Contents

(A) Leontief (Fixed Proportions) Technology.

(B) Activity Analysis Technology

(C) Putty-Clay Technology

The concept of marginal productivity relies upon the idea of adding a
particular factor to a production process while letting all other factors remain constant.
This translates, in production theory, to changing the *technique* of production,
which is a little more than a fancy term for the factor proportions employed in a
particular production process. In the canonical smooth, two-input, one-output production
function, Y = ｦ (K, L), the *technique* of production at
any particular point is merely the capital-labor ratio, K/L. Consequently, refer to *technology*
as the set of feasible techniques available to the producer. If the producer wishes to
produce a particular desired level of output, Y*, then the technology he has available are
all the capital-labor ratios matching the points on the relevant isoquant.

**(A) Leontief (Fixed-Proportions) Technology**

The smooth curvature of the isoquants in our previous section implies that
we assumed that there was an *infinite* number of techniques available and that the
producer could substitute continuously among them. Indeed, in order to be able to *define*
the marginal product as a derivative, we need this assumption.

However, we earlier spoke of the possibility
that technology may be such that factors *must*, for some reason or other, be used in
fixed, constant proportions. Taking Cassel's
(1918) example, in digging a pit, adding an extra worker alone may not increase output if
we do not also give him a spade and certainly, adding a spade alone will not increase
output unless we also increase labor. In this case, labor and capital must work together
in a constant proportion: one man, one spade. As we can see, there is *no*
possibility of substituting a spade for a man or a man for a spade. This translates, in
our context, to arguing that the technology facing the producer is composed entirely of a *unique*
technique, i.e. capital and labor must be used in a fixed proportion.

Suppose that our particular production function Y = ｦ
(K, L) exhibits this fixed proportions property. In other words, let us suppose that to
produce a single unit of output, we *need* v units of capital *and* u units of
labor. There is no flexibility in technique here. The coefficients v and u are the fixed
input requirements in order to produce a single unit of output. Consequently, if we want
to produce Y units of output, we need vY units of capital and uY units of labor. In other
words, K = vY are the capital requirements and L = uY are the labor requirements. As a
result, the only technique is L/K = u/v. In other words, there is a *particular*
fixed proportion of capital and labor required to produce output. An increase in either
one of the factors by themselves without increasing the other proportionally will lead to
absolutely no increase in output.

The implied L-shaped isoquants of such a production function are shown in
Figure 4.1. Such a technology is referred to alternatively as "Fixed
Proportions" or "No Substitution", or "Marx-Leontief" or
"Walras-Cassel" or "Input-Ouput" technology (or some iteration
thereof). At any particular output level Y*, there is a necessary level of K* and L* which
cannot be substituted. Note that these levels are determined purely technologically.
Increasing only labor inputs (from L* to L｢ for instance) will
*not* result in any higher output. Rather, the extra labor, without the extra capital
to work with, will be entirely wasted. The implication is that fixed-proportions
technology is "no less than a formal rejection of the marginal productivity theory.
The marginal productivity of any [factor] ... is zero." (Leontief, 1941, p.38).

Figure 4.1- Leontief (No Substitution) Isoquants

The production function for a no-substitute case can be written as:

Y = min(K/v, L/u)

which is also referred to as a *Leontief production function* - as
this form was introduced by Wassily Leontief
(1941). Notice that if K is at K* and L is at L｢ , then K*/v
< L｢ /u. Thus, Y = K*/v. If so, then the technically
efficient level of labor would, by definition, be where K*/v = L/u or L = (u/v)K* which,
as is obvious in Figure 4.1, is at L*. As a result, then we can easily note that the
following holds all along the ray from the origin:

Y/L = (1/v)K/L.

This implies that the *intensive* production function, y = f (k) where y = Y/L and k = K/L is effectively a straight line with
slope 1/v up to the capital-labor ratio k* = K*/L* and is horizontal thereafter, as shown
in Figure 4.2. This constrasts sharply with smooth concave
intensive production function we considered earlier in Figure 3.4.

