Download Production Functions

Definitions and Basic Concepts
• Managers make resource allocation decisions
about:
– Production operations
– Marketing
– Financing
– Human Resource
• Production decisions determine the type and
amount of inputs such as land, labour, raw and
processed materials, factories, machinery,
equipment and managerial talent – to be used in
the production of a desired quantity of output.
• The objective is to minimize cost for a given
output or to maximize output for a given input
budget
• In microeconomics, production is the
economic process by which inputs are
transformed or converted into outputs
• Production uses resources to create goods and
services that are suitable for exchange
INPUTS
(Resources)
PROCESS
(Inputs c verted/
transf med)
(Suitable for
exchange)
• Creation of wealth which adds to society’s
welfare
• Some economist define production broadly to
be all economic activity other than
consumption
• The vital link in the process of satisfying wants which
are almost unlimited relative to resources available
• It is therefore important that the production process
utilises the limited resources efficiently in order to
create maximum possible welfare
• Subject to production possibilities the basic questions
considered are what, how and for whom to produce
• Some production decisions are left to the
private sector (firms) and the market
mechanism while others are taken up by the
government.
• The total level of output in an economy is the
sum of the output of all the individual firms
• A production process can be defined as any
activity that increases the similarity between the
pattern of demand for goods and services, and
the quantity, form, and distribution of these
goods and services available to the market place.
• Inputs and outputs of the production process
may be either intangible (service) or tangible
(goods)
• Primary sector
– Includes that part of an economy whose activities
are directly related to natural resources such as
agriculture and fishing
• Secondary sector
– Encompasses that part of an economy whose
activities are concerned with construction, energy
and manufacturing including the processing of the
output of the primary sector
• Tertiary sector
– Includes that part of an economy whose activities
are concerned with the provision of services such
as banking, insurance and tourism
• INPUTS
– Factors of production
– Anything that a firm buys for use in its production or other
process
– Any good or service that is used in the process of
production
• OUTPUT
– Any commodity which a firm produces or processes for
sale
– Any good or service that comes out of pro duction process
Fixed Inputs
• The quantity employed in the production
process remains constant over a given period
of time regardless of the quantity of output
produced.
• The cost will be paid whether production is
operated at a high or low rate of output
– Plant, buildings, machinery
• The supply of fixed inputs remains inelastic in
the short run
Variable Inputs
• Input whose quantity employed in the
production process changes depending on
desired quantity of output
– Labour, materials
• The supply of variable inputs is elastic in the
short run
Short Run
• A period of time in which the supply of certain
factors of production is fixed/inelastic
• Production of commodity can be increased by
increasing the variable inputs such as labour and
raw materials
• Only a subset of the total input combinations is
available to the firm because some of the inputs
are fixed
• Short run does not refer to any fixed time period
Long Run
• A period of time when the supply of all factors
of production is elastic but not long enough to
permit a change in technology
• Production of commodity can be increased by
employing more of both variable and fixed
inputs
• All possible input combinations are available
to the firm
Very Long Run
• Refers to a period of time in which the
technology of production is subjected to change.
• The production function changes
• The technological advances means that a larger
output can be created with a given quantity of
input
Factors of Production
Factors of Production
• Resources employed to produce goods and
services
• Used to create value and allow economic
performance
Factors of Production
• Scarcity of factors of production poses
humanity economic problems leading to
choice between competing goals
• Economist typically place the many different
factor inputs into four categories viz land,
labour, capital and enterprise
Land
• Includes minerals, water, forests and all natural
resources that are used in production
• Space occupied to carry out any production
process
– Factory space, office
• Basic resources within land, sea or air which can
be extracted for productive use
– Metal ores, coal, oil
Peculiarities of Land
•
•
•
•
Nature’s gift to man
Fixed in quantity (has no supply price)
Permanent
Infinite variation of degree of fertility and
situation so that no two pieces of land are exactly
alike
• Should land be regarded as a form of capital?
Labour
• All human attributes, physical and mental,
that are used in production
• Labour is not homogeneous
• Trained labour is sometimes referred to as
“human capital”
• Some economist argue that labour is the
ultimate production factor since nothing
happens without the intervention of labour
Peculiarities of Labour
•
•
•
•
Labour is inseparable from the labourer himself
Labour has to sell his labour in person
Labour is perishable – no reserve price
Changes in price of labour reacts rather curiously
on its supply
• Should labour be treated differently from a
commodity?
