Engineering Economics.docx

Dr. Raza Ullah
Economics for Engineers (AE-602)
1
AE-602 ECONOMICS FOR ENGINEERS 3(3-0)
By
Dr. Raza Ullah
Institute of Agricultural and Resource Economics
University of Agriculture, Faisalabad
Course Contents
Review of concepts and definitions of economics, wants, scarcity, choice, demand and supply,
price determination and elasticities. Production function, total, average and marginal physical
product curves and their interaction. Cost structure, types of economic costs and their
interactions. Interest and interest rates, Time value of money, Net Present Value (NPV), Present
worth comparison, Cash-flow diagram, Minimum Attractive Rate of Return (MARR), Internal
Rate of Return (IRR), Depreciation, Break-even analysis.
Book(s)
Riggs, J. L., D. D. Bedworth, and S. U. Randhawa. 1998. Engineering Economics (fourth
edition). The McGraw-Hill Companies, Inc.

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By Dr. Raza Ullah
Institute of Agricultural and Resource Economics
University of Agriculture, Faisalabad
Economics
Economics is the study of allocating scarce resources among different alternatives to fulfill
maximum of unlimited human wants.
Scarce resources:
Refers to the finite/limited quantity of resources that are available to meet society’s needs.
There are three types of scarce resources:
i. Natural and biological resources
a. Examples of natural resources are land and mineral deposits
b. Examples of biological resources are livestock, wildlife and different genetic varieties
of crops
ii. Human resources (examples are labour and management)
iii. Capital or Manufactured resources (machines, equipment and structures)
Resource scarcity forces consumers and producers to make choices. Sometimes the choices we
make are constrained not only by resource scarcity but also by non-economic consideration.
These forces may be political, legal or moral.
Human wants:
Human wants mean the desire to get and use goods or services, which provide utility. For
example, a hungry person wants food, bare footed person wants shoes, a patient want medical
care. There are two types of human wants.
a. Economic wants: These are the wants which can be satisfied only by spending money
b. Non-economic wants: The desires of the people which are possible to satisfy without
the use of money, e.g. desire for friendship or love etc.
Characteristics of human wants
i. Unlimited
ii. Re-appear: most of the wants, once satisfied, appear again
iii. Compete with each other (due to limited resources, all wants can bot be satisfied at
the same time)
iv. Difference in importance e.g. food is more important than computer/mobile
v. Alternative means of satisfaction e.g. visiting Karachi by air or train, eat bread or rice.
vi. Wants are complementary e.g. car and petrol)
vii. Present wants appear to be more important than future wants.

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viii. Wants change with time, place, income, age, fashion, climate, occupation, social
status and prices.
Scope and Subject Matter of Economics
Economics, as a subject, is a combination of arts and science, which contains the principles and
laws with the help of which the functioning of the economy and its variants takes place. The
following basic facts define the existence of an economy-
1. Consumers have unlimited wants for goods and
services.
2. There is a scarcity of the productive resources
which helps to produce goods and services to
satisfy human wants.
Thus, Economics is a study of how to produce
goods and services from scarce resources so as to
satisfy the human wants and needs and how
sustainably we consume these goods and services.
Subject Matter of Economics:
Economics can be studied through a) traditional
approach and (b) modern approach.
a) Traditional Approach: Economics is studied
under five major divisions namely consumption,
production, exchange, distribution and public
finance.
1. Consumption: The satisfaction of human
wants through the use of goods and services is called consumption.
2. Production: Goods that satisfy human wants are viewed as “bundles of utility”. Hence
production would mean creation of utility or producing (or creating) things for satisfying human
wants. For production, the resources like land, labour, capital and organization are needed.
3. Exchange: Goods are produced not only for self-consumption, but also for sales. They are
sold to buyers in markets. The process of buying and selling constitutes exchange.
4. Distribution: The production of any agricultural commodity requires four factors, viz., land,
labour, capital and organization. These four factors of production are to be rewarded for their
services rendered in the process of production. The land owner gets rent, the labourer earns
wage, the capitalist is given with interest and the entrepreneur is rewarded with profit. The
process of determining rent, wage, interest and profit is called distribution.
5. Public finance: It studies how the government gets money and how it spends it. Thus, in
public finance, we study about public revenue and public expenditure.
b) Modern Approach: The study of economics is divided into: i) Microeconomics and ii)
Macroeconomics.
1. Microeconomics analyze the economic behavior of any particular decision making unit such
as a household or a firm. Microeconomics studies the flow of economic resources or factors of
production from the households or resource owners to business firms and flow of goods and
services from business firms to households. It studies the behavior of individual decision making
Goods and Services
Goods and services are the outcome of human
efforts to meet the wants and needs of people.
Goods are tangible and physical economic output
that can be seen and touched e.g. books, pens,
folders etc. while services are intangible and are
provided by other people such as doctors,
dentists, barbers etc. Consumption of goods and
services is assumed to
provide utility (satisfaction) to the consumer.
There are three main groups of consumable
products;
i. Durable goods e.g. cars, washing
machines etc.
ii. Non-durable goods e.g. meat, milk etc.
iii. Services such as haircuts, education
etc.

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unit with regard to fixation of price and output and its reactions to the changes in demand and
supply conditions. Hence, microeconomics is also called price theory.
2. Macroeconomics studies the behavior of the economic system as a whole or all the decision-
making units put together. Macroeconomics deals with the behavior of aggregates like total
employment, gross national product (GNP), national income, general price level, etc. So,
macroeconomics is also known as income theory.
Positive economics: an approach to economics that seeks to
understand behavior and operation of systems without making any
judgement. It describes what it is and how it works. Positive
economics deals with observed economy.
Normative economics: an approach to economics that analyses
outcomes of economic behavior, evaluates them as good or bad
and may suggest improvements. Normative economics is also
called policy economics. Normative economics deals with desired
economy.
Economic System
Economic system consists of institutions chosen or accepted by people or nation or group of
nations as the means through which resources are utilized for the satisfaction of human wants.
Types of Economic systems
a. Capitalism (market economy): economic system where means of production (resources)
are owned and managed by private individuals and institutions. Any profit that is
generated by the use of that private property belong to the owner.
b. Socialism (command economy): an economic system where means of production
(resources) are controlled and managed by the state. Private ownership of means of
production is not allowed. People can have personal property which is transferable and
inheritable.
c. Communism: in communism resources are controlled and managed by a Central State
authority and there is also a restriction on the ownership of personal property in
communism. Individuals are assigned work by the state and they are given a bit
remuneration of their services.
d. Mixed Economy: mixed economy combines the features of capitalism and socialism. A
number of industries are owned and managed by the state while private enterprise is also
allowed and even encouraged to operate a large number of industries and to own the
various means of production.
Demand
The quantities that the consumers are willing and able to buy from the market at various prices.
Economy
The large set of inter-related
economic production and
consumption activities which aid
in determining how scarce
resources are allocated. The
economy encompasses everything
related to the production and
consumption of goods and services
in an area.

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Demand Curve:
A graphic representation of the demand schedule. A negative sloped line showing the inverse
relationship between the price and quantity demanded.
Price of
good
Quantity
Demanded
24 200
20 300
16 400
12 500
8 600
Figure 1: Demand Curve
Law of Demand:
There is a negative or inverse relationship between the price of any good or service and the
quantity demanded, holding other factors constant. As price increases quantity demanded
decreases and vice-versa.
Individual and Market Demand:
Individual demand is the schedule of quantities demanded by a single person or a single
household at various prices. The market demand is the total demand of all person in the market
and is denoted by adding the individual demands.
Price of Milk (Rs/liter) Individual Demand (Liters
per day)
Daily Market Demand (Liters)
A+B+C
A B C
16 4 0 2 6
12 6 2 3 11
8 8 4 4 16

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Figure 2: Individual and Market Demand
Change in quantity demanded and change in demand:
Change in Quantity demanded: refers to the movement from one point to another point- from
one price quantity combination to another on a fixed demand curve.
Change in Demand: refers to a shift in the entire demand curve either to the right (an increase in
demand) or to the left (a decrease in demand).
Non-price determinants of demand or factors that shift the demand curve (Demand Shifters):
i. Change in income: an increase in income increases the demand for normal goods,
while reducing the demand for inferior goods (as cabbage, turnips, used clothing etc.)
ii. Change in taste and preferences
iii. Change in the prices of related goods: a decrease in the price of coffee reduces the
demand for tea (substitute goods). For substitutes the change in the price of one
causes a shift in demand for another in the same direction as the price change. A
decline in the price of coffee increases the demand for cream (complimentary goods).
For complimentary goods a change in the price of one causes an opposite shift in the
demand for the other.
iv. Change in expectations regarding future prices, future incomes and future product
availability.
v. Population (the number of consumers in the market).
Supply
Supply may be defined as the quantities that the producers are willing and able to put on the
market at various prices during some period of time.
Supply Curve:
A graphic representation of the supply schedule. A positive sloped line showing the direct
relationship between the price and quantity supplied.

