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

Truong Dang Thuy

The primal vs duality approach

Derivation of cost and profit function

Introduction

 optimal input uses

 optimal output supply

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

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

Specification of the cost function

 Cost min problem

Specification of the profit function

 Profit max problem

 FOCs

 Solve FOCs to get

 Substitute into to obtain

Properties

Issues in estimation

The cost function

Properties of the cost function

c w y

x w y w

j i

Issues in estimating the cost function

 Factor cost shares sum to 1

 Homogeneity

 Monotonicity

 Concavity

 Symmetry

Factor cost share

 The factor cost shares sum to 1

 For the translog cost function

 The cost share equations are

 ,    , 

,

c

i i i

Homogeneity of the cost function

 Proportional changes in input prices leave factor demand unchanged

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

 ,   ,  0

c tw y  tc w y t 

1

i i

Monotonicity

 The cost function must be increasing in w

 For the translog cost function

Concavity

 The cost function must be concave in w

 The Hessian matrix must be negative semi-definite

j i

Uses of the cost function

, ln

i

j ij

j i

c w y

c w y MES

w w







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

 Aggregate level data

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

 revenue shares

 declining substitutability between capital and labor

 price elasticity increase over time for all inputs

 technical change rate 1.8% per year

Properties

Issues in estimation

The profit function

Properties of the profit function

p w

y p w p

p w

x p w w

Issues in estimating the profit function

Issues in estimating the profit function

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

Example: Alpay et al (2002)

 Title: Productivity growth and environmental

regulations in Mexican and U.S food manufacturing

 Objective: compare productivity growth of Mexican and U.S food manufacturing and the impacts of

environmental regulations

Example: Alpay et al (2002)

 Methodology

 profit function + revenue share equations + expenditure share equations

 profit: short-run profit (capital fixed)

 functional form: translog profit

 homogeneity and symmetry: imposed

 pollution abatement expenditure

 time trend included for productivity growth

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

Primal or dual?

Primal and dual, what can they do?

 estimate factor demand

 estimate output supply

 factor substitution

 technical changes

 efficiency (not covered in this lecture)

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)

 dual functions are more convenient to analyze

economic impacts

Disadvantages of duality

 Prices are also co-linear

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

 are not usually satisfied

 are usually imposed