Capital
• Goods which are not for current consumption
but which will assist consumer goods to be
produced in the future
• Physical Capital
– Factory buildings, tools, equipment and machines
Capital
• Human Capital
– Accumulated skill, knowledge and experience
without which physical capital cannot achieve its
full productive potential
• Financial Capital
– Funds usually needed to acquire and develop
physical and human capital
Enterprise
• An entrepreneur is a person who organizes
the production of goods and services.
• Co-ordinates and correlates the other factors
of production
• Should an entrepreneur be part of human
capital?
Production Theory
• Production Theory seeks to answer the
following questions: – What effect will changes in the inputs have on the
outputs
– How is optimum technique of production chosen
Production Theory
• Most of production theory is based upon
efficiency i.e.
– Producing the maximum output possible with a
given level of input usage
– Producing a given level of output at the lowest
possible price
Discussion Question
• State the four factors of production and
discuss the distinctive features of each of the
main factors of production?
Production Functions
Production Function
• Analysis tool to explain the input-output
relationship i.e. it is the link between levels of
input usage and attainable output levels
• The relationship between the quantities of
inputs used and the maximum quantity of
output produced, given current knowledge
about technology and organisatio n
Production Function
• With a given state of technology, the attainable
quantity of output depends on the quantities of
the various inputs employed in production
• A schedule (graph or table or mathematical
equation) that specifies the maximum feasible
quantity of a commodity that can be produced
from any specified set of inputs given the existing
technology
Production Function
• Many different inputs are used in production.
In the most general case, maximum out, Q,
can be defined as a function of the level of
usage of the various inputs, X
Q = f (X1, X2,…, Xn)
Production Function
• Assuming production output (Q) is a function
of labour (L), land (LB), capital (K), time (T) and
managerial efficiency (M), the production
function would be expressed as: Q=f(L, LB, K, T, M)
Production Function
• The simplified algebraic production function
using two inputs of Labour and capital
Q= f(L, K)
Where L = Labour and K = Capital
Production Function
• Increasing production (Q) would require
increasing input K and or L
• Whether both K and L can be increased
depends on planning horizon i.e. whether
short run or long run
Production Function
• In the short run therefore a firm can increase
its production by increasing only labour since
the supply of capital is fixed
• In the long run the firm can employ more of
both labour and capital.
Production Function
• The firm will therefore have two production
functions: -
• Short run production function (Single Variable
Production Function)
Q = f(L)
• Long run production function
Q = f(K,L)
• Production is more efficient when the same
output results from less input.
• Some economists (e.g. Leibenstein) argue that
production processes tend to be inherently
inefficient due to satisficing behaviour.
Production Efficiency
• Technical Efficiency
– When a firm produces the maximum level of
output that can be obtained from a given
combination of inputs
– It is impossible to obtain higher levels of output
given existing technology
Production Efficiency
• Economic Efficiency
– Using the combination of inputs that permits a
firm to produce a given amount of output at the
lowest possible cost
– Economic ally efficient method of producing a
given level of output depends on the prices of the
inputs
Assumptions of Production Function
• Perfect divisibility of both inputs and output
• Limited substitution of one factor for another
• Constant technology
Assumptions of Production Function
• Inelastic supply of fixed factors in the shortrun
• Technical efficiency
Production in the Short Run
Q = f (L)
Production in the Short Run
• The short run production function is written
as:
Q = f(L)
Short Run Production Function Graph
Output (Q)
TPL
L
Labour (L)
Single variable Production Function
Total Product
• The curve gives the maximum amount of
output that can be produced by each amount
of the variable input (labour) when combined
with the fixed input (capital)
• See illustration
Marginal Product
• MP is the addition to the total product
resulting from a very small change in the
variable input
• The laws of returns are concerned with the
relation between marginal change in input
and the resulting marginal change in output
Marginal Product
• Marginal Product (MPL ) of labour (the variable
factor) may be defined as the change in
output (Q) resulting from a very small change
(∂L) in labour employed, other factors held
constant
• MPL = ∂Q/∂L
Marginal Product
• The incremental change in total output that
can be obtained from the use of one more
unit of an input in the production process
while holding constant all other inputs
Average (Physical)Product
• The ratio of total output to the amount of the
variable input used in producing the output
•
• APL = Q/L = f(L)/L
Product Curves
• Total Product Curve
– No output can be produced with zero workers
– Total product/Output first increases at an
increasing rate when MP is increasing
– Total product then increases at a decreasing rate
when MP is decreasing
– The maximum possible total product is reached
beyond which output will decline indicating a
negative MP.
