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Theories of costs

When a firm produces any commodity the services of different factor of production are required.

In order to secure the services of the factor of production, a firm makes payments of these

factors. The amount which is paid by a firm for this purpose is cost of production. A cost of

production is the amount of money paid out or otherwise, sacrifices by a firm in order to secure

the productive services with which output is produced.

Basic Concepts of theory cost

Fixed and variable costs

i. Fixed costs (FC) - costs that do not change as production is changed (increased
or decreased). They have to be paid in advance of production. They exist even if
output is zero.

Note “These costs remain unchanged as the output level of the firm changes. It does not
matter what level of output the firm produces (even zero output makes no difference), any
cost which is a fixed cost will remain the same. Common examples of fixed costs are as
follows” Rent, Interest on loans, Insurance, Depreciation

fig 1 cost

fc

Quantity

ii. Variable costs (VC) - costs that change/vary with output.

Note: “Any cost which varies directly with the level of output would be classified as a variable
cost. Varying directly means that the total variable cost will be dependent on the level of
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output”. Common examples of variable costs are as follows: Direct labour, Raw
materials and components, Packaging costs, heating and lighting

costs

vc

Quantity

Total, average and marginal cost

 Total costs (TC) - the sum of fixed costs and variable costs at a particular level of
output. So, TC = TFC + TVC.
 Marginal costs (MC) - the cost of one more unit of output. In other words, MC is
the increase in total cost from producing one more unit of output.
 Average costs (AC) - total costs divided by the level of output. There are three
aspects of average cost:
 Average total cost (ATC) which is total cost divided by the level of
output,
 Average fixed cost (AFC) which is total fixed cost divided by the level of
output and
 Average variable cost (AVC) which is total variable cost divided by the
level of output.

Average Fixed Cost Definition

The average fixed cost (AFC): is the fixed cost that does not change with the change in the

number of goods and services produced by a company. The average fixed cost (AFC) is the fixed

cost per unit and is calculated by dividing the total fixed cost by the output level. Therefore, in a

short run when the units of production increase, the average fixed cost per unit decreases:

likewise, when the firm produces less units, the average cost increases per unit. Examples of
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average fixed cost are the salaries of permanent employees, the mortgage payment on machinery

and plant, rent, and more.

The average fixed cost (AFC) curve is a rectangular hyperbola. The area under the

curve is constant because TFC is constant at all levels of output.

Average cost curve shape

Average cost curves are typically U-shaped.

Average total cost starts relatively high, because at low levels of output total costs are dominated by the

fixed cost; the denominator (output) is so small that average total cost is large. Average total cost then

declines, as the fixed costs are spread over an increasing quantity of output. But as output expands still

further, the average cost begins to rise. At the right side of the average cost curve, total costs begin rising

more rapidly as diminishing returns gain momentum as well explained by the law of variable proportions

(law of diminishing return)

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Figure “a” stands for a short run average cost while figure “b” stands for long run average cost

Average variable cost

Average variable cost (AVC) is calculated by dividing variable cost by the quantity produced.

The average variable cost curve lies below the average total cost curve and is typically U-

shaped or upward-sloping

Opportunity/alternative cost: this cost is the cost lost due to scarcity of the resources; this is an

opportunity cost or alternative cost. We know that resources available to a person, firm or

society are scarce but have alternative uses with different returns. Income maximizing resources

owners put their scarce resources to their most productive use and thus, they forego the income

expected from the second-best use of the resource.

The opportunity cost may be defined as the returns expected from the second-best use of the

resource that are foregone due to the scarcity of the resources.

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For more elaboration of the opportunity cost, suppose a firm has a sum of TZS 100,000 for

which it has two alternative uses. It can buy either a printing machine or photocopy machine

both having a life span of 10 years. From the printing machine the firm expects the annual

income of TZS 20,000 and from the photocopy machine the firm expect the annual income of

TZS 15,000.

The investing firm will invest its money in printing machine and foregoe the income from

photocopy machine. Therefore, the opportunity cost of the printing machine is TZS 15,000

which is the income expected from photocopy.

