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Applications of Derivatives in Economics
B.PrahalathanSenior Lecturer
Department of CommerceFaculty of Management Studies & Commerce
University of Jaffna
1. Introduction
This article provides a simple introduction to the general topic of calculus. Calculus is concerned with the mathematical study of change - The techniques of calculus actually provide a quick way of performing calculations.
In the last half of the 18th century Isaac Newton [1642-1727] and G.W. Leibniz [1646-1716] lay the foundations for what was to become known as “calculus.” This was a powerful tool to deal with infinitesimal problems of arc lengths, tangents to curves, area and volumes. The Bernoulli brothers, Jackob [1654-1705] and Johann [1667-1748] extended the tool to deal with the processes of change. During the mid 19th century (from about 1825 to 1875) a growing number of writers applied calculus to the problems of changes in the realm of economics. Johann Heinrich von Thünen [1783-1850], Hermann Heinrich Gossen [1810-1858], Jules Dupuit [1804-1866] and Augustin Cournot [1801-1877] applied mathematical tools to economic problems. In the 1870’s W.S. Jevons [1835-1882], Carl Menger [1840-1921] and Léon Walras [1834-1910] initiated what has come to be known as the “Marginalist Revolution” in economics. The Marginalist Revolution established the importance of calculus in economic analysis. During the 19th and 20th centuries the role of mathematics in economics has expanded.
This article will show how differentiation can be used to find out the maximum and minimum values of a function. Since, the derivative provides information about the gradient or slope of the graph of a function. In many applications, a scientist, engineer, or economist for example, will be interested in such points for obvious reasons such as maximising power, or profit, or minimising losses or costs.
2.0 Functions & Economic Relationships
Models of economic relationships are abstractions that explain many of the forces that shape economic behaviour. Models can be in a narrative or mathematical format. In a mathematical format, the variables may be quantified or measured.
A function is a relationship (or mapping) between two or more variables whereby the value of one variable (the dependent variable) is determined by the values taken by the other variables (the independent variables).
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The general notation for a function is a follows:
Y f X Z ( , )
Where
Y : dependent variable
X, Z : independent variables
f ( ) : means “is a function of”
This is the most general form of a function. It simply states that the variable Y is a function of
the variables X and Z.
Economic ExamplesThe following are examples of commonly used functions in economic models.
QD = f (P) : Demand function
C = f (YD) : Consumption function
U = f (X, Y) : Utility function
Q = f (L, K) : Production function
C = f (Q) : Total Cost function
2.1 Linear Function
A two variable linear function can be written as follows:
Y a bX
Where, a and b are two parameters which can be assigned fixed values. The values of a and b determine the exact form of the linear function.
Example: Assume a = 10, b = 2. Thus we have the following specific linear function;
Y X 10 2
We can now construct a table and then graph this function.
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Figure - 1
To draw the graph of a linear function we only need to calculate and plot two points and then simply join them up with a straight line.
Economic ExamplesSome common examples of linear functions used in economic analysis are:
Q a bP cY
Q g hP
C a cY
M Y r
D
S
D
Demand function
Supply function
Keynesian consumption function
Demand for money function
Note the common use of upper case letters for variables and lower case English or Greek letters for parameters.
2.2 Polynomial Functions
A two variable general polynomial function is shown below:
Y a b X b X b X b Xnn 1 2
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3 ......
This is often referred to as polynomial of degree n. The most commonly used polynomial functions are those of degree n = 2 (the quadratic function), and degree n = 3 (the cubic function).
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X Y
0 10
2 14
4 18
6 22
8 26
10 30
10
Y=10+2X
5
15
20
2 4 6 8 10X
25
Y
2.3 Quadratic Function
A quadratic function is a polynomial of degree 2; i.e. Y a b X b X 1 22
It has three parameters a, b1 and b2 - values for these determine the specific form and graph of the function.
The graph of a quadratic function is either a U-shape (convex) or an inverted U-shape (concave). Which shape the function has depends on the value of the parameter b2
If b2 < 0 the function is concave
If b2 > 0 the function is convex
A concave or convex function has only one turning point - a maximum if concave, a minimum if convex.
