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(1)

THE DUALITY APPROACH:


COST AND PROFIT



FUNCTIONS




(2)

The primal vs duality approach



Derivation of cost and profit function




(3)

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




(4)

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

min

st

 


c



x

wx

y

f x






(5)

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

)




(6)

Specification of the cost function



Cost min 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




(7)

Specification of the profit function



Profit max problem



FOCs



Solve FOCs to get



Substitute

into

to obtain



 



max



x

pf x

wx



 

0 i



i

i



pf x

wx



,

unconditional factor demand



x

x p w



 



pf x

wx





,




x

x p w



p w

,



 




(8)

Properties



Issues in estimation




(9)

Properties of the cost function


1


2


3


4


5


6



7 Shephard lemma



Symmetry by Young theorem



,

0 for ,

0



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

0



c w




2

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










(10)

Issues in estimating the cost function




Factor cost shares sum to 1


Homogeneity




(11)

Factor cost share



The factor cost shares sum to 1



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

,

1



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










(12)

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






ij

0



i





i

0



i






1



ln

ln

ln

ln

ln

ln



2



i

i

ij

i

j

i

i



i

i

j

i




(13)

Monotonicity



The cost function must be increasing in w


For the translog cost function




ln

ln

0



i

i

ij

j

i



j

i



i

i

i



c

c

c



s

w

y

i



w

w

w










(14)

Concavity



The cost function must be concave in w




(15)

Symmetry



Cross price effects of factor demand are equal






2

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










(16)

In empirical studies




Cost shares: estimated simultaneously with the cost



function (system of equations)



Homogeneity, monotonicity, convavity and symmetry



are either:




(17)

Uses of the cost function



Factor demand



Output supply



Morishima elasticity of substitution


,

,



c


i



i



c w y



x

w y



w









,

1



,

,



c w y



mc w y

p

y

mc

w p



y







,


ln


,


ln


i


j


ij


j


i



c w y




c

w y




(18)

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




(19)

Example: Ray (1982)



Data



2 outputs



livestock


crop



5 inputs



hired labor



capital (real estate, motor vehicles and machinery)


fertilizers




purchased feed, seed and livestock


miscellaneous inputs




(20)

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




(21)

Example: Ray (1982)



Technical change in the cost function





ln

c w y

,



t







(22)

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




(23)

Properties



Issues in estimation




(24)

Properties of the profit function


1


2


3


4


5



6 Hotelling lemma



7 Symmetry




p w

,

0





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











(25)

Issues in estimating the profit function



Homogeneity



Monotonicity



Convexity: Hessian matrix positive semi-definite



Symmetry



tp tw

,

t

p w

,

t

0




,


0


k


p w


p






,


0


i


p w


w







,

,



k

i

i

k



p w

p w



p w

w p










(26)

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




(27)

Example: Alpay et al (2002)



Title: Productivity growth and environmental



regulations in Mexican and U.S. food manufacturing



Objective: compare productivity growth of Mexican




(28)

Example: Alpay et al (2002)



Methodology



profit function + revenue share equations +


expenditure share equations



profit: short-run profit (capital fixed)


functional form: translog profit




(29)

Example: Alpay et al (2002)



Data: aggregate




output: restricted short-run profit


Inputs



labor


material



pollution abatement expenditure




(30)

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







(31)

(32)

Primal and dual, what can they do?




estimate factor demand


estimate output supply


factor substitution



technical changes




(33)

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




(34)

Disadvantages of duality



Prices are also co-linear



Properties/restrictions of the dual functions



(homogeneity, monotonicity, concavity and


symmetry)




(35)

(36)

The data – rice production activity




(37)

Preparing data




* GENERATING VARIABLE COST



gen cost = urea * p_urea + npk * p_npk


gen lcost = ln(cost)



* Generating log-var


gen lp_urea = ln(p_urea)


gen lp_npk = ln(p_npk)


gen loutput = ln(output)



* GENERATING INTERACTION TERMS


gen lp_urea2 = lp_urea * lp_urea


gen lp_npk2 = lp_npk * lp_npk



gen lp_urea_npk = lp_urea * lp_npk


gen loutput2 = loutput*loutput




(38)

The Cobb-Douglas production function




_cons -2.443404 .386535 -6.32 0.000 -3.201219 -1.685588
loutput .9030405 .014217 63.52 0.000 .8751676 .9309133
lp_npk .5981721 .2857531 2.09 0.036 .0379434 1.158401
lp_urea .5704618 .2956906 1.93 0.054 -.0092499 1.150173

lcost Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 5900.28906 4163 1.41731661 Root MSE = .84356
Adj R-squared = 0.4979


