The Production Function for Wilson Company

l-t i

"f trt

!::

n

--i :

:: f

'-' T}i

Chapter 7: Production Economics 263

1o. Extension of the cobb-DougJas Production Functiorz-The cobb-Douglas produc- tion function (Equation 7.16) can be shown to be a special case of a larger class of linear..homogeneous production functions having the following mathematical form,Il

Q = Yfdr-r + (1 - a)r-Pl*/P

where y is an effrciency parameter that shows the output resulting from given quantities of inputs; d is a distribution parameter (0 < a < 1) that indicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in lhe pro- duction process; and v is a scale parameter (v > 0) that indicates the type of re- turns to scale (increasing, constant, or decreasing). Show that when v = l, this function exlibits constant returns to scale. lHint: lnlease capital K and labor L each by a factor of 1., or K* = (tr)K and l* = (1,)L, and show that output Q also increases by a factor of 1,, or Q- = (I)(Q).]

11. Lobo Lighting Corporation currently employs 100 unskilled laborers, S0 factory technicians, 30 skilled machinists, and 40 skilled electricians. Lobo feels that the marginal product of the last unskilled laborer is 400 tights per week, the marginal product of the last factory technician is 450 lights per week, the marginal product of the last skilled machinist is 550 lights per week, and the marginal product of the Iast skilled electrician is 600 lights per week. Unskilled laborers earn $400 per week, factory technicians .urn $sob per week, machinists earn $700 per week, and electricians earn $750 per week.

Is Lobo using the lowest cost combination of workers to produce its targeted output? If not, what recommendations can you make to assist the company?

THE PRODUCTION FUNCTION FOR WILSON COMPANY Economists at the wilson company are interested in developing a production func- tion for fertilizer plants. They collected data on 15 different piutrir tnut produce fertil- izer (see the following table).

0uestions 1. Estimate the Cobb-Douglas production function Q = qL0 ;g9,, where e = output;

I = labor input; K = capital input; and o, p1, and p2 are the puru*.t.r, to be estimated.

2, Test whether the coefficients of capital and labor are statistically significant. 3. Determine the percentage of the variation in output that is "expiained" by the

regression equation.

IlSee R. G' Chambers, Applied Production Analysis (Cambridge: Cambridge University Press, 1988).

I Part 3: Production and Cost

4,, Deterrnine.the labor andicapital estimated,parameters,',and,grve"an, economic in-

terpretation of eachvalue ' "" : ': ': ':' S. Determine' whether,thii production,'function exhlbits, increaeing .decreasing r , eon-stant returns to scale, (Ignore frre :issue of statistical srg4ificance)' ,

, ,:tA3OR (,oo0 woRKaR

HOU.RS).PLANT OUTPUT.

{0oo TONS} CAPlTAL ' ($ooo)

1

1

3

4

5

: 8

o

10

1I t2 t3 l4 15

605.3

566,1

647.\

525.7

7t2,3

487.5

76.L.6,

:442-5

,821.1

397:8

$96.7

359.3

,9V9'.L

33r.7

7{164:9

18,891,

19,201

2A;655

15,08?'

20,300

t6,079

24,19-4

1,1,504

2597a ,fi,127

,25,,62?'

12477 '24,002

8,042

2T,972

vtoS 651.&

822.9

550,3

859.0

613:0,

851.3

6554

900,6

550,4

a42.2'

540.5

949.4

575i 925:8