Figure 4.2 -Intensive Leontief Production Function

The reasoning for the shape is clear enough. In intensive production
function representations, it is *as if* we are holding labor constant and just
increasing capital.. Now, if the capital-labor ratio is precisely k* = K*/L*, we are
sitting on the ray in Figure 4.1, thus the best one can do is produce y*. If we attempt to
increase capital above K* (and thus increase the capital labor ratio above k*), output
does not increase at all. Thus, the output-labor ratio remains unchanged at y*. However,
if the capital labor ratio falls *below* k*, it is as if we reduced capital while
leaving labor the same. As we know from Leontief production functions, we must reduce
output. Alternatively, we could a decline in k below k* as leaving capital unchanged and
just increasing the amount of labor. In either case, the output labor-ratio declines below
y*. Thus, starting from k = 0, then up to k*, the output-labor ratio increases at rate
1/v, while after k* it remains unchanged.

**(B) Activity Analysis Technology**

A moderately more flexible version of this type of situation is the *activity
analysis* form of production technology, introduced into economics largely by John von Neumann (1937) and taken up enthusiastically by
the Neo-Walrasians (e.g. Koopmans, 1951). In addition to the already cited
texts, activity analysis production theory is summarized in W.J. Baumol (1958, 1961), R. Dorfman (1958), R. Dorfman, P.A. Samuelson and R. M. Solow (1958) and D. Gale
(1960).

In activity analysis, producers can choose among a small, finite number of
distinct *production processes* or *activities*. We do not use the word
"technique" for reasons that will be clear soon.* *This is shown in Figure
4.3, where we are given three possible activities, represented by the rays A_{1},
A_{2} and A_{3}. In activity analysis, each of these rays is commonly
referred to as an *activity ray* or a *production process ray*. Each of the
activity rays has a different slope representing the different fixed proportions of
capital and labor used in that particular activity. In Figure 4.3, we denote these slopes
by the terms u_{i}/v_{i}.

The terms u_{i}, v_{i} are the *unit input coefficients*
for the ith activity. Specifically, to produce a single unit of output using technique A_{i},
we need v_{i} units of capital and u_{i} units of labor, thus to produce
Y* units of output using that activity, the capital requirements are K_{i} = v_{i}Y*
and the labor requirements are L_{i} = u_{i}Y*. Thus, the slope of the ith
activity ray will be L_{i}/K_{i} = u_{i}/v_{i}. Notice
that this implies that, in Figure 4.3, u_{1}/v_{1} > u_{2}/v_{2}
> u_{3}/v_{3}, which implies that A_{1} the most
labor-intensive (and least-capital intensive) of the three activities, while A_{3}
is the most capital-intensive (and least labor-intensive) process.

Figure 4.3- Activity Analysis Isoquants

Now, we can achieve a particular level of output, Y*, by undertaking
production processes A_{1}, A_{2} or A_{3} or some combination of
these. As a result, the activity analysis isoquant is kinked in Figure 4.3. Thus, if we
choose to use process A_{2}, then our labor-capital ratio must be u_{2}/v_{2},
as is forced upon us by the slope of the activity ray A_{2}. However, in order to
produce Y*, we need to be at point e_{2} on the activity ray A_{2}. At e_{2},
the inputs are K_{2} = v_{2}Y* and L_{2} = u_{2}Y*. If we
wished to produce more output (say Y｢ ) but without changing
technique, we would move radially up the activity line A_{2} from e_{2} to
e_{2｢ }in Figure 4.3. Note that the labor-capital
ratio is the same at both these points.

The advantage of activity analysis models over our earlier
Leontief ones is that we are not constrained to using only one activity. We can, for
instance, use two activities at the same time. Suppose, as shown in Figure 4.3, we choose
to produce output Y*, but by using a combination of activities A_{2} and A_{3}.
In other words, we produce some of our output using process A_{2} and some of it
using process A_{3}. As a result, we end up at point b on the isoquant in Figure
4.3. Note that b lies on a straight line between e_{2} and e_{3} thus we
can say it is some convex combination of the two activities:

b = l e

_{2}+ (1-l )e_{3}

where l ﾎ (0, 1).
Thus, the level of use of activity A_{2} is scaled down from e_{2} to l e_{2} on activity ray A_{2}, while the level of use
of activity A_{3} is scaled down from e_{3} to (1-l
)e_{3} on activity ray A_{3}.

