Production Theory and Estimation the Firm and its Technology, Ch 6 Plus ideas we need from Ch 7, OptimalInput Combinations and Cost Functions.

Mansfield and Yohe

C o rn

To ta lPro d uc t

Fe rtilize r

A

B

C

D

E

F

DF

DQ

0

.

The long run and returns to scale:

1. increasing returns2. constant returns3. decreasing returns

This slope is called the "Marginal Rate

of Technical Substitution"

Optimal utilization of an input:

MRPY = MEY

The Optimal Combination of Inputs

MPa/(Pa) = MPb/(Pb) = … = MPn/(Pn)

This is demonstrated by LaGrangean

method.

To ta l C o st

Q ua ntity o fO utp ut

To ta l C o sts

0

oFC

We c a n d e rive this to ta l c o stfunc tio n fro m the p ro d uc tio nfunc tio n we stud ie d p ro vid e d tha t the inp ut p ric e s a re kno wnto us. Wha t d o e s "FC " re p re se nt?

Q ua ntity0

Ave ra g e C o st

AC

The a ve ra g e c o st c urve is a lso "U sha p e d ."

Q ua ntity

Ave ra g eFixe d C o st

AFC

An a d va nta g e o f la rg e sc a le istha t fixe d c o sts c a n b e sp re a d o ve rm a ny m o re wid g its.

Issues regarding the estimation of

production functions and the practical

applications of the estimates--these follow.

.

.

.

.

.

.

.

ln Q

lnL

v1v2

v3

v4

v5

v6

v7

The e stim a tio n p ro c e ss, using o rd ina ryle a st sq ua re s, is typ ic a lly d o ne a line a rize d ve rsio n o f the m o d e l. Fo re xa m p le , the lo g /lo g ve rsio n o f theC o b b -Do ug la s p ro d uc tio n func tio n is line a r.

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.

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.

.

.

Q

Input

Prod uc tionfunc tion

u1

u2

u3

u4

u5

u6

Ac uta l d a ta p o ints ra re ly fit p e rfe c tly to a c urve . We c om m o nly d e fine the b e st c urve a s the one tha t m inim ize sthe sum o f the squa re d "e rro rs", he re the se a re the sq ua re s o f the ui.

The n we tra nsfo rmthe re sults fo r thelo g /lo g ve rsio nb a c k to the o rig ina l c urve dfo rm to g e t thee stim a te d a c tua l p ro d uc tio nfunc tio n.

OutputsLand Labor Equipment Materials Other ResourcesApples Q24.65242 77.73192 30.41621 130.5623 209.9966 587618.79478 27.437 16.49433 76.94727 236.7154 36354.659752 79.88781 0.338023 8.765742 192.9683 87984.55803 99.29876 44.22924 17.93959 152.0199 297765.57001 34.28046 44.85364 94.28036 237.8825 623184.22263 89.57336 12.10467 136.9953 134.4489 750271.5573 21.10881 44.5215 60.10136 233.6126 4760

85.39414 30.80439 35.19438 45.01644 200.1886 449776.97167 51.06259 8.47093 16.0292 23.01955 227859.88652 6.856842 42.5177 92.36735 45.71781 517866.60145 62.17344 20.85112 109.35 5.283063 604414.02002 26.44556 38.34991 16.11898 95.68023 171611.74053 43.01037 38.98331 134.1327 126.2095 466724.89105 25.95466 4.631264 86.85035 194.2379 399376.18268 21.83509 43.14855 106.8624 141.6165 650711.27326 63.57194 6.418483 85.2708 66.18858 356535.79691 61.19172 0.431686 134.1159 168.7767 4400

82.607 85.01718 19.09457 62.12133 166.6409 52124.50921 74.86749 22.30187 3.19602 70.72382 644

6.079836 94.79541 21.13041 57.69959 69.32605 270131.2036 6.807599 36.24315 117.287 21.84704 4737

71.65889 78.55763 3.515965 124.9571 50.48423 620183.06356 35.6512 24.79884 133.3848 16.35334 706588.32544 51.55598 18.87963 21.98655 96.84964 296065.24533 52.43446 26.36403 79.52929 91.02759 558095.50766 79.94227 0.666809 26.76433 32.68259 26081.87991 34.14628 33.23958 102.0363 146.3933 2549

83.72708 71.71039 25.3888 8.321572 179.4968 17534.824828 88.84756 7.913311 135.7395 88.3941 36695.772931 93.03085 11.13234 47.5269 89.20939 245717.09222 77.12693 35.21731 31.26825 176.1242 260083.80141 84.72119 8.742205 95.05901 86.37202 583634.76315 67.97825 6.707286 84.12513 150.2786 447830.0013 35.5497 1.76852 82.51957 176.6183 3596

63.85156 54.78346 8.145172 134.3211 138.6146 65887.301041 49.46101 38.23907 117.0737 104.2174 3889

46.611 56.645 21.707 78.24 122.673 4161.889

Apple production process data set.