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Supply Schedule
Price of eggs
(Rs./dozen)
Quantity supplied
(dozen)
16 400
20 600
24 800
28 1000
32 1200
Figure 3: Supply curve
Law of Supply:
There is a positive relationship between price and quantity supplied, holding other factors
constant. As price increases quantity supplied increases and vice versa.
Change in quantity supplied and change in supply:
Change in quantity supplied: refers to the movement from one point to another on the same
supply curve.
Change in supply: when the entire supply curve shifts either to right or left is called change in
supply.
Non-price determinants of supply or supply shifters:
i. Cost of inputs
ii. Technology
iii. Taxes and subsidies
iv. Producer price expectation
v. The number of suppliers
vi. Weather
Elasticity
Elasticity is used to measure the effect of changes in price on quantity demanded (supplied). It is
a good way to measure consumer response.
Price elasticity of demand:
The degree of responsiveness of demand to change in price is called price elasticity of demand.
Percentage change in quantity demanded divided by percentage change in price.
Ed =
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒
Ed =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑
÷
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑝𝑟𝑖𝑐𝑒
Mid-point formula for calculating elasticity:

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Ed =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑
(𝑠𝑢𝑚 𝑜𝑓 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑖𝑒𝑠)/2
÷
(𝑠𝑢𝑚 𝑜𝑓 𝑝𝑟𝑖𝑐𝑒𝑠)/2
Price/unit Quantity demanded Total revenue Elasticity co-efficient
5 2000 10000
4 4000 16000 Ed =
2000
(6000)/2
÷
1
(9)/2
= 3
3 7000 21000 Ed =
3000
(11000)/2
÷
1
(7)/2
= 1.91
2 11000 22000 Ed =
4000
(18000)/2
÷
1
(5)/2
= 1.11
1 16000 16000 Ed =
5000
(27000)/2
÷
1
(3)/2
= 0.56
Ed = 3 means that 1 percent change in price will change the quantity demanded by 3 percent.
Determinants of price Elasticity of Demand:
i. Substitutability: Goods having close substitute makes its demand highly elastic.
(Coke, Pepsi).
ii. Proportion of Income: A good taking a very small part of our budget has inelastic
demand (safety matches).
iii. Luxuries versus Necessities: Demand is inelastic for necessities (bread) and demand
for luxuries is elastic.
Cross elasticity of Demand:
The change in price of a commodity does not only effect its own demand but also the
demands for many other related commodities (e.g. meat and fish).
Cross price elasticity of demand represents the responsiveness of a commodity to change in
price of a related good. Mathematically;
Edxy =
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑 𝑜𝑓 𝑔𝑜𝑜𝑑 𝑥
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑔𝑜𝑜𝑑 𝑦
Example:
Price of wheat (Rs/Maund) Py Quantity demanded of Rice (Maunds) Qx
200 1000
300 1200
Percentage change in price of wheat =
100
200
x100 = 50
Percentage change in quantity demanded of Rice =
200
1000
x 100 = 20
Edxy =
20
50
=
2
5
= 0.4

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Rice is a substitute for wheat so Edxy of demand for rice has a positive value i.e. if price of
wheat rises, people increases the consumption of rice.
For complement commodities the value of cross elasticity will come out to be negative.
Income elasticity of demand:
Ei =
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑐𝑜𝑚𝑒
Example:
Monthly income of Mr. A (Rs.) Monthly demand for Meat (kg) of Mr. A
15000 4
20000 5
Percentage change in quantity demanded of meat =
1
4
x 100 = 25
Percentage change in income =
5000
15000
x 100 =
1
3
x 100 = 33.33
Ei =
25
33.33
= 0.75
 A negative income elasticity of demand is associated with inferior goods: an increase in
income will lead to a fall in the demand and may lead to changes to more luxurious
substitute.
 A positive income elasticity of demand is associated with normal goods; an increase in
income will lead to a rise in demand. If income elasticity of demand of a commodity is
less than 1, it is a necessity good. If the elasticity of demand is greater than 1, it is a
luxury good or a superior good.
 A zero income elasticity (or inelastic) demand occurs when an increase in income is not
associated with a change in the demand of a good. These would be sticky goods.
Elasticity of Supply
The degree of responsiveness of supply of a commodity to change in its price. When a small
rise in price causes a large expansion in supply or a small decrease in price leads to a large
contraction in supply, the supply of the commodity is said to be elastic on the other hand if
supply shows little variation in response to a given change in price, the supply is known as
inelastic.
Es =
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒
or
Es =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑
÷
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑝𝑟𝑖𝑐𝑒

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Es =
𝛥𝑞
𝑞
÷
𝛥𝑃
𝑃
Example:
Price (P) Quantity Supplied (qs)
3 100
4 110
Es =
10
100
÷
1
3
=
3
10
= 0.3
Determinants of price elasticity of supply:
i. Availability of raw materials
ii. Length and complexity of production
iii. Mobility of factors
iv. Time to respond
v. Excess capacity
vi. Inventories
When the supply is represented linearly the co-efficient of elasticity of any linear supply
curve that
i. Passes through the origin is unit elastic.
ii. Cuts the y-axis is greater than 1 (elastic).
iii. Cuts the x-axis is less than 1 (inelastic).
Market Equilibrium or Determination of Market Price
Prices of commodities are determine by the interaction of two forces i.e. demand and supply.
Demand has inverse relation with price and on the other hand supply has direct relation with
price. It is the equality of these two forces which settles the price of a commodity at a particular
level in the market.
Market equilibrium refers to a situation where forces of demand and supply balance each other.
How the forces of demand and supply bring about equilibrium can be illustrated by simple
example.
Price of wheat
(Rs./kg)
Quantity
demanded
(million tons)
Quantity
supplied (M.T)
Market
Condition
Direction of
Price change
2 18 0 D>S Upward
4 16 4 D>S Upward
6 14 8 D>S Upward
8 12 12 D=S Equilibrium
10 10 16 D<S Downward
12 8 20 D<S Downward

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Figure 4: Market Equilibrium
The Production Function
The technical relationship between inputs and output indicating the maximum amount of output
that can be produced using alternative amounts of variable inputs in combinations- with one or
more fixed inputs under a given state of technology.
The production function is the mathematical relationship describing the way inputs and outputs
are physically correlated. Notationally a common, or general, form for the mathematical
relationship is
Y = f(X1, X2, …, Xn)
Where y is the output resulting from the production process, the X’s identify the different inputs
used in the production process to produce Y, and f() represents the mathematical relationship
between the inputs and output, called the production function. Notationally, a vertical bar ( l ) is
used to distinguish between the variable and fixed factors of production in the representation of
the production function. For example
Y = f (X1 l X2, … , Xn)
Where X1 is the variable input (i.e. fertilizer, number of employers, etc.) and X2 to Xn are the
fixed factors of production (i.e., land, building, etc.).
Example:
A wheat producer knows that if s/he doesn’t apply any fertilizer his/her wheat yield will be 20
Kgs per kanal, due to the existing nutrients in the soil, and that for every pound of fertilizer
applied wheat yield will increase by 0.4 Kgs.

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Amount of X used (fertilizer) Amount of Y produced (Kgs of wheat)
0 20
10 24
20 28
30 32
40 36
50 40
Mathematically, this production function can be expressed as;
Y = 20 + 0.4X
Where X is the Kgs of fertilizer applied per Kanal and Y is the Kgs per Kanal yield of wheat.
This production function can be graphically represented as;
Figure 5: Production Function
Types of Production Responses
The amount of output produced in any physical production process depends on the level of
inputs used and the production function. As the level of input use changes, the level of output
also changes, with the rate of change in output dependent on the technical relationship between
inputs and output. There are four possible production response relationships, or rates of change,
between input use levels and the resultant output. These are discussed as follow;
i. Constant Returns
In this type of response each additional unit of input is as productive as the previous unit. In
other words, for each additional units of input used, output increases at a constant rate (the rate
of change in output remains constant). The production function Y = 2X depicts this relationship.