Economic rent/economic profit

Associated with the opportunity cost is the concept of economic rent/ economic profit. From the

example given above the economic rent/profit of the printing machine is the excess of its

earnings over the income expected from photocopy. Then the economic rent of the printing

machine will be TZS 5000. The implication of this is that investing in printing machine is

preferable so long as the economic rent of the business is greater than zero.

Explicit vs Implicit costs

The explicit/actual cost are those costs which are actually incurred by a firm in items like

payment of labour, materials, plant, building, machinery, equipment, travelling and transport,

advertisement etc. These costs are recorded for accounts purposes.

The implicit costs are the costs which do not take form of the cash outlays. Nor do appear in

accounting systems. Such costs are known as implicit/ imputed costs. Opportunity cost is an
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important example of the implicit costs. For example, if an entrepreneur doesn’t spend his

service in his own business and works as a manager in some other firm on a salary basis. If set

up his business, he foregoes his salary as a manager. This loss of salary as a manager is the

opportunity cost of income from his own business and is the implicit cost of his own business.

Note: Implicit cost is not taken into accounts while calculating loss and gains but they are of

important considerations in deciding whether or not to retain a factor in its present use.

TRANSACTION COST

Transaction cost is the expense one incurs by engaging in economic exchange of any kind. They

represent the trade expenses that one needs to cover to aid the exchange of goods and services in

a market. Examples of typical transaction costs are labour-cost, transportation, broker fees, bank

charges, commissions.

Types of transaction costs

There are three main types of transaction costs. Let us explore each with the help of an
example. Suppose a person is trying to buy a car. Here the transaction costs include

Search Costs: These are the costs of searching for information on sellers or availing services
that can help a person find the best commodity. Some examples are

 Communication and
 Consulting fees,
 Advertising expenses,
 Travel expenses, etc.

The individual trying to buy a car pays a car dealer agent to help them find the perfect property.
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Bargaining costs: These are the costs related to negotiation and conflicting interests regarding
trading until the transacting parties agree. An individual has to spend time and labor looking at
different cars and negotiating the price with the owners. Here they are incurring bargaining costs.

Enforcement costs are the expenses one incurs to verify and ensure that the parties involved in a
contract comply with its terms. It relates to the time, money, and effort taken by a person to
ensure that everyone meets the terms of the agreement. The person finally planning to purchase
hires a lawyer to draw up the paperwork for sale. In this case, they are paying for the
enforcement costs.

Types of transaction cost

SEARCH COST: BARGANIG COSTS: ENFORCEMENT COST:

Cost incurred for searching Costs associated with Costs related to ensuring
and gathering information, negotiations under two all parties comply to the
recognizing potential parties, settle on final contract and policies
sources of supply agreement

How transaction costs work (hidden cost)

To buy or sell any commodity in the market, a person must find a seller/dealer who can provide
the commodity. The person then negotiates the price they are willing to buy or sell the
commodity, leading to a bargain. They finally agree on a price that is acceptable to both parties.
The person will then need to draw up a contract for the sale and hire legal help to ensure that all
parties abide by the contract terms. Here, the transaction cost includes the expense for availing of
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the dealer's and the lawyer's services. It is an additional charge the buyer has to pay besides the
commodity’s price. As such, these costs inflate the total cost involved in a purchase. They are
also hidden costs because one often can’t see the results of these expenses in terms of a physical
product or service.

“Hence, one can calculate the transactional cost of a particular purchase as the
difference between the price a buyer paid and the seller received”

The cost-output relations

Depending on whether cost analysis pertains to short run or long run, there are two kinds of

costs. i) Short run cost function ii) long run cost function

Short run cost-output relations: the basic analytical tools used in the analysis of cost behavior

are Total, Average and Marginal cost.

The short run TC is composed of two elements which TFC and TVC

Thus, i) TC = TFC + TVC

TC
ii) AC = ATC =
Q
TFC TVC
AC = +
Q Q
AC= AFC + AVC
δ TC
iii) MC=
δQ
δ TFC δ TVC
MC = +
δQ δQ
δ TFC
But = 0
δQ
δ TVC
Therefore, MC =
δQ
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Linear cost functions: a linear cost function takes the following form.

