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Chapter 9 page 1
Chapter 9 page 1
Topic 4 Extra page1
Topic 4: The Production Function 1) The Production Function 2) Changing Factors of Production in the Short and Long
runs 3) The Short-Run Production Function Total Product Average Product Marginal Product
Chapter 9 page 2
Production functions◦ Average product, marginal product and related concepts
Properties of the production technologies◦ No free lunch, Non reversibility, convexity, Free
disposability
Production in the short run (SR) ◦ Idea of Diminishing returns
Production in the long run (LR)◦ Isoquants
◦ Marginal rate of technical substitutions (MRTS)
2Ch 9: Production
Chapter 9 page 3
Topic 4 Extra page2
Objective: To examine how firm and industry supply curves are derived.
Introduction: Up to this point we have examined how the market demand function is derived. Next, we will examine the supply side of the market. We will explore how firms minimize costs and maximize productive efficiency in order to produce goods and services. By effectively combining labour and capital, the firm develops a production process with the objective of efficient resource allocation and cost minimization. The firm is assumed to produce a given output at minimum cost.
Chapter 9 page 4
Topic 4 Extra page3
The Production Function Definitions: Factors of Production: are factors used to produce output. Example: Labour Capital - machines -buildings Land Natural resources State of technology: consists of existing knowledge about method of production.
Chapter 9 page 5
Topic 4 Extra page4
The quantity that a firm can produce with its factors of production depends on the state of technology. The relationship between factors of production and the output that is created is referred to as the production function.
“The production function describes the maximum quantity of output that can be
produced with each combination of factors of production given the state of technology.”
Chapter 9 page 6
Topic 4 Extra page5
For any product, the production function is a table, a graph or an equation showing the maximum output rate of the product that can be achieved from any specified set of usage rates of inputs. The production function summarizes the characteristics of existing technology at a given time; it shows the technological constraints that the firm must deal with.
Chapter 9 page 7
What is a production function?◦ A production function is the relationship by which inputs
are combined to produce output.
◦ Naturally we need some technology that efficiently converts inputs into outputs.
Some common assumptions on technology◦ No free lunch: without input there is no output.
◦ Non-reversibility: Can’t run the production process in reverse.
◦ Free disposability: Throw away the excess without any cost or using more inputs.
◦ Convexity:
3Ch 9: Production
Chapter 9 page 8
Mathematically,
Let x1 and x2 be inputs then output is produced by the following:
Y = F(x1,x2).
Frequently encountered inputs: Capital (K) and Labor (L)
4Ch 9: Production
Chapter 9 page 9
The Production“Mountain”
5Ch 9: Production
Chapter 9 page 10
Important production functions
1. The Leontief technology (fixed-proportions
technology)
Y(x1,x2) = min(ax1,bx2)
Example: Y(K,L) = min(1/6L,K)
2. The Cobb-Douglas technology (More realistic)
Example Cobb-Douglas p.f.:
3. Perfectly substitutable inputs p.f.:
Y(K,L)=aK+bL
2/12/1 LKY
6Ch 9: Production
Chapter 9 page 11
Topic 4 Extra page6
Model Assumption: The quantity produced per period is ‘q’, and the two factors of production are labour and capital.
Notation: The Production Function of the Firm: q=f(L, K)
where f , the function, describes the relationship between the inputs L, K and the output each different combination produces per period.
Chapter 9 page 12
Topic 4 Extra page7
Changing Factors of Production in the Short and Long Runs
We must make a distinction between the short and long run. In the short run, a firm is able to change some of the factors of production, but at least one factor is fixed. In the long run, all factors of production can be varied
Chapter 9 page 13
Topic 4 Extra page8
The Short-Run Production Function Model: consider the simplest case where there is one input whose quantity is fixed and one input whose quantity is variable.
Suppose that the fixed input is the number of machines (capital) and the variable input is labour.
In the short run the firm cannot change the number of
machines quickly without incurring a high cost.
With one fixed input the short-run production function shows how total output changes as the variable factor changes.
Chapter 9 page 14
Topic 4 Extra page9
The Total, Average and Marginal Product of Labour Total Product Function: expresses the relationship between the variable input and the total output. →The total product function of labour: TPL: shows the various amounts of output that is produced when the amount of labour is varied with a given fixed amount of capital.
Chapter 9 page 15
Topic 4 Extra page10
TPL(L,K*)=f(L, K*) Quantity TPL(L,K*) Labour The diagram illustrates what occurs when the amount of the variable input increases.
Chapter 9 page 16
Topic 4 Extra page11
The total product (output) increases when the amount of labour increases, holding the amount of capital fixed at K*. Quantity increases initially at an increasing rate, but eventually quantity increases at a decreasing rate when more labour is employed.
