and present and future values of a continuous stream of payments, to name but a few. We will try to present, in this chapter, some simple and useful examples of applying the integration notion to problems from different areas of economics.

As also understood from the information given, the answer is E. All of the statements in the options related to the aplications of integrals in economics are correctç

One important use of calculus in business and economics is marginal analysis. Marginal cost, marginal revenue and marginal profit all involve the rates of change of the profit, revenue and cost with respect to the number of units of a certain commodity produced or sold. Thus, they are defined to be the first derivative of a “total function”. Since integration is finding the antiderivative of a given function, i.e. finding the function whose derivative is given the total functions may be found without much difficulty by integrating the marginal functions.

As also understood from the information given, the answer is E. All of the statements in the options related to derivation of total function from marginal function are correct.

One of the most fundamental concepts in economics is the law of supply and demand for a certain commodity or a service in a freemarket environment. The quantity of the commodity produced and sold may be described by two curves called the supply and demand curves for the particular commodity in question. We defined supply as the amount of a product per unit of time that producers are willing to sell at various given prices when all other factors are held constant. Similarly, the demand is defined as the quantity of a commodity or a service that people are willing or able to buy at a certain price when all factors are held constant. Supply is the relation between the quantity of a product and the factors determining this quantity, and since there is a direct proportion, the supply function is an increasing function. This function is denoted by S. On the other hand, the demand is in inverse proportion with the quantity supplied so the demand function is a decreasing function. This function is denoted by D.

As also understood from the information given, the answer is E. All of the statements in the options related to consumer and producer surplus are correct.

A certain product or a service is valued at different prices by different consumers. As a consequence of this, the quantities to be bought vary according to different prices the product is sold. Some consumers may be willing to pay more for a certain product than other consumers or other products that they are willing to buy considering the purchase more important than the others. Some consumers, on the contrary, may prefer to wait until the product gets cheaper. Although different consumers are willing to pay different prices to a particular item, every product has a fixed price in the free-market.  If consumer may buy a product under a price they are willing to pay, this may be considered as a gain. This gain is called as consumer surplus (CS).  In other words, if a product is bought under a price that is desired to be paid, the unpaid part of the total amount to be paid is the consumer surplus.

As also understood from the information given, the answer is E. All of the statements in the options related to the consumer surplus are correct.

Consumer surplus is derived whenever the price a consumer actually pays is less than they are prepared to pay. A demand curve indicates what price consumers are prepared to pay for a hypothetical quantity of a good, based on their expectation of private benefit.

In a certain market, if a producer who is willing to sell a product at a certain price may sell the product higher than the price he asks, then he has increased his revenue. This extra amount gained is called the producer surplus (PS). In other words, the producer surplus is the difference of the price the producer is willing to sell and the actual price he sells his product.

As also understood from the information given, the answer is E. All of the statements in the options related to the consumer surplus and the producer surplus are correct.

In order to measure the distribution of income, various methods have been developed. One of the most commonly used method is the Lorenz curve.  Developed in 1906 by Max Lorenz, the Lorenz curve is a graphical representation of the distribution of wealth, or of income.

To construct the Lorenz curve, first, people in society or the households are sorted from the lowest income group to the highest. Then, the total portion received from the total income by a number of a group at an arbitrary income level or lower is calculated. If this process is repeated for each income level, the graph of the curve can be constructed. The function y = L(x) which is referred to this curve is called the Lorenz Function. Typically, a Lorenz curve is a continuous function connecting the points (0, 0) and (1, 1) which is both increasing and convex. It should be mentioned at this point that the line y = x, in the context of income distribution, corresponds to the perfect equality of the wealth. Now that the line y = x, passes through the points (0, 0) and (1, 1) it is always above the graph of the Lorenz curve. This implies that as the inequality of income increases in a community, graph of the Lorenz curve diverges away from the perfect equality.

As also understood from the information given, the answer is E. All of the statements in the options related to the Lorentz curve are correct.

To construct the Lorenz curve, first, people in society or the households are sorted from the lowest income group to the highest. Then, the total portion received from the total income by a number of a group at an arbitrary income level or lower is calculated. If this process is repeated for each income level, the graph of the curve can be constructed. The function y = L(x) which is referred to this curve is called the Lorenz Function. Typically, a Lorenz curve is a continuous function connecting the points (0, 0) and (1, 1) which is both increasing and convex. It should be mentioned at this point that the line y = x, in the context of income distribution, corresponds to the perfect equality of the wealth. Now that the line y = x, passes through the points (0, 0) and (1, 1) it is always above the graph of the Lorenz curve. This implies that as the inequality of income increases in a community, graph of the Lorenz curve diverges away from the perfect equality. Note that L(x) is an increasing function on the interval [0, 1] and has the following properties:

As also understood from the information given, the answer is E. All of the properties in the options are properties of the Lorentz curve.

Another important measure of income inequality is the Gini Index. Gini index is twice the area between the line of perfect equality (y = x) and the Lorenz curve L(x). The Gini index is always between 0 and 1. When it is zero it means that everyone has the same income. In the extreme case of Gini index being 1, a single individual receives all the income.

As also understood from the information given, the answer is E. All of the statements in the options related to the Gini index are correct.

The consumer surplus is the difference between the price a consumer is willing to pay and the actual price he pays. A decrease in the equilibrium price increases the consumer surplus. While consumers pay less than they were willling to pay, new producers also join the market. On the contrary, as the market price increases the consumer surplus decreases.

As also understood from the information given, the answer is E. All of the statements above related to the consumer surplus are correct.

The total surplus is the difference between the total value of a commodity on the consumer side and the total cost of the producers. In other words, it is the sum of the consumer surplus and the producer surplus. A decrease in the total surplus corresponds to a decrease in prosperity. Increasing the trading volume, i.e. increasing the market equilibrium, signifies an increase in the public welfare.

As also understood from the information given, the answer is E. All of the statements above related to the total surplus are correct.

As we know that total cost function is the ingegral of marginal cost function:

TC=

Since the cost of first product is 28:

TC(1)=6+2+C=28

C=20

So the total cost function is given by TC=20+6x²+2x

First we have to find the price when the amount is 5kgs.

So for q=5 D(q)=40-q²=40-25=15

So price is 15.

Consumer surplus is given by:

PS=

So in this case for P=15 and q=5, the producer surplus will be:

Firstly we have to find the production amount for the given price. So if P=S(q)=18:

18=q²+2

4=q

Producer surplus is defined as:

Gini index is defined as:

Present value is defined as: