Tarun Das Lecture Productivity-1

8/9/2019 Tarun Das Lecture Productivity-1

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UNSIAP TARUN DAS SESSION 17-18 PRODUCTIVITY 1

Production Functions and

Productivity MeasuresProfessor Tarun Das

Economic Adviser, MOF, India


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Contents

1. Production Function2. Cobb Douglas Production Function3. Equilibrium conditions4. Total Factor Productivity5. Solow Residual and Growth Accounting


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1.1 Production Function: Definition A technical relation which connects factor inputs and

outputs represents technology of a firm, industry or the economy includes all the technically efficient methods of production

a) Method of production: Combination of inputs (factors)required to produce one unit of output.

b) Technically efficient method of production: When there

are two methods A & B, A is said to be more technicallyefficient than B, if A uses less of at least one factor and nomore of other factors compared to B


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1.2 Isoquant: Let us assume that a firm uses two factors Labor (L) andCapital (K) and produces a single Output (Q). Then the production function

is given by Q = f (K, L). An isoquant is the locus of all feasible mix of

inputs (K, L) which produce the same level of output (Q). Q = f (K, L)

a) Linear isoquant

Perfect Substitutability

b)

Perfect complementarity of inputs


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c) Smooth curve isoquant:

Continuous substitutability of K and L over a

certain range, beyond which factors cannotsubstitute each other.

Isoquant

0

100

200

300

400

0 5 10

Capital

Labor

L


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General Production Function

1.3 General form of a production functionX = f (L,K,, )

Where X= value added

L= labour input

K= capital input

= Returns to scale parameter

= efficiency parameter (reflectingentrepreneurial- organizational aspects)


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1.4 Important concepts involved in production functions

There are two broad concepts of productivity averageproduct (AP) and marginal product (MP).Averageproduct measures the output per unit of an input, whereasmarginal product means rate of change in output due to

change in input by one unit.

Average product of factors APL = Q / L, APK = Q / K

Marginal product of factorsMPL = X/ L, MPK = X/ K


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1.5 A Numerical Example

L Y APL = Y/L MPL =dY/dL

1 631 631 631

2 956 478 325

3 1220 407 263

4 1450 362 230

5 1657 331 208

6 1849 308 192

7 2028 290 1798 2197 275 169

9 2358 262 161


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1.6 Important concepts involved in production functions

I)Average product is always positive. But, marginal product may bepositive, zero or negative. However, production theory concentrates

only on the efficient part of the production function (MPL>0 and MPK>0).

Also, the production theory concentrates only on the diminishing (but

positive) part of the marginal product. That is

MPL>0 and ( MPL )/ L=2X/ L2 < 0MPK>0 and ( MPK )/ K=

2X/ K2 < 0

ii)Marginal Rate of Substitution (MRS): The slope of the isoquant -K/

L is called the MRS or the rate of technical substitution. It defines the

degree of substitutability of factors.

-K/ L = (X/ L) / (X/ K) = MPL/ MPK


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1.7 Factor Intensity:

(iii) Factor intensity: Slope of the line joining origin to the isoquant

gives the factor intensity. The lower part of the isoquant is more

labour intensive while the upper part is more capital intensive.


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1.8 Some Important Concepts

iv) Elasticity of substitution: MRS depends upon the units ofmeasurement. Elasticity of substitution is a unit-free measure and is

defined as follows

= [(K/L)/(K/L)] / [(MRS)/(MRS)]

v) Product Lines:A product line shows the physical movement fromone isoquant to another, which is the same as the change in output

from level to another, resulting from change in one or both of the

factors of production.

vi) Isocline: Isocline is the locus of points on different isoquantswhere the MRS is constant. For homogenous functions, the isoclines

are straight lines passing through the origin and the K/L ratio (factor

intensity) is constant along the isocline. However, in case of non-

homogenous production functions, isoclines are not straight lines and

the K/L ratio (factor intensity) is not constant along the isocline.


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1.8 Applications

Cobb-Douglas Production Function

X=b0Lb1K

b2

i) Marginal Product of Factors

MPL = X/ L = b0b1L

b

1

-1

K

b

2 = b1/L[b0L

b

1K

b

2] = b1/L. X = b1. APL

Similarly, MPK = X/ K = b2/K. X = b2. APK

ii) MRS

--K/ L = MPL/ MPK = [b1/L. X] / [b2/K. X] = [b1/ b2]. [K/ L]

iii) Elasticity of Substitution

= [(K/L)/(K/L)] / [(MRS)/(MRS)]=

[(K/L)/(K/L)]/[[b1/ b2]. [K/ L]/ [b1/ b2]. [K/ L]] = 1


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1.9 Homogeneityand Returns to Scale

Suppose the production function is X= f (L, K) and we increase the inputs

P times, then the new output is X* = f (PL, PK).

If X*= f (PL, PK) = PF. f (L,K) = PF. X, then the production

function is said to be homogenous of degree F.

Otherwise it is a non-homogenous production function.

There are three possible cases of a homogenous production functions.

F=1; Constant returns to scale => Output increases in the same

proportion as inputs

F Output increases less than

proportionately with inputs

F>1; Increasing returns to scale => Output increases more than

proportionately with inputs


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2.1 Cobb-Douglas Production Function

In economics, the Cobb-Douglas functional form ofproduction function is the most oswidely used function torepresent the relationship of an output to inputs. It was

proposed by Kunt Wicksell (1851-1926), and testedagainst statistical evidence by Paul Douglas and Charles

Cobb in 1928. For production, the function isY = ALK,

where: Y= output; L = labor input ; K=capital inputand A, and are constants determined by technology.