Residual 2960.26013 4160 .711600993 R-squared = 0.4983
Model 2940.02893 3 980.009644 Prob > F = 0.0000
F( 3, 4160) = 1377.19
Source SS df MS Number of obs = 4164
. reg lcost lp_urea lp_npk loutput



(39)

Linear homogeneity



Prob > F = 0.2546


F( 1, 4160) = 1.30


( 1) lp_urea + lp_npk = 1


. test lp_urea + lp_npk = 1




(40)

The translog cost function




_cons -20.02878 5.772784 -3.47 0.001 -31.34653 -8.711038
lout_npk -.2041232 .2942038 -0.69 0.488 -.7809201 .3726737
lout_urea .0451066 .3052942 0.15 0.883 -.5534335 .6436467
lp_urea_npk -7.073271 10.24913 -0.69 0.490 -27.16705 13.02051
loutput2 .0086156 .0091541 0.94 0.347 -.0093314 .0265625
lp_npk2 .8205222 5.340899 0.15 0.878 -9.650498 11.29154
lp_urea2 3.172856 5.40346 0.59 0.557 -7.420817 13.76653
loutput 1.11335 .4502833 2.47 0.013 .2305538 1.996146
lp_npk 15.02131 8.356314 1.80 0.072 -1.361537 31.40416
lp_urea 1.189279 8.8623 0.13 0.893 -16.18557 18.56413

lcost Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 5900.28906 4163 1.41731661 Root MSE = .84238


Adj R-squared = 0.4993
Residual 2947.68538 4154 .70960168 R-squared = 0.5004
Model 2952.60369 9 328.067076 Prob > F = 0.0000
F( 9, 4154) = 462.33
Source SS df MS Number of obs = 4164



(41)

Cobb-Douglas or Translog?



Prob > F = 0.0071


F( 6, 4154) = 2.95


( 6) lout_npk = 0



( 5) lout_urea = 0


( 4) lp_urea_npk = 0


( 3) loutput2 = 0


( 2) lp_npk2 = 0


( 1) lp_urea2 = 0




(42)

Testing linear homogeneity of the


Translog cost function



Prob > F = 0.0018


F( 3, 4154) = 5.03


( 3) lout_urea + lout_npk = 0



( 2) lp_urea2 + lp_npk2 + lp_urea_npk = 0


( 1) lp_urea + lp_npk = 1



> */ (lout_urea + lout_npk = 0)




> */ (lp_urea2 + lp_npk2 + lp_urea_npk = 0) /*


. test (lp_urea + lp_npk = 1) /*




(43)

Imposing linear homogeneity on the


Translog cost function



constraint 1 lp_urea + lp_npk = 1



constraint 2 lp_urea2 + lp_npk2 + lp_urea_npk = 0


constraint 3 lout_urea + lout_npk = 0



cnsreg lcost lp_urea lp_npk loutput lp_urea2




(44)

Imposing linear homogeneity on the


Translog cost function




_cons -1.257867 .5809441 -2.17 0.030 -2.396828 -.1189055
lout_npk -.1345858 .2770172 -0.49 0.627 -.6776877 .4085162
lout_urea .1345858 .2770172 0.49 0.627 -.4085162 .6776877
lp_urea_npk -5.672884 10.25086 -0.55 0.580 -25.77006 14.42429
loutput2 .0106869 .0083695 1.28 0.202 -.0057219 .0270957
lp_npk2 1.937082 5.07915 0.38 0.703 -8.020767 11.89493
lp_urea2 3.735802 5.236984 0.71 0.476 -6.531487 14.00309
loutput .7192963 .1365959 5.27 0.000 .4514953 .9870973
lp_npk 6.290339 3.880915 1.62 0.105 -1.31833 13.89901
lp_urea -5.290339 3.880915 -1.36 0.173 -12.89901 2.31833

lcost Coef. Std. Err. t P>|t| [95% Conf. Interval]


( 3) lout_urea + lout_npk = 0


( 2) lp_urea2 + lp_npk2 + lp_urea_npk = 0
( 1) lp_urea + lp_npk = 1



(45)

Translog cost with cost share equation




snpk 4164 3 .3089158 0.0133 57.27 0.0000
lcost 4164 9 .8416739 0.5001 10431.41 0.0000

Equation Obs Parms RMSE "R-sq" chi2 P

Three-stage least-squares regression


> */ (snpk lp_npk lp_urea loutput), constraint(4 5 6 7)


. reg3 (lcost lp_urea lp_npk loutput lp_urea2 lp_npk2 loutput2 lp_urea_npk lout_urea lout_npk) /*
. constraint 7 [snpk]loutput = [lcost]lout_npk


