Review of Production Economics

1

• Review of Production Economics

2
Agenda

• Introduction
• Production Functions
• Cost Functions
• Revenue Functions
• Profit Functions
• Distance Functions
• Econometric Estimation of Production Functions

3
Introduction

• Assumptions
• Single period
• prices are known with certainty

4
Agenda

• Introduction
• Production Functions
• Cost Functions
• Revenue Functions
• Profit Functions
• Distance Functions
• Econometric Estimation of Production Functions

5
Production Functions

• Consider a firm that uses N inputs to produce a
  single output
• Technological possibilities can be summarized in
  a production function
• Properties
• Nonnegative
• Weak Essentiality
• Nondecreasing in X
• Concave in X

6
Single Input Production Function

7
Multiple input case

8
Multiple input case

9
Quantities of Interest

• Production function has to be twice-continuously
  differentiable, the we can calculate the
• Marginal Product
• Marginal rate of technical substitution
• Output elasticity
• Direct Elasticity of Substitution

10
Marginal Product and Marginal Rate of Technical
Substitution

• Marginal Product
• Marginal Rate of Technical Substitution
• Implicit function how much of is
  required to produce a fixed output when we use
  certain amounts of the other inputs
• Measures the slope of an isoquant

11
Output Elasticity and Direct Elasticity of
Substitution

• Output Elasticity
• Direct Elasticity of Substitution
• DES measures the percentage change in the input
  ratio relative to the percentage change in the
  MRTS
• Measures he curvature of the isoquant

12
Elasticities of Substitution

• MRTS ? slope of an isoquant
• DES ? curvature of an isoquant
• DES 0 no substitution is possible
• MRTS perfect substitutes
• Einscannen Bild 2.4

13
AES and MES

• Allen partial elasticity of substitution (AES)
  and Morishima elasticity of substitution are long
  run elasticities
• ? Allow all input to vary
• DES is a short term elasticity because it
  measures substitutability between two inputs
  while holding the others fixed.
• DES AES in the two input case.

14
Returns to scale

• Marginal product
• Measures the output response when one input is
  varied and all other inputs are hold fix.
• When all inputs are varied simultaneously CRS,
  DRS and IRS
• When kgt1
• DRS
• CRS
• IRS

15
Returns to scale

• DRS
• If a proportionate increase in all inputs results
  in a less than proportionate increase in output
• CRS
• If a proportionate increase in all inputs results
  in the same proportionate increase in output
• CRS
• If a proportionate increase in all inputs results
  in a more than proportionate increase in output

16
Elasticity of scale

• Elasticity of scale (total elasticity of
  production)
• Where E is the output elasticity
• The production function exhibits locally DRS, CRS
  or IRS as the elasticities of scale is less than,
  equal to or greater than 1.

17
An Example

• Cobb Douglas production function with two inputs

18
An Example
19
Duality production

• Output supply curve, Labor input demand, Capital
  input demand , Profit maximization
• Cost minimization
• Normal case in PSM
• Easier
• Input Demand Function
• First order derivatives with respect to
  quantities and the Lagrange parameter
• Marginal input demand functions

20
Agenda

• Introduction
• Production Functions
• Cost Functions
• Revenue Functions
• Profit Functions
• Distance Functions
• Econometric Estimation of Production Functions

21
Cost Functions

• Firms decide on the mix of inputs they wish to
  use in order to minimize costs
• No influence on input prices, perfectly
  competitive
• With T as the transformation function
• Where w is a vector of input prices.
• Search over all technically feasible input-output
  combinations and find the input quantities that
  minimize the cost of producing the output vector
  q

22
An Example

• Cobb Douglas production function with two inputs
• Minimize the function with respect to x1

23
Conditional Input Demand Functions
24
Final Cost Function

• All Cobb Douglas Functions are self dual (cost
  function and production function have the same
  functional form)
• Nondecreasing and linearly homogeneous in prices
  and nondecreasing in output
• Nonnegative, Nondecreasing in w, Nondecreasing
  in q, Homogeneity, Concave in w

25
Cost Minimization

26
Deriving Conditional Input Demand Equations

• When we have more than a few inputs or less
  tractable production functions
• More common to derive conditional input demand
  equations by working back from well behaved cost
  functions
• Shepards Lemma says
• Once a well behaved cost function has been
  specified or estimated econometrically we can
  obtain the conditional input demand equations.

