Models in economics provide simplified representations of real-world situations to better understand them. A key assumption in models is that all other relevant factors remain unchanged. The law of demand states that as price increases, quantity demanded decreases, and vice versa. Demand can be shown through schedules, curves, and functions relating price and quantity. Profit is maximized where marginal revenue equals marginal cost, satisfying both the first- and second-order conditions through the slopes of the total revenue and total cost curves.

1. Economic Models:
Basic Mathematical Tools applied in
economics

2.  Simplified representations of reality play a crucial role
in economics.
Why models?

3. Models in Economics
•A model is a simplified representation of a real
situation that is used to better understand real-life
situations.
 Create a real but simplified economy
 Simulate an economy on a computer
 Ex.: Tax models, money models
•The “other things equal” assumption means that
all other relevant factors remain unchanged.

4. Demand
A relation showing how
much of a good consumers
are willing and able to buy
at each possible price during
a given period of time, other
things held constant

5. Law of Demand
• A decrease in the price of a good, all other
things held constant, will cause an increase in
the quantity demanded of the good.
• An increase in the price of a good, all other
things held constant, will cause a decrease in
the quantity demanded of the good.

6. Demand schedule Demand curve
Price
Quantity
Demanded
1.25
$ 8
1.00
$ 14
0.75
$ 20
0.50
$ 26
0.25
$ 32
$-
$0.25
$0.50
$0.75
$1.00
$1.25
2 8 14 20 26 32
Quantity
Price
D
The Demand Curve
The demand curve
slopes downward
because of the law
of demand

7. Functional Relationships
• Relationship between two variables, for e.g. price and
output sold, expressed in various ways
 Table or graph
 Use of equations – Quantity sold depends on the price,
in other words quantity sold is a function of price.
 P is the independent value and Q is the dependent
value
p
p
f
Q 5
200
)
( 



8. Linear Demand Function
QX = a0 + a1PX + a2N + a3I + a4PY + a5T
PX
QX
Intercept:
a0 + a2N + a3I + a4PY + a5T = 200
Slope:
QX/PX = a1 =-5

9. Non-linear functions
• For e.g. Total Revenue
• TR=PQ
• Marginal Revenue (Slope of Total revenue)
• MR=Change in TR associated with change in Q

10. Tabular form Representation
Q P=100-10Q TR=100Q-10Q2 AR MR
0 100 0 - 0
1 90 90 90 90
2 80 160 80 70
3 70 210 70 50
4 60 240 60 30
5 50 250 50 10
6 40 240 40 -10

11. Graphical Representation & Concept of
Slope & Curvature
TR
TR
Q
O
A
B
C

12. • Slope of TR Curve at a particular point
represents MR at a particular output, i.e.,
change in TR for an infinitesimal change in
output level
• Implication of slope for any variable implies
marginal value of the same variable
• Curvature depends on changes in slope or
changes in marginal value
Changes in Slope

13. Changes in Curvature
• Linear Curve – Marginal value constant, no
change in curvature
• Curve Convex to the origin – Marginal value
(Slope) changing at an increasing rate
• Curve Concave to the origin – Marginal value (
Slope) changing at a decreasing rate

14. Average and Marginal
• Graphically Average value can be derived from the total value
curve.
• Average at a point on the Total value curve is equal to the
slope of the ray from the origin to that particular point
• To increase (decrease) the average value, Average value
should be less (more) than the Marginal value
• Average Value constant implies its equality with Marginal
Revenue

15. Find out from Total Cost,
Average, & Marginal Cost
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q
MC = TC/Q

16. Average Cost (AC)
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q

17. Total, Average, and
Marginal Cost
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q
MC = TC/Q

18. Total, Average, and Marginal Cost
0
60
120
180
240
0 1 2 3 4
Q
T C ($)
0
60
120
0 1 2 3 4 Q
AC , M C ($)
AC
M C

19. Optimization Techniques
• In Economics different optimization techniques as a solution
to decision making problems
• Optimization implies either a variable is maximized or
minimized whichever is required for efficiency purposes,
subject to different constraints imposed on other variables
• E.g. Profit Maximization, Cost Minimization, Revenue
Maximization, Output Maximization
• A problem of maxima & minima requires the help of
differential calculus

20. Profit Maximization
Q TR TC Profit
0 0 20 -20
1 90 140 -50
2 160 160 0
3 210 180 30
4 240 240 0
5 250 480 -230

21. Profit Maximization
0
60
120
180
240
300
0 1 2 3 4 5
Q
($)
MC
MR
TC
TR
-60
-30
0
30
60
Profit

22. Marginal Analysis to profit maximization
• Marginal Analysis requirement for profit
Maximization,
Marginal Revenue = Marginal Cost
(MR) (MC)
• Marginal Value represents slope of Total value
curves,
• Thus slopes of TR &TC should be equal

23. Conditions of Profit Maximization
• MR=MC is a necessary condition for Maximization, not a
sufficient one as this condition also hold for loss maximization
• Sufficient condition requires that reaching a point of
maximization, profit should start declining with any further
rise in output, i.e. Slope of TC should rise & Slope of TR must
fall after reaching the point of Maximization,
• Change in MC>Change in MR

24. Concept of the Derivative
The derivative of Y with respect to X
is equal to the limit of the ratio
Y/X as X approaches zero.

25. Rules of Differentiation
Constant Function Rule: The derivative of a
constant, Y = f(X) = a, is zero for all values of a
(the constant).
( )
Y f X a
 
0
dY
dX


26. Rules of Differentiation
Power Function Rule: The derivative of
a power function, where a and b are
constants, is defined as follows.
( ) b
Y f X aX
 
1
b
dY
b aX
dX

 

27. Rules of Differentiation
Sum-and-Differences Rule: The derivative
of the sum or difference of two functions U
and V, is defined as follows.
( )
U g X
 ( )
V h X

dY dU dV
dX dX dX
 
Y U V
 

28. Rules of Differentiation
Product Rule: The derivative of the product
of two functions U and V, is defined as
follows.
( )
U g X
 ( )
V h X

dY dV dU
U V
dX dX dX
 
Y U V
 

29. Rules of Differentiation
Quotient Rule: The derivative of the ratio
of two functions U and V, is defined as
follows.
( )
U g X
 ( )
V h X

U
Y
V

   
2
dU dV
V U
dY dX dX
dX V



30. Rules of Differentiation
Logarithmic Function:
The derivative of a log function of X is defined as
follows.
d(log X)/dX= 1/X

31. Rules of Differentiation
Chain Rule: The derivative of a function that is a
function of X is defined as follows.
( )
U g X

( )
Y f U

dY dY dU
dX dU dX
 

32. Using derivatives to solve max and min problems
Optimization With Calculus
To optimize Y = f (X):
First Order Condition:
Find X such that dY/dX = 0
Second Order Condition:
A. If d2Y/dX2 > 0, then Y is a minimum.
OR
B. If d2Y/dX2 < 0, then Y is a maximum.

33. CENTRAL POINT
The dependent variable is maximized when its
marginal value shifts from positive to
negative, and vice versa

34. The Profit-maximizing rule
Profit(p) = TR – TC
At maximum profit
dp/dQ = dTR/dQ - dTC/dQ = 0
So,
dTR/dQ = dTC/dQ (1st.O.C.)
=> MR = MC
d2TR/ dQ2 = d2TC/dQ2 (2nd O.C.)
==> dMR/dQ < dMC/dQ
This means
slope of MC is greater than slope of MR function

35. Constrained Optimization
To optimize a function given a
single constraint, imbed the
constraint in the function and
optimize as previously defined