Microeconomics: Production Theory

Production Theory
The Production Function
Total, Average, and Marginal Products
The Production Function in the Long Run

1. Production Theory
KZ AXDAD

Dr. Manuel Salas-Velasco
University of Granada, Spain

Page 1 Dr. Manuel Salas-Velasco

2. The Production Function

Production refers to the transformation of inputs into outputs
(or products)

+ An input is a resource that a firm uses in its production process
for the purpose of creating a good or service

+ A production function indicates the highest output (Q) that a
firm can produce for every specified combinations of inputs
(physical relationship between inputs and output), while holding
technology constant at some predetermined state

+ Mathematically, we represent a firm's production function as:

Assuming that the firm produces only one type of output with two inputs,
labor (L) and capital (K)

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3. The Production Function
DES]

+ The quantity of output is a function of, or depend on, the
quantity of labor and capital used in production

+ Output refers to the number of units of the commodity
produced

e Labor refers to the number of workers employed

+ Capital refers to the amount of the equipment used in
production

e We assume that all units of L and K are homogeneous or
identical

e Technology is assumed to remain constant during the period of
the analysis

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4. Production Theory
BY DABA

The Production Function in the
Short Run

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5. The Short Run

e The short run is a time period in which the quantity
of some inputs, called fixed factors, cannot be
increased. So, it does not correspond to a specific
number of months or years

e A fixed factor is usually an element of capital (such
as plant and equipment). Therefore, in our
production function capital is taken to be the fixed
factor and labor the variable one

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6. Total, Average and Marginal Products

e Total product (TP) is the total amount that is
produced during a given period of time

e Total product will change as more or less of the
variable factor is used in conjunction with the given
amount of the fixed factor

e Average product (AP) is the total product divided by
the number of units of the variable factor used to
produce it

e Marginal product (MP) is the change in total product
resulting from the use of one additional unit of the
variable factor

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7. Output with Fixed Capital and Variable

Labor

Quantity of labor (L)

Total product (TP)

Marginal product (MP)

Average product (AP)

0

0

50

50

50.00

60

55.00

130.00

130

130.00

60

116.00

50

105.00

20

92.86

0

81.25

1
2
3
4
5
6
7
8
9

-10

71,11

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Dr. Manuel Salas-Velasco

8. Total Product, Average Product and
Marginal Product Curves

9. Relationships among Total, Marginal an
Average Products of Labor
jC

TP E H + With labor time continuously
product divisible, we can smooth TP, MP,
and AP, curves

+ The TP curve increases at an
increasing rate up to point A;
past this point, the TP curve
rises at a decreasing rate up to
Labor Point C (and declines thereafter)

APL

+ The MP, rises up to point A,
MP,

becomes zero at C, and is
negative thereafter

Marginal product,
+ The AP, raises up to point B
and declines thereafter (but
remains positive as long TP is

Labor positive)

Average product

Page 9 Dr. Manuel Salas-Velasco

10. Law of Diminishing

TP By

Returns

Total
product

Labor

APL
MPL

Marginal “bal

>

Average product, !

Labor

La

C
Lo LA

+ This law states that as
additional units of an input are
used in a production process,
while holding all other inputs
constant, the resulting
increments to output (or total
product) begin to diminish
beyond some point (after A, in
the bottom graph)

+ As the firm uses more and more
units of the variable input with the
same amount of the fixed input,

each additional unit of the variable
input has less and less of the fixed
input to work and, after this point,
the marginal product of the variable
input declines

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Dr. Manuel Salas-Velasco ]

11. Stages of Producti

on

APL
MPL

[ei
! Total
! product
f Labor
A
Margin onan
Averagf cet
Labor

The relationship between the MP,
and AP, curves can be used to
define three stages of production
of labor (the variable input)

Stage I of labor

Is the range of production for which
increases in the use of a variable input
cause increases in its average product

EZ] Stage II of labor

Is the range for which increases in the
use of a variable input causes decreases
in its average product, while values of its
associated marginal product remain
nonnegative