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A Production Schedule with Constant Returns
X Y Added output for each added unit of input
0 0
1 2 2
2 4 2
3 6 2
4 8 2
5 10 2
The increase in production is also depicted by the slope of the production function. When we
have constant returns the slope (change in output (Y) over the change in input (X), is always
constant. In this example the slope is ΔY/ ΔX = 2/1 = 2.
ii. Increasing Returns
Increasing returns occur when each additional unit of input added to the production process
yields more additional product than the previous unit of input.
A Production Schedule with Increasing Returns
0 0
1 1 1
2 3 2
3 6 3
4 10 4
5 15 5
When increasing returns are observed, the slope of the production function is getting steeper or
increasing, as additional units of input are used. This implies that output (Y) increases at an
increasing rate for each additional unit of input (X) that is used.

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Figure 6: Types of Production Responses
iii. Decreasing Returns
The decreasing returns response s seen when each additional unit of input increases the
production level, but with a smaller change than the previous inputs.
Production Schedule with Decreasing Returns
0 0
1 5 5
2 9 4
3 12 3
4 14 2
5 15 1
When decreasing returns are observed, the level of production continues to increase but at a
slower rate. The slope of the production function s getting flatter as the change in increased
production divided by change in input becomes smaller.
iv. Negative Returns
Negative returns occur when each additional unit of input added to the production process
decreases the production level. That is, there is a negative relationship between the additional
use of input and the resultant output.
Production Schedule with Negative Returns

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7 17
8 15 -2
9 12 -3
10 8 -4
11 3 -5
Negative returns typically occur after the maximum production level is reached and reflect
the constraints fixed factors impose on production.
The Typical Production Function
The production function represents the physical correlation between inputs and the resultant
output. It tells us the amount of output that can be obtained from a combination of alternative
amounts of variable inputs with one or more fixed factors. For the sake of simplicity, we
assume that only one product is produced using only one variable inputs while other inputs
are assumed to be fixed.
Product Curves
Three concepts, referred to as product curves, are commonly used to study the typical
production function. These are;
i. Total Physical Product (TPP)
TPP curve illustrates the relationship that exists between output and one variable input,
holding all other inputs constant. Total physical product is measured in physical terms and
represents the maximum amount of output brought about by each level of input use. Since
TPP represents total output, it is often referred to as Y.
ii. Average Physical Product (APP)
APP shows how much production, on average, can be obtained per unit of the variable input
with a fixed amount of other inputs. It is calculated as the output at each level (TPP) divided
by that level of input (X), or;
APP = TPP/X
iii. Marginal Physical Product (MPP)
MPP represents the amount of additional (marginal) total physical product obtained from
using an additional unit of variable input. It is computed as the change in output divided by
change in input use:
MPP = ΔTPP ÷ ΔX
we can put these three concepts, which comprise the typical production function, into action
using the following example. The manager of UAF Sandwich shop can use the number of
employees to increase the number of sandwiches prepared every hour. In this example, the

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variable input (X) is the number of employees and the output (Y) is the number of
sandwiches made per hour.
X (Input) Y (TPP) APP (Y ÷X) MPP (ΔTPP ÷ ΔX)
0 0
1 2 2 2
2 5 2.5 3
3 9 3 4
4 11 2.75 2
5 12 2.40 1
6 11 1.83 -1
Figure 7: Product Curves (TPP, APP and MPP)
We can also identify the production responses observed earlier. The production function is
our example exhibits increasing returns through the third unit of input (X=3) as TPP is
increasing at increasing rate (MPP is increasing). Decreasing returns occur when each
additional unit of input is less productive than the previous unit. In our case decreasing
returns are set in from the third unit to the fifth unit of input (X=5) as TPP is increasing but at
a slower rate (MPP is declining, but still positive). Negative returns occur past the fifth unit
of input (MPP is negative).
Relationship between Product Curves
The shape of TPP curve shows that as input use increases, the quantity produced increases
but the rate of increase changes. The TPP curve exhibits increasing returns up to point (F) in
Figure 8 below, meaning output is increasing at an increasing rate. At this point, called the

Dr. Raza Ullah
17
inflection point or the point of diminishing returns, TPP changes from increasing at an
increasing rate to increasing at decreasing rate. The TPP curve exhibits decreasing returns,
where each additional unit of input generates less additional (marginal) product than the
previous unit, from point (F) to point (H) where TPP reaches its maximum. Beyond point (H)
the total physical product actually begins to decrease with the use of additional input as
negative returns set in.
Figure 8: Relationship between Product Curve and Stages of Production
APP and MPP are derived directly from the input-output relationship of the total TPP curve.
Thus, the product curves are related to each other and these relationships are important. The
MPP curve describe the slope of TPP curve. As in Figure 8, the MPP increases up to point
(F), the range in which TPP exhibits increasing returns. At point (F), the point of diminishing
returns (where the slope of the TPP changes from increasing at an increasing rate to
increasing at decreasing rate), the MPP reaches its maximum. MPP is declining, but still
positive, from point (F) to point (H) indicating that output is increasing but at a slower rate,
which corresponds to the range in which TPP exhibits decreasing returns. MPP is equal to
zero at point (H), which corresponds directly to the maximum TPP. MPP becomes negative
after point (H), corresponding to the decreasing production as negative returns are observed.
APP curve describes the average productivity of the variable inputs. The APP curve initially
increasing reaching a maximum at point (S) and then declines. The high point of APP, point

Dr. Raza Ullah
18
(S), is located where a line drawn out of the origin is tangent (barely touching) to the TPP.
The slope at each point on this line is the ratio of output to input, a ratio that is the definition
of APP. Where this line is tangent to the TPP, point (S), identifies the maximum ratio of
output per unit of input, which is the definition for maximum of APP, for this production
function. At that point where APP reaches its maximum, MPP crosses APP, or MPP = APP.
As long as MPP is greater than APP, average physical product continues to increase because
the marginal increase in output raises the average for all inputs at a given level.
Stages of Production
The product curves, in the production function (Figure 8), provide enough information to
identify a profitable range of production. These regions are referred to as stages of
production and are classified as rational or irrational. Stage I begins where no input is
utilized, point (0), and ends where APP is at its maximum (APP = MPP), at point (S) in the
above figure. If we decide to produce at all we will use inputs at least to this point because
every succeeding unit of input up to point (S) has a marginal product that raises the average
(MPP > APP) even though the added output per added unit of input (MPP) is decreasing.
Stage II begins at point (S), the input use level where APP is maximum (MPP = APP), and
ends at point (H), the input use level where TPP is maximum (MPP = 0). Somewhere in this
region of production profits will be maximum, making Stage II the rational stage of
production. For each additional unit of input that is used within Stage II, TPP is increasing
but APP is decreasing, resulting in a trade-off between increased production and decrease
productivity for each variable input. This trade-off is the core of economic decision making
and requires the skill of the decision maker or manager.
Stage III begins where TPP is at its maximum, point (H). If we continue to use additional
units of input, we actually decrease the total production level. Since output decreases as we
increase input use, it is impossible to maximize profits in this production region. Thus, Stage
III is obviously an irrational area of production.
Cost of Production
Economic Cost or Cost of Production: The payments a firm must make to attract inputs
(resources) and keep them from being used to produce other outputs are referred to as
economic cost or cost of production.
Economic costs can be either implicit or explicit. Explicit costs are the normal “out-of-
pocket” or “cash” costs of inputs used in production. Implicit costs are the costs associated
with inputs owned by the firm. Since the firm owns the inputs there is no need to make cash
payment to be able to use the input, but use is not free. There is an opportunity cost
associated with using firm-owned inputs. The opportunity cost of using an input is the value
of the contribution that input could make in its highest valued alternative use (i.e., land used
to produce corn versus soybeans; farmer stays on own farm versus working in factory).

