Technical Change

Technical Change

Definition

Technical change is defined as any shift in the production possibility set. Let t denote a technology index. Then, the production possibility set can be written as F(t), where z ≡ (-x, y)  F(t) means inputs x  Rn can be used to produce outputs y  Rm under technology t.

Technical change is defined as any shift in the technology index t, say from t to t’, such that F(t)  F(t’).
Technical progress is defined as a shift in the technology index from t to t’ such that F(t)  F(t’).

Technical progress from t to t’ means that it becomes possible to produce more outputs y using the same inputs x, or alternatively to produce the same outputs y using less inputs x.

Feasibility of the netput vector z ≡ (-x, y)  F(t)  Rn+m can be alternatively expressed using the corresponding output possibility set Y(x, t) ≡ {y: (-x, y)  F(t)}, or the input requirement set X(y, t) ≡ {x: (-x, y)  F(t)}. The production technology can be represented by:

the shortage function S(z, t, g) = infs {: (z - s g)  F(t)} (where g is a reference netput bundle satisfying g  R, g  0),
the output distance function DO(z, t) = inf {: (y/)  Y(x, t)},
the Farrell input distance function DF(z, t) = inf {: ( x)  X(y, t)},
or the Shephard input distance function DI(z, t) = sup {: (x/)  X(y, t)} = 1/ DF(z, t).

We know that z ≡ (-x, y) being feasible implies that S(z, t, g)  0, DO(z, t)  1, DF(z, t)  1, and DI(z, t)  1. Also, we know that, under free disposal, the boundary of the feasible set is given by S(z, t, g) = 0, DO(z, t) = 1, DF(z, t) = 1, and DI(z, t) = 1. These functions can provide a basis for measuring technical change. Consider the case where an increase from t to t’ is associated with technical progress: F(t)  F(t’), with t’ > t. Technical progress can be then assessed as follows:

S (z, t, g) - S(z, t’, g) > 0, where S(z, t, g) is decreasing in t, and [S(z, t, g) - S(z, t’, g)] measures the number of units of the reference netput bundle g that can be obtained under technical progress from t to t’. And under differentiability, |S(z, t, g)/t| provides a local measure of technical change.
ln(DO(z, t)) – ln(DO(z, t’)) > 0, where DO(z, t) is decreasing in t, and [ln(DO(z, t)) – ln(DO(z, t’))] measures the proportional increase in all outputs y that can obtained, given inputs x, due to technical progress from t to t’. And under differentiability, |ln(DO(z, t))/t| provides a local measure of technical change.
ln(DF(z, t)) – ln(DF(z, t’)) > 0, where DF(z, t) is decreasing in t, and [ln(DF(z, t)) – ln(DF(z, t’))] measures the proportional decrease in all inputs x that can obtained, given outputs y, due to technical progress from t to t’. And under differentiability, |ln(DF(z, t))/t| provides a local measure of technical change.
ln(DI(z, t’)) – ln(DI(z, t)) > 0, where DI(z, t) is increasing in t, and [ln(DI(z, t’)) – ln(DF(z, t))] measures the proportional decrease in all inputs x that can obtained, given outputs y, due to technical progress from t to t’. And under differentiability, ln(DI(z, t))/t provides a local measure of technical change.

Measuring Productivity
Partial productivity indexes

Commonly used partial productivity indexes are:

- land productivity, as measured by the evolution of crop yield (production per acre or per hectare).

- labor productivity, as measured by the evolution of production per worker or per unit of labor time.

These productivity indexes are called "partial" productivity indexes because they focus on the input-output relationship only one input at a time. They are relatively easy to estimate. However, the fact that they do not control for all input use suggests that they may not provide much information of the existence and nature of technical change. For example, an increase in yield does not necessarily imply technological change. It may simply be associated with an increase in the use of non-land inputs (e.g. fertilizer), i.e. a move along a given production frontier.