The use of capital at point b would be:

K

_{b}= l v_{2}Y* + (1-l )v_{3}Y*

where v_{2} is the capital-output ratio in activity A_{2}
and v_{3} the capital-output ratio in activity A_{3}. Similarly the
employment of labor implied by b is:

L

_{b}= l u_{2}Y* + (1-l )u_{3}Y*

Thus, the use of capital and labor will be partly allocated to one process
and partly allocated to another. In other words, the producer at b will use two processes
(A_{2} to A_{3}), allocating proportion l of
the total activity to process A_{2} and the rest (1-l )
to process A_{3}. As l approaches 1, we will move from
less use of process A_{3} to more use of process A_{2} to produce the
desired level of output, Y*, (i.e. the b would be closer to e_{2} than e_{3}).

As we see, activity analysis isoquants allow us not only to choose among
several different production processes, but also any combination of production processes.
Thus, compared with the Leontief case, the activity analysis production technology allows
for a (modest) degree of substitability between different factors by allowing producers to
choose combinations of different activities. More strictly, combinations of A_{1}
and A_{2} (such as at c in Figure 4.3) and combinations of A_{2} and A_{3}
(such as at point b) are allowed. However, note that *no* combinations of processes A_{1}
and A_{3 }will be contemplated to produce output Y*. This is not due to lack of
possibility but rather because of the lack of efficiency of any such combination. In
particular, a convex combination of e_{1} and e_{3} would lie above the
isoquant formed by e_{1}e_{2}e_{3}, but this does *not* imply
a higher level of output but rather higher uses of inputs to produce the same output.
Thus, combinations of A_{1} and A_{3} are inefficient. (cf. Koopmans, 1951).

Note that while there is subsitutability between production processes,
there is still no guarantee that factor prices will be decisive in determining factor
proportions. For instance, if the isocost rests on the A_{1}A_{2} segment,
we might choose R, but we can also very well choose any other point on that segment. Thus,
factor price ratios will only determine "more or less" the activity (or
activities) we will use, but will not determine the precise proportions of capital and
labor employed. While substitution exists, it is "moderate". It is not
"substitution" between factors, but rather substitution between activities.

In order to compare activity analysis production functions with other
production functions, it may be useful to visualize the resulting kinked intensive
production function y = f (k), as shown in Figure 4.4. The
labelling corresponds to the three-activity scenario we had earlier in Figure 4.3. The
logic for this shape is simple. Let the output-labor ratio for the ith process used be
denoted y_{i} = (Y/L)_{i} = 1/u_{i} and the capital-labor ratio is
k_{i} = (K/L)_{i} = v_{i}/u_{i}. Thus:

y

_{i}/k_{i}= (1/u_{i})/(v_{i}/u_{i}) = 1/v_{i}

As this is true for a particular activity, we can impose three new rays, B_{1},
B_{2} and B_{3}, with slopes y_{i}/k_{i} = 1/v_{i}
for each of the three activities representing the three activities in Figure 4.4. Now,
from Figure 4.3, we know that v_{1} < v_{2} < v_{3} as A_{3}
is the most capital-intensive and A_{1} is the least capital-intensive. Thus, B_{1}
is the steepest ray and B_{3} is the flattest, as we see in Figure 4.4.

Figure 4.4- Intensive Activity Analysis Production Function

Let us now trace out the resulting intensive production function in Figure
4.4. Capital-labor ratios k_{1}, k_{2} and k_{3} on the horizontal
axis correspond to v_{1}/u_{1}, v_{2}/u_{2} and v_{3}/u_{3}
respectively. Consequently, if the capital-labor ratio is *precisely* k_{1},
then the most efficient thing to do is adopt activity A_{1} and produce as much as
we can. To see why this is the case, consider our earlier Figure 4.3 again. Suppose the
capital labor endowment is K_{1} and L_{1}, so that the capital-labor
ratio is k_{1} = K_{1}/L_{1}. Now, if the producer uses the
labor-intensive process A_{1} he can produce at activity level e_{1｢ }and achieve output Y｢ . However,
if we force him to use the more capital-intensive process A_{2}, he will be
produce at activity level e_{2} and thus achieve the lower output Y*. This is
because the more capital-intensive process will be able to absorb the entire amount of
capital, K_{1}, but it can only use L_{2} of labor, thus the remaining
labor, L_{1}-L_{2}, will have to be thrown away. Thus, it is *not*
efficient to use process A_{2} when facing a low capital-labor ratio such as k_{1}
= K_{1}/L_{1}.