27
Shepards Lemma

• Cost function defined
• Econometric estimation
• ? Input demand function
• First order derivatives with respect to prices
• Symmetry Condition
• EXAMPLE see Coelli
• Primal approach
• Dual approach

28
Demonstration

• Cost Function
• First order derivatives with respect to prices
• Equations are identical to the input demand
  equations
• Nonnegative
• Nonincreasing in w
• Nondecreasing in q
• Homogeneity
• Symmetry

29
Economies of Scale and Scope

• Measures of returns to scale are available in the
  multi-output case, can be defined in terms of the
  cost function.
• Overall scale economies
• Cost savings from producing different numbers of
  output (economies of scope)
• 
  c(w,qm) denotes the cost of producing the m th
  output only, and c(w,qM-m) denotes the cost
  of producing all outputs except the mth
  output

30
Analysis of Cost Elasticities

• Source Berichmann Public Transport
• Short-run given area variable labor
• INPUT Employees, capital, energy
• OUTPUT Bus kilometer and others
• Analysis of Cost Elasticity µ(C)
• if MCgt(lt)AC, µ is bigger (smaller) than one

31
Different Types of Cost Elasticities (I)

• A) Economies of Scale
• 1-Output SE1-µ(C) mu(C)lt1
• Multi-Output with j1,, n Outputs
• B) Economies of Densitiy
• C(w,y,T) T fixed factor of traffic
  density, e.g. km Network size
• Return to Traffic Density (RTS)
• with y Output N Network density k unit
  colum vector
• C) Economies of Capital Stock Utilization
• CC(w,y,k_fixed)
• Returns to utilization (RTU)
• with C is the cost function k unit colum
  vector
• if gt1 it is better to use capital more
  intensively

32
Different Types of Cost Elasticities (I)

• D) Economies of Scope
• Y(y1,,yn)
• Y decomposed into the two vectors Yx,Yn-X
• Degree-of-Scope-Economics (DSC)
• E) Economies of Network
• Return-to-Network Scale (RTN)
• If RTN gt 1 there are positive network effects
• with j1,,N network variables, e.g. routes

33
Agenda

• Introduction
• Production Functions
• Cost Functions
• Revenue Functions
• Profit Functions
• Distance Functions
• Econometric Estimation of Production Functions

34
Revenue Functions

• Cost Functions determine the minimum cost of
  producing a given output vector q.
• Revenue Function determine the maximum revenue
  obtained form a given input vector x
• Maximizing revenue mirrors the problem of
  minimizing cost
• Both variants of the profit function
• Micro economics cost function are widely used
• Macro economics revenue function

35
Revenue Maximization problem

• Multiple input, multiple output
• r(p,x) max pq such that t(q,x) 0
• p is a vector of output prices over which the
  firm has no influence
• properties nonnegative, nondecreasing in p,
  nondecreasing in x, convex in p, homogen

36
Example

• Maximizing revenue subject to the technology
  constraint
• Cobb Douglas production function with two inputs
• Revenue maximization problem
• Because there is only one output, technology
  defines the short run conditional output supply
  function

37
Revenue Function

• Nondecreasing in prices and input quantities
• Conditional output supply function
• Differentiating the revenue function with respect
  to outputs

38
Agenda

• Introduction
• Production Functions
• Cost Functions
• Revenue Functions
• Profit Functions
• Distance Functions
• Econometric Estimation of Production Functions

39
Profit Functions

• Cost and Revenue Function how firms use input
  and output price information to choose levels of
  either inputs or outputs, but not both
• Choose inputs and outputs simultaneously
  Profit Functions
• Decision in order to maximize Profit (Revenue
  Cost)
• Maximum profit varies with p and w

40
Example

• Production Function, Cobb Douglas

41
Input Demand Function

• Substituting this result back into the production
  function yields the output supply function
• And the profit function
• Nonnegative
• Nondecreasing in p
• Nonincreasing in w
• Homogeneity
• Convex in (p,w)
• Generalizations of the properties of cost and
  revenue functions

42
Profit Maximizing Solution

• -
• Single output case
• With the first order condition
• LMR LMC
• Long run profit maximizing level of output is the
  level that equates long run marginal revenue with
  long run marginal cost.

43
Input Demand and Output Supply Equations

• Shepards Lemma, obtain conditional demand
  equations directly from the cost function,
  without having solved an optimization problem
• General case for profit function Hotellings
  Lemma

44
Illustration

• Applying Hotellings Lemma

45
Hotellings Lemma

• Used to establish the following properties of the
  input demand and output supply functions
• With regard to x
• Nonnegative
• Nonincreasing
• homogen
• symmetry
• With regard to y
• - nonnegative
• nondecreasing
• homogen
• symmetry

46
Hotellings Lemma

• Hotellings Lemma Production Function
• First order derivatives with respect to prices
• Due to symmetry condition Youngs Theorem

47
Agenda

• Introduction
• Production Functions
• Cost Functions
• Revenue Functions
• Profit Functions
• Distance Functions
• Econometric Estimation of Production Functions

48
Efficiency Measurement using stochastic frontiers
Distance Functions

• Multi-output Production and Distance Function
• Output Distance Function
• Input Distance Function
• Use of Distance Functions

49
Multi-output Production and Distance Functions

• Single output production function
• Cobb Douglas and Translog
• To accommodate multiple output situations
• Specify a multi-output production function The
  Distance Function

Is a function, d h(x,y), that measures the
efficiency wedge for a firm in a multi-input,
multi-output production context. It is thus a
generalization of the concept of the production
frontier
50
Multi-input multi-output production technology