2.2.Productivity indexes

Given a technology index t, productivity indexes can be based on either the output distance function DO(z, t) or the Farrell input distance function DF(z, t). Recall that DO(z, t) = DF(z, t) under constant return to scale (CRTS), but DO(z, t)  DF(z, t) under non-CRTS. Each function has a different interpretation: DO(z, t) measures the proportional rescaling in all outputs y that gets point z ≡ (-x, y) to the production frontier, while DF(z, t) measures the proportional rescaling in all inputs x that gets point z ≡ (-x, y) to the production frontier. We have seen that the distance functions DO(z, t) and DF(z, t) can each be interpreted as a technical efficiency index.

In general, under the technology index t, the distance functions DO(z, t) and DF(z, t) have the following properties:

DF(z, t)  1 or DO(z, t)  1 implies that z ≡ (-x, y) is feasible,
DF(z, t) = DO(z, t) = 1 implies that z ≡ (-x, y) is on the frontier technology,
DF(z, t) < 1 or DO(z, t) < 1 implies that z ≡ (-x, y) is technically inefficient (as it is below the frontier technology),
Finally, DF(z, t) > 1 or DO(z, t) > 1 implies that z ≡ (-x, y) is “super efficient”: it is used under a technology that is “better” than technology t, placing point z ≡ (-x, y) above the frontier technology associated with the technology index t. Under technical change, this can happen if z ≡ (-x, y) is chosen under an “improved” technology t’ where F(t)  F(t’).

Consider a technological change from t to t’. To compare t and t’, define the following productivity indexes

PI(z) = DF(z, t)/DF(z, t’), for input-based index,

and

PO(z) = DO(z, t)/DO(z, t’), for output-based index.

As noted above, PI(z) = PO(z) under CRTS, but PI(z)  PO(z) under non-CRTS. This means that the choice between the input-based index and the output-based index is irrelevant under CRTS, although it becomes important under non-CRTS.

For simplicity, we will discuss the generic productivity index

P(z) = D(z, t)/D(z, t’)  0,

where D(z, t) is either DF(z, t) or DO(z, t), with the understanding that the index P(z) can be either PI(z) = DF(z, t)/DF(z, t’) or PO(z) = DO(z, t)/DO(z, t’).

Note: In the case where z ≡ (-x, y) is on the production frontier of technology t, then D(z, t) = 1, and the above productivity index becomes P(z) = 1/D(z, t’). In such a situation, the productivity index P(z) is the inverse of the distance function D(z, t’).

Alternatively, in the case where z ≡ (-x, y) is on the production frontier of technology t’, then D(z, t’) = 1, and the above productivity index becomes P(z) = D(z, t). In such a situation, the productivity index P(z) is equal to the distance function D(z, t).

In general, the productivity index P(z)  0 can be any magnitude.

Definition 1: The shift from technology t to t’ exhibits (weak) global technical progress if

P(z) > 1 ( 1) for all z ≡ (-x, y).

Global technical progress implies that D(z, t) > D(z, t’) for all z ≡ (-x, y). In other words, any input-output vector z ≡ (-x, y) appears “technically less efficient” under technology t’ (compared to technology t). This can be interpreted to mean that the production frontier is higher under t’ than t.

Under technical progress, the productivity index P(z) can be interpreted as follows:

[100  (PI(z) - 1)] is the percentage reduction in resource use x (or in production cost) that can be achieved by switching from technology t to t’ while producing outputs y.
[100  (PO(z) - 1)] is the percentage increase in output y (or in revenue) that can be achieved by switching from technology t to t’ while using inputs x.

In general, the productivity index P(z) can be estimated in an industry by observing firm behavior. These observations can take three forms: time series observations, cross-section observations, and panel observations (i.e., cross-section data over time).

Time series data can be used to evaluate productivity change in industry subject to significant technical change over time (e.g., involving large output increases and/or large input decreases). However, this approach works well only if the extent of technical inefficiency is small. Indeed, if technical inefficiency is common and significant, then it becomes difficult to identify a change in technical efficiency from a change in productivity. For example, from time series data alone, it is not possible to tell whether a technically inefficient firm sees its D index rise over time because it adopted a new technology, or because it became more efficient using the “old technology”. For that reason, productivity analysis based on time series data often assumes technical efficiency.