Let us return to Figure 4.4. By what we have just said, when capital-labor
ratio is k_{1}, we will produce with process A_{1} and achieve
output-labor ratio y_{1}. Using the same logic, if the capital-labor ratio is
precisely k_{2}, the most efficient avenue is to produce with process A_{2}
and achieve output-labor ratio y_{2}. Simlarly, if the capital-labor ratio is
exactly k_{3}, we produce with process A_{3} and achieve output-labor
ratio y_{3}.

Now, by an analogous logic to that of the earlier Leontief case, if the
capital-labor ratio is so low that it lies below k_{1}, then the most efficient
thing to do is to produce output completely in accordance with the most labor-intensive
process available. There will be waste, of course, as labor is so plentiful that some of
it will go unused, but it is the best that can be done given that only these three
activities are available. Consequently, for capital-labor ratios below k_{1},
activity A_{1} will be used exclusively, consequently the output achieved will be
according to the B_{1} ray for the same reasoning we gave in the Leontief case.
Indeed, as we saw in Figure 4.3, the isoquant above the A_{1} ray is vertical,
precisely as in Leontief.

If, in contrast, the capital-labor ratio is very high (above k_{3})
then the best thing to do is to adopt the most capital-intensive process available, namely
A_{3}. Again, there will be waste of capital, but it is the best we can do. Note
that, again, in Figure 4.3, the isoquant beyond the A_{3} ray is horizontal, as in
Leontief, thus the reasoning for a horizontal portion of the production function above k_{3}
is the same. In other words, increasing capital-labor ratio *beyond* k_{3}
results in no increase in the output-labor ratio.

The interesting portion of the production function in Figure 4.4 is that
between k_{1} and k_{3}. The points on the production function between the
rays B_{1} and B_{2} correspond to combinations of processes A_{1}
and A_{2}, as in our earlier Figure 4.3. Similarly, points on the production
function between B_{2} and B_{3} correspond to combinations of processes A_{2}
and A_{3}. We can see immediately here why a combination between activities A_{1}
and A_{3} is inefficient: a chord connecting the kinks of the production function
at b_{1} and b_{3} would lie entirely below the production function, thus
it is inefficient. In sum, the intensive activity analysis production function shows that
increasing from labor-intensive to capital-intensive techniques, we move from low
capital-labor (and low output-labor) ratios to higher capital-labor (and high
output-labor) ratios.

To understand the activity analysis production used in von Neumann and
Neo-Walrasian models, it is useful to introduce some more general notaiton. Let there be k
different production processes or activities. Let there be m factors of production. A
production process is characterized by a *unit intensity vector*, denoted a_{i}.
In general, a unit intensity vector is an m-dimensional vector:

a

_{i}= [a_{1i}, a_{2i}, ..., a_{mi}]｢

where a_{ji} is a *unit input coefficient* denoting to the
amount of factor j required by activity i when activity i is run at "unit
intensity". Note that these unit input coefficients in activity analysis are *different*
from those used in input-output analysis. A unit input coefficient a_{ji} does *not*
indicate the amount of input j required to produce a unit of output via process i. Rather,
it indicates the amount of input j required to produce whatever units of output are
produced when process i is run at so-called "unit intensity". Thus, a_{ji}
is *not* the general analogue of our earlier coefficients u_{i} or v_{i}.

To understand the meaning of unit intensity, refer to Figure 4.5. Here we
have two factors (capital K and labor L) and two activities represented by their activity
rays from the origin, A_{1} and A_{2} respectively. Notice that there is a
curve connecting points (1, 0) and (0, 1) is the "normalizing" curve (it is part
of the circumference of a circle with radius 1). Activity i is run at *unit intensity*
if the vector representing it is on the this curve. Thus, a_{1} is the unit
intensity vector of activity A_{1} and a_{2} is the unit intensity vector
of activity A_{2}. Notice that a_{1} = [a_{K1}, a_{L1}]
and a_{2} = [a_{K2}, a_{L2}], thus the slope of the activity ray A_{1}
is a_{L1}/a_{K1} and the slope of activity ray A_{2} is a_{L2}/a_{k2}.

