Concepts of Production functions

1
Concepts of Production functions

• Input Output
• Rate of resource transformation into products
• Rates vary different products different
  conditions
• Many ways to produce a product, given fixed
  factors
• E.g. quantity of fertiliser is a function of soil
  type, climate, irrigation, technology, seed type
• Different representations of the production
  process, such as
• graphic,
• tabular,
• symbolic,
• diagrammatic,
• written,
• algebraically

2
Symbolic representation

• Y ? output vector
• X ? input vector
• ? functional form of the relationship that
  transforms X into Y
• Does not specify contribution or importance
• Lists factors
• Does not specify fixity

3
One input one output case

• This is the starting point
• While we evaluate the impact of one input, the
  others are held fixed but NOT disregarded
• InputOutput
• From biological and physical experiments
• Farms records
• Empirical process
• Practical, observed, pragmatic, seen,
  experimental, experiential, heuristic
• More replication or observation ? higher accuracy
• Sampling procedure is important, i.e.
  representative sample of the population

4
Reality meets theory

• Despite biological complexity of production
  process, input-output relationships are
• real and
• observable
• Yet, biological laws of growth are NOT UNIFORM
• I.e. functional representation in case 1 may be
  different from that of case 2.
• Depends on given factors and on the product
• E.g. fertiliser ? maize is different from feed ?
  milk, etc.

5
Practical limitations

• Considering all factors in the process will be
  prohibitive, due to
• Lack of information on all factors
• Degrees of freedom limits in estimation
• Representativeness
• Simulation possibilities
• Graph in 2 to 3 dimensions
• Tabulation limited
• Total variability renders results of sensitivity
  analysis scenarios non-comparable

6
What do we do in practice?

• Causal relationships are tested and investigated
  when the effects of limited numbers of variables
  are isolated
• We never know the true functional form (), thus
• Approximate with known forms
• Represent through tables and graphs
• Estimation of production functions requires
  effort
• Production functions are highly specialised
• Production relations are based on experience

7
Assumptions

• Perfect certainty
• Decision making process for the FUTURE
• Management decisions based on production function
  is only as accurate as the fitted function
• RISK vs. UNCERTAINTY
• Assumption simplifies analysis and allows for
  clear development of the basic principals
• Combined with assumption of no time discounting ?
  assumption of static, instantaneous production

8
Assumptions cont.

• Level of technology
• Assumption that the farmer uses the most
  efficient process available to him, i.e. one that
  yields the highest output from a given amount of
  input
• Not necessary, but reasonable
• Method used is called
• level of technology or
• state of the arts
• Length of the time period
• Long run opportunity to change level of all input
  usage in the short run, not
• Very short run
• Short run
• Long run

9
Classical Production f(x)

• General principals, applicable to all situations
• Total Physical Product yf(x)TPP
• Marginal Physical Product MPP
• Average Physical Product APP
• Elasticity of production
• Shape of classical production function
• Characteristics necessary for study of production
  functions
• describes change in Y as successive units of X
  are added to set of fixed factors (technical unit
  and associated technologies)

10
Implicit additional assumptions

• Continuous curve
• only when inputs and outputs are perfectly
  divisible
• Opposite is indivisible or lumpy inputs (e.g.
  cows, tractors, labourers)
• Convenient, not necessary
• Homogeneity of inputs and outputs
• All units of an output or an input are
  homogeneous and of the same marketable quality,
  i.e. the 20th bag of fertiliser is quality-wise
  identical to the 8th bag
• Necessary and difficult to relax!

11
Total Physical Product
TPP describes how Y changes as extra units of X
are added to a bundle of fixed factors.yield,
output, Y
y
TPP
Yf(x)3x2x2-0.1x3
x
12
Average Physical Product APP

• Divide TOTAL output by TOTAL input I
• Average rate at which an input is transformed
  into a product. Measures the efficiency of the
  production process

APPy/x32x-0.1x2
y
APP
x
13
Marginal Physical Product MPP

• MPP is the change in output resulting from a 1
  unit change in input. It represents the slope of
  the TPP curve

dy/dx34x-0.3x2
y
MPP
x
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Notes on MPP

• When MPP is increasing, output is increasing at
  an increasing rate
• When MPP is decreasing, output is still
  increasing but at a decreasing rate
• When MPP is zero output is at a maximum and when
  it is negative, total output decreases

15
DMR

• Law of Diminishing Marginal Returns if
  increasing amounts of one input are added to a
  production process while all the others are held
  constant the amount of output added per unit of
  variable input will eventually decrease.

16
Stages of ProductionStage I

• Stage I MPPgtAPP. APP reaches a maximum at the
  end of stage I
• Average rate at which X is transformed to Y is
  increasing

Y
MPP
APP
X
17
Stages of ProductionStage II

• Stage II MPP ltAPP and decreasing but greater
  than zero.
• Variable input efficiency at a max at BEGINNING
  of stage II
• Efficiency of fixed input is max at END of stage
  II

Y
MPP
APP
X
18
Stages of ProductionStage III

• Stage III MPP is less than zero.
• Too much variable input, output begins to
  decrease.

Y
MPP
APP
X
19
Stages of ProductionTPP
Y
Inflection point
TPP
III
I
II
X
20
In what stage should we produce?

• In order to determine the most profitable point
  of production, input and output prices need to be
  known.
• BUT some recommendations can be given without
  prices!
• If output has value input use should continue
  until stage II
• If input is free we should not use so much as to
  end up in stage III
• Production should be somewhere in stage II

21
Stages of ProductionAlgebraically

• Stage II begins where
• As this indicates where APP is at a maximum.
  Why??
• Stage III begins where MPP0 WHY??

22
Elasticity of production

• Elasticity of production measures the degree of
  responsiveness between output and input
• epMPP/APP
• Where is stage I?
• Where is stage II?
• Where is stage III?

23
For tomorrow

• Why is the relationship between output and
  variable input one of diminishing returns?