Cross-section data can be used to evaluate technical efficiency in an industry. However, this approach works well only if technical change is moderate and/or technological adoption is fast. Indeed, if technical change is significant and technological adoption is slow, then it becomes difficult to identify productivity changes from technical inefficiencies. For example, from cross-section data alone, it is not possible to tell whether a firm is technically inefficient because it is using an old technology (i.e., it is a late adopter of a new technology), or because it is an early adopter that is technically inefficient in the use of the new technology.

Panel data can be used to evaluate both technical efficiency and productivity growth in an industry. It can rely on each cross-section data to estimate a time-specific production frontier. This production frontier can be used to estimate technical efficiency indexes. And shifts in the estimated production frontier across periods can be used to estimate productivity growth.

Consider a firm choosing the input-output vector zt≡ (-xt, yt) under technology t. When there is technical change from t to t’, there are two possible productivity indexes measuring productivity growth for this firm:

P(zt) = D(zt, t)/D(zt, t’), (using zt as the reference point)

= 1/D(zt, t’) if the firm is technically efficient under technology t

< 1/D(zt, t’) if the firm is technically inefficient under technology t,

P(zt’) = D(zt’, t)/D(zt’, t’), (using zt’ as the reference point)

= D(zt’, t) if the firm is technically efficient under technology t’

> D(zt’, t) if the firm is technically inefficient under technology t’.

Other related indexes have been proposed in the literature. They include the Malmquist indexes

M(t) = D(zt’, t)/D(zt, t), (based on technology t)

= D(zt’, t) if the firm is technically efficient under technology t

> D(zt’, t) if the firm is technically inefficient under technology t,

and

M(t’) = D(zt’, t’)/D(zt, t’), (based on technology t’)

= 1/D(zt, t’) if the firm is technically efficient under technology t’

< 1/D(zt, t’) if the firm is technically inefficient under technology t’.

Note that, if technical efficiency holds under both technologies t and t’, then

M(t) = D(zt’, t) = P(zt’),

and

M(t’) = 1/D(zt, t’) = P(zt).

Thus, under technical efficiency, the Malmquist indexes M(t) and M(t’) are identical to the productivity indexes P(zt’) and P(zt), respectively. More generally, the following relationships hold

M(t) = D(zt’, t)/D(zt, t)

= [D(zt’, t’)/D(zt, t)]  [D(zt’, t)/D(zt’, t’)]

= [D(zt’, t’)/D(zt, t)]  P(zt’),

and

M(t’) = D(zt’, t’)/D(zt, t’)

= [D(zt’, t’)/D(zt, t)]  [D(zt, t)/D(zt, t’)]

= [D(zt’, t’)/D(zt, t)]  P(zt),

This shows that the Malquist indexes M(t) and M(t’) can each be written as the product of two terms, the first term [D(zt’, t’)/D(zt, t)] measuring technical efficiency effects, and the second term P() measuring technical change. It follows that

[M(t)  M(t’)]1/2 = [D(zt’, t’)/D(zt, t)]  [P(zt’)  P(zt)]1/2

This shows that the geometric mean of the two Malmquist indexes [M(t)  M(t’)]1/2 can be decomposed into two terms: the term [D(zt’, t’)/D(zt, t)] measuring technical efficiency effects, and the term [P(zt’)  P(zt)]1/2 measuring technical change.

The rate of technical change

Consider output-based measurements based on DO(z, t), the output distance function under technology t.

Definition 2: Assuming that DO(z, t) is differentiable in t, the rate of technical change associated with an increase in the technology index t is defined as: -ln(DO(z, t))/t.

The rate of technical changehas the following interpretation: it measures the proportional rate of increase in all outputs y that can be generated due to technical progress, given inputs x.

Above, we proposed the productivity index PO(z) = DO(z, t)/DO(z, t’) measuring technical change from t to t’ (with t’ > t). Taking logarithm, this gives

ln(PO(z)) = ln(DO(z, t)) – ln(DO(z, t’)).