Product Curves
• Average and Marginal Product Curve
– When MP is greater than AP, AP is increasing
– When MP is less than AP, AP is falling
– When AP is at its maximum i.e. neither rising nor
falling, MP equals AP
Laws of Production
• The traditional theory of production studies
the marginal input-output relationships under
– (i) Short Run
– (ii) Long – Run
Laws of Production
• Short run input-output relations are studied
with one variable while other inputs are held
constant. The laws of production under these
assumptions are called The Laws of Returns to
Variable Proportions
Laws of Production
• In the Long run the input – output relations
are studied assuming all the inputs to be
variable. The input-output relations are
studies under The Laws of Return to Scale
Laws of Variable Proportions
• In the short run, firms can employ only a
limited or fixed quantity of fixed factors and
unlimited quantity of variable factors
• Firms can employ in the short run varying
quantities of variable inputs against a given
quantity of fixed factors
• This kind of change in input combinations
leads to variation in factor propor tions
Laws of Variable Proportions
• Also known as Law of Diminishing Marginal
Returns
– If more and more units of a variable input are
applied to a given quantity of fixed inputs, the
total output may initially increase at an increasing
rate, but beyond a certain level, output increases
at a diminishing rate.
Laws of Variable Proportions
• Marginal increase in total output eventually
decreases when additional units of variable
factors are applied to a given quantity of the
fixed factors
Diminishing Returns
• 1. Diminishing Total returns
• Implies reduction in total product with every additional unitof input.
• 2. Diminishing Average returns
• Refers to the portion of the AP curve after its intersection with MP curve.
• 3. Diminishing Marginal returns (DMR)
• The point where the MP curve starts to slope down and travels all theway
down to the x-axis and beyond.
• Putting it in a chronological order, at first the marginal returns start
to diminish, then the average returns, followed finally by the total
returns.
Exercise
• Assume the labour (L)- output (Q) relationship
is given by the following production function: – Q = 10L2 + 20L1 – L3
• Find the following: – TP, MP and AP schedules
– TP where MP = AP
– Labour (L) required to maximize output
Three Stages in Production
• Stage 1
– MPL continues to increase making TPL increase at
an increasing rate
– Increasing Returns due to: • Underutilization of the fixed factor i.e. capital
• Advantages of division of labour
Three Stages in Production
• Stage 2
– MPL starts falling so that TPL increases at a decreasing rate
– Diminishing marginal returns is due to : • Decreasing labour-capital ratio
– Each additional worker has less and less tools and equipment to work with
and eventually the productivity of the marginal worker decreases
• Limited number of ways to achieve labour specialization
• Crowding effect
• Stage 3
– MPL is negative and TPL falls
– Some work is just more difficult to accomplish when
superfluous personnel are present
Assumptions of DMR
• State technology does not change in the
course of application of the law
• Input prices remain unchanged
• Variable factors are homogeneous
Short-Run Cost of Production
C = f (Q)
Costs of Production in Short Run
Opportunity Cost
(What Owners give up to use
a resource)
Explicit Opportunity Cost
(Resources owned by others)
Implicit Opportunity Cost
(Resources owned by the Firm)
Costs of Production in Short Run
• Explicit Costs
– Monetary payment made by a firm for the use of
an input owned or controlled by other individuals
– Also referred to as accounting costs
• Implicit Costs
– The foregone return the firm’s owners could have
received had they used their own resources in
their best alternative use
Costs of Production in Short Run
• The total cost of using resources for
production is the sum of all explicit costs and
all implicit costs.