Economic Example: Quadratic Function
Total Revenue function
Start with the linear demand function facing a price-taking firm (Perfect competitive firm)
Q a bP
Invert it to make price (P) a function of quantity sold (Q)
bp a Q
Pa
b
Q
b
Recall that Total Revenue = Price times Quantity, i.e. TR = PQ
So,
TR PQa
b
Q
bQ
Expanding the bracket gives us the quadratic total revenue function:
TRa
bQ
Q
b
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3.0 Slope and Rate of Change4
The slope of a line measures the change in the dependent variable associated with a change in the independent variable.
3.1 Slope of a Straight Line
When the function is linear, the slope can be calculated as
=
Slope of upward line Slope of downward line Slope of the horizontal line
Given a function P = 10 – 2Q both a table and a graph can be constructed in following table and figure.
Quantity (Q) 0 1 2 3 4 5
Price (P) 10 8 6 4 2 0
The slope of the line in the graph is the change in price caused by a change in quantity, for every one-unit change in Q, the value of price (P) will change by 2 units in the opposite direction.
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2
6
8
1 2 3 4 5Quantity
10
Price
Y
X
Y
X
XY=0
Panel = A Panel = B Panel = C
Y
X
Y
X
Y
X
Figure 2
Figure – 3
It may be useful to notice that the function P = 10 – 2Q can be expressed where Q = f (P). The equation would be Q = 5 - 0.5P. Notice that the slope of the function Q= f (P) is the reciprocal of the slope of the equivalent function P = f (Q).
3.2 Slope of the Nonlinear Function
Not all functions in economics are linear. The slope of the curves can be derived by measuring the slope of the tangent.
When the function is nonlinear the slope can be imagined as the slope of an arc between two
points on the function or as the slope of a tangent at a point on a function. Generally,
suggests the slope of a straight line or the slope of an arc between two points on a nonlinear function.
In the following figure a nonlinear function Y= f(X) is graphed. If the value of X changes from 4 to 10 then ΔX is +6. The change in Y (ΔY) is +3. The slope of the arc between points
“a” and “b” is = = 0.5.
Figure - 4Notice that as the change in X (ΔX) approaches 0, the tangent tt’ represents the change in Y “caused” by a change in X. If the change in X (ΔX) approaches the limit of 0(ΔX→0), the
6
3
6
4 10
t’
Y=f (X)
t
b
X
Y
notation for a derivative, is used. The slope of tt’ is calculated by taking the first
derivative of the function, Y =f(X). If you use , the slope is an estimate of the “average”
slope between points a and b that lie on the function. The value of is the slope of the
tangent at a point on the function. If there are several independent variables, each influencing
the value of the dependent variable, the notation for a partial derivative, is used.
4.0 Derivative of Function
The slope of the graph of a function is called the derivative of the function. This slope of function is usually referred to as the derived function. An alternative notation for the derived
function is . (dee y by dee x).
4.1 Differentiation
The process of finding the derived function symbolically is known as differentiation.
4.2 Differential Coefficient
Consider the function.
Y = f (x)
x = independent variable.
Y = dependent variable.
By differentiate y with respect to x, we can get ‘first differential coefficient of y with respect
to x’. This is denoted by . That is, is the first derivative of y with respect to x. By
differentiate again with aspect to x, we can get the ‘second derivative of y with respect to x’.
This denoted by .
Note: Function is f (x)
1st derivative of f (x) is denoted by f’ (x) 2nd derivative of f (x) is denoted by f” (x)
4.3 A Simple Rule for a DerivativeSimple rule to calculate the first derivative for a function Y = aXb is
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=a (b) X (b – 1) .This is the equation for the tangent line at any value of X.
For exampleApplying this rule to the equation, Y = 2 + 1X - .05X2 The equation can be written asY = 2X0 + 1X1 - .05 X2, This will make
= 2(0) X (0-1) + 1(1) X (1-1) - .05(2) X (2-1) = 1 - .1X.
If X is 10 the value of Y is 7. [Y = 2+1(10) - .05(10)2 = 12 - .05(100) = 12-5 = 7]. That is, When X is 10 and Y is 7, the slope of the tangent to the function Y = 2 + 1X - .05X2
will be = 1 - .1X or = 1 - .1X = 1 - .1(10) = 0. = is the change in Y caused by a
very, very, small change in X (ΔX→0).