If the distance function is differentiable in t, for a small change in t (with dt = t’ – t), this can be written as

ln(PO(z)) = [-ln(DO(z, t))/t] dt,

or when dt = 1,

ln(PO(z)) = -ln(DO(z, t))/t,

PO(z) = exp[-ln(DO(z, t))/t].

This implies that the rate of technical change, -ln(DO(z, t))/t, equals the logarithm of the productivity index, PO(z). Alternatively, the productivity index PO(z) equals the exponential of the rate of technical change, -ln(DO(z, t))/t.

The single product case

Consider the single output case where m = 1. The technology can then be represented by the production function

f(x, t) = maxy {y: (-x, y)  F(t)},

= maxy {y: y  Y(x, t)},

where Y(x, t) = {y: (-x, y)  F(t)} is the output possibility set given inputs x and technology t.

With DO(z, t) = y/f(x, t) and using definition 2, it follows that the rate of technical change associated with an increase in the technology index t is:

ln(f(x, t))/t.

And the associated output-based productivity index is:

PO(z) = exp(ln(f(x, t))/t).

The rate of technical change thus measures the relative change in output y due to the partial effect of one unit increase in the technology index t.

The rate of technical change can be measured:

from the production function, by estimating y = f(x, t) and deriving the value of [ln(f(x, t))/t] from the regression equation.
from the cost function, by estimating the cost function C(r, y, t), where C(r, y, t) = minx {r  x: y  f(x, t)}. The associated Lagrangean is L = r  x +  [y - f(x, t)]. Using the envelope theorem, note that C(r, y, t)/t = -[C(r, y, t)/y] [(f(x, t)/t]. It follows that the rate of technical change can be measured from the cost function as follows:

ln(f(x, t))/t = [f(x, t)/t]/y,

= -[C(r, y, t)/t]/[y C(r, y, t)/y],

= -[ln(C(r, y, t))/t]/[ln(C(r, y, t))/ln(y)].

(Note that, under constant return to scale, [ln(C(r, y, t))/ln(y)] = [C(r, y, t)/y]/[C(r, y, t)/y] = 1, implying that the rate of technical change is: ln f(r, t)/t = -[ln(C(r, y, t))/t], where a 1% upward shift in the production function is equal to a 1% decrease in the cost of production.

from the profit function, by estimating the profit function (p, r, t), where (p, r, t) = maxx {p f(x, t) - r  x}. Using the envelop theorem, note that (p, r, t)/t = [p f(x, t)/t]. It follows that the rate of technical change can be measured from the profit function as follows:

ln(f(x, t))/t = [f(x, t)/t]/y = [(p, r, t)/t]/[p y],

= [ln((p, r, t))/t]/[p y/(p, r, t)], for (p, r, t) > 0.

Measuring technical change as a residual

For simplicity, consider the single product case where m = 1 (all the arguments presented below can be easily extended to the multi-product case where m > 1). Assuming technical efficiency, consider the production function y = f(x, t). Total differentiation of this equation with respect to x and t gives

d ln(y) =  [ln f(x, t)/xi] d xi + [ln f(x, t)/t] d t.

It follows that the rate of technical change can be written as

ln f(x, t)/t = d ln(y)/[d t] - {[ln f(x, t)/xi] [d xi]/[d t]}. (1)

This illustrates that the rate of technical change is the rate of output change that cannot be explained by the change in inputs. Note that this implicitly treats technical change as a residual measure.

Total factor productivity indexes

Following on the implications of (1), denote by ri the input price for the i-th input. Under cost minimization, note that ln f(x, t)/xi = [f(x, t)/xi]/y = ri/[y C(r, y, t)/y], where we used the first order condition for cost minimization: f(x, t)/xi = ri/[C(r, y, t)/y], i = 1, 2, ..., n. It follows that, under cost minimization, equation (1) becomes:

ln f(x, t)/t = [d ln y]/[d t] -  {[ri/(y C(r, y, t)/y)] [d xi]/[d t]},

= [d ln y]/[d t] - {[ri xi/C(r, y, t)]/[ln C(r, y, t)/ln y] [d ln xi]/[d t]},(2)

where [ln C(r, y,t)/ln y] = [C(r, y, t)/y]/[C(r, y, t)/y] and [d ln xi] = [d xi]/xi. But, under constant return to scale, [ln C(r, y, t)/ln y] = [C(r, y, t)/y]/[C(r, y, t)/y] = 1. Thus, under constant return to scale, equation (2) takes the form:

ln f(x, t)/t = [d ln y]/[d t] - {Si [d ln xi]/[d t]},(3)

where Si = [ri xi/C(r, y, t)] denotes the i-th cost share, i = 1, 2, ..., n.