Dr. Raza Ullah
19
Economic costs are usually categorized in fixed costs and variable costs.
Fixed Costs: are those costs which do not vary with the level of production, and are the costs
associated with the fixed factors of production. As the fixed factors of production must be
maintained even if no production occurs; hence, the costs associated with these factors must
be paid even if production is zero.
Variable Costs: Variable costs of production do vary as output level changes and are the
costs associated with using the variable factors of production. If no variable input is used,
variable costs are zero; as input use increases, resulting in increased output, variable costs
increase.
Cost Relationships in Production
Costs on Total Output Basis
Three costs are used to measure the costs of production on the basis of total output produced:
total fixed costs, total variable costs, and total cost.
Total Fixed Cost (TFC): Sometimes called overhead costs, include both implicit and explicit
costgs of nputs that are fixed in the short-run and do not change as output level changes.
Because fixed factors of production must be maintained even if production is zero, fixed
costs must be paid at all production levels, even zero.
Total Variable Cost (TVC): include both implicit and explicit costs of inputs that are
variable, or changeable, in the short-run and change as the level of output changes. Total
variable costs represent the total dollar amount paid for each variable input used to produce
each output level. Total variable costs are calculated by summing the cost of each variable
input used (equal to the units of input used multiplied by the input’s price);
TVC = (PX1 x X1) + (PX2 x X2) + …. + (PXn x Xn)
Where Xi represents the units of variable input i used, and PXi is price of variable input Xi.
Total cost (TC): is simply the sum of total fixed cost and total variable cost at each level of
output.
TC = TFC + TVC
Total cost is the overall cost of producing each level of output.
Recall the sandwich shop example, where the variable input is the number of employees and
output is the number of sandwiches prepared per hour. Assume that the store manager can
hire an additional worker for $4/hour (Px = $4) and that the TFC of the shop is $10 (TFC =-
$10). The total cost curves associated with this production function are presented in the table
below.
Total Cost Curves for Sandwich Shop: TFC = $10 and Px = $4

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20
X (Input) Y (TPP) TFC ($) TVC (Px x X) TC (TFC + TVC)
0 0 10.00 $0.00 $10
2 2 10.00 4.00 14.00
2 5 10.00 8.00 18.00
3 9 10.00 12.00 22.00
4 11 10.00 16.00 26.00
5 12 10.00 20.00 30.00
6 11 10.00 24.00 34.00
Figure 9: Cost Curves
Total fixed cost do not change as output changes but remain at a constant level of $10. Total
variable cost increase as input use, and output levels, increase. At each output level TVCs are
calculated by multiplying the units of input used to produce that output level by the price of
the input. Total cost is the sum of total fixed cost and toal variable cost at each level of
output. For example, when Y = 2, the total cost is $14 ($10 + 4); when Y = 12, total cost is
$30 ($10 + 20). Graphically, as Figure 8 shows, the distance between TVC and TC at each
output level is equal to TFC. Figure 9 also shows that as we enter Stage III (where X = 5 and
Y = 12) both TVC and TC curve back toward the vertical axis indicating that although output
falls as input use increases (Stage III) the costs of using those inputs continue to increase.
Costs on Per-Unit of Output Basis
Four cost curves are used to measure costs of production on a per-unit of output basis. Just
as APP and MPP are derived from TPP, the per-unit costs are derived from the total cost
curves. The per-unit costs include three average costs and a related cost concept, which is of
great importance in economic analysis, marginal cost.

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21
Average Fixed Cost (AFC): is the average cost of the fixe inputs per-unit of output, and is
calculated by dividing the total fixed costs by each output level;
AFC = TFC ÷ TPP
Average Variable Cost (AVC): is the average cost of the variable inputs per-unit of output
and is calculated by dividing the total variable costs by each output level;
AVC = TVC ÷ TPP
Average Total Cost (ATC): is the average total cost per-unit of output. It is calculated by
dividing the total cost at each output level by the output level;
ATC = TC ÷ TPP
Marginal Cost (MC): is the increase in total cost necessary to produce one more unit of
output. It is calculated as the change in total cost divided by the change in output;
MC = ΔTC ÷ ΔTPP
Marginal cost is the addition to total cost incurred to produce an additional unit of output.
Since TFC does not change with output level, TFC does not impact MC. Thus we can
express MC as;
MC = ΔTC/ ΔY = Δ(TFC + TVC)/ ΔY = ΔTVC/ ΔY
We can illustrate the calculation of the four per-unit cost curves using our sandwich shop
example. The per-unit cost curves associated with this production function are provided in
table below.
Per-Unit Cost Curves for Sandwich Shop: TFC = $10 and Px = $4
X Y TFC TVC TC AFC AVC ATC MC
0 0 10 $0. $10 --- --- --- ---
1 2 10 4 14 5 2 7 2
2 5 10 8 18 2 1.60 3.60 1.33
3 9 10 12 22 1.11 1.33 2.44 1
4 11 10 16 26 0.91 1.45 2.36 2
5 12 10 20 30 0.83 1.67 2.50 4
6 11 10 24 34 0.91 2.18 3.09 -4

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22
Figure 10: Per-Unit Cost Curves
In Figure 10 the ATC, AC and MC curves are all U-shaped, first decreasing at lower levels of
output, where increasing returns are typically experienced, and increasing later on where
decreasing returns begin to occur. The U-shaped curves reflect the increasing/decreasing returns
exhibited by the TPP, from which these curves are derived.
Graphically the ATC curve is the vertical summation of the AFC and AVC curves, thus the
vertical distance between TC curve reaches a minimum (or low point) at a larger output level
than AVC because in the output range beyond the AVC minimum AFCs are declining faster than
the increase in AVC; hence ATC continues to decrease for short time. When the increase in
AVC is greater than the decrease in AFC, ATC begins to increase.
The MC curve crosses the AVC and ATC curves from below and at the minimum point on each
of these curves; this is a mathematical relationship to which there are no exceptions. As long as
MC is below AVC and ATC, both curves must be failing, because less is being added to the total
cost for each successive unit of output than the average of all previous units. As long as MC is
above AVC and ATC, both must be rising, because the MC of each added unit of output is larger
than the average of the previous unit.
Cost Related to Production Function
The cost curves are derived directly from the physical production function process. Thus, there is
a direct relationship between the physical aspect of production (the product curves) and the
economic aspects of production (the cost curves).

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23
Figure 11: Relationship between product curves and per-unit cost curves
Although the axis are different, the physical characteristics of the production function can be
transferred directly to the economic values of the cost curves. It is evident from figure 11 that the
APP and AVC, and the MPP and MC curves are mirror images of one another.
The shape of the MC curve is a direct result of the increasing/decreasing returns exhibited by the
TPP and since MPP indicates the slope of the TPP, MPP and MC are related. When MPP is
increasing (there are increasing returns), MC is decreasing. Increasing returns means that each
additional unit of input is more productive than the last. If the cost of using additional unit of
input is constant, and each additional unit of input produces more output, then the additional cost
of producing an additional unit of output (MC) must be decreasing Thus, when we have
increasing returns (MPP increasing), MC will be declining. When MPP reaches its maximum,
point (1) in Figure 11, MC reaches its minimum. When MPP is decreasing (we have decreasing
returns), MC is increasing. Decreasing returns means that each additional unit of input is less
productive than the last. Because the cost of using another unit of input is constant, and each
additional unit of input produces less output, then the additional cost of producing an additional
unit of output (MC) must be increasing. Thus, when we have decreasing returns (MPP
decreasing), MC will be rising. As we enter Stage III, MPP becomes negative, point (3) in Figure
11, MC also becomes negative. However, since Stage III is an irrational stage of production, this
portion of the curve is typically not depicted.

Dr. Raza Ullah
24
APP curve represents the average productivity of each unit of input. As the average productivity
of the variable input used increases (APP is increasing), the AC of using the variable input to
produce output decreases (AVC is decreasing). At point (2) where APP reaches its maximum,
the pint of maximum physical efficiency, AVC reaches its minimum. As APP decreases the
average productivity of the variable input is decreasing; thus the average cost of using the
variable input to produce output is increasing (AVC is increasing).
In Figure 11 MPP crosses APP from above where APP is at its maximum; MC crosses AVC
from below where AVC is at its minimum. This intersection point, point (2), is the beginning of
Stage II. Thus, if it is possible to produce at all, the firm will produce up to the minimum point
on AVC (maximum point on APP).
These relationships are summarized in table below.
Relationships between Product Curves
Physical or Technical Economic
1. When MPP > APP, APP ↑ 1. When MC < AVC, AVC ↓
2. When MPP = APP, APP is maximum 2. When MC = AVC, AVC is
minimum
3. When MPP < APP, APP ↓ 3. When MC > AVC, AVC ↑
These same relationships can be shown algebraically. The relationship between AVC and APP is
as follows;
AVC = TVC/Y = (Px x X)/Y = Px x (X/Y) = Px/(Y/X) = Px/APP
Since Px is a constant, as APP increases, AVC will decrease. The shape of AVC depends on the
production function, but the amount, or level, of AVC depends on the input price.
The relationship between MC and MPP is
MC = ΔTC/ ΔY = Δ (Px x X)/ ΔY = (ΔX x Px)/ ΔY = (ΔX/ ΔY) x Px = Px / (ΔY/ ΔX) = Px/MPP
Since Px is a constant, as MPP increases, MC will decrease.
To illustrate these relationships, the product curves and the per-unit cost curves for our sandwich
shop are presented in table below. APP reaches its maximum at X = 3 (Y = 9), which is the point
where AVC reaches its minimum at Y = 9 (X = 3). The maximum of MPP is between X = 2 and
X = 3 (Y = 5 and Y = 9), which corresponds to where MC reaches its minimum between Y = 5
and Y = 9 (X = 2 and X = 3). The beginning of Stage II where APP is maximum corresponds to
where AVC is minimum at Y =9 (X = 3). Stage III begins where TPP is maximum, which
corresponds to where MC becomes negative at Y = 12 (X = 5).
Per-Unit Cost Curves for Sandwich Shop: TFC = $10 and Px = $4
X Y APP MPP AFC AVC ATC MC
(Y/X) (ΔY/ ΔX) (TFC/Y) (TVC/Y) (TC/Y) (ΔTC/ ΔY)