If + = 1, the production function has constant returnsto scale

If + < 1, returns to scale are decreasing, and

If + > 1, returns to scale are increasing.


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3.1 Equilibrium of the Firm

a)Maximization of output subject to a cost constraint

Max X=f(L,K) subject to C = wL+rK

C-wL-rK=0 => Max X = Max = X+(C-wL-rK) where is theLagrangian multiplier

/ K =X/ K r = 0 => = 1/r. X/ K

/ L =X/ L w = 0 => = 1/w. X/ L

Equating the values of we get [X/ L]/[X/ K] = MPL/ MPK = w/r


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3.2 Minimization of cost subject to a output constraint

Min C = wL+rK subject to X=f(L,K)

X-f(L,K)=0 => Min C = Min = wL+rK+(X-f(L,K))

/ K =r-.X/ K = 0 => = r/[X/ K].

/ L =w-.X/ L = 0 => = w/[X/ L]

Equating the values of we get [X/ L]/[X/ K] = MPL/ MPK = w/r

Therefore the conditions for equilibrium of the firm are

i) MPL/ MPK = w/r

ii) The isoquants are convex to the origin


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3.3 Choice of the optimal expansion path

The optimal expansion path in the long run is the locus of

points of tangency of isocost lines and successive isoquants. If the

production function is homogenous, the expansion path is a straight

line through the origin with the slope being equal to the ratio of

factor prices. In the short run, it is a straight line parallel to the axis

of the variable factor

Cost Function from Production Function

Production Function: X = f(I)

Isocost Line: C = g(I)

X = f(I) => I=f-1(X) => C=g(f-1(X)) => C=h(X) (Cost Function)


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3.4 Constant Elasticity of Substitution (CES)

Production Function

CES production function has got two major characteristics.

i) It is homogenous of degree one

ii) It has a constant elasticity of substitution ()

PF s that lack one or both these features do not belong to the classofCES. For instance a P.F. of the form q = A. Kb1L

b2 have a

constant unit elasticity of substitution. However, they are

homogenous of degree one only if b1+b2 =1 (Cobb-Douglas with

constant returns). Therefore, none of the PF s of this general form,

except Cobb-Douglas, can be classified as CES production

functions.

CES P.F. = A[b1K-+ (1-b1)L

-] -1/

Where A>0 and 0


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3.5 CES Production FunctionThe marginal product of the factors is

q / K = b1/A .(q/K) 1+ and q / L = (1-b1)/A .(q/L) 1+, which arepositive in the domain x1, x2 >0

The MRS = b1/(1-b1). (L/K)1+.

Elasticity of substitution is given by = 1/(1+ ) => = (1- )/. The MRS

is decreasing and the isoquants are convex for >-1=> >0.

The equilibrium condition

MRS = b1/(1-b1). (L/K)1+. But, 1+ = 1/. Substituting this and equating

MRS to the ratio of factor prices we get,

MRS = b1/(1-b1). (L/K)1/ = r/w

L/K = a (r/w) where a= b1/(1-b1) ------------------(*)

(*) shows that in CES production function, the input ratio is a simple power

function of the input price ratio.


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3.6 CES Production Function

Applying logarithms on both sides of (*) we get

log (L/K) = log a + log (r/w) ------------(**)

It can be seen that (**) is simple log-linear regression. When you run this

regression (**), the slope coefficient gives you the elasticity of substitution.


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4.1 Total Factor Productivity

Take Cobb-Douglas Production Function

Y = ALK,

where: Y= output; L = labor input ;

K=capital input and A, and areconstants determined by technology.

Assuming perfect competition, and can

be shown to be labour and capital's share ofoutput, and undert constant returns to scale

+ = 1.


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The Cobb-Douglas function form can be estimated as

a linear relationship using the following expression:

Log Y = log A + E log L + F log K

Here, an increase in national income Y is explained by

an increase in the capital available (K), an increase

in the labor force (L), or an improvement in the

productivity used (A).The Production Function

shows that there are two factors involved in

economic growth

viz.factor accumulation andimprovements in efficiency.

Total-factor productivity (TFP) addresses any effects

in total output not caused by inputs or productivity.


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The levels of national income, the capital stock, and the sizeof the labor force can all be estimated through widelyavailable economic statistics.

A regression line can then be estimated to explain the levelof national income in terms of labor, capital and a residual.

A change in the residual, total factor productivity, represents

the change in national income that is not explained bychanges in the level of inputs (capital and labor) used.

Total Factor Productivity can be measured by A=Q/(Lx Ky)

This is normally taken as a measure of the level oftechnology employed. The annualized growth rate of A iscalled the Solow Residual. Over longer periods of time, it

may be used as a measure of technological change. Overshorter periods of time, it could reflect the effect of thebusiness cycles.


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5.1 Solow Residual

The Solow residual is a number describing empiricalproductivity growth in an economy from year to year anddecade to decade. Robert Solow defined rising

productivity as rising output with constant capital andlabor input.

It is a residual" because it is the part of growth thatcannot be explained through capital accumulation. TheSolow Residual is procyclical and is sometimes called therate of growth of total factor productivity.

Solows model is given by

Ln (Y(t)) = ln (A(t) + E ln (K(t)) + (1 - E ) ln (L(t)) + I

Or, ln (A(t)) = Ln (Y(t)) - E ln (K(t)) - (1 - E ) ln (L(t))


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5.2 Solow Growth Accounting Model


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5.3 Griliches-Jorgenson Dual Approach


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Thank you

Have a Good Day

Tarun Das Lecture Productivity-1

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