Now consider a change from t = 0 to t = 1. Denote by xit and yt the observed value taken respectively by xi and y at time t, t = 0, 1. Then, we have the following discrete approximations:

[d ln y]/[d t] = ln y1 - ln y0 = ln (y1/y0),

[d ln xi]/[d t] = ln xi1 - ln xi0 = ln (xi1/xi0), i = 1, 2, ..., n,

Si = ½ [Si0 + Si1],

where Sit = [rit xit]/[rit xit] is the i-th input cost share at time t, i = 1, 2, ..., n, t = 0, 1.

Using these approximations, equation (3) becomes

ln f(x, t)/t = ln[y1/y0] - {½ [Si0 + Si1] [ln (xi1/xi0)]}.(4)

Equation (4) provides an empirically tractable measure of the rate of technical change from t = 0 to t = 1.

Note that IO [y1/y0] can be interpreted as an output quantity index for the observation at t = 1, using t = 0 as a base. Also, note that II exp[{½ [Si0 + Si1] [ln (xi1/xi0)]}] is the Theil-Tornquist index of input quantity for the observations at t = 1, using t = 0 as a base. The index II provides a single measure of all the inputs used in the production process. This index is commonly used in empirical work. First, it can be shown to be a superlative index. (An index is said to be superlative if it is an “exact” index associated a “flexible” production function, i.e. a production function that does not impose a priori restrictions on the Allen elasticities of substitution). Second, it is an “exact” index associated with the translog (flexible) production function, a functional form often used in econometric work.

The rate of technical change given in (4) can thus be written as

ln f(x, t)/t = ln [IO] - ln [II] = ln [IO/II]. (5)

And we have shown that the output-based productivity index PO(z) can be obtained as the exponential of the rate of technical change,

PO(z) = exp(ln(f(x, t))/t). (6)

Combining (5) and (6) gives the following productivity index

PO(z) = IO/II, (7)

where IO is an output quantity index and II is an input quantity index. The productivity index in (7) is often called a total factor productivity (TFP) index. It has the intuitive interpretation of being an input-output ratio (in a way similar to the partial productivity indexes), except that the input is an aggregate input measured by the index II reflecting all n inputs.

From equations (4) and (6), the total factor productivity (TFP) index can be measured as:

PO(z) = IO/II = exp{ln[y1/y0] - {½[Si0 + Si1] [ln (xi1/xi0)]}}.(8)

Equation (8) is the Christensen-Jorgenson TFP index commonly used in productivity analysis.

Note 1: Other quantity indexes (besides the Theil-Tornquist quantity index) can also be used. Choosing different quantity indexes would generate different formulas for calculating a TFP index.

For example, equation (3) was derived from (1) under cost minimization and constant return to scale. As an alternative, consider using profit maximization. Let p denote the competitive market price for output y. Under profit maximization, note that ln f(x, t)/xi = [f(x, t)/xi]/y = ri /[p y], where we used the first order condition for profit maximization: f(x, t)/xi = ri/p, i = 1, 2, ..., n. It follows that, under profit maximization, equation (1) yields:

 ln f(x, t)/t = [d ln y]/[d t] - {[ri/(p y)] [d xi]/[d t]},

= [d ln y]/[d t] - {[ri zi/(p y)] [d ln xi]/[d t]}. (2’)

Then, equation (3) takes the form:

 ln f(x, t)/t = [d ln y]/[d t] - {si [d ln xi]/[d t]},(3’)

where si = [ri xi/(p y)], i = 1, 2, ..., n. The rate of technical change in (4) then becomes:

ln f(x, t)/t = ln[y1/y0] - {½[si0 + si1] [ln (xi1/xi0)]},(4’)

where sit = [rit xit]/[pt yt], pt being the price of yt, t = 0, 1. And the corresponding TFP index is:

TFP = exp{ln[y1/y0] - {½[si0+si1] [ln (xi1/xi0)]}}.