TC= bQ + a

WHERE, TC= total cost, Q = UNITS OF OUTPUTS, a = TFC, and b = change in TVC

TC b Q a
AC = = +
Q Q Q

a
AC = b + Q

δTC
MC =
δQ

MC = b

Quadratic cost function:

TC = a + bQ + Q2

AC = TC/Q

a/Q + bQ/ Q + Q2/Q

AC = a/Q + b + Q

MC = δTC/δQ

δ a/δQ + δbQ/δQ +δQ2/δQ

MC = b + 2Q

Cubic function: TC = a +bQ + cQ2 + Q3

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AC = a/Q + bQ/Q +CQ2/Q + Q3/Q

AC = a/Q + b + cQ + Q2

MC = δTC/Q

δ a/δQ + δbQ/δQ+ δcQ2/δQ+δQ3/δQ

MC = b+ 2cQ+3Q2

Seminar: with mathematical illustration show how that MP is inversely proportional to MC

Cost minimization: given TC = 8 + Q2 + Q

Find level of output that minimize AVC

NOTE: TFC = 8 and

TVC = Q + Q2

Then the AFC will be 8/Q

2
Q +Q
AVC =
Q
AVC = Q + 1

The critical values of AVC is one that will minimize AVC and this is accomplished by

differentiating the AVC equation with respect to Q and equate to zero as follows:

Critical values of Q = δAVC/δQ = 0

δ (6 -0.9Q + 0.05Q2)/ δQ

0.1Q -0.9 = 0

critical value of Q = 9.

This is the level of output that minimizes the AVC of the firm.
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Minimization of AC: The objective of the businesses firms is to minimize AC of their product.

The level of output that minimizes AC can be obtaining by differentiating the AC equation.

Given the TC = 10 + 6Q -0.9Q2 + 0.05Q3

Then the AC = 10/Q + 6 - 0.9Q + 0.05Q2

δ AC/ δ Q = δ(10/Q + 6 -0.9Q + 0.05Q2)/δQ

-10/Q2 -0.9 +0.1Q

Multiply by Q2 throughout

-10 -0.9Q2 +0.1Q3

Find the critical values of Q By differentiating and equating the equation to zero

-10 -0.9Q2 +0.1Q3= 0

For simplicity of the equation multiply by 10 throughout

Q3 -9Q2 - 100 = 0

Then solve the above equation quadratically or factorization method to have value of Q.

Then the critical value of Q = 10

Then you can calculate the minimum AC substitution the value of Q

AC = 10/Q + 6 - 0.9Q + 0.05Q2 and Q = 10

10
= + 6 – (0.9*10) + (0.05*102)
10
= 1 + 6 – 9 + 0.05 * 100
=7–9+5
= 3 Units
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Output optimization in the SR:

In economics optimizing behavior is, selecting the best out of the available options with the

objective of maximizing gains from given resources. Thus, economics is thus a social science,

which study human behavior in relation to optimizing allocation of available resources to

achieve the given ends/results.

Let us suppose a short run cost function is given as

TC = 200+5Q+2Q2

NOTE: The level of output is optimized at a level of production at which MC = AC

AC = 200/Q +5 +2Q

MC = 5 +4Q

Equate MC = AC

200/Q + 5 + 2Q = 5 +4Q

200/Q =2Q

Multiply by Q throughout

200 = 2Q2

divide by 2 both side

100 =Q2 then the optimum value of Q = 10

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The cost of production include the following elements

 Wages of labour

 Rent paid for land

 Interest for capital

 Cost for raw materials

 Repair charges

 Publicity expenses etc

Production cost may also be categorized into fixed cost and variable costs. Fixes cost is the cost

which doesn’t vary with the level of production. Whether the production output is zero the fixed

cost remains the same. While variable cost is that cost which vary with the level of production.
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Short run and long run cost function curves

The relationship between Average cost and Marginal cost

 When the average cost is falling the marginal cost is less than the average cost

 When the average cost is minimum the marginal cost is equal to average cost

 When the average cost is rising, the marginal cost is greater than the average cost

Total cost

SMC Curve

SAC curve

Output

During short run we can change the quantity of variable factors only and fixed factors remain the

same. On other hand, during long run period, no factor remains fixed. The short run AC curve

and MC curve are U-shaped, because when firms produces more and more units of commodities,

in the beginning the AC goes down then reaches minimum point and beyond that it begins to

rise.