• The total cost (explicit and implicit costs) must
be considered when making decisions
Practice Question
• Until recently you worked as an Administrator
earning USD30,000 annually. Then you inherit a
piece of commercial real estate bringing in
USD12,000 in rent annually. You decided to leave
your job and operate a video rental store in the
office space you inherited. At the end of the first
year, your books showed total revenue of
USD60,000 and total cost of USD30,000 for video
purchases, utilities, taxes and supplies.
– What is your economic profit for the ye ar?
Short Run Total Costs
• Total Fixed Costs (TFC)
– Amounts paid for fixed inputs
– Do not vary with output
• Total Variable Costs (TVC)
– Amounts paid for variable inputs
– Vary with variations in output e.g. increase with
increase in output
• Total Cost (TC) = TFC + TVC
Average Costs
• Average Fixed Cost (AFC)
AFC = TFC/Q
– AFC approaches zero as output increases
– Irrelevant for decision making
• Average Variable Cost (AVC)
AVC = TVC/Q
– AVC falls then increases
• Average Total Cost (ATC) = TC/Q or AVC + AFC
Short Run Marginal Costs (SMC)
• Marginal cost is the incremental increase in total
cost that results from a one-unit increase in
output. In other words, it is the cost of marginal
unit produced.
• SMC – the change in total cost divided by the
change in total output
SMC = ∆TC = ∆TVC
∆Q
∆Q
Exercise
• Suppose fixed costs for a company S are equal
to $200, and the company’s variable costs are
given by the following relationship (where Q =
output):– VC = 60Q – 3Q2 + 0.10Q3
Derive the following cost functions
• TC
• ATC
Costs Function
• Cost-output relations are expressed through a
Cost Function
• Cost function is a mathematical model,
schedule, or graph that shows costs of
producing various quantities of output
Costs Function
• Cost Functions depend on: – Production function (specifies the technical
input-combination and its relation to the
output)
– Market-Supply function of inputs
• Production function combined with supply
function of inputs or prices of inp uts
determine the Cost function of th e firm
Cost Function
C = f(Q, T; Pf, K)
Where C = Total Cos
Q = Quantity Produced
T = Technology
Pf = Factor Price
K = Capital (the fixed factor)
Short Run Cost Function
• Since in the short run all determinants of cost
other than Q are constant, the short run cost
function may be specified as: C = f(Q)
Relations Between Short Run Costs
and Production
• TVC is derived directly from the short run production
function (Total Product Curve - TPC)
• AVC derived from AP
• SMC derived from MP

When MP (AP) is increasing SMC (AVC) is decreasing and when
MP (AP) is decreasing SMC (AVC) is increasing
• See illustration
Relations Between Short Run Costs
and Production
• As output (Q) increases: – TFC remains constant (by assumption)
– TVC increases but at varying rates. It
increases at a decreasing rate initially and
then at an increasing rate
– TC increases first at a decreasing rate then
at an increasing rate
Relations Between Short Run Costs
and Production
• As output (Q) increases: – AFC decreases continuously. AFC would
approach zero if Q became extremely large
– AVC initially decrease then increases
– SMC initially decreases then increases. SMC
equals AVC and ATC at their respective
minimum levels
– ATC becomes increasingly close to AVC
Decision Making in the Short Run
• Short-run cost-output relationships help
managers to plan the most profitable level of
output given the resources that are
immediately available
• Decisions points: – Whether to produce or not
– Optimal level of output
Optimum Output and Cost Curves
• Optimum level of output is one which can be
produced at a minimum average cost, given the
technology.