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25
0 0 --- --- --- --- --- ---
2 2 2 2 5 2 7 2
2 5 2.5 3 2 1.60 3.60 1.33
3 9 3 4 1.11 1.33 2.44 1
4 11 2.75 2 0.91 1.45 2.36 2
5 12 2.40 1 0.83 1.67 2.50 4
6 11 1.83 -1 0.91 2.18 3.09 -4
From the table it is also easy to show that MC = Px ÷ MPP and that AVC = Px ÷ APP. For
example, the average variable cost of producing 2 sandwiches is $2, which is equal to Px ($4)
divided by the APP (2.00) when X = 1.
Engineering Economics
Engineers are planner and builders. They are also problem solvers, managers and decision
makers. Engineering economics touches each of these activities. Plans and production must be
financed. Problems are eventually defined by dollar dimensions, and decisions are evaluated by
their monetary consequences. Much of the management function is directed toward economic
objectives and is monitored by economic measures.
Engineering economics is closely aligned with conventional microeconomics, but it has a history
and a special flavor of its own. It is devoted to problem solving and decision making at the
operations level. An engineering economist draws upon the accumulated knowledge of
engineering and economics to identify alternative uses of limited resources and to select the
preferred course of action. Evaluations rely mainly on mathematical models and cost data, but
judgment and experience are pivotal inputs.
Engineering Decision Makers
The following general questions are representative of those that as engineer might encounter;
i. Which of several competing engineering designs should be selected?
ii. Should the machine now in use be replaced with a new one?
iii. With limited capital available, which investment alternative should be funded?
iv. Would it be preferable to pursue a safer conservative course of action or to follow a
riskier one that offers higher potential returns?
v. How many units of production have to be sold before a profit can be made? This area
is commonly called break-even analysis.
vi. Among several proposals for funding that yield substantially equivalent worthwhile
results but have different cash flow patterns, which is preferable?
vii. Are the benefits expected from a public service project large enough to make its
implementation costs acceptable?
Two characteristics of the questions above should be apparent. First, each deals with a choice
among alternatives; second, all involve economic considerations.
Interest and Interest Rate

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26
Interest is the cost of using capital. Interest represents the earning power of money. It is the
premium paid to compensate a lender for the administrative cost of making a loan, the risk of
non-repayment, and the loss of use of the loaned money. A borrower pays interest charges for the
opportunity to do some-thing now that otherwise would have to be delayed or would never be
done.
Its history extends as far back as the recorded transactions of humanity. In earliest times, before
money was coined, capital was represented by wealth in the form of personal possessions, and
interest was paid in kind. For example, a loan of seed to a neighbor before planting was returned
after harvest with an additional increment.
Today, there are many credit instruments, and most people use them. Business and government
are the biggest borrowers. Businesses seek the use of capital goods to increase productivity.
Governments borrow against future tax revenues to finance highways, welfare programs, and
public services. Households also borrow to make purchases in excess of their current cash
resources. Such borrowers, and the corresponding lenders, must acknowledge the time value of
their commitments.
The following example reveals the significance of interest in economic transactions and confirms
the importance of understanding how interest operates, whether it pertains to personal finances or
to professional practices.
Case 1 Case 2
Repayment
period,
years
Monthly
payment,
$
Total
interest,
$
Interest
rate, %
Monthly
payment,
$
Total
interest,
$
15 984.51 77,210 7.5 699.21 151,722
20 867.57 108,217 8.5 768.63 176,707
25 804.96 141,485 9.5 841.15 202,806
30 768.63 176,707 10 877.28 215,812
In case 1, a $100,000 loan is taken at an interest rate of 8.5 % with four repayment periods while
in case 2, a $100,000 loan is taken for 30 years at four different interest rates. The two cases
revealed that a shorter repayment period at a given interest rate or a lower interest rate for a
given loan period gives a significant saving.
Simple Interest
In simple interest the interest earned is directly proportional to the capital involved in the loan.
The interest earned I through several time periods is found by the following equation.
I = PiN
Where
P = present amount of principal amount

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27
I = interest rate per period
N = number of interest periods (usually years)
Since the principal, or amount borrowed, P is a fixe value, the annual interest charged is
constant. Therefore, the total amount a borrower is obligated to pay a lender is
F = P + I = P + PiN = P(1 +iN)
Where F is a future sum of money to be paid. When N is not a full year, there are two ways to
calculate the simple interest earned during the period of the loan. When ordinary simple interest
is used, the year is divided into twelve 30-day periods, or a year is considered to have 360 days.
In exact simple interest, a year has precisely the calendar number of days, and N is the fraction of
the number of days the loan is in effect that year.
An example of simple interest as the rental cost of money is loan of $1000 for 2 months at 10
percent. With ordinary simple interest, the amount to be repaid is
F = P(1+ iN)
Where N is 2/12 year, giving
F = $1000(1 + 0.01667) = $1016.67
With exact simple interest when the two months are January and February in a non-leap year, the
future sum is
F = P[1 + (𝑖) (
31+28
365
)]
and
F = $1000 (1 + 0.01616) = $1016.16
Compound Interest
Again, assume a loan of $1000, this time for 2 years at an interest rate of 10 percent compounded
annually; the pattern of interest compounding is shown in Table.
Future value of $1000 loan when interest is due on both the principal and unpaid
interest
Year Amount owed at
beginning of year, $
Interest on amount
owed, $
Amount owed at
end of year, $
1 1000 1000 x 0.10 = 100 1000 + 100 = 1100
2 1100 1100 x 0.10 = 110 1100 + 110 = 1210
The amount to be repaid with simple interest is
$1000 [1 + 0.1(2)] = $1200
Thus, the amount to be repaid for the given loan is now $10 greater for compound interest than
for simple interest ($1210 - $1200). The $10 difference accrue from the interest charge on the
$100 earned during the first year that was not accounted for in the simple-interest calculation.
The formula for the calculations in table above, using previously defined symbols, is
F2 = P + Pi + (P + Pi)i

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28
F2 = Compound amount due in 2 years
P = Amount borrowed
Pi = Year 1 interest
i = interest rate
F2 = P + Pi + (P + Pi)i
= P(1 + i + i + i2
)
= P(1 + i)2
= $1000(1 + 0.10)2
= $1000(1.21) = $1210
The key equation in the development above is F2 = P(1 + i)2
. Generalized for any number of
interest periods N, this expression becomes
FN = P(1 + i)N
and (1 + i)N
is known as the compound-amount factor.
Nominal Interest Rates
Interest rates are normally quoted on an annual basis. However, agreements may specify that
interest will be compounded several times per year: monthly, quarterly, semiannually etc. For
example, 1 year divided into four quarters with interest at 2 percent per quarter is typically
quoted as 8 percent compounded quarterly. Stated in this fashion, the 8 percent rate is called a
nominal annual interest rate. The future value at the end of 1 year for $200 earning interest at 8
percent, compounded quarterly is developed as
F3 months = P + Pi = $200 + $200(0.02) = $204
F6 months = P + Pi = $204 + $ 204(0.02) = $208.08
F9 months = P + Pi = $208.08 + $208.08(0.02) = $212.24
F12 months = P + Pi = $212.24 + $212.24(0.02) = $216.48
The result of the nominal interest rate is to produce a higher value than might be expected from 8
percent compounded annually. At 8 percent compounded annually, the $200 mentioned above
would earn in 1 year
F12 months = P + Pi = $200 + $200(0.08) = $216
Which is $0.48 less than the amount accrued from the nominal rate of 8 percent compounded
quarterly. An interest rate of 1.5 percent per month is also a nominal interest rate that could
appear quite reasonable to the uninitiated. Using the compound-amount factor to calculate how
much would have to be repaid on a 1-year loan of $1000 at a nominal interest rate of 18 percent
compounded monthly (1.5 percent per period with 12 interest periods per year) gives
F12 months = $1000(1 + 0.015)12
= $1195.62
This can be compared with the future value of the same loan at 18 percent compounded
semiannually (9 percent per period with two interest periods per year):

