Bài 7: Hàm chi phí và lợi nhuận

reg3 (lcost lp_urea lp_npk loutput lp_urea2 lp_npk2 loutput2 lp_urea_npk lout_urea lout_npk) /*. * Generating cost share for npk[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 min 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

  0 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  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



  

 

Issues in estimating the cost function

 Factor cost shares sum to 1  Homogeneity

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   

 

    

   

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

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   

 



       

Concavity

 The cost function must be concave in w

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



  

 

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

 



 ,    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

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

 2 outputs

 livestock  crop

 5 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  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

 

   

 

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

 

 



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)

 dual functions are more convenient to analyze

Disadvantages of duality

 Prices are also co-linear

 Properties/restrictions of the dual functions

(homogeneity, monotonicity, concavity and symmetry)

The data – rice production activity

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

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 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

Linear homogeneity

Prob > F = 0.2546 F( 1, 4160) = 1.30 ( 1) lp_urea + lp_npk = 1 test lp_urea + lp_npk = 1

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 328.067076 Prob > F = 0.0000 F( 9, 4154) = 462.33 Source SS df MS Number of obs = 4164

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

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) /*

Imposing linear homogeneity on the Translog cost function

 constraint lp_urea + lp_npk = 1

 constraint lp_urea2 + lp_npk2 + lp_urea_npk = 0  constraint lout_urea + lout_npk = 0

 cnsreg lcost lp_urea lp_npk loutput lp_urea2

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 =

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

Translog cost with cost share equation

snpk 4164 .3089158 0.0133 57.27 0.0000 lcost 4164 .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 7)

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

constraint [snpk]lp_urea = [lcost]lp_urea_npk constraint [snpk]lp_npk = [lcost]lp_npk2/2 constraint [snpk]_con = [lcost]lp_npk

gen snpk = npk*p_npk/cost

* Generating cost share for npk

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

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) /*

Imposing homogeneity

snpk 4164 .3089154 0.0133 69.54 0.0000 lcost 4164 .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 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 =

constraint [lcost]lp_urea2 + [lcost]lp_npk2 + [lcost]lp_urea_npk = constraint [lcost]lp_urea + [lcost]lp_npk =

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