• The minimum ATC is determined by the point of
intersection between ATC and SMC curves
• Optimum level of output is not necessarily the
maximum-profit output. Profits cannot be known
unless firm’s revenue curves are kno wn
• The optimal level of output is that level of
output that maximises the objective function
– Profit = TR - TC
• Increase output as long as MR (P) > MC
• Decrease output as long as MR (P) < MC
• Optimal production is therefore where
MR(P)=MC
• A firm should produce a positive output and
suffer a loss only if that loss is smaller than the
firm would incur by producing nothing (loss
minimizing output level)
• When a firm earns a positive economic profit, it is
earning more than a normal rate of return
Discussion Question
• “When a manager is using a technically
efficient input combination, the firm is also
producing in an economically efficient
manner.” Evaluate the statement (20 marks)
PRODUCTION AND COSTS IN THE
LONG RUN
Production Theory in the Long Run
• In the long run the supply of all inputs is
elastic i.e. all factors of production are
variable
• A firm is able to employ larger quantities of all
inputs which increases the scale of production
• Long run Production Theory is also called Laws
of Returns to Scale
Production Function in the Long Run
• Given that all inputs are variable (elastic) in the
long run, the production function in the long run
(assuming only 2 inputs) and given the
production technology, is given as: Q = f(K,L)
Where Q = Output
K = Capital
L = Labour
Production in the Long Run
• Objectives when no inputs are fixed: – output maximization
– cost minimization
• Important tools of analysis when two inputs
are variable: – Production Isoquant
– Isocosts
Production Isoquant
• Also called Equal Product Curve or Production
Indifference Curve
• A curve (locus of points) showing all possible
combinations of the inputs physically capable
of producing a given (fixed) level of output
• See Table
Isoquant Map
C
Q3
a
d
Q2
Q1
L
• Each isoquant (Q1,
Q2, Q3) shows
different
combinations of
labour and capital
which when used
efficiently can
produce a given
level of output
Production Isoquant
• Isoquant concept implies that it is possible to
substitute some amount of one input for
some of the other, say labour for capital, while
keeping output constant
• Each point on the isoquant is technically
efficient i.e. the maximum possible output is
that associated with the given iso quant
Characteristics of Isoquants
• Isoquants cannot intersect or be tangent to
each other ; – each capital – labour
combination can be on only one isoquant
• Input combinations other than those on a
given isoquant can be used to produce the
given level of output, but such a combination
would not reflect the maximum-amount-ofoutput
Characteristics of Isoquants
• Isoquants have a negative slope over the relevant
range (economic region/ profit maximization
region). This indicates that if a firm decreases the
amount of capital employed, more labour must
be added in order to keep the rate of output
constant
• For any combination along the isoquant, if usage
level of either input is reduced and the other is
held constant, output must decline
Characteristics of Isoquants
• Isoquants are convex to the origin implying
the diminishing marginal rate of technical
substitution.
• A group of isoquants is called an isoquant map
• On an isoquant map all isoquants lying above
and to the right of a given isoquant indicate
higher levels of output
Marginal Rate of Technical Substitution
• The rate at which one marginal unit of input is
substituted for another along an isoquant
(without changing the level of output)
MRTS = -∆K/ ∆L
• MRTS decreases because by assumption no
factor is a perfect substitution for another
Marginal Rate of Technical Substitution
• MRTS diminishes over the relevant range of
production
• MRTS declines along an isoquant as labour
increases and capital decreases
Laws of Return to Scale
• Explain the simultaneous and proportionate
increase in output that results from a given
proportionate increase in all the input factors
employed in the production process
• Three possibilities: – Increasing returns to scale
– Constant returns to scale
– Diminishing returns to scale
Increasing Returns to Scale
• An increase in total output of a firm is greater
than proportional increase in inputs.
– A 10% increase in factor inputs produce a 30%
increase in production output
• Increasing returns to scale are attributed to
economies of scale
Economies of Scale
• Economies of scale occurs when long run average
cost falls as output increases. This could be as a
result of: – Labour economies
• Indivisibility
• Higher degree of specialisation
– Technical economies
• Indivisibility
• Specialised capital equipment
Economies of Scale
– Marketing economies
– Financial economies
– Distribution and transport economies
– Managerial economies
– Dimensional relations
Constant Returns to Scale
• Increase in total output is proportional to
increase in inputs
– A 10% increase in inputs (K and L) produce a 10%
increase in output
• Occur where factors of production are
perfectly divisible
Constant Returns to Scale
• Happens when economies of scale disappear
and diseconomies of scale are yet to set in the
production process
Diminishing Returns to Scale
• Increase in total output is less than
proportional to increase in inputs
– A 10% increase in inputs (K and L) produce a 5%
increase in output
• Attributed to diseconomies of scale
Diseconomies of Scale
• Occur when long run average cost rises as
output increases
• Mostly caused by: – Managerial diseconomies
– Limitedness or exhaustibility of the natural
resources
– Labour inefficiency
– Pecuniary diseconomies
Exercise
• Suppose a production function is given as Q = f
(K, L). When inputs K and L are increased by
factor y, production function reads as follows:
s Q = f (yK, yL)
What kind of returns to scale does this
production function reveal if (i) s = y, (ii) s < y and
(c) s > y?