Dr. Raza Ullah
29
F2 periods = $1000(1.09)2
= $1188.10
Thus, more frequent compounding with a nominally stated annual rate does indeed increase the
future worth.
Effective Interest Rate
The effective annual interst rate is simply the ratio of the interst charge for 1 year to the principal
(amount loaned or borrowed). For the 1-year loan of $1000 at a nominal annual interest rate of
18 percent compounded monthly,
Effective annual interest rate =
𝐹−𝑃
𝑃
=
$1196−$1000
$1000
=
$196
$1000
(100%) = 19.6%
For the same loan at 18 percent compounded semiannually,
Effective annual interest rate =
$1188−$1000
$1000
=
$188
$1000
(100%) = 18.8%
The effective annual interest rate can be obtained without reference to the principal amount.
Based on the same reasoning utilized previously and by using the following equation, the
effective annual interest rate ieff for a nominal interest rate r of 18 percent compounded
semiannually is
ieff = (1 +
𝑟
𝑚
)
𝑚
– 1 = (1 +
0.18
2
)
2
− 1
= (1 + 0.09)2
-1 = 1.188-1
= 0.188 or 18.8 %
This means that a nominal annual interest rate of 18 percent compounded semiannually is
equivalent to a compound-interest rate of 18.8 percent on an annual basis.
Continuous Compounding
The ultimate limit for the number of compounding periods in 1 year is called continuous
compounding. Under this accrual pattern, m approaches infinity as interest compounds
continuously, moment by moment. The effective interest rate for continuous compounding for a
nominal interest rate r is developed as follows;
The interest period are made infinitesimally small:
ieff = lim
𝑚→∞
(1 +
𝑟
𝑚
)
𝑚
- 1
The right side of the equality is rearranged to include r in the exponent.
(1 +
𝑟
𝑚
)
𝑚
– 1 = [(1 +
𝑟
𝑚
)
𝑚/𝑟
]
𝑟
− 1
The term inside the square brackets is recognized as the value of the mathematical symbol e [e =
2.718 is the value of (1 + 1/n)n
, as n approaches infinity]:
lim
𝑚→∞
(1 +
𝑟
𝑚
)
𝑚/𝑟
= e
By substitution,

Dr. Raza Ullah
30
𝑖∞ = lim
𝑚→∞
[(1 +
𝑟
𝑚
)
𝑚/𝑟
]
𝑟
− 1 = er
– 1
As an example of continuous compounding, when the nominal interest rate is r = 18.232 percent,
𝑖∞ = er
– 1 = e0.18232
-1= 0.20 or 20%.
and, correspondingly, when the effective annual interest rate is 22.1 percent,
0.221 = er
– 1
1.221 = er
r = in 1.221 = .20 = 20%
The most obvious computational effect of using continuous interest is that it produces a larger
future amount than does the same rate compounded discretely; the effective interest when r = 20
percent instead of i = 20 percent realizes a 10.5 percent increase in value:
0.221 − 0.20
0.20
(100%) = 10.5%
The rationale for using continuous interest in economic analyses is that the cash flow in certain
situations is best approximated by a continuous pattern; i.e., cash transactions tend to be spread
out over 1 year in a more or less even distribution rather than being concentrated at particular
dates. Some mathematical models are also facilitated by the assumption of continuous
compounding rather than periodic compounding.
In actual practice, however, interest rates are seldom quoted on a continuous basis, and the vast
majority of organizations use discrete compounding periods in their economic studies. The
reason for this is probably custom or the familiarity that makes it easier to understand periodic
interest as end-of-year values and financial experiences with annual tax, insurance, or mortgage
payments contribute to thinking in terms of discrete periods. Yet continuous and discrete
compounding are both only approximations of true cash flow, because each neither flows like a
free stream of water nor gushes like a geyser at given intervals. Receipts and disbursements are
often irregular in amount and in timing.
Conversion symbols
There are seven basic interest factors for discrete compounding. Each factor is described by a
name and a notational form where a functional symbol suggests the use of the interest factor, as
in (F/P, i%, N), again for the compound-amount factor that is used to find F given P, given a
specific interest rate i% and N. Time value conversion and associated factors are summarized in
table below
The symbols for the first six time-value conversions are abbreviations for the equivalent values
sought (future worth F, present worth P, or uniform series amount A) and the data given (F, P, or
A with its associated interest rate i% and number of compounding periods N). In the equation
F = $1000(F/P, 10, 2)
$1000 is the known present amount, the interest rate i is 10 percent, and F is the equivalent
future worth after two periods (N = 2). The whole symbol stands for the numerical expression (1
+ 0.10)2
.

Dr. Raza Ullah
31
Interest factors for discrete cash flow with end-of-period compounding
Factor To
find
Given Symbol Formula
Compound Amount F P (F/P, i%, N) F = P(1 + i)N
Present Worth P F (P/F, i%, N) P = F (1/(1 + i)N
Sinking Fund A F (A/F, i%, N)
A = F [i/{(1 + i)N
– 1}]
Series Compound amount F A (F/A, i%, N) F = A
(1 + i)𝑁−1
𝑖
Capital Recovery A P (A/P, i%, N) A = P [
𝑖(1+𝑖)𝑁
(1+𝑖)𝑁−1
]
Series Present Worth P A (P/A, i%, N) P = A [
(1+𝑖)𝑁−1
𝑖(1+𝑖)𝑁 ]
Arithmetic Gradient Conversion A G (A/G, i%, N) A = G [
1
𝑖
−
𝑁
(1+𝑖)𝑁−1
]
F = Future Worth, P = Present Worth, A = Annuity Amounts, G = Uniform change in Amount
The conversion descriptions and symbols indicate that certain factors are reciprocals of one
another;
(F/P, i%, N) =
1
(
𝑃
𝐹
, i%, N)
(A/F, i%, N) =
1
(
𝐹
𝐴
, i%, N)
(A/P, i%, N) =
1
(
𝑃
𝐴
, i%, N)
Other relationships are not so apparent from the abbreviations but are useful in understanding
conversion calculations. The following equalities will be verified during the development of the
conversion symbols;
(F/P, i, N) x (P/A, i, N) = (F/A, i, N)
(F/A, i, N) x (A/P, i, N) = (F/P, i, N)
(A/F, i, N) + i = (A/P, i, N)
Example 1: Find the future worth of $1500 invested in an enterprise for 2 years that yield 10%
profit.
F = ? P = $1500, N = 2, i = 0.10
F = $1,500 (F/P, i%, N)
F = F = P (1 + i)N
F = 1,500 (1 + 0.10)2
F = 1,500 (1.21)

Dr. Raza Ullah
32
F = $1,815
Example 2: Find the present value of a future amount $1000 to be received after 3 years keeping in view
the prevailing interest rate of 8 percent.
P = ? F = $1000 N = 3 i = 0.08
P = $1000 (P/F, i%, N)
P = F (1/(1 + i)N
P = 1000 (1/(1 + 0.08)3
P = 1000 (1/1.259712)
P = 1000 (0.7938)
P = $793.83
Example 3: Find the annuity A if a future amount of $5,866 is received after 5 years with the prevailing
interest rate of 8 percent.
A = ? F = $5,855 N = 5 i = 0.08
A = (A/F, i%, N) = A = F [i/{(1 + i)N
– 1}]
A = (A/F, 8, 5) = 5866 {0.08/(1 + 0.08)5
– 1}
A = 5866 (0.17046)
A = $1000
Compound Amount of a uniform series of Payments
Time of Payment (end of year) Amount A of payment, $ Future Worth at end of each Year,
$
1 1000 1000(1.08)4
= 1360
2 1000 1000(1.08)3
= 1260
3 1000 1000(1.08)2
= 1166
4 1000 1000(1.08)1
= 1080
5 1000 1000(1.08)0
= 1000
Annuity value F at the end of year 5 = $5866
Example 4: Find the future worth of the annuity composed of five annual payments of $1000, each
invested at 8 percent compounded annually.
F = ? A = $1000 N = 5 i = 0.08
F = $1000(F/A, 8, 5)
F = A
(1 + i)𝑁−1
𝑖
F = $1000 {(1 + 0.08)5
– 1}/ 0.08
F = $1000 (5.866)
F = $5866

