Microeconomics: Production Theory

Production Theory

Dr. Manuel Salas-VelascoUniversity of Granada, Spain

Dr. Manuel Salas-VelascoPage 1

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:

Q = f (L, K)

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


The Production Function

• 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

• Labor refers to the number of workers employed

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

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

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

Q = f (L, K)


Production Theory

The Production Function in the Short Run


The Short Run

• 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

• 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

),( KLfQ


Total, Average and Marginal Products

• Total product (TP) is the total amount that is

produced during a given period of time

• Total product will change as more or less of the

variable factor is used in conjunction with the given

amount of the fixed factor

• Average product (AP) is the total product divided by

the number of units of the variable factor used to

produce it

• Marginal product (MP) is the change in total product

resulting from the use of one additional unit of the

variable factor


Output with Fixed Capital and Variable Labor

Quantity of labor (L) Total product (TP) Marginal product (MP) Average product (AP)

0 0

1 50 50 50.00

2 110 60 55.00

3 390 280 130.00

4 520 130 130.00

5 580 60 116.00

6 630 50 105.00

7 650 20 92.86

8 650 0 81.25

9 640 -10 71.11


Total Product, Average Product and Marginal Product Curves

0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7 8 9

Quantity of labor per time period, L

To

tal p

rod

uc

t (u

nit

s p

er

tim

e p

eri

od

)

-40

-20

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

0 1 2 3 4 5 6 7 8 9 10

Quantity of labor per time period, L

AP

an

d M

P (

un

its

pe

r ti

me

pe

rio

d)TP

MPL

APL


Relationships among Total, Marginal an Average Products of Labor

Total product

Labor

Labor

Marginal product

Average product

APL

MPL

TP

LA LB LC

A

A

B

B

C

C

• With labor time continuously divisible, we can smooth TP, MPL

and APL curves

• The MPL rises up to point A, becomes zero at C, and is negative thereafter

• The APL raises up to point B and declines thereafter (but remains positive as long TP is positive)

• 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 point C (and declines thereafter)

A = inflection point


Law of Diminishing Returns

Total product

Labor

Labor

Marginal product

Average product

APL

MPL

TP

LA LB LC

A

A

B

B

C

C • 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


Stages of Production

Total product

Labor

Labor

Marginal product

Average product

APL

MPL

Stage III of labor

Stage I of labor

Stage II of labor

TP

3 4 8

A

A

B

B

C

CThe relationship between the MPL

and APL curves can be used to define three stages of production of labor (the variable input)

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

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

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


In Terms of Calculus …

• Marginal product of labor = change in output/change in labor

input =

L

QMPL

),( KLfQ

Example. Let’s consider the following production function:

Q = 8 K1/2 L1/2

If K = 1

Cobb-Douglas production function

21

21

218

LKMPL 2

1

4

LMPL

21

8 LTPL

• 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:


The Production or Output Elasticity of Labor

Q

L

L

QEL

Q

LMPE LL

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


Production Theory

The Production Function in the Long Run


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)

• 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:

),( KLfQ


Production Isoquants

),( KLfQ

L

K

Units per time period

Units

per

tim

e p

eriod

K1

K2

K3

L1 L2 L3

An isoquant is a set of input combinations that can be used to produce a given level of output

This curve indicates that a firm can produce the specified level of output from input combinations (L1, K1), (L2, K2), (L3, K3), …

a

b

c

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


Marginal Rate of Technical Substitution

),( KLfQ

L

K

Units per time period

Units

per

tim

e p

eriod

K1

K2

K3

L1 L2 L3

a

b

c

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

L

KMRTS

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

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


MRTS When Labor and Capital Are Continuously Divisible

KdK

QLd

L

QQd

),( KLfQ

KdK

QLd

L

Q

0

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

Setting this total differential equal zero (since output does not change along a given isoquant):

LdL

QKd

K

Q

K

Q

L

Q

Ld

Kd

K

L

MP

MP

Ld

Kd

K

L

MP

MP

Ld

Kd

K

L

MP

MPMRTS

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)


A Numerical Example

32

31

3 LKQ

32

32

313

KL

K

QMPK

Assume the production function is:

The marginal product functions are:

31

31

323

LK

L

QMPL

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:

L

K

LK

LK

MP

MPMRTS

K

L 22

32

32

31

31

unitsKKK 333339 31

32

32

31

2

3

32MRTS This result indicates that the firm can substitute 2 units of capital for 1

unit of labor and still produce 9 units of output


Production Theory

Econometric Analysis of Production Theory


The Cobb-Douglas Production Function

ueKLAQ 21

• Q = output

• L = labor input

• K = labor capital

• 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


The Cobb-Douglas Production Function

21ˆˆˆ

KLAQ

• Our problem is to obtain an estimated function:

• However, if we take logarithms in both sides, we have:

ueKLAQ 21

uKLAQ lnlnlnln 21

β0

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


The Properties of the Cobb-Douglas Production Function

1. The estimated coefficient β1 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

2. Likewise, the estimated coefficient β2 is the elasticity of output

with respect to the capital input, holding the labor input

constant

uKLQ lnlnln 210

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


The Properties of the Cobb-Douglas Production Function

3. The sum of the estimated coefficients β1 and β2

gives information about the returns to scale

• 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

• If the sum is greater than 1, there are increasing returns

to scale; e.g. doubling the inputs will more than double

the output

uKLQ lnlnln 210


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

• The log-linear model:

• Regression by the OLS method:

iiii uXXY 33220 lnln

iii XXY 32 ln490.0ln499.1338.3ˆln

output elasticity of labor

output elasticity of capital

1.989

increasing returns to scale

d = 0.891

** **

** Significant at 5%-level


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

2

Zone of

indecision

Zone of

indecision

0 4dL dU

No autocorrelation

4 – dU

Evidence of

positive

autocorrelation

Evidence of

negative

autocorrelation

4 – dL

d = 0.8910.946 1.543 2.457 3.054

Regression including a time variable:

TIMEXXY iii 064.0ln469.0ln878.0348.9ˆln 32

0.814 1.750 2.250 3.186

d = 1.939

** ***

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

d = 1.939


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

errorLMKY jjjj lnlnlnln 3210

• The log-linear model:

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

jjjj LMKY ln202.0ln373.0ln401.0461.1ˆln

** Significant at 5%-level

** ** ** **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


Problems in Regression Analysis

• 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)


Detection of Heteroscedasticity: The Breusch-Pagan Test

• Step 1. Estimate the model using OLS and obtain the residuals, ûi

• Step 2. Obtain the maximum likelihood estimator of σ2: ∑ûi2/n

• Step 3. Construct the variable pi: divide the squared residuals obtained

from regression (ûi2) by the number obtained in step 2

• Step 4. Regress pi on the X’s and obtain ESS (explained sum of

squares)

• Step 5. Obtain the B-P statistic (ESS/2) and compare it with the critical

chi-square value

critical chi-square value

There is heteroscedasticityThere is not heteroscedasticity

B-P B-P


Education

Microeconomics: Production Theory