A function shows the relationship between two or more variables known as dependent and independent variables. In a given market and in a given time period, the demand function for a commodity is the relation between the amount of the commodity demanded and its determinants like the price of the commodity, price of related goods, the income of the consumer, taste, and preferences, distribution of income, size and composition of the population and so on. Demand Function shows the functional relationship between quantity demand and various determinants of demand. There are different types of demand functions showing the relationship between demand and its determinants.

According to the words of Mark Hirschey and James L. Pappas, *“The market demand function for a product is a statement of the relation between the aggregate quantity demanded and all factors that affect this quantity”.*

Mathematically, the functional relationship between the quantity demand of the product and determinants of demand is;

**Q _{dx}=f (P_{x}, P_{R}, Y, E, TP, A, Z)**

Where,

Q_{dx}= Quantity demand of commodity X; f= Functional relationship; P_{x}= Price of commodity X; P_{R}=Price of related goods (the price of substitute and complementary goods); Y= Money income of the consumer; E= Expectations of price changes; TP= Taste and preferences; A= Advertisement expenditure; Z= other factors affecting demand

In our above function, all the variables on the right-hand side are independent variables, and the variable which is on the left side is the dependent variable. Among all the determinants of demand price of the product is the most influencing factor. So for simplicity demand function is expressed as the function of the price of the commodity as it is the major determinant of demand. Thus, in short, the demand function is;

**Q _{X}= f (P_{X})**

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### Classification of Demand Function

The major types of demand functions are as follows;

**Short Run/Single Variable Function**

This function establishes a functional relationship between one of the determinants of demand as an independent variable for a commodity and demand for that commodity, other things remaining the same. Examples of single variable demand functions are as follows;

**Price demand function: Qx=f (Px)**

Where Qx= demand for good X; f=function; P_{x}= price of good X

**Income demand function: Q _{X}= f(Y)**

Where Qx= demand for good X; f=function; Y= income

**Cross demand function: Q _{X}=f (P_{Y})**

Where Qx= demand for good X; f=function; P_{Y}= price of good Y or related goods.

**Multi-variable/Long Run Function**

The multiple demand function establishes the functional relationship between all the determinants of demand and demand for a commodity. It explains the composite effect of all determinants of demand on demand for a commodity. Mathematically it can be expressed as;

**Q _{dx}**=f (P

_{x}, P

_{R}, Y, E, TP, A, Z)

Where Q_{dx}= Quantity demand of commodity X; f= Functional relationship; Px= Price of commodity X; PR=Price of related goods (the price of substitute and complementary goods); Y= Money income of the consumer; E= Expectations of price changes; TP= Taste and preferences; A= Advertisement expenditure; Z= other factors affecting demand

**Linear Demand Function**

Based on the slope of the demand curve there are two types of demand functions. If the slope of the demand curve remains constant throughout its length, it is called the linear demand function. It means in the case of linear function the rate of change of the dependent variable and independent variable is the same or a constant rate. Here in our case, the rate of change in the price of the commodity and quantity demand for the same commodity remains at a constant rate, and thus the demand function becomes linear.

Mathematically it can be expressed as;

**Q _{X}=a-bP_{x} **

Where

Q_{X}=Demand for good X; P_{x}= Price of good X; a= demand intercept or autonomous demand or demand at zero price; b= slope of demand function or rate of change in demand with respect to changes in price.

Given the demand function, if values of a and b are known, the total demand function for a commodity at any given price (Px) can easily be measured. For instance, let us assume that a=60 and b=4. Now the demand function can be written as;

Q_{X}=60-4P_{x}

This demand function also can be written as:

4P_{x}=60-Q_{X}

P_{x}= (60-Q_{X})/4

So P_{x} = 15-0.25Q_{X} which is known as an* inverse demand function*.

**Non-linear Demand Function**

If the slope of demand curves changes all along the demand line then it is said to be non-linear or curvilinear. It means if the independent variable (the price of the commodity) and dependent variable (demand for the same commodity) change at different rates, the demand function will be non-linear. A non-linear demand function is generally expressed in power function as follows;

**Q _{x}= aP_{x}^{-b}**

As a rectangular hyperbola form;

Where,

Q_{x}= Demand for X good; P_{x}= Piece of good X; a= Autonomous demand or demand at zero price of demand intercept; b= rate of change in demand with respect to changes in demand or slope of the demand curve. Here ‘a’ and ‘b’ should be positive and C=0.

**Interpretation of Demand Function**

Consider the demand for, say good J. Using the given notations; q_{J}= the quantity demand of good J; p_{J}=the own price; p_{K,} p_{L} = price of related goods; Y=income and T=other factors. The demand function for good J in numerical form is;

**q _{J}=f(p_{J, }p_{K, }p_{L, }M,T)**

How various factors affecting the quantity demand of a commodity determine the properties of demand function f is explained below;

The law of demand refers to the statement that other things remain constant, as p_{J }changes, q_{J }changes in the opposite direction. Hence the partial derivative of q_{J} w.r.t. p_{J} is negative in sign. Suppose good K is the substitute goods for good J. Then change in p_{K }causes q_{J} to change in the same direction. Therefore partial derivative of q_{J} w.r.t. p_{K} is positive. Suppose that good J and good L are complementary to each other. The partial derivative of q_{J} w.r.t p_{L} is negative.

The partial derivative w.r.t M depends on whether good J is normal or inferior. It is negative if god J is inferior and positive if good J is a normal good. The sign of partial derivative w.r.t T depends on the nature of T.

If we ignore or suppress the other factors T, an example of a demand function will be;

**q _{J}= 100-2 p_{J}+0.5 p_{K}– p_{L}+1.5 M.**

It means, for instance, that if there is a unit increase in its own price, the quantity demand will fall by 2 units. Suppose if the price is measured in hundreds of rupees. Then starting from 100 if p_{J} increases by Rs. 100 then q_{J} will decrease by 2 units.

If we are only concerned with the change in the price of good A only, that is p_{K}, p_{L,} and M are unchanged, for simplicity, we can also suppress them and can write the demand function very simply as;

**q _{J}= f(p_{J})**

Therefore the dependence of demand for a particular good on several factors can express algebraically in the form of a demand function. And there are different types of demand functions showing such a relationship between demand and its determining factors.

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