A function is a mathematical relationship in which the values of a dependent variable are determined by the values of one or more independent variables.

**Functions**

A
function is a mathematical relationship in which the values of a dependent
variable are determined by the values of one or more independent variables.

Functions
with a single independent variable are called Simple Univariate functions.
There is a one to one correspondence. Functions,with more than one independent
variable, are called Multivariate functions. The independent variable is often
designated by X. The dependent variable is often designated by Y. For example,
Y is function of X which means Y depends on X or the value of Y is determined
by the value of X. Mathematically one can write Y = f(X).

A
statement of relationship between two quantities is called an equation. In an
equation, if the largest power of the independent variable is one, then it is
called as Linear Equation. Such equations when graphed give straight lines. For
example Y = 100-10X.

For a
straight line, there are two variables namely X and Y. X is called independent
variable and Y is called dependent variable.

When ‘X’
value increases by one unit, then the corresponding change in the ‘Y’ value is
called as the slope of the line. Slope of the line is obtained by the formula,

**m ****=**** slope
(marginal change)**

Where (X_{1},
Y_{1}) and (X_{2}, Y_{2}) are two arbitrary points

Slope or Gradient of the line represents the ratio of the
changes in vertical and horizontal lines.

The
formula for constructing a straight line is

**(Y ****-**** Y _{1})
**

If the
two points are (0, 0) and (X, Y) then the formula is Y = Mx

__Example 12.1__

Find the
equation of a straight line which passes through two points (2, 2) and (4, -8) which are (X_{1}, Y_{1}) and (X_{2}, Y_{2})
respectively.

**Note: **For drawing a straight line, at
least two points are required. Many straight lines** **can pass through a single point.

2(y-2) = -10(x-2)

2y-4 = -10x +20

2y = -10x+24

**y=-5x ****+****12**

-5 is slope, denoted by **m**

12 is **y** intercept, or constantdenoted by **c** This is of the form Y = mX + c

y = 12-5x when x = 0; y = 12

When y = 0; x = 12/5 = 2.4 (This line looks likes
demand line in micro economics )

By
applying the above method, the demand and supply functions are obtained.

**Demand Function**: Q_{d}** **=** **f (P_{x}** **) where Q_{d}** **stands for Quantity demand of a
commodity** **and P_{x} is the
price of that commodity.

**Supply Function: **Qs** **=** **f (P_{x})
where ‘Q_{s}’** **stands for
Quantity supplied of a commodity** **and
Px is the price of that commodity.

In the
example 12.1 the equation Y -5X + 12 has been obtained. It is a linear function.

Since
slope is negative here, this function could be a demand function.

Price-quantity
relationship is negative in demand function.

Q_{d}
= 12-5 *X* or Qd = 12-5*P*. If P = 2, Qd = 2.

When P
assumes 0, only 12 alone remains in the equation. This is called Intercept or
Constant, if P = 0 and Qd
= 12.

In
Marshallian analysis, money terms measured in Y-axis and physical units are
measured in X-axis. Accordingly, price is measured in Y-axis and quantity
demanded is measured in X-axis

__Example: 12.2__

Find the
supply function of a commodity such that the quantity supplied is zero, when
the price is ** Rs.**5 or below and the supply (quantity)
increases continuously at the constant rate of 10 units for each one rupee rise
when the price is above ** Rs.**5.

__Solution:__

To
construct the linear supply function atleast two points are needed. First data
point of supply function is obtained from the statement that the quantity
supplied is zero, when the price is ** Rs.**5, that is (0, 5). The second and third
data points of

When p = 5, supply is zero. When p = 6, supply is 10 and so on. When p is less than 5, say 4, supply
is -10, which is possible in mathematics. But it is meaningless in Economics.
Normally supply curve originates from zero, noting that when price is zero,
supply is also zero.

The
equation of the straight line joining two data points (10, 6) and (20, 7) is
given as The equation of the straight line is

substituting
the values of (x1, y1) (x2, y2) by (10, 6) , (20, 7) respectively,

10(Y - 6) = X - 10

10Y - 60 = X - 10

10Y - 60 + 10 = X

10Y - 50 = X

-X = -10Y + 50

Multiplying
both sides by minus (-), we get

X = -50 + 10Y

Considering
X as quantity supplied and Y as price (p)

Then X = 10p - 50 (or)

X = -50 + 10 p

If Price
= 0; Q = -50

If Q = 0;
P = 5

**Note: **The coefficient of ‘p’ is** **-** **in demand function.** Note: **The coefficient of ‘p’ is** **-** **in demand function.

The
coefficient of ‘p’ is + in
supply function.

**4. Equilibrium**

The point
of intersection of demand line and supply line is known as equilibrium. The
point of equilibrium is obtained by using the method of solving a set of
equations. One can obtain the values of two unknowns with two equations. At
equilibrium point,

*Demand = Supply*

(These
are hypothetical examples)

100-10p = 50 + 10 p

100-50 = 20p

50 = 20p

p = 2.5

When p = 2.5, Demand = 100-10 (2.5) = 75

When P = 2.5, Supply = 50 +10 (2.5) = 75

__Example: 12.3__

Find the
equilibrium price and quantity by using the following demand and supply
functions Qd = 100-5P and Qs = 5P
respectively.

__Solution:__

Equilibrium
is attained when,

Qs = Qd

5 P = 100 - 5 P

10P = 100

P = 10

When P = 10

In supply
function

Qs = 5 P = 5 x 10 = 50

In demand
function,

Qd = 100 - 5 P = 100 - 5(10) = 50

Hence at

**P**** ****=**** ****10, Qd****
****=****
****50, Qs****
****=****
****50.**

Quantity
demanded is equal to supply at 50 units when price is ** Rs.**10

__Example: 12.4__

The
market demand curve is given by D = 50 - 5P. Find the maximum price beyond which nobody will buy the commodity.

__Solution :__

Given

Qd = 50 - 5P

5P = 50 - Qd

5P = 50 when Qd is zero.

P=50/5

P = 10 When P = 10,
Demand is 0

Hence P = 10, which is the maximum price beyond which nobody will demand the
commodity.

__Example: 12.5__

The
demand for milk is given by

Find the
linear demand function and its slope.

__Solution:__

Equation
of demand function joining two data points (100, 1) and (50, 2) are (x1, y1)
and (x2, y2) respectively.

-50 (y - 1) = 1 (x - 100)

-50y + 50 = x - 100

-50y + 50 + 100 = x

-50y + 150 = x

x = 150 - 50y

Hence the
demand function is

**Qd ****=**** 150 –
50P and Slope m ****=**** – 50**

**Think and Do for Water Management in your area**

Try to find the demand function for water in your street and the
daily total demand for water in litre for all purposes.

Tags : Definition, Linear Equation, Application, Equilibrium, Example, Solution , 11th Economics : Chapter 12 : Mathematical Methods for Economics

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11th Economics : Chapter 12 : Mathematical Methods for Economics : Functions - Mathematical Methods for Economics | Definition, Linear Equation, Application, Equilibrium, Example, Solution

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