Dr. Raza Ullah
33
Present Worth of a uniform series of payments
Time of Payment (end of year) Amount A of payment, $ Present Worth of payments at
beginning of year 1, $
1 1000 1000(1.08)-1
= 926
2 1000 1000(1.08)-2
= 857
3 1000 1000(1.08)-3
= 794
4 1000 1000(1.08)-4
= 735
5 1000 1000(1.08)-5
= 861
Present Worth P of 5-year annuity = $3993
Example 5: Determine the amount of each annual payment made for 5 years in order to repay a debt of
$3993 bearing 8 percent annual interest rate.
A = ? P = $3993 N = 5 i = 0.08
A = $3993(A/P, 8, 5)
A = P [
𝑖(1+𝑖)𝑁
(1+𝑖)𝑁−1
]
A = $3993[
0.08(1+0.08)5
(1+0.08)5−1
]
A = $3993 (0.25046)
A = $1000
Example 6: Find the present worth of five year-end payments of $1000 received by a firm at 8
percent interest rate.
P = ? A = $1000 N = 5 i = 0.08
P = $1000(P/A, 8, 5)
P = A [
(1+𝑖)𝑁−1
𝑖(1+𝑖)𝑁 ]
P = $1000 [
(1+0.08)5−1
0.08(1+0.08)5]
P = $1000 (3.99271)
P = $3992.71
Time Value of Money
The time value of money (TVM) is the idea that money available at the present time is worth
more than the same amount in the future due to its potential earning capacity. This core principle
of finance holds that, provided money can earn interest, any amount of money is worth more the
sooner it is received. TVM is also referred to as present discounted value. Money deposited in
a savings account earns a certain interest rate. Rational investors prefer to receive money today
rather than the same amount of money in the future because of money's potential to grow in

Dr. Raza Ullah
34
value over a given period of time. Money earning an interest rate is said to be compounding in
value.
Depending on the exact situation in question, the TVM formula may change slightly. The most
fundamental TVM formula takes into account the following
variables:
FV = Future value of money
PV = Present value of money
i = interest rate
n = number of compounding periods per year
t = number of years
Based on these variables, the formula for TVM is:
FV = PV x (1 + (i / n)) (n x t)
For example, assume a sum of $10,000 is invested for one year at 10% interest. The future value
of that money is:
FV = $10,000 x (1 + (10% / 1) (1 x 1)
= $11,000
The formula can also be rearranged to find the value of the future sum in present day dollars. For
example, the value of $5,000 one year from today, compounded at 7% interest, is:
PV = $5,000 / (1 + (7% / 1) (1 x 1)
= $4,673
The number of compounding periods can have a drastic effect on the TVM calculations. Taking
the $10,000 example above, if the number of compounding periods is increased to quarterly,
monthly or daily, the ending future value calculations are:
Quarterly Compounding: FV = $10,000 x (1 + (10% / 4) (4 x 1)
= $11,038
Monthly Compounding: FV = $10,000 x (1 + (10% / 12) (12 x 1)
= $11,047
Daily Compounding: FV = $10,000 x (1 + (10% / 365) (365 x 1)
= $11,052
This shows TVM depends not only on interest rate and time horizon, but how many times the
compounding calculations are computed each year.
Time-Value Equivalence
Two things are equivalent when they produce the same effect. The effective interest rate
computed for a nominally stated interest rate is an equivalent expression of the interest charge.
Both interest charges produce the same effect on an investment. In considering time-value
conversion, the equivalent numerical values of money are determined, not values with equivalent
purchasing power. The amount of goods that can be purchased with a given sum of money varies
up and down as a function of special localized circumstance and nationwide or worldwide
economic conditions.
If $1000 were sealed and buried today, it would have a cash value of $1000 when it was dug up
2 years from now. Regardless of changes in buying power, the value remains constant because
The method uses to know the
future value of a present amount is
known as Compounding. The
process of determining the present
value of the amount to be received
in the future is known
as Discounting.

Dr. Raza Ullah
35
the earning power of the money was forfeited. It was observe earlier that $1000 deposited at 10
percent interest compounded annually has a value of $1210 after 2 years. Therefore, $1000 today
is equivalent to $1210 in 2 years from now, if it earns at a prevailing rate of 10 percent
compounded yearly. Similarly, to have $1000 in 2 years from now, one need only deposit
$1000
1
(1 + 0.10)2
= $826.45
today. In theory, then, if 10 percent is an acceptable rate of return, an investor will be indifferent
between having $826.45 in hand or having a trusted promise to receive $1000 in 2 years.
Consider other facets of the earning power of money. The $1000 could have been used to pay
two annual $500 installments. The buried $1000 could be retrieved after 1 year, an instalment
paid, and the remaining $500 reburied until the second payment became due. if the $1000 is
instead deposited at 10 percent, $1100 would be available at the end of the first year. After the
first $500 installment was paid, the remaining $600 would draw interest until the next payment.
Paying the second $500 installment would leave
$600(1.10) - $500 = $660 - $500 = $160
in the account. Because of the earning power of money, the initial deposit could have been
reduced to $868 to pay out $500 at the end of each of the 2 years:
First year:
$500
1.10
= $455
Second year: $455 +
$500
(1.10)2 = $455 +
$500
1.21
= $868
The concept of equivalence is the cornerstone for time-value-of-money comparison. To have a
precise meaning, income and expenditures must be identified with time as well as with amount.
A decision between alternatives having receipts and disbursements spread over a period of time
is made by comparing the equivalent outcomes of the alternatives at a given date.
Cash Flow Diagrams
A cash flow diagram is a picture of a financial problem that shows all cash inflows and outflows
plotted along a horizontal time line. It can help you to visualize a financial problem and to
determine if it can be solved using TVM methods.
During the construction of a cash flow diagram it is usually advantageous to first define the time
frame over which cash flows occur. This establishes the horizontal scale, which is divided into
time periods, often in years. Receipts and disbursements are then located on the time scale in
adherence to problem specifications. Individual outlays or receipts are designated by vertical
lines; relative magnitudes can be suggested by the heights of the lines, but exact scaling wastes
time. Whether a cash flow is positive or negative (positive above the axis and negative below the
axis) depends on whose viewpoint is portrayed.

Dr. Raza Ullah
36
Figure 12a and b: Cash Flow Diagram
Figure 12a and b represents the same transaction: a loan paid off in three installments. From the
borrower’s viewpoint, the receipt of the loan is a positive inflow of cash, whereas subsequent
installment payments in the Figure 12a represent negative outflows. Flows are reversed when
viewed from the lender’s perspective in Figure 12b.
Although cash flow diagrams are simply graphical representations of income and outlay, they
should exhibit as much information as possible. It is useful to show the interest rate, and it may
be helpful to designate the unknown that must be solved for in a problem. Figure 12 is redrawn
in Figure 13 to represent specific problems. In Figure 13a, the three equal payments are
represented by a convenient convention that indicates a 3-year annuity in which the payment size
A is unknown. Amount A may be circled to indicate the needed solution. The given interest rate
(i = 10 percent) is entered in a conspicuous space, and the amount of the loan is placed on the
arrow at time 0. The style of a cash flow diagram is important only insofar as it contributes to
clarity.
Figure 13a and b: Different Versions of Cash Flow Diagram

Dr. Raza Ullah
37
In Figure 13b, numbered years are replace with dates, and the size of A is given. The problem is
to find the interest rate that makes the annuity equivalent to the loan value. Sometimes it may
clarify the situation to put in dashed lines for arrows that represent cash flows of unknown
magnitude. This or a similar tactic is especially useful when a problem comprises several
separate cash flow segments, each of which must be replaced by an equivalent value; these, in
turn, are converted to a single sum representing all the segments. The obvious requirements for
cash flow diagramming are completeness, accuracy, and legibility. The measure of a successful
diagram is that someone else can understand the problem fully from it.
Calculation of Time-Value Equivalences
Concepts concerning the time value of money will now become working tools. Compound-
interest factors for both discrete and continuous interest will be applied to a variety of cash
flows. The purpose of the calculations is to develop skills in converting cash flow patterns to
equivalent sums that are more useful in comparing investments. The use of cash flow diagrams
will be emphasized to portray receipts and disbursements associated with an economic situation.
Single-Payment Cash Flow
Translation of a future amount to its present worth, or the reverse from present to future, has
already been demonstrated for both discrete and continuous compounding. Sometimes both
present and future amounts are known, so the problem is to find the value of i or N that makes
them equivalent.
Example
At what annual interest rate will $1000 invested today be worth $2000 in 9 years?
Solution
Given P = $1000, F= $2000, and N = 9 years, find i.
𝐹
𝑃
= (F/P, i, 9)
$2000
$1000
= 2 = (F/P, i, 9)
= (1 + i)9
= 2
= i = 21/9
– 1= 0.08 or 8%
Present-Worth Analysis
Present-worth comparisons are made only between co-terminated proposals, to ensure equivalent
outcomes. Co-termination means that the lives of the involved assets end at the same time. When
assets have unequal lives, the time horizon for an analysis can be set by a common multiple of
asset lives or by a study period that ends with the disposal of all assets.
Conditions for Present-Worth Comparison
i. Cash flows are known: The accuracy of cash flow estimates is always suspect
because future developments cannot be anticipated completely. Transactions that
occur now, at time 0, should be accurate, but future flows become less distinct as the
time horizon is extended.