As regards to long run curves, these are flatter than the short run curves.it is stated that in the

long run all factors are variable and it is possible to change the quantity of all factors according

to the circumstances. During LR fixed costs can be varied to the considerable extent. In the SR

reduces output rises the AC because fixed costs remain the same. In the LR period, FC can be
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reduced so as AFC may be lowered in the LR than in the SR thus why the LR cost curve is flatter

than SR cost curve.

cost

LMC

LAC

Output

Another reason for flatter long run average cost curve is the fact that factors of production can be

divided into smaller parts. if all factors of production can be used in varying proportions, it

means the scale of operations of the firm can be changed.

SAC4

COST SMC1 SAC1 SMC2 SAC2 SAC3 LAC

SMC3 SMC4

0 OUTPUT
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Exercises on production and cost theories.

Question one: (5 Marks)

(a) i) Why is short run average cost curve is U-Shaped?
ii) Illustrate and explain the relationship between SAC and SMC.
(b) Suppose a cost function is given as TC = 100+5Q +Q2. find
i) equations for AC and MC
ii) AC and MC for 10 units of outputs
iii) the value of Q at which AC=MC
Question two: (3 Marks)

Suppose cost function of Kisasa-Dodoma super Mills is given as follows

TC = 500+ 1OQ+5Q2

a) Find the output that minimizes average cost

b) Suppose the firm is producing only 8 units of output. Should the firm increase or decrease
the output so as to minimize AC.
c) What is the optimum level of output

Question three: fill the following table accordingly (3 Marks)

Output Q TFC =60 TVC = 10Q TC AC MC

1 60 10 70 70.00 -
2 60 20 80 40.00 10
3 60 30 90 30.00 10
4 60 40 100 25.00 10
5 60 50 110 22.00 10
6 60 60 120 20.00 10
7 60 70 130 18.57 10
8 60 80 140 17.50 10
9 60 90 150 16.67 10
10 60 100 160 16.00 10

Question four: (3 Marks)

Suppose a short run production function is given as follows:

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Q = 24L2 - 2L3. Where Q = output, L = variable inputs; find the following

a) Marginal product function

b) Average product function
c) the value of L that maximizes Q

Question five: (3 Marks)

suppose a short run production function is given as follows:

Q = 10L +15L2-L3 where: Q=Output, L=labour employed per unit of time

i) Derive the MPL and APL functions

ii) Find the outputs at which APL = MPL
iii) Find L for producing 600 units of Q

Question six: (3 Marks)

a) What is the law of diminishing return?

b) With the aid of the diagrams explain three categories return to scale and their causative
reasons

c) Clearly show the relevant return to scale revealed by the following production functions

i) Q = L 0.7 K 0.3 ii) Q = L 0.4 K 0.3 ii) Q = L 0.7 K 0.5

d) What is the main reason for Long-run average total cost curves, U-shape?

Question 7: Differentiate the following functions (15 Marks)

1. s = 5 t3 + 4t 2
2. f(x)=4 x 3-18 x 2-4x + 33
3. f(x)= x 3−4 x 2−11x−190
4. f(x)= (x ¿¿ 2+25)¿ x
6
5. f(x)= x

2
6. f(x)= 2
x +¿
x
2
7. f(x)= 2 x +3
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3
4L
8. TP = 5
- 2 L2
1 5
9. y = 3x
3 - 2x

10. (x – 1) ( x 2+ x+1 ¿
11. ( x 2−5 x ¿ x
2

2 x +5
12. z= 3 x −2

2 x +1
13. z= 2
x −1
9
14.y= x 4

( ) ( x− 1x +3 x )
1
15.y= x + x
2