Figure 4.5- Activity Analysis Vectors

Any particular point along an activity ray A_{i} can be
represented as a scaling up or down of the unit intensity vector. Thus, any vector on A_{i}
can be denoted z_{i}a_{i} where z_{i} is referred to as the *intensity*
of activity i. This is *not* normalized and does not represent a "fraction"
in any sense. Rather, if z_{i} > 0, it means that activity A_{i} is run
at more-than-unit intensity (moves up along ray A_{i} above a_{i}) and if
z_{i} < 0, then activity A_{i} is run at less-than-unit intensity
(moves down along ray A_{i} below a_{i}). Thus, in our earlier Figure 4.3,
e_{2}, l e_{2} and e_{2｢
}can all be expressed as scalings up (or down) of the unit-intensity vector a_{2}.

The factor requirements of a particular activity A_{i }run at a
particular intensity z_{i} can be defined accordingly: specifically, the
requirement of factor j by activity A_{i} is z_{i}a_{ji}. Suppose,
as in Figure 4.5, activity A_{1} is run at intensity z_{1} and activity A_{2}
is run at intensity z_{2}. Therefore, z_{1}a_{K1} represents the
capital requirements by process 1 and z_{2}a_{K2} represents the capital
requirements by process 2, so that total capital requirements are z_{1}a_{K1}
+ z_{2}a_{K2}. Similarly, total labor requirements are z_{1}a_{L1}
+ z_{2}a_{L2}. Thus the input combination used in the economy is
represented by vector z_{1}a_{1} + z_{2}a_{2}, which is
shown in Figure 4.5 as the sum of the vectors z_{1}a_{1} and z_{2}a_{2}.

As z_{1}, z_{2} > 0 in Figure 4.5, then the total
inputs required when we run both activities at vector z_{1}a_{1} + z_{2}a_{2}
is more than just running one of the activities at z_{1}a_{1} and that, in
turn, is greater than that activity at its unit-intensity level, at a_{1}.
Consequently, we would expect the output produced by a_{1} is less than that of z_{1}a_{1}
and that, in turn, is less than the output achieved at z_{1}a_{1} + z_{2}a_{2}.
Thus, the activity analysis isoquants we had earlier drawn out in Figure 4.3 can be
overlaid on Figure 4.5.

We can denote by y_{i} the level of output produced by running
activity i at unit intensity. Thus, y_{1} corresponds to the output achieved by
vector a_{1} and y_{2} represents the output achieved by vector a_{2}.
Consequently, z_{1}y_{1} is the output achieved by vector z_{1}a_{1}
and, of course, z_{1}y_{1} + z_{2}y_{2} is the output
corresponding to factor input combination z_{1}a_{1} + z_{2}a_{2}.
Thus, in general, we can write the activity analysis production function as:

y = ・/font>

_{i=1}^{p}z_{i}y_{i}

where z_{i} is the intensity of the ith activity and y_{i}
is the unit intensity output of process i. This production function is what is commonly
found in activity analysis.

There are a few points worth clarifying here. First, as has been noticed,
we are assiduously avoiding using the term "technique" with regard to our
activities A_{1}, A_{2} and A_{3}. This is because, as we saw,
while only three "activities" are available, we were nonetheless able to
accommodate a whole series of capital-labor ratios by *combining* activities. Indeed,
as we saw in Figure 4.4, every capital-labor ratio between k_{1} and k_{3}
is available to the producer, thus the number of *techniques* available, which are
defined by the factor proportions available, are indeed infinite. Thus, when moving from
any capital-labor ratio to another, the producer is indeed changing *technique* in
the strictest sense, even though all he might be doing is changing the relative degree of
intensity of use of the same two activities, A_{1} and A_{2}. Thus, there
are an *infinite* number of techniques but only a *finite* number of activities.