Note 2: In the multi-output case where y = (y1, y2, ..., ym) with corresponding prices p = (p1, p2, ..., pm), the Theil-Tornquist output quantity index (under constant return to scale) is: IO = {½[Si0 + Si1] [ln (yi1/yi0)]}, where Sit = [pit yit]/[i pit yit] is the i-th revenue share at time t, i = 1, 2, ..., m, t = 1, 2. Substituting this index in (6) and (9) provides the generalization of the above productivity analysis to a multi-output situation.

The nature of technical change

We have seen that technical progress takes place when there is an upward shift in the production frontier, or equivalently an inward shift in the isoquant map. The exact nature of this shift is of interest. Below, given z ≡ (-x, y), we present the analysis in the multi-output case based on the Shephard input distance function DI(z, t) = sup {: (x/)  X(y, t)}, where DI(-x, y, t) being linear homogeneous in x, and DI(z, t) = 1 represents the boundary of the feasible set associated with technology t.

7.1.Hicks neutral technical change

Definition 3: Technical change exhibits implicit Hicks input neutrality if the Shepard input distance function is weakly separable in the following sense

DI(-x, y, t) = D(h(-x, y), y, t),

where h: Rn+m R.

Note that DI(-x, y, t) being linear homogeneous in x, we can take h(-x, y) to be linear homogeneous in x, with DI(-x, y, t) = h(-x, y) D(y, t).

Under differentiability and assuming that DI/x > 0, implicit Hicks input neutrality implies that the marginal rate of substitution between any two inputs satisfies

MRSij = (DI/xi)/(DI/xj), for any i  j,

= [(D/h)(h/xi)]/[(D/h)(h/xj)],

= (h/xi)/(h/xj),

= homogeneous of degree zero in x, and independent of t.

Thus, under implicit Hicks input neutrality, the marginal rate of substitution between any inputs is independent of the technology index t.

Assuming that D(h, y, t) is strictly increasing in h, consider the implications of Hicks input neutrality for cost minimizing behavior. Under implicit Hicks input neutrality, we have

C(r, y, t) = minx {r  x: DI(-x, y, t)  1},

= minx {r  x: D(h(-x, y), y, t)  1}, if DI(-x, y, t) = D(h(-x, y), y, t),

= minH,x {r  x: D(H, y, t)  1}, H = h(-x, y)},

= minH {minx {r  x: H = h(-x, y)}: D(H, y, t)  1},

= minH {H c(r, y): D(H, y, t)  1}, where [H c(r, y)] = minx {r  x: H = h(-x, y)} since h(-x, y) is linear homogeneous in x,

= G(y, t) c(r, y),

where H = G(y, t) is the solution of D(H, y, t) = 1 for H.

From Shephard lemma, it follows that xic(r, y, t) = G(y, t) c(r, y)/ri, i = 1, …, n. This implies that xic(r, y, t)/xjc(r, y, t) = [c(r, y)/ri]/[c(r, y)/ri] is independent of t for all i  j. Thus, under implicit Hicks input neutrality, relative cost-minimizing input demands are independent of the technology index t. This means that the relative cost share {ri, xic(r, y, t)/[rj xjc(r, y, t)]} is also independent of t for all i j. Letting Si(r, y, t)  ri xic(r, y, t)/C(r, y, t) denote the i-th cost share, it follows that the cost shares Si(r, y, t) are independent of the technology index t under implicit Hicks input neutrality for all i = 1, ..., n. Intuitively, under implicit Hicks input neutrality, technical change does not affect the relative contribution of each input to the production process.

7.2.The single output case

In the single output case (where m = 1) and under free disposal, the Shephard input distance function can be used to obtain the associated production function: solving DI(-x, y, t) = 1 for output y yields the production function y = f(x, t).