The Duality Approach - Cost and Profit Functions

 Properties/restrictions of the dual functions (homogeneity, monotonicity, concavity and symmetry).  are not usually satisfied  are usually imposed.[r]

THE DUALITY APPROACH: COST AND PROFIT

FUNCTIONS

The primal vs duality approach

Derivation of cost and profit function

Production Economics

 optimal allocation of resources in the production of goods and services given

 technology

 resource constraints

 output demand (and thus prices of outputs)  prices of inputs

 Basic issues:

 optimal input uses

The primal vs dual approach  Primal approach

 optimal input and output levels are obtained by solving the optimization problem

 Dual approach

 Inputs demand and output supply functions can be derived from the dual functions

 

max

x pf x  wx st  

c

x wx y  f x

 

Problems of the primal approach

 endogeneity and simultaneity of the production function

(need instrumental variables, and more advanced techniques to fix)

 multicollinearity of inputs in the production function (may

result in incorrect estimates, sometimes unable to obtain the estimates)

 for some functional forms, it is hard to obtain input demands

and output supply (the optimization is not always easy)

Specification of the cost function  Cost problem

 Lagrangian function  FOCs

 Solving FOCs to obtain

 Cost function

 

min st c

x wx y  f x

  c  

L x  wx  y  f x 

   

0 i

i i

L x

w f x

x 



   



 ,  conditional factor demand

c

x  x w y

 ,   , 

c

c  wx w y  c w y

Specification of the profit function  Profit max problem

 FOCs

 Solve FOCs to get

 Substitute into to obtain

 

max

x   pf x  wx

  i

i i

pf x  wx  

 ,  unconditional factor demand

x  x p w

 

pf x wx

  

 , 

x  x p w

 p w, 

 

Properties

Issues in estimation

Properties of the cost function      

 Shephard lemma

 Symmetry by Young theorem

 ,  for ,

c w y  w y 

 ,  is non-decreasing in

c w y w

 ,  is non-decreasing in

c w y y

 ,  is linearly homogenous in

c w y w

 ,  is continuous and concave in

c w y w

 ,   

,

c i i

c w y

x w y w



 

 , 0

c w 

        2 , , , , so c c j i

i j j i j i

x w y

c w y c w y x w y

w w w w w w



  

 

Issues in estimating the cost function  Factor cost shares sum to

Factor cost share

 The factor cost shares sum to

 For the translog cost function

 The cost share equations are

 ,    , 

,

c i i i

w x w y s w y

c w y

 i  , 

i

s w y 



1

ln ln ln ln ln ln

2

i i ij i j i i

i i j i

c     w   w w   w y

ln

ln ln

ln

i

i i ij j i

j i

i i

w

c c

s w y

w c w   

 

    

   

Homogeneity of the cost function

 Proportional changes in input prices leave factor demand unchanged

 For the translog cost function, linear homogeneity is satisfied if

 ,   , 

c tw y  tc w y t 

1

i i

 

 ij

i

 

 i

i

 



1

ln ln ln ln ln ln

2

i i ij i j i i

i i j i

Monotonicity

 The cost function must be increasing in w  For the translog cost function

ln ln

i i ij j i

j i

i i i

c c c

s w y i

w w w   

 

      

 

Concavity

 The cost function must be concave in w

Symmetry

 Cross price effects of factor demand are equal

       

2

,

, , ,

or

c c

j i

i j j i j i

x w y

c w y c w y x w y

w w w w w w



  

 

In empirical studies

 Cost shares: estimated simultaneously with the cost function (system of equations)

 Homogeneity, monotonicity, convavity and symmetry are either:

Uses of the cost function  Factor demand

 Output supply

 Morishima elasticity of substitution

 ,   ,  c

i

i

c w y x w y

w

 



 ,     

, ,

c w y

mc w y p y mc w p

y            , ln , ln i j ij j i

Example: Ray (1982)

 Title: A translog cost function analysis of U.S agriculture 1939-1977

 Objectives

 measure elasticity of substitution

 measure price elasticity of factor demand  measure technical change

Example: Ray (1982)  Data

 outputs  livestock  crop

 inputs

 hired labor

 capital (real estate, motor vehicles and machinery)  fertilizers

 purchased feed, seed and livestock  miscellaneous inputs

Example: Ray (1982)  Estimated equations:

 cost function

 cost share equations

 revenue share equations

 Functional form: translog cost function  Dependent variables:

 farm production expense (index)  cost shares

Example: Ray (1982)

 Technical change in the cost function

 

ln c w y,

t



Example: Ray (1982)

 Treatment for properties of the cost function  homogeneity: imposed

 monotonicity: ignored  concavity: ignored  symmetry: ignored  Findings

 declining substitutability between capital and labor  price elasticity increase over time for all inputs

Properties

Issues in estimation

Properties of the profit function     

 Hotelling lemma

 Symmetry

 p w, 

 

 p w,  non-decreasing in p



 p w,  non-increasing in w



 p w,  linear homogeneous in  p w, 



 p w,  continuous and convex in  p w, 

  ,    , k k p w

y p w p      ,    , i i p w

x p w w       ,   ,   ,   , 

so i

k i i k i k

p w p w y p w x p w

p w w p w p

 

   

 

Issues in estimating the profit function

 Homogeneity

 Monotonicity

 Convexity: Hessian matrix positive semi-definite

 Symmetry

tp tw,  t  p w,  t

     ,  k p w p       , i p w w      ,   , 

k i i k

p w p w

p w w p

 

 



Issues in estimating the profit function

 Although not required, profit function is usually estimated together with the revenue share

equations

and the input expenditure share equations

     

ln , ,

, ln

y

k k k

k

k k

p w p w p y p

s p w

p p                 

ln , ,

, ln

x

i i i

i

i i

p w p w w x w

s p w

Example: Alpay et al (2002)  Title: Productivity growth and environmental

regulations in Mexican and U.S food manufacturing  Objective: compare productivity growth of Mexican

Example: Alpay et al (2002)  Methodology

 profit function + revenue share equations + expenditure share equations

 profit: short-run profit (capital fixed)  functional form: translog profit

Example: Alpay et al (2002)  Data: aggregate

 output: restricted short-run profit  Inputs

 labor  material

 pollution abatement expenditure

Example: Alpay et al (2002)

 Dual productivity growth from the profit function

 The primal productivity growth could be derived from the dual productivity growth

 technical changes that are unaffected by prices

 

ln p w,

t





Primal and dual, what can they do?  estimate factor demand

 estimate output supply  factor substitution

 technical changes

Advantages of duality approach  sometimes it’s hard to solve the optimization

problem for the primal production function

 in production function, inputs are very likely to be co-linear (more than prices)

Disadvantages of duality  Prices are also co-linear

 Properties/restrictions of the dual functions (homogeneity, monotonicity, concavity and symmetry)