Theory of Production

COM 3305Economic Analysis for Managers

Theory of Production

Production - a process through which factor inputs are made into output that directly or indirectly satisfy consumer demand

2

PRODUCTION

SHORT RUNAT LEAST ONE

FIXED FACTOR &GIVEN TECHNOLOGY

LONG RUNALL FACTOR INPUTSVARIABLE BUT NOT

technology

VERY LONG RUNALL FACTOR INPUTS

AS WELL AS TECHNOLOGY VARY

Short Run Production

3

L3L1

L3

Rate of Output Q

Rate of Labour input L

Rate of Labour input L

TPL

MPL

L1 L2

APL

Increasing Marginal Returns

Diminishing Marginal Returns

Negative Marginal Returns

Rate of Output Q

• Short run: Production with one variable input. maximum rate output obtainable from a

given combination of fixed capital and labour input Diminishing Marginal returns prevails Optimal input of variable factor of production, labour

is obtained when OR

When value marginal product of labour is equal to market money wage rate, or marginal product of labour equals to real wage rate. When product market is imperfectly competitive, this condition changes to i.e. marginal revenue product of labour equals to money wage rate

4

),( LKfQ

xL

x

Lx

x

L

PWMP

PMPP

PVMP

.WVMPL

WMRPL LxL MPMRMRP .

Determination of optimal variable factor input in the Short run

LVMP

5

Wage

2w

3w

Labour

LMRP

8136

Marginal revenue product function or value marginal product function of labour is the demand curve for labour

LD

• The marginal product of labor function for Helamuthu Millers is given by the equation,

MPL= 10(K/L)0.5

• Currently the firm is using 100 units of capital and 121 units of labor. Given the very specialized nature of the capital equipment, it takes six to nine months to increase the capital stock, but the rate of labor input can be varied daily. If the price of labor is Rs. 10 per unit and the price of output is Rs.2 per unit is the firm operating efficiently in the short run? If not explain why and determine the optimal rate of labor input.

• Long run: Production with Two variable inputs maximum rate output obtainable

from a given combination of capital and labour input which are variable

A firm faces with the problem of efficient allocation of resource in production, i.e., produce output that maximizes profits.

Two analytical tools are used: Production isoquant and production isocost.An isoquant shows all different combinations of

capital and labour input that produce a given level of output

Isocost line defines all different combinations of capital and labour that can be purchased for a given outlay or expenditure/ budget

),( LKfQ

7

Determination of optimal variable factor inputs in the Long run - Isoquants

8

Labour (L)

Capital (K)

200Q

346Q

490Q

4

4

2

2

1

1

21

21

100 LKQ

Slope of an isoquant is equal to the negative of the ratio of marginal products of labour and capital, i.e., K

LLK MP

MPMRTS

Determination of optimal variable factor inputs in the Long run -Isocost

9

13.3

Labour (L)

Capital (K)

30C40C

50C

16.7

2520

10

15

LwKrC ..

Slope of an isocost is equal to the negative of the ratio of wage and rent, i.e.,

Lrw

rCK

LK32

350

rw

32 rw and

Optimal variable factor inputs in the Long run-Isoquant and Isocost

10

13.3

Labour (L)

Capital (K)16.7

2520

10

15

LwKrC ..

Optimal inputs of factors of production is obtained when isoquant is tangent to the isocost line, i.e., when

Lrw

rCK

LK32

350

rw

MPMP

K

L

Production expansion path

Optimal variable factor inputs in the Long run: Effect of change in input prices, example – Wage rate of labour increases or alternatively rent for capital increases

21

21

100 LKQ

LAKQ

0L

11

Labour (L)

Capital (K)16.7

25

LwKrC ..

Increased wage reduces input of labour and increases input of capital Increased rent reduces input of capital and increases input of labour

Lrw

rCK

LK32

350

Capital labour ratio K/L

1K

2K

1L 2L

0K

Japan’s Minister Calls American Workers “Lazy”

12

Isocost in Japan

16001090

Isocost in United States ● A

● B

Q = 100

Capital

Labour

Higher capital-labour ratio in Japan

Lower capital-labour ratio in US

US uses more labour intensive production technology; Labour relatively cheap in US Japan uses more capital intensive production technology; Capital relatively cheap in Japan

Returns to ScaleRefers to the magnitude of the change in the rate of output relative to the change in scale.

• Constant Returns to Scale– When Increase input by 100%– Output also increases by 100%

• Increasing Returns to Scale– When inputs are increased by 100% – Output will increased by greater than 100% (i.e. 120%)

• Decreasing Returns to Scale– When inputs are increased by 100% – Output increases by less than 100% (i.e 50%)

13

Output elasticityIt represents the change in the quantity produced as the change in with respect to the inputs

• Eo = % change in output (ΔQ%)

% change in inputs (ΔX%) or

= ΔQ x X ΔX Q

Decision criteria: when Eo > 1 it reflects an increasing returns to scale, where Eo < 1 it reflects a decreasing returns to scale and Eo = 1 constant returns to scale.

Specification of the production function

• Steps to be followed– Data collection using cross sectional data or time

series– Production specification: Deciding and

conceptualizing a functional form (linear or non linear form)

– Estimating the parameter values using the method selected

Production specification• Q = f(L,K) this is the basic production function that we can

estimate using two categories of inputs. In order to analyze the behavior of these parameters we can use regression analysis.

• Q = aK + bL• There are certain limitations in this linear production function.I. It is not necessary to use both inputs to achieve an expected

output levelII. In a linear curve it is harder to reflect the marginal rate of

substitutionIII.In a linear equation marginal product of labor(MPL) is constant

To eliminate these weaknesses it is recommended to use none linear form or the non linearity in the parameters.

• Y = aXbZc

• this functional form is important in calculating and estimating the values for the parameters of b and c.

• b= % change in y / % change in x• C = % change in y / % change in ZThe values in these two parameters (b and c)

represents the elasticities.

Cobb-Douglas Production Function • This has been developed by two economists “

Charles Cobb” and “Paul Douglas” in 1920 in order to reflect the behavior of inputs and outputs.

• Q = A K αLβ

• Benefits of this function– Can calculate marginal product of labor and capital– Can analyze output elasticities– Can estimate economies of scale

Multiple output cost functions• Economies of Scope: when total cost of

producing and is less than total cost of producing and separately

• or • alternatively measure of ES

• Cost Complementarity: when marginal cost of producing one output declines when output of another product is increased

19

1Q 2Q1Q 2Q

),()()( 2121 QQCQCQC

),(),()()(

21

2121

QQCQQCQCQC

S

The production function of a firm is given by,Q = 2K0.5L0.5

Where Q is the level of output K and L are capital and labor respectively. Assume that the capital stock is fixed and (K) = 9 units. The price of output, (P) = Rs. 6 per unit, and the wage rate (W) = Rs. 2 per unit.

(a)What is the profit maximizing number of labor to be hired?

(b)What is the optimal number of labor if the wage rate increased to (W) = Rs. 3 per minute.

The production process of the Uni-lever ltd. Is given by the following Cobb-Douglas function,

Q = 100K0.6L0.5

Where K is capital inputs and L is labor inputs. (a)Calculate the output elasticity of each input. (b)What is the returns to scale shown by the

above production function?(c)What is the increase quantity of output if the

company increases input of both capital and labor by 8 percent?

The Cobb-Douglus production function estimated by the XYZ company is as follows.

Q = 100K0.6L0.4

where, Q = Units of outputK= Capital unitsL= Labor units

(a)Calculate output elasticities of labor and capital.(b)What type of returns to scale does this production

indicate?