. constraint 6 [snpk]lp_urea = [lcost]lp_urea_npk
. constraint 5 [snpk]lp_npk = [lcost]lp_npk2/2
. constraint 4 [snpk]_con = [lcost]lp_npk


. gen snpk = npk*p_npk/cost


. * Generating cost share for npk



(46)

Translog cost with cost share equation





lp_urea_npk lout_urea lout_npk


Exogenous variables: lp_urea lp_npk loutput lp_urea2 lp_npk2 loutput2
Endogenous variables: lcost snpk



_cons 1.116778 .1409364 7.92 0.000 .8405478 1.393008
loutput .0089511 .0051982 1.72 0.085 -.0012372 .0191393
lp_urea -.3792256 .065678 -5.77 0.000 -.5079522 -.2504991
lp_npk .0304985 .0428719 0.71 0.477 -.0535289 .1145258
snpk



_cons -20.84974 5.387569 -3.87 0.000 -31.40918 -10.2903
lout_npk .0089511 .0051982 1.72 0.085 -.0012372 .0191393
lout_urea -.1963716 .1491281 -1.32 0.188 -.4886574 .0959141
lp_urea_npk -.3792256 .065678 -5.77 0.000 -.5079522 -.2504991
loutput2 .0064543 .0085096 0.76 0.448 -.0102242 .0231328
lp_npk2 .0609969 .0857438 0.71 0.477 -.1070579 .2290517
lp_urea2 -2.505675 .7265642 -3.45 0.001 -3.929715 -1.081635
loutput 1.238252 .4167321 2.97 0.003 .421472 2.055032
lp_npk 1.116778 .1409364 7.92 0.000 .8405478 1.393008
lp_urea 14.70696 3.820055 3.85 0.000 7.219785 22.19412
lcost



(47)

Testing homogeneity



Prob > chi2 = 0.0009


chi2( 3) = 16.38




( 3) [lcost]lout_urea + [lcost]lout_npk = 0



( 2) [lcost]lp_urea2 + [lcost]lp_npk2 + [lcost]lp_urea_npk = 0


( 1) [lcost]lp_urea + [lcost]lp_npk = 1



> */ (lout_urea + lout_npk = 0)



> */ (lp_urea2 + lp_npk2 + lp_urea_npk = 0) /*


. test (lp_urea + lp_npk = 1) /*




(48)

Imposing homogeneity




snpk 4164 3 .3089154 0.0133 69.54 0.0000
lcost 4164 6 .8430911 0.4984 10572.17 0.0000

Equation Obs Parms RMSE "R-sq" chi2 P

Three-stage least-squares regression


> */ (snpk lp_npk lp_urea loutput), constraint(4 5 6 7 8 9 10)


. reg3 (lcost lp_urea lp_npk loutput lp_urea2 lp_npk2 loutput2 lp_urea_npk lout_urea lout_npk) /*
. constraint 10 [lcost]lout_urea + [lcost]lout_npk = 0


. constraint 9 [lcost]lp_urea2 + [lcost]lp_npk2 + [lcost]lp_urea_npk = 0
. constraint 8 [lcost]lp_urea + [lcost]lp_npk = 1



(49)

Imposing homogeneity





lp_urea_npk lout_urea lout_npk


Exogenous variables: lp_urea lp_npk loutput lp_urea2 lp_npk2 loutput2
Endogenous variables: lcost snpk



_cons 1.16657 .1309565 8.91 0.000 .9099001 1.42324
loutput .0084214 .0051685 1.63 0.103 -.0017087 .0185515
lp_urea -.388852 .065142 -5.97 0.000 -.516528 -.2611761
lp_npk .0201149 .0410971 0.49 0.625 -.0604339 .1006637
snpk



_cons -1.5325 .4902529 -3.13 0.002 -2.493378 -.5716221
lout_npk .0084214 .0051685 1.63 0.103 -.0017087 .0185515
lout_urea -.0084214 .0051685 -1.63 0.103 -.0185515 .0017087
lp_urea_npk -.388852 .065142 -5.97 0.000 -.516528 -.2611761
loutput2 .0082655 .0077823 1.06 0.288 -.0069875 .0235185
lp_npk2 .0402298 .0821942 0.49 0.625 -.1208678 .2013275
lp_urea2 .3486222 .0628382 5.55 0.000 .2254615 .4717829
loutput .7699614 .1236369 6.23 0.000 .5276375 1.012285
lp_npk 1.16657 .1309565 8.91 0.000 .9099001 1.42324
lp_urea -.1665701 .1309565 -1.27 0.203 -.4232402 .0900999
lcost






×