Dr. Raza Ullah
38
ii. Cash flows are in constant-value dollars: The buying power of money is assumed to
remain unchanged during the study period.
iii. The interest rate is known: Different interest rates have a significant effect on the
magnitude of the calculated present worth. The rate of return i required by an
organization is a function of its cost of capital, attitude toward risk, and investment
policy. Alternative courses of action for the same proposal are normally compared by
using the same interest rate, but different proposals may be evaluated at different
required rates of return.
iv. Comparison are made with before-tax cash flows: Inclusion of income taxes greatly
expands the calculation effort for a comparison and correspondingly increases reality.
v. Comparison do not include intangible considerations: intangibles are difficult-to-
quantify factors that pertain to a certain situation. For instance, the “impression”
created by a design is an intangible factor in evaluating that design and an important
one for marketing, but it would not be included in a present-worth comparison unless
its economic consequences could be reasonably estimated (if dollar value is assigned,
a factor is no longer intangible.).
vi. Comparison do not include consideration of the availability of funds to implement
alternatives: It is explicitly assumed that funds will be found to finance a course of
action if the benefits are large enough. Although financing is not a direct input for
computations, the output computed can be appraised with respect to available
funding. For instance, an old, inefficient machine could be kept in operation because
there appears to be insufficient capital available in the organization to afford a
replacement, but an engineering economic analysis might point out that the savings
from replacing the outdated machine would be so great that the organization could
not afford to not find funds for replacement.
Inflation and the constant dollar assumption
A house bought around 1990might have cost $18,000.
Today, that same house is selling for at least $80,000.
Inflation causes prices to rise and decreases the purchasing
power of money with the passage of time. A common way
to eliminate inflation effects is to convert all cash flows to
money units that have constant purchasing power, called
constant, or real, dollars. This approach is most suitable for
before-tax analysis, when all cash flow components inflate
at uniform rates. Another way to handle the situation is to
actually perform the analyses with the cash flows in the
estimated amount of money exchanged at the time of transaction. These money units are called
future or actual dollars.
In inflation, there is a percentage compounding effect over a period of years. If f is the inflation
rate over the next few years, actual dollars in year N can be converted to constant dollars by
Constant dollar =
𝐴𝑐𝑡𝑢𝑎𝑙 𝑑𝑜𝑙𝑙𝑎𝑟𝑠
(1+𝑓)𝑁
Since 1/(1 + i)N
is the single-payment PW factor, we can rewrite the conversion equation as
Inflation
A sustained increase in the general
level of prices for goods and
services. It is measured as an
annual percentage increase.
As inflation rises, every dollar you
own buys a smaller percentage of a
good or service.

Dr. Raza Ullah
39
Constant dollars = (actual dollars) (P/F, f, N)
The reverse occurs if we want to convert constant dollars to actual dollars at time period N:
Actual dollars = (constant dollars) (F/P, f, N)
Now we can show a cash flow series in actual dollars and the cash flow series after we have
removed the inflationary effects:
End of Year Actual Dollars 4% Annual inflation Constant dollars
0 -5000 x 1/1.040
= 5,000
1 1000 x 1/1.041
= 962
2 800 x 1/1.042
= 740
3 600 x 1/1.043
= 533
4 400 x 1/1.044
= 342
5 200 x 1/1.045
= 164
If we were to perform a present-worth analysis under our constant-dollars assumption, we would
use the rightmost column of cash flow data, not the second. The minimum attractive rate of
return (MARR) is used by firms to evaluate their economic investments includes the effect of
inflation. Use of this rate, referred to as the market interest rate, requires that all cash flows be in
actual dollars. The MARR used with constant dollars is the inflation-free interest rate that
represents the earning power of capital when inflation effects have been removed.
Basic Present-Worth Comparison Patterns
The present worth of a cash flow over time is its value today, usually represented by as time 0 in
a cash flow diagram. Two general patterns are apparent in present-worth calculations. These are
i) Present-worth equivalence and ii) net present worth.
i. Present-worth equivalence: One pattern determines the present-worth equivalence of
a series of future transactions. The purpose is to secure one figure that represents all
the transactions. This figure can then be compared with a corresponding figure that
represents transactions from a competing option, or it can be compared with the
option of doing nothing. Often there is a go/no go situation where each alterantive is
selectively weighed to decide whether it is worth exercising. For instance, a series of
expenses that will occur in the future can be discounted to obtain its PW, and then a
decision can be made about whether an investment of the PW amount should be made
now to avoid the expenses. Similar reasoning guides the equivalence comparison in
the following example.
Example: Equivalent PW of an Option
An investor can make three end-of-year payments of $15,000, which are expected to generate
receipts of $10,000 at the end of year 4 that will increase annually by $2500 for the following 4
years. If the investor can earn a rate of return of 10 percent on other 8-year investments, is this
alternative attractive?

Dr. Raza Ullah
40
From the cash flow diagram, it is apparent that the receipts and disbursements each constitute an
annuity, one positive and the other negative. Both are discounted to time 0 at 10 percent;
PW = -$15,000 (P/A, 10,3) + [$10,000 + $2500(A/G, 10, 5)] (P/A, 10, 5) (P/F, 10, 3)
= -$15,000(2.48685) + [$10,000 + $2500(1.81013)] (3.79079) (0.75131)
= -$37,303 + $41,369 = $4066
ii. Net present Worth
The second general pattern for PW calculations has an initial outlay at time 0 followed by a
series of receipts and disbursements. This is the most frequently encountered pattern, which
leads to the fundamental relation.
Net present worth = PW (benefits) – PW (costs)
The criterion for choosing between mutually exclusive alternatives is to select the one that
maximizes net present worth or simply the one that yields the larger positive PW. A negative PW
means that the alternative does not satisfy the rate-of-return requirement.
In addition to mutually exclusive alternative where we can select only one alternative, we could
have independent alternatives where more than one alternative may be selected. The criterion for
consideration of independent alternatives is that they should have a PW which is equal to or
greater than zero.
Example: Net present-worth comparison
Two devices are available to perform a necessary function for 3 years. The initial cost (negative)
for each device at time 0 and subsequent annual savings (positive), both in dollars are shown in
the following table. The required interest rate is 8 percent.
Year
0 1 2 3
Device A 9,000 4,500 4,500 4,500
Device B 14,500 6,000 6,000 8,000
Solution
<<Figure>>
PW (Device A) = -$9,000 + $4,500(P/A, 8, 3)
= - $9,000 + $4,500(2.5771) = $2597
PW (Device B) = -$14,500 + $6,000(P/A, 8, 2) + $8,000(P/F, 8, 3)
= -$14,500 + $6000(1.78326) + $8,000(0.79383)
= $2550

Dr. Raza Ullah
41
Both alternative meet the minimum acceptable rate of return, because both PW are positive, and
their net present worths are close in value. In this case, other considerations must be involved in
the choice, such as the availability of the extra $5,500 needed to purchase device B.
Rate of Returns
In financial analysis, return is a profit on an investment. The rate of return can be define in many
ways. i) The rate of return is a percentage that indicates the relative yield on different uses of
capital. ii) A rate of return is the gain or loss on an investment over a specified time period,
expressed as a percentage of the investment’s cost. iii) Rate of return is a profit on an investment
over a period of time, expressed as a proportion of the original investment. The time period is
typically a year, in which case the rate of return is referred to as annual return.

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