Also let input prices be: 250, 133, 450, 200, and 100.

Means at bottom.

Dependent Variable is lnQ, ie lnCrop

Coefficients Standard Error t StatIntercept, lnA 5.005880605 0.084948788 58.92821705lnLand 0.224808232 0.006630809 33.90359192lnLabor 0.061995483 0.012365185 5.013712359lnEquipment 0.065023475 0.00600921 10.82063664lnLivestock 0.484313926 0.008114067 59.68818549lnOtherRes 0.013996119 0.008782322 1.593669535

Regression StatisticsMultiple R 0.997407407R Square 0.994821535Adjusted R Square0.993928696Standard Error 0.043294338Observations 35

ANOVAdf F Significance F

Regression 5 1114.223 3.35-32

Summary log regression output for Apple firm data.

T h e C o b b / D o u g l a s P r o d u c t i o n f u n c t i o n : u ieKKKLAKQ 44

33

22

11

N o t i c e t h a t I h a v e c a l l e d L a n d , K 1 , h e r e L i s l a b o r o f c o u r s e , a n d E q u i p m e n t , M a t e r i a l s a n d O t h e r R e s o u r c e s b e c a m e K s , 2 t h r u 4 . F i n a l l y , r e c a l l t h a t " e " i s t h e m a t h e m a t i c a l c o n s t a n t f r e q u e n t l y u s e d i n c a l c u l u s , a n d u i i s t h e e r r o r t e r m f o r e a c h f i r m i . A s a n e x a m p l e o f m a r g i n a l p r o d u c t s , t h e f o l l o w i n g i s t h e m a r g i n a l p r o d u c t o f K 1 i n a n a l y t i c a l t e r m s :

44

33

22

1111

11

KKKLAKK

QMP K

N o t i c e t h a t I d r o p p e d t h e e t e r m , t h i s i s b e c a u s e I i n t e n d t o e v a l u a t e t h e m a r g i n a l p r o d u c t s a t t h e m e a n s o f t h e v a r i a b l e s , a n d t h e m e a n v a l u e o f u i i s z e r o ( h i n t : e t o t h e z e r o p o w e r i s u n i t y ) .

Given the regression results we know that the alphas and betas are LnA = 5.0 To find A, we take e5.0 = 148.41

1 = 0.2248; = 0.0620; 2 = 0.0650; 3 = 0.4843; 4 = 0.0140. From the Excel program, the means of the inputs are: K1Mean = 46.6; L= 56.6; K2Mean = 21.7; K3Mean = 78.2; K4Mean = 122.7. So to calculate the marginal product of K1. Plug in the values. MPK1 = .2248(148.41)(46.6)-.8752(56.6).0620(21.7).0650(78.2).4843(122.7).0140 = 16.02

What do people do with such results?

It is a warning bell: If marginal products per dollar aren't approximately equal across inputs, then you have a big problem.

16.02342 This is the marginal product for K1

3509.755 This is the estimated total product at mean inputs

Quick way: to get the next one

3.925262 marginal product for Labor10.92238 marginal product for Equipment K221.76506 marginal product for Materials K30.421492 marginal product of Other Res K4

"Quick way" refers to the fact that you don't need to recalculate the entire expression each time. Find Q evaluated at mean input levels. Then, MP1 = 1 Q/K1.

Input Marg. Prod. Input Price MargProd/InpPriceLand 16.02 250 0.06408Labor 3.925 133 0.029511278Equipment 10.922 450 0.024271111Materials 21.765 200 0.108825Other Res. 0.4215 100 0.004215

The Apples Problem data.

Here Labor and Equipment give the best performance on the margin, while Other Resources and Land are relatively useless in this project on the margin.