• Technology set S is then defined as
• X non negative K1 input vector
• Y non negative M1 output vector
• Set of all input output vectors (x,y) such that x
  produce y

51
Distance Functions

• Allow one to describe a multi-input, multi-output
  production technology, without the need to
  specify a behavioral objective (cost
  minimization, profit maximization)
• May specify both
• Input Distance Function
• Output Distance Function

52
Output Distance Function

• Maximal proportional expansion of the output
  vector, given an input vector!
• Production technology defined by the set S,
  equivalently defined using output sets, P(x)
• Properties inaction is possible, non zero output
  levels cannot be produced from zero levels of
  inputs, strong disposability of outputs, strong
  disposability of inputs, P(x) is closed, bounded
  and convex.

53
Output Distance Function

• Output distance function is defined on the output
  set, P(x)
• Properties of d0
• Non decreasing in y and increasing in x
• Linearly homogeneous in y
• If y belongs to the production possibility set of
  x then the distance is
• Distance is equal to unity if y belongs to the
  frontier of the production possibility set

54
Output Distance Function and Production
Possibility Set

• y2

B
Y2
C
A
PPC-P(x)
y1
Y1
55
Input Distance Function

• Characterizes the production technology by
  looking at a minimal proportional contraction of
  the input vector given an output vector
• Defined on the input set L(y)
• L(y) represents set of all input vectors x which
  can produce output vector y

56
Distance Function Specification (Coelli (1998),
p. 66)
x2
A
L(x)
x2A
B
C
Isoq- L(x)
x1A
x1
57
Use of Distance Function

• Can be used to define a variety of Index Numbers
• (Malmquist Index)
• Can also directly estimated by either econometric
  (SFA) or mathematical programming methods (DEA)
• Estimated distance functions have been used
  seeking measures of shadow prices.

58
Agenda

• Introduction
• Production Functions
• Cost Functions
• Revenue Functions
• Profit Functions
• Distance Functions
• Econometric Estimation of Production Functions

59
Production cost and Profit Functions

• Overview of econometric methods for estimating
  economic relationship.
• A single dependent variable in a function of
  one or more explanatory variables (Production
  function, Cost function)
• y f(x1, x2, , xn)
• Specify algebraic form
• Common functional forms (flexible, linear in
  parameters, regular, parsimonious,

60
Some Common Functional Forms

61
Accounting for Technological change

• Economic relationship may vary over time
• Account for technological change time trend
• Linear
• Cobb
• -Douglas
• Translog

62
Time Trend

• Implicit assumptions about the nature of
  technological change
• Percentage change in y in each period due to
  technological change
• Derivative of lny with respect to t
• linear
• Cobb-
• Douglas
• Translog

63
Neutral and Non-neutral Technical Change

64
Estimate Distance Functions using Econometric
Methods

• Specify translog functional form, estimate
  unknown parameters of the distance function
• Input distance function in log form
• Impose homogeneity of degree one in inputs
• We obtain
• Function to estimate, ML or COLS, Frontier 4.1

65
Input Distance Function Specification(Coelli
(2002), p. 12 sq.)

• The original Translog Form of an Input
  Distance-Function with M outputs and K inputs and
  D as distance function value is given by
• Restrictions required for homogeneity of degree
  1 in inputs are
• And those for symmetry are
• The level of inefficiency can be estimated from a
  stochastic frontier production function of the
  form y f(x)v-u, where v is the error term
  (assumed to be N0, s ) and u is the one-sided
  inefficiency term. The level of efficiency is
  estimated by exp(-u)). Consequently, lnD0i-ui.

66
Input Distance Function Specification(Coelli
(2002), p. 12 sq.)

• Imposing the homogeneity restrictions (by
  dividing the whole equation by an optional input)
  results in
• Where lnDI can be interpreted as inefficiency
  term (ui)
• given the stochastic error (vi) this model is
  formulated in the common SFA form and can be
  estimated with conventional SFA software.
• For estimation purposes, the negative sign on the
  dependent variable can be ignored. This results
  in the signs of the estimated coefficient being
  reversed.

67
Input Distance Function Estimation(based on
Coelli (2002), p. 13 and Bjorndal (2002), p. 8)

• For for I (i 1, 2, , I) firms, this
  econometric specification with lnDi -ui, in its
  normalized form is expressed by
• For estimation the sign of the explained variable
  is not of importance. If one uses lnx1 rather
  than lnx1, the estimated coefficients are
  reversed. However this is more consistent with
  the expected signs of conventional production
  functions (Coelli and Perelman 1996). Further it
  provides a convenient means of qualitatively
  assessing the model. As for the Error Component
  Model in SFA, a distribution for ui has to be
  assumed. Again normal distribution truncated at
  zero, uj N (µ,s2) and a half-normal
  distribution truncated at zero, uj N (0, s2)
  are most common.