EEE Stage III of labor

Is the range for which the use of a
variable input corresponds to negative
values for its marginal product

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Dr. Manuel Salas-Velasco |

12. In Terms of Calculus ...

+ Marginal product of labor = change in output/change in labor
input =AQ/AL

e If we assume that inputs and outputs are continuously or
infinitesimally divisible (rather than being measured in discrete
units), then the marginal product of labor would be:

m =f

Example. Let's consider the following production function:
Q=8K!?L!? Cobb-Douglas production function
TP, =8 L*

MP, =8K?4L> IK=1>

MP, =4L*

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13. The Production or Output Elasticity of Labor

The elasticity of output with respect to the labor
input measures the percentage change in output for a 1
percent change in the labor input, holding the capital
input constant

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14. Production Theory
BY DABA

The Production Function in the
Long Run

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15. The Long Run

+ The long run is a time period in which all inputs may be varied
but in which the basic technology of production cannot be
changed

+ The long run corresponds to a situation that the firm faces
when is planning to go into business (to expand the scale of its
operations)

e Like the short run, the long run does not correspond to a
specific length of time

+ We can express the production function in the form:

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16. Production Isoquants

Units per time period

This curve indicates that a firm can produce the
specified level of output from input combinations
(Lu Ky), (Lor Ko), (Lar Ks), ---

As we move down from one point on an
isoquant to another, we are substituting
one factor for another while holding
output constant

Q=f(LK)

Units per time period

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Dr. Manuel Salas-Velasco

17. Marginal Rate of Technical Substitution

+ The marginal rate of technical substitution (MRTS)
measures the rate at which one factor is substituted for
another with output being held constant

e Since we measure K on the vertical axis, the MRTS
represents the amount of capital that must be
Kyp---\a sacrificed in order to use more labor in the production
process, while producing the same level of output:
AK/AL (the slope of the isoquant which is negative)

+ We multiply the ratio by -1 in
order to express the MRTS as a
positive number

Q =f (L,K)
.

Units per time period

Units per time period

Page 17 Dr. Manuel Salas-Velasco

18. MRTS When Labor and Capital Are
Continuously Divisible

Let's take the total differential of the production function:

00 2Q
dQ= —dL+—dkK
2 OL OK
Setting this total differential equal zero (since output does not change along a
given isoquant):

©
o= Lars ur ax Bar KL GE MR
OL OK OK OL dl _%0 dL MP,
OK

he
dE. MF Ba

In production theory, the marginal rate of technical substitution is equal to the
ratio of marginal products (in consumer theory, the marginal rate of substitution is
equal to the ratio of marginal utilities)

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Dr. Manuel Salas-Velasco

19. A Numerical Example

Assume the production function is:
2
3

Q=3K'L
The marginal product functions are:
00 _3Kt21t MP, = 31345

MP, ==
* OL

I no
MP, 2-K=-L2-—2K-
MRTS = L = =
MP, KL L

If we specify the level of output as Q = 9 units, and the firm uses 3 units of labor, then

the amount of capital used is:
9=3K'3) > 3-ŸK > K=3units

=2 This result indicates that the firm can substitute 2 units of capital for 1

MRTS = 26)
3 unit of labor and still produce 9 units of output
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20. Production Theory
a AID

Econometric Analysis of
Production Theory

Page 20 Dr. Manuel Salas-Velasco

21. The Cobb-Douglas Production Function

+ Q= output

+ L = labor input

+ K = labor capital

e u = stochastic disturbance term

+ e= base of natural logarithm

+ The parameter A measures, roughly speaking, the scale of
production: how much output we would get if we used one unit
of each input

+ The parameters beta measure how the amount of output
responds to changes in the inputs

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22. The Cobb-Douglas Production Function

Q= A LA K* e"
e Our problem is to obtain an estimated function:

= A LÀ Kr
+ However, if we take logarithms in both sides, we have:
InQ =ÍnA)* A InL + B, InK +u
Bo

