Bài 7: Hàm chi phí và lợi nhuận

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(1)<div class='page_container' data-page=1>

THE DUALITY APPROACH:

COST AND PROFIT

FUNCTIONS

</div>
(2)<div class='page_container' data-page=2>

The primal vs duality approach

Derivation of cost and profit function

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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

</div>
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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





</div>
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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

)

</div>
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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

L x

w

f x

x





 









,



conditional factor demand

c

x



x

w y



,

 

,



c



wx

w y



c w y

</div>
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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

pf x



wx







,



unconditional factor demand

x



x p w

 

pf x

wx









,



x



x p w



p w

,



 



</div>
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Properties

Issues in estimation

</div>
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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

c w y

x

w y

w









, 0



0 c w



















2 ,

so

c

j

i

j

i

j

i

x

w y

c

w y

c

w y

x

w y

w w

w





</div>
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Issues in estimating the cost function



Factor cost shares sum to 1



Homogeneity

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Factor cost share



The factor cost shares sum to 1



For the translog cost function



The cost share equations are



,







,





,

c

i

w x

w y

s w y

c w y



i



,



1 i

s w y





1 ln

2 i

ij

i

j

i

j

i

c

 







w







w







w

y

ln

i

ij

j

i

j

i

w

c

s

w

y

w c

w

















</div>
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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







ij

0 i







i

0 i







1 ln

2 i

ij

i

j

i

j

i

</div>
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Monotonicity



The cost function must be increasing in w



For the translog cost function

ln

0 i

ij

j

i

j

i

c

s

w

y

i

w

























</div>
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Concavity



The cost function must be concave in w

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Symmetry



Cross price effects of factor demand are equal

















2 ,

or

c

j

i

j

i

j

i

x

w y

c

w y

c

w y

x

w y

w w

w





</div>
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In empirical studies



Cost shares: estimated simultaneously with the cost

function (system of equations)



Homogeneity, monotonicity, convavity and symmetry

are either:

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Uses of the cost function



Factor demand



Output supply



Morishima elasticity of substitution



,





,



c

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

</div>
(18)<div class='page_container' data-page=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

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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

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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

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Example: Ray (1982)



Technical change in the cost function





ln

c w y

,

t



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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

</div>
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Properties

Issues in estimation

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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

p w

y

p w

p











,







,

i

p w

x

p w

w





 





,





,





,





,



so

i

k

i

k

i

k

p w

y p w

x p w

p w

w p

w

p







</div>
(25)<div class='page_container' data-page=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

k

p w

w p







</div>
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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

p w

p w p

y p

s

p w

p





















ln

,

ln

x

i

p w

p w w

x w

s

p w

</div>
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Example: Alpay et al (2002)



Title: Productivity growth and environmental

regulations in Mexican and U.S. food manufacturing



Objective: compare productivity growth of Mexican

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Example: Alpay et al (2002)



Methodology



profit function + revenue share equations +

expenditure share equations



profit: short-run profit (capital fixed)



functional form: translog profit

</div>
(29)<div class='page_container' data-page=29>

Example: Alpay et al (2002)



Data: aggregate



output: restricted short-run profit



Inputs



labor



material



pollution abatement expenditure

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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





</div>
(31)<div class='page_container' data-page=31></div>
(32)<div class='page_container' data-page=32>

Primal and dual, what can they do?



estimate factor demand



estimate output supply



factor substitution



technical changes

</div>
(33)<div class='page_container' data-page=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

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Disadvantages of duality



Prices are also co-linear



Properties/restrictions of the dual functions

(homogeneity, monotonicity, concavity and

symmetry)

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(36)<div class='page_container' data-page=36>

The data – rice production activity

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(37)<div class='page_container' data-page=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

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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

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Linear homogeneity

Prob > F = 0.2546

F( 1, 4160) = 1.30

( 1) lp_urea + lp_npk = 1

. test lp_urea + lp_npk = 1

</div>
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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

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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

</div>
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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) /*

</div>
(43)<div class='page_container' data-page=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

</div>
(44)<div class='page_container' data-page=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

</div>
(45)<div class='page_container' data-page=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

</div>
(46)<div class='page_container' data-page=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

</div>
(47)<div class='page_container' data-page=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) /*

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(48)<div class='page_container' data-page=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

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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

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Bài 7: Hàm chi phí và lợi nhuận

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

<i>x</i>

<i>pf x</i>



<i>wx</i>

min

st

 

<i>c</i>

<i>x</i>

<i>wx</i>

<i>y</i>



<i>f x</i>





Problems of the primal approach



endogeneity and simultaneity of the production function

(

<i>need instrumental variables, and more advanced techniques to </i>

<i>fix</i>

)



multicollinearity of inputs in the production function (

<i>may </i>

<i>result in incorrect estimates, sometimes unable to obtain the </i>

<i>estimates</i>

)



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

and output supply (

<i>the optimization is not always easy</i>

)

Specification of the cost function



Cost min problem



Lagrangian function



FOCs



Solving FOCs to obtain



Cost function

 