Where the slope is 0, the function has a maximum, a minimum or an inflection point. To determine which, the second derivative (or second order condition) is used.
5. Stationary pointsAny point at which the tangent to the graph is horizontal is called a stationary point.
We can locate stationary points by looking for points at which = 0. Consider the graph of
the function, Y(x), shown in the following figure.
Figure - 5
If, at the points marked A, B and C, we draw tangents to the graph; note that these are parallel to the X axis. They are horizontal. This means that at each of the points A, B and C the gradient of the graph is zero.
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The gradient of a graph is given by .Consequently, = 0 at points A, B and C. All of
these points are known as stationary points.
6. Turning Points
In figure 6, Note that at points A and B the curve actually turns. These two stationary points are referred to as turning points. Point C is not a turning point because, although the graph is flat for a short time, the curve continues to go down as we look from left to right. So, all turning points are stationary points. But not all stationary points are turning points (e.g. point
C). At a turning point = 0. Not all points where = 0 are turning points, i.e. not all
stationary points are turning points.
7.0 Distinguishing Maximum Points from Minimum Points
7.1 Maxima
A function f (x) said to be maximum at points where it changes from an increasing to decreasing function. Point A in Figure 6 is called a local maximum (maximum) because it is the highest point, and so represents the greatest or maximum value of the function.
7.2 Minima
A function f (x) is said to be minimum at points where it changes from a decreasing to an increasing function. Point B in Figure 7, is called a local minimum (minimum) because it is the lowest point, and so represents the least, or minimum, value of the function.
Figure – 67.3 The Conditions for Minimum
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In figure – 7, goes from negative through zero to positive as x increases. In the left of the
minimum point is negative because the tangent has negative gradient. At the minimum
point, = 0. To the right of the minimum point is positive. Since here the tangent has a
positive gradient. So goes from negative, to zero, to positive as x increases.
If is increasing near the stationary point then that point must be minimum. That is, if
the derivative of is positive then the is increasing. So the stationary point is a
minimum. The derivative of called the second derivative, is written as . So, if is
positive at a stationary point, then that point must be a minimum turning point.
Figure - 7
Conditions for a Minimum Value
: Necessary Condition
: Sufficient Condition
7.4 The Conditions for Maximum
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Left of the maximum point is positive because the tangent has positive gradient. At the
maximum point, .To the right of the maximum point is negative because the
tangent has negative gradient. So when the function is positive goes from positive, to
zero, to negative as x increases.
Figure - 8
If is decreasing near the stationary point then that point must be maximum. That is, if
the derivative of is negative then the is decreasing; so the stationary point is
maximum. The derivative of called the second derivative, is written as . So, if is
negative at a stationary point, then that point must be a maximum turning point.
Conditions for a Maximum Value
: Necessary Condition
: Sufficient Condition
7.5 Testing a Function for Maximum and Minimum
Step (I)
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Find the First Derivative of the function – f’(x) OR. .
Step (II)
Set the First Derivative Equal to Zero and find the real roots of the equation f ’(x) = 0 OR
in order to find the stationary points of the independent variable (x).
Step (III)
Find the Second Derivative of the function f” (x) OR .
Step (IV)
Substitute each Stationary Point for the Independent Variable in the Second Derivative . If
result is negative [ ], then the function is a maximum for that stationery point, If the
result is positive, [ ], then the function is a minimum for that stationary point.
Change of Stationary Point
0 - Maximum Point
0 + Minimum Point
Example :- Given a Y= x2 – 4x + 5, test the function whether it is minimum function or maximum function.
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Step (I)
Find the First Derivative of the function – .
= 2x – 4
Step (II) Set the First Derivative Equal to Zero and find the real roots of the equation f ’(x) = 0 OR
in order to find the stationary points of the independent variable (x).
= 0
2x – 4 = 0
x = 2.
If, x = 2, then
Y = 1
Step (III)
Find the Second Derivative of the function - .
= 2x – 4
Here,
Therefore the function is minimum function.