This is the log-log, double-log or log-linear model

Page 22 Dr. Manuel Salas-Velasco

23. The Properties of the Cobb-Douglas
Production Function

InQ = B, + BinL+ B, InK +u

Fitting this equation by the method of least squares, we have:

+

The estimated coefficient £, is the elasticity of output with
respect to the labor input; that is, it measures the percentage
change in output for a 1 percent change in the labor input,
holding the capital input constant

Likewise, the estimated coefficient Z, is the elasticity of output
with respect to the capital input, holding the labor input
constant

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24. The Properties of the Cobb-Douglas
Production Function

InQ = B, + 4, InL + B, InK +u

3. The sum of the estimated coefficients £, and £

gives information about the returns to scale

e If this sum is 1, then there are constant returns to
scale; that is, doubling the inputs will double the output,
tripling the inputs will triple the output, and so on

+ If the sum is less than 1, there are decreasing returns to
scale; e.g. doubling the inputs will less than double the
output

e If the sum is greater than 1, there are increasing returns
to scale; e.g. doubling the inputs will more than double
the output

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25. Cobb-Douglas Production Function: The
Agricultural Sector in Taiwan (1958-1972)

+ The log-linear model:

In¥, =f, +B, InX,,+ B, Xy, +4,

e Regression by the OLS method: d = 0.891

In¥, =-3.338 41.499 x +0490m X,

output elasticity output elasticity
of labor of capital

1.989
** Significant at 5%-level increasing returns to scale

Page 25 Dr. Manuel Salas-Velasco

26. Cobb-Douglas Production Function: The
Agricultural Sector in Taiwan (1958-1972)

Evidence of No autocorrelation Evidence of
positive negative
| autocorrelation | | autocorrelation |
Indecision Indecision
0 d dy 2 4-dy 4-d 4
d = 0.891 d = 1.939

0.946 1.543 2.457 3.054
0.814 1.750 2.250 3.186

Regression including a time variable: d = 1.939

In} =9.348 + 0.878 In X,, — 0.469 In X,, + 0.064 TIME
OK

kk * *

** Significant at 5%-level
* Significant at 10%-level

Page 26 Dr. Manuel Salas-Velasco

27. Cobb-Douglas Production Function: The
U.S. Bell Companies (1981-82)

+ The log-linear model:

InY, =f, + fP, MK, + £, nM, + $, nL, + error

e Regression by the OLS method: d = 1.954 (no autocorrelation)

InY, =1.461+0.401Ink, + 0.373 Ina (0203)n L,
** Ak **
output elasticity
of labor

A 1 percent increase in the labor input
led on the average to about a 0.2
percent increase in the output

** Significant at 5%-level

Page 27 Dr. Manuel Salas-Velasco

28. Problems in Regression Analysis

e Regression analysis may face two main econometric problems
when we use cross-sectional data (data on economic units for a
given year or other time period):

— Multicollinearity
— Heteroscedasticity

+ This arises when the assumption that the variance of the error term is
constant for all values of the independent variables is violated

+ This situation leads to biased standard errors

+ When the pattern of errors or residuals points to the existence of
heteroscedasticity, the researcher may overcome the problem by using
logs or by running a weighted least squares regression

+ Nowadays, several computer packages (STATA, LIMDEP, ...) present
robust standard errors (using White’s procedure)

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29. Detection of Heteroscedasticity: The
Breusch-Pagan Test

e Step 1. Estimate the model using OLS and obtain the residuals, à;

+ Step 2. Obtain the maximum likelihood estimator of 0?: >û?/n

e Step 3. Construct the variable p;: divide the squared residuals obtained
from regression (ú2) by the number obtained in step 2

+ Step 4. Regress p, on the X's and obtain ESS (explained sum of
squares)

e Step 5. Obtain the B-P statistic (ESS/2) and compare it with the critical
chi-square value

B-P B-P
There is not heteroscedasticity There is heteroscedasticity

critical chi-
square value

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Dr. Manuel Salas-Velasco