• Discuss the factors that cause increasing and
decreasing returns to scale?
Cost of Production in the Long
Run
C = wL + rK
Cost of Production in the Long Run
• Consideration of relative input prices is
important in order to find the least-cost
combination of inputs to produce a given level
of output
• Isocost curve is the useful tool used to analyse
the cost of purchasing inputs
Isocost Curves
• Shows the various combinations of inputs that
may be purchased for a given level of
expenditure at given input prices
• Isocost curves play a key role in finding the
combination of inputs that produce a given
level of output at the lowest possible total
cost
Input Price
• Market determined prices through the forces
of demand and supply of that input
• Bargained prices due to e.g. bulk purchases
Cost Function
• If we continue to denote the quantities of
capital and labour by K and L and denote their
respective prices by r and w, the total cost will
be
C = wL + rK
• Total cost is the sum of the cost of L units of
labour at w dollars per unit and of K units of
capital at r dollars per unit
Illustration
• A manager must pay USD25.00 for each unit
of labour services and USD50.00 for each unit
of capital services employed. He wishes to
know what combinations of labour and capital
can be purchased for USD400.00 total
expenditure on inputs
Cost Function
• At constant input prices, w and r for labour (L)
and capital (K) respectively, a given
expenditure on inputs (E) will purchase any
combination of labour and capital given by the
following equation (called an isocost curve)
K = (E/r) – (w/r)L
Characteristics of Isocost Curves
• If the constant level of total cost associated with
a particular isocost curve changes, the isocost
curve shifts parallel.
• An increase in cost, holding input prices constant,
leads to a parallel upward shift in isocost curve.
• A decrease in cost, holding input prices constant,
leads to a parallel downward shift in isocost
curve.
Characteristics of Isocost Curves
• The slope of the isocost curve is equal to the
negative of the relative input price ratio, -w/r.
• The ratio indicates how much capital must be
given up if one unit of labour is purchased. In the
example
-w/r = -$25/$50 = -1/2
If the manager wishes to purchase one more unit of
labour at USD25, ½ unit of capital, which cost USD50,
must be given up in order to keep the total cost of
input combination constant
Optimal Combination of Inputs
• Profit – Maximization
– If profit maximization is the primary goal,
then managers are concerned with
searching for the least-cost combination of
inputs to produce a given (profitmaximizing) output
– Decide how much output to pro duce
Optimal Combination of Inputs
– How to produce – given that any level of
output can be produced using many
combinations of inputs (illustrated by
isoquants)
– If a manager wants to produce the given level
of output at the lowest possible total cost, the
manager will choose the combination on the
desired isoquant that costs the lea st
Optimal Combination of Inputs
• Output Maximization e.g. non-profit
organisation
– Manager has a budget or fixed amount of
money available for production and wish to
maximize output
Optimal Combination of Inputs
– There are different amounts of input
combinations that can be purchased for a
fixed amount of expenditure on inputs
(illustrated by isocosts)
– The manager must choose the combination
on the isocost curve that lies on the
highest isoquant
Production of a Given Output at
Minimum Cost
• The input combination where the isocost is
tangent to the desired isoquant is the cost
minimizing equilibrium
• In cost minimizing equilibrium, MRTS = w/r
• To minimise the cost of producing a given level
of output, the manager employs the input
combination for which MRTS = w/ r
Production of Maximum Output at
with a Given Level of Cost
• Whether a manager is searching for input
combination that minimises cost for a given
level of output or maximises total production
for a given level of expenditure on resources,
the optimal combination of inputs to employ
is found using the same rule
Production of Maximum Output at
with a Given Level of Cost
• In case of two variable inputs, L and K, the
manager of a firm maximises output for a given
level of cost by using the amounts of L and K such
that the MRTS equals the input price ratio (w/r).
• In terms of a graph this condition is equivalent to
choosing the input combination where the slope
of the given isocost curve equals the slope of the
highest attainable isoquant