Secondly, it is noticeable that one of the great casualties of activity
analysis when there are a finite number of activities is the very concept of marginal
productivity. Strictly speaking, kink points such as b_{1} or b_{2} in
Figure 4.4, are simply not differentiable and thus marginal productivity is not defined,
at least as a derivative. One of the solutions pursued by economists is to use a
generalized concept of derivative, known as the *Clarke Normal Cone*, as introduced
into economics by Bernard Cornet (1982). However, this takes us a bit too far afield, for
the moment. For a review of the analysis of non-smooth derivatives, see the excellent
review by Donald J. Brown (1991).

Thirdly, one of the main lessons of the activity analysis production function is that we can conceive of the conventional, smooth concave production production function and smooth isoquants as a limiting case of activity analysis production functions. Specifically, if we increase the number of activities, the number of kinks in the isoquants and the production function would increase. If we had an infinite number of activities available - in fact, a continuum of them - so that. every capital-labor ratio would be represented by its own activity, there would be no kinks at all: the isoquant and the production function would be nice, smooth and differentiable, as commonly stipulated in conventional production theory. In such a case, one need no longer speak of substitution between activities as such and just talk about substitution between techniques directly.

Another way of reconciling smooth production technology with activity
analysis or even Leontief technology, would be via the Marshallian time dimension.
Specifically, Marshall (1890) had argued that
the degree of substitution between factors is not immutable over different time horizons.
As he argued, *in the long-run*, all factors are variable and can be substituted with
little difficulty, but *in the short-run*, the degree of substitution between factors
is more limited. Thus, "fixed capital" is fixed in the short-run and variable in
the long-run.

This implies that one can think that in the long-run, we face a smoother
production technology than in the short run. One can incorporates this idea via the notion
of "putty-clay" production technology, as pursued by Leif Johansen (1959), W.E.G. Salter (1960) and Robert M. Solow (1962). The basic idea is that *before*
production begins, an entrepreneur is faced with a wide variety of possible techiniques of
production he can use. However, once he decides upon a given technique and sets production
in motion, he has *fewer* possibilities of further substitution: the machines are in
place, contracts are in force, etc., thus he cannot easily fire-and-hire at will. Thus the
term "putty-clay" technology: putty can be molded as we wish before baking, but
once baked, it becomes hardened clay and we cannot change its shape.

Thus, putty-clay models distinguish between *ex ante* (or
"putty") and *ex post* (or "clay") technology. Heuristically, *ex
ante* production technology will tend to have high substitution possibilities, and thus
is probably closer to the smooth production functions of conventional Neoclassical models.
However, *ex post* production technology has far fewer substitution possibilities,
and thus is closer to the Leontief case.

To see the implications, consider the example depicted in Figure 4.6.
Suppose that labor is fluid but capital, once decided upon, stays in place in the
short-run. The entrepreneur wishes to produce Y* and his *ex ante* technological
possibilities are all the points on the smooth isoquant Y*. Suppose the entrepreneur
happens to choose the technique indicated by the activity ray A_{1}. The *ex
post* technology thus becomes the Leontief isoquant formed at that point with capital
requirements K_{1} = v_{1}Y* and labor requirements L_{1} = u_{1}Y*.
In constrast, had he chosen a different technique, e.g. A_{2}, then his *ex post*
technology becomes the Leontief isoquants formed by the ray A_{2}.

Figure 4.6- Putty-Clay Technology

The situation depicted in Figure 4.6, thus, represents the most extreme
form of putty-clay model: complete *ex ante* substitution (represented by smooth
isoquant) and absolutely no *ex post* substitution (represented by the Leontief
isoquants that emerge after the choice).

Putty-clay technology is not easily representable by an all-encompassing
production function precisely because *ex post* technology is dependent on the
initial choice of technique. Nonetheless, it is useful to envisage production technology
in putty-clay terms, as it provides another way of thinking about the smooth technology
used in conventional production functions. Specifically, with putty-clay in mind, we need
not necessarily think of smooth production functions as the idealistic
"limiting" case of activity analysis technology but rather as merely
"long-run" production technology. In other words, smooth production technology
should not be expected to apply in reality at any time or over short-periods of time (a
few years), but it ought to be the case when one examines technological possibilities over
a long stretch of time (say, a few decades or even a century).

Back |
Top |
Selected References |