Factor Bias, Technical Change, and Valuing Research

Lecture XXIV

Mathematical Model of Technical Change

If we start from the quadratic production function specified as

assuming an output price of p and input prices of w1 and w2 for inputs x1 and x2, respectively, the derived demands for each input can be expressed as

1 2 0 1 1 2 2

2 211 1 12 1 2 22 2

,

1 22

f x x a a x a x

A x A x x A x

* 12 2 22 1 22 1 12 21 1 2 2

11 22 12

* 12 1 11 2 12 1 11 22 1 2 2

11 22 12

, ,

, ,

A a p A a p A w A wx p w w

p A A A

A a p A a p A w A wx p w w

p A A A

In order to analyze the possible effect of technological change, we hypothesize an input augmenting technical change similar to the general form of technological innovation introduced by Hayami and Ruttan.

Specifically, we introduce two functions

where γ1(Ψ) and γ2(Ψ) are augmentation factors and Ψ is a technological change

*1 1 1

*2 2 2

x x

x x

Hence, γ1(Ψ), γ2(Ψ)≥1 for any Ψ. Thus, technological change increases the output created by each unit of input. Integrating these increases into the forgoing production framework, the derived demands for each input becomes:

*1 1 2

12 2 1 2 22 1 1 2 22 2 1 12 1 2

2 211 22 12 1 2

*2 1 2

12 1 1 2 11 2 1 2 12 2 1 22 1 22

11 22 12 1 2

, , ,

, , ,

x p w w

A a p A a p A w A w

p A A A

x p w w

A a p A a p A w A w

p A A A

In order to simplify our discussion, we assume that the new technology does not affect the effectiveness of x2 , or γ2(Ψ) → 1 . Under this assumption the derived demand for each input becomes

*1 1 2

12 2 1 22 1 1 22 1 12 1 2

2 211 22 12 1

*2 1 2

12 1 1 11 2 1 12 1 22 1 2

11 22 12 1

, , ,

, , ,

x p w w

A a p A a p A w A w

p A A A

x p w w

A a p A a p A w A w

p A A A

In order to examine the effect of the technological change on each derived demand, we take the derivative of each of the demand curves with respect to Ψ as γ2(Ψ) → 1 yielding

1

1

*1 1 2 12 2 22 1 22 1 12 2 1

211 22 121

12 2 22 1 12 2 1

211 22 12

*2 1 2 12 1 11 2 12 1 11 2 1

211 22 121

12 1 11 2 11 2 1

211 22 12

, , , 2

, , ,

x p w w A a p A a p A w A w

A A A p

A a p A a p A w

A A A p

x p w w A a p A a p A w A w

A A A p

A a p A a p A w

A A A p

Valuing State Level Funding for Research: Results for Florida

The most basic definition of productivity involves the quantity of output that can be derived from a fixed quantity of inputs. For example, most would agree that a gain in productivity has occurred if corn yields increased from 70 bushels per acre to 75 bushels per acre given the same set of inputs (i.e., pounds of fertilizer, or hours of labor).

Aggregate agricultural outputs and inputs could be computed based on Divisia quantity indices. Specifically, Yt let be the aggregate output index computed as:

where rit is the revenue share of output i.

n

i ititt yrY1

Similarly, the aggregate input index can be computed as Xt

where sit is the cost share of input i.

Equating aggregate output with aggregate input yields

m

j jtjtt xsX1

t t tY X

Rearranging slightly yields

The rate of technical change can the derived from the log change in both sides:

tt

t

Y

X

1

1 1

ln ln lnt t t

t t t

Y Y

X X

TFP in the Southeast

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Tota

l Fac

tor

Prod

uctiv

ity

Alabama Florida Georgia South Carolina

TFP Growth Versus R & D Stock

16.4

16.6

16.8

17

17.2

17.4

17.6

17.8

18

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Nat

ural

Log

arith

m o

f 19

96 D

olla

rs

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Nat

rual

Log

arith

m o

f T

FP

R & D Stock Total Factor Productivity

The Johansen (1988) approach involves estimating a vector error-correction mechanism expressed as

where xt is a vector of endogenous variables, ∆xt denotes the time-difference of that vector, Dt is a vector of exogenous variables, εt is a vector of residuals, and Π , Γi , and Φ are estimated parameters.

11

k

t t i t i t ti

x x x D

If a long-run relationship (e.g., cointegrating vector) exits, the Π matrix is singular (Π=α’β ). The β vector is the cointegrating vector or long-run equilibrium.

The statistical properties of the cointegrating vector are determined by the eigenvalues of the estimated Π matrix.

Denoting λi represent the ith eigenvalue (in descending order of significance), the test for significance of the cointegrating vector can be written as

1 11

ˆ2 ln ln 1p

ii r

Q H r H p T

which tests the hypothesis that r cointegrating vectors are present, H1(r) , against the hypothesis that p cointegrating vectors are present, H1(p)

The existence of a cointegrating vector in this framework implies that the linear combination (zt) of the natural logarithm of TFP and research and the natural logarithm of research and development stocks (RDt ) is stationary, or a long-run equilibrium between these two series exists.

While this cointegrating vector is not uniquely identified, the long-run relationship can be expressed as

Building on this expression, the long-run relationship can be expressed as

ln1.00013.433

ln0.794t

tt

TFPz

RD

ln 0.794ln 13.433t t tTFP RD z

Manipulating this result further, yields

Using the geometric mean of both TFP and research and development stocks, TFP increases 0.0302 with a one million dollar increase in the research and development stock. This number appears small, but it represents 113 percent of the average annual increase in productivity observed in the state.

0.794t t

t t

d TFP TFP

d RD RD

In order to understand the possible causes of the lack of a long-run equilibrium between agricultural profitability and productivity, I express the change in profit over time as:

t t tD D F D

where πt denotes profit in period t, Ft denotes Total Factor Productivity in time t, and Ψt denotes the change in relative price ratio in time t.

In order to derive this relationship, we start with agricultural profit defined as:

1 1

n m

t it it jt jti jp y w x

Differentiating both sides yields:

Rewriting this expression using logarithmic differentiation yields

1 1

1 1

n n

t it it it iti i

m m

jt jt jt jtj j

d dp y p dy

dw x x dw

1 1

1 1

n n

t it it it iti i

m m

jt jt jt jtj j

D r D p r D y

s D w s D x

This expression can be rearranged to yield

1 1

1 1

n m

t it it jt jti j

n m

it it jt jti j

D r D y s D x

r D p s D w