Summary The “Marginalist Revolution” of the late 19th century was the result of the application of calculus to economic phenomena. The word “marginal” refers to the rate of change in a dependent variable “caused” by a change in an independent variable. Marginal can refer to
the rate of change as the slope of a line or the slope of a tangent to a nonlinear function at
a point .The concept of a “marginal” is fundamental to choosing among alternatives.
References
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1. Edward,T.D.(2004).Introduction to Mathematical Economics,3rd Edition,Tata
McGraw – Hill Publishing Company Ltd,NewDelhi,India.
2. Jacques,I. (1994).Mathematics for Economics and Business,2nd Edition, Addition –
Wesley Publishing Company, United Kingdom.
3. Karunaratna,K.R.M.T.(2003).Business Economics: Mathematical Application, 1st
Edition, Tharanjee Printers,Maharagama,Colombo,SriLanka.
4. Soni,R.S. (1999).Business Mathematics with Applications in Business and Economics,
3rd Edition, Piyush Printers Publishers Pvt Ltd,New Delhi, India.
Marginal Functions
1. Marginal Revenue
Marginal Revenue is the derivative of total revenue with respect to demand.
MR = .
Example:-
If the total revenue function is TR = 100 Q - 2Q2
MR = = 100 – 4Q
MR = 100 – 4Q.
2. Marginal Cost
Marginal cost is the derivative of total cost with respect to input
MC =
Example:-
If the total cost function is TC = 2Q2 + 6Q + 13
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MC = = 4Q + 6
MC = 4Q + 6.
3. Marginal Product
Marginal product of an input is the derivative of output with respect to that particular input. For example, Marginal product of labor is
MPL =
That is marginal product of labor is the derivative output with respect to labor.
Example:-
If the production function is Q = 300 L2 – 4L
MPL =
MPL = 600 L – 4
4. Marginal Propensity to Consume
Marginal propensity to consume is the derivative of consumption with respect to income.
MPC =
Example:-
It the consumption function is C = 0.01Y2 + 0.2Y + 50
MPC = = 0.02Y + 0.2
MPC = 0.02Y + 0.2
5. Marginal Propensity to Save
Marginal propensity to save is the derivative of savings with respect to income.
MPS =
Example:-
If the saving function is S = 0.02Y2 – Y+ 100
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MPS = = 0.04 – 1.
MPS = 0.04Y – 1.
Practical Example
The demand equation of a commodity is given by P + Q = 30The total cost function is given by TC = ½ Q2 + 6Q + 7
i. Find the level of output which maximizes total revenue.ii. Find the level of output which maximizes profit. Calculate MR and MC at this value
of Q. What do you observe?
1. Find the level of output which maximizes total revenue.
Finding the Total Revenue Function
P + Q = 30Rearranging price function asP = 30 – QTotal revenue is defined byTR = PQHenceTR = (30 – Q) QTR = 30Q – Q2
Find the level of output which maximizes total revenue
At a stationary point
= 0
30 – 2Q = 0
So,
Q = 15 Units
Find the second derivative of the function
The value is negative.
Conditions for a Maximum Value is
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: Necessary Condition
: Sufficient Condition
Here the conditions are satisfied, therefore the total revenue is maximised at the
output level of 15 units.
2. Find the level of output which maximizes profit
Finding the Profit Function
The profit function is defined by
= TR – TC
Here
TR = 30Q – Q2
TC = ½ Q2 + 6Q + 7
= (30Q – Q2) – (½ Q2 + 6Q + 7)
= - Q2 + 24Q – 7
At a stationary point
= 0
- 3Q + 24 = 0
So,
Q = 8 Units
Find the second derivative of the function
The value is negative.
Conditions for a Maximum Value is
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: Necessary Condition
: Sufficient Condition
Here the conditions are satisfied, therefore the profit is maximised at the output level of 8 units.
3. Calculation of MR and MC at the output level of 8 units which maximises the profit.
If
TR = 30Q – Q2
MR =
MR = 30 – 2Q
When Q = 8 Units
MR = 14
If
TC = ½ Q2 + 6Q + 7
MC =
MC = Q + 6
When Q = 8 Units
MC = 14
The observation is, when the Q = 8 the values of MR and MC are equal. That is at the point of maximum profit MR = MC.
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