Algebraically: ∂
∂
2
2 0TP L K
LL ( , )*
> at first;
and then eventually, ∂
∂
2
2 0TP L K
LL ( , )*
< (becomes negative). At some point, adding more labour units no longer increases output.
Chapter 9 page 17
Topic 4 Extra page12
We can derive the average and marginal product function of labour from the total product function.
The average product function, APL, measures output per unit of labour:
Average product of labourTotal product of labourNumber of labour units
AP (L,K )TP (L,K )
LL* L
*
=
=
Average product of labour is the measure of productivity of labour. Average product of labour, APL, measures output per unit of labour.
Chapter 9 page 18
Topic 4 Extra page13
It is the slope of a ray drawn from the origin to any point on the TPL function. The average product of labour for any given level of employment is equal to the slope of a straight line drawn from the origin to the total product function at that employment level. Generally, the APL increases at first as labour is increased. I.e. the output per worker increases initially. Further increases in labour reduce APL. APL declines when employment increases.
Chapter 9 page 19
Topic 4 Extra page14
The marginal product of an input is the addition to total output resulting from the addition of the last unit of the input when the amount of other inputs used is held constant. The marginal product function of labour, MPL, measures the change in quantity due to a change in the labour input, or the slope of the total product function of labour:
Marginal product of labour in total product of labour in number of labour units
MP (L,K ) TP (L,K )LL
* L*
=
= =
ΔΔΔ
Δ∂∂TPKL
L
Chapter 9 page 20
Topic 4 Extra page15
There is a distinct relationship between marginal product and average product: When: MP>AP , AP is increasing MP<AP , AP is decreasing MP=AP , AP is constant and at a maximum The law of diminishing returns describes the eventual decline in the marginal product of the variable factor as the variable factor increases with other factors held constant. The law of diminishing returns applies only to situations where one factor is increasing and the other factors are fixed.
Chapter 9 page 21
Topic 4 Extra page16
“The law of diminishing marginal returns: if equal increments of an input are added, and
the quantities of other inputs are held constant, the resulting increments of product will decrease beyond some point; that is, the marginal product of the input will diminish.”
Chapter 9 page 22
Topic 4 Extra page17
Output Labour Output/L Labour
TPL
Average Product
Marginal Product
Chapter 9 page 23
Topic 4 Extra page18
Chapter 9 page 24
Marginal product of an input ◦ Marginal = “extra”
◦ Compare this to the concept of Marginal Utility
Guiding thought: ◦ “By how much does production increase if we increase an
input by a small amount?”
◦ Example: the marginal product of capital =
“small amount” = infinitesimally small amount, i.e. ◦ but loose interpretation is often 1 unit change.
K
LKFMPK
),(
K
7Ch 9: Production
Chapter 9 page 25
Total Product: ◦ The amount of output produced at given the level of inputs.
Marginal Product: ◦ The change in the total product that occurs in response to a
unit change in the variable input (all other inputs held fixed).
Example,
Average Product (of a variable input): ◦ is defined as the total product divided by the quantity of that
input.
Example, AP of labor=Y/L
◦ Geometrically, it is the slope of the line joining the origin to the corresponding point on the total product curve.
K
LKFMPK
),(
8Ch 9: Production
Chapter 9 page 26
Short run:
◦ the longest period of time during which at least one of the
inputs used in a production process cannot be varied.
Assuming K is fixed at , the short run production
function is then
Law of diminishing return:
◦ if other inputs are fixed, the increase in output from an
increase in the variable input must eventually decline. In other
word, the marginal product will increase then decrease as
more variable input is added. This is a short run phenomenon.
0KBLKLKY 0),(
9Ch 9: Production
Chapter 9 page 27
10Ch 9: Production
Chapter 9 page 28
11Ch 9: Production
Chapter 9 page 29
12Ch 9: Production
Chapter 9 page 30
Some key things to remember:
◦ When marginal product curve lies above the average
product curve, the average product curve must be rising
and vice versa.
◦ Allocate the resource so that its marginal product is the
same across different activities.
◦ Always allocate resource to the activity that yields higher
marginal product.
13Ch 9: Production
Chapter 9 page 31
Long run:
◦ The shortest period of time required to alter the amounts of
all inputs used in a production process.
◦ In other words, all inputs are variable in the LR.
Isoquants:
◦ is the set of all input combinations that yield a given level
of output. Similar to the indifference curve in consumer
theory.
14Ch 9: Production
Chapter 9 page 32
1) The Long-Run Production Function ►Marginal Rate of Technical Substitution ►Returns to Scale ►Cobb-Douglas Production Function
Chapter 9 page 33
The Long-Run Production Function
In the long run, all factors of production are variable. Substitution Among Factors Similar to the notion of substituting between goods to maintain constant utility along an indifference curve, firms usually can produce the same output quantity by substituting between factors of production.
Chapter 9 page 34
The important question that needs to be addressed is:
“What combination of factors should be used to
produce this output?”
This question is difficult to answer because there is more than one way to produce the product.
Chapter 9 page 35
This can be illustrated with the aid of isoquant analysis.
The amount of capital is on the vertical axis and number of labour units is on the horizontal axis. The curve with an output of ‘q0’ is called an isoquant.
“Iso” means equal
“quant” means quantity.
The amount produced is the same along the isoquant. The points along the isoquant q0 represent the different factor combinations that can produce q0 units per period.
Chapter 9 page 36
“An isoquant shows the different combinations of factors of production that can produce a
given quantity of output.” Capital
K1
K2 q0 (Isoquant)
0 L1 L2 Labour
Chapter 9 page 37
15Ch 9: Production
Chapter 9 page 38
Marginal rate of technical substitution◦ It is the rate at which one input can be exchanged for
another without altering the total level of output.
16Ch 9: Production
Chapter 9 page 39
MRTS=- slope of the isoquant
Derivation:
L
K
L
K
MP
MPMRTS
MP
MP
L
FK
F
dK
dL
dLL
FdK
K
FcLKF
isoquant Slope
isoquant Slope
0),(
17Ch 9: Production
Chapter 9 page 40
The Marginal Rate of Technical Substitution
The marginal rate of technical substitution (MRTS) measures the rate of substitution of one factor for another along an isoquant.
“The marginal rate of technical substitution is the rate at which a firm can substitute capital
and labour for one another such that the output is constant.”
MRTSKLconstant
==
ΔΔKL q where Δ ΔK L is the slope
between two points on an isoquant.
Chapter 9 page 41
Note: An isoquant cannot have a positive slope.
An increase in one factor of production causes output to increase. Hence this increase in one factor must be offset by a decrease in other factor in order to keep output at the same level.
As the firm moves along the isoquant from left to right, the slope increases. The firm substitutes labour for capital, but at a diminishing rate. When this occurs there is a diminishing marginal rate of technical substitution.
Chapter 9 page 42
The MRTS and MP of Both Factors of Production The MRTS and marginal product of labour and capital are related. Suppose the firm decides to increase the amount of capital it employs, holding the amount of labour constant. Output will increase by the amount ΔqK because of the increase in capital ΔK . The increase in output is approximated by
Δ Δq MP KK K=
Chapter 9 page 43
If the firm holds capital constant and decreases the amount of labour it employs, output decreases by ΔqL , where the amount of labour decreases by ΔL.
The decrease in output is approximated by Δ Δq MP LL L= .
Along a given isoquant, output must be constant. If the firm increases capital by ΔK , labour must decrease by
an amount ΔL such that: ΔqK + ΔqL =0 in order to remain on the same isoquant.
Chapter 9 page 44
Substituting in these expressions, the condition becomes:
MP K MP LK LΔ Δ+ = 0
Rearranging such that we have an expression for the MRTS in terms of the marginal products of the two factors:
MRTS
KL
MPMPKL
L
K≡ = −
ΔΔ
The MRTS of K for L equals the negative of the ratio of the marginal product of L and K.
Chapter 9 page 45
Returns to Scale In general, the level of a firm’s productivity changes as the quantity produced by the firm changes. Returns to scale refers to the percentage change in output to a percentage change in inputs.
Chapter 9 page 46
Returns to scale: It tells us that what happens to output when all inputs are
increased by exactly the same proportion.
Guiding thought: What happens if we doubled all inputs?
Now, if
Output doubles too constant returns to scale
Output less than doubles decreasing returns to scale
Output more than doubles increasing returns to scale
18Ch 9: Production
Chapter 9 page 47
Three Cases:
1) When the percentage increase in inputs is smaller than the percentage increase in output, there are increasing returns to scale.
2) When the percentage increase in inputs leads to the
same percentage increase in output, there are constant returns to scale.
3) When the percentage increase in inputs is larger than
the percentage increase in output, there are decreasing returns to scale.
Chapter 9 page 48
Returns to Scale and the Cobb-Douglas Production Function A common production function is the Cobb-Douglas production function. Algebraically: q=ALaKb where A, a and b are constant and greater than zero.
Chapter 9 page 49
To determine the returns to scale for this function, we could change labour and capital by a factor ‘m’ and then determine if output changes by more than, equal to or less than ‘m’ times. q=A (mL)a(mK)b q=AmaLam b K b q=ma+b[ALaK b] Since originally output was q=ALaKb, we can determine if output will increase by either less than m times if a+b<1 (because ma+b<m), by exactly m times if a+b=1 or by more then m times if a+b>1.
Chapter 9 page 50
