Specialization

The Extended Static Models

Suppose that a complete set of tasks indexed by are performed in order to produce the one good in the economy. In the production process, the complete set of -tasks may be partitioned into disjoint non-empty subsets of -tasks such that is the complete set of -tasks and that each subset
of -tasks may be carried out independently of each other. Labor and capital
, which includes both human and physical capital, must be allocated to each set of -tasks in order to produce output from that set. We assume that labor is homogeneous; so the labor aggregate is simply the sum of over
Following Becker (1965), we interpret labor inputs as amounts of time spent in production.
Capital is heterogeneous; and the capital aggregate is an aggregate of the specialized capital allocated to set , , using a linearly
homogeneous aggregator function


(9)
where An example of such an aggregator function is

the weighted-sum function as used in Lancaster's (1966) linear-characteristics model:

(10)

Another example is the Cobb-Douglas function:

(II)
where and Yet another example is the CES function:
(12)


where The capital aggregate may also be constructed by Divisia indexing procedures. See Griliches and Jorgenson (1966), Jorgenson and Griliches (1967), Hulten (1973, 1990), and Jorgenson (1990) for discussions on the estimation of aggregator functions of the type given by (9). Unlike Dixit and Stiglitz (1977) and Ethier (1982), we do not assume that there are intrinsic benefits from using a larger variety of capital inputs. In our model, different varieties and different amounts of capital are considered equivalent if they lead to the same capital aggregate.
Output is produced using labor input and effective capital input which is the total amount of capital inputs used in production, according to the following constant-returns-to-scale aggregate production function:
(13)

In per-capita terms, equation (13) is

(14)


where is per-capita output and is per-capita effective capital input. Notice that, if is the total amount of capital inputs used in the set of
tasks, since the aggregator function is linearly homogeneous,
— where —

, and for any . Also note that the per-capita amount of labor allocated to set is denoted by ; and that Equation (14) may

also be written as


(15)
The following discussion will be in per-capita terms.

Suppose that the identical human agents in the economy have names on the closed unit interval [0, 1]. Each agent has a unit of labor which is supplied inelastically and may be allocated in any way to each of the tasks. Each agent is also endowed with amounts of the capital aggregate. Each agent has preferences for the good represented by and Since every agent is the same in every way, we may use a representative agent in the following discussion. The representative agent's utility-maximization problem requires the maximization of output given per- capita labor endowment and per-capita capital endowment If the representative agent produces in autarky, then the per-capita effective capital input

is and the per-capita output is given by


(16)
It tuns out, as in Section 3, that production in autarky is not optimal. The source of the inefficiency of production in autarky is that, when each individual is spending time in set , he is using only capital , and all the other sets of specialized capital inputs are not being used.
When there are sets of -tasks and types of specialized capital, it follows from the results in Section 3 that it is possible for agents to coordinate such that each of sets of specialized capital may be fully utilized. We now show that the

specialization of capital a coordination technology that allows each agent to use as much of the capital allocated to a set of -tasks as possible while working on that set of -tasks. A set of agents may use the specialization of capital to coordinate the use of a set of specialized capital by multiple shifts of workers, either within the firm or in a rental market, in order to increase the effective amount of capital each agent may work with in each set of -tasks. In order to avoid job fatigue and to facilitate the coordination of the use of each of the sets of specialized capital by the agents, the representative agent in our aggregate model will allocate labor uniformly to the sets of -tasks so that

— (17)

This is because, since there are sets of -tasks, agents, and different sets of specialized capital, if for , then no two agents need to work on the same task at the same time. Since it is possible that no two agent work on the same task at the same time, it is also possible that each agent may use all the capital allocated to the set when working on the set of -tasks. We now give a simple time schedule that will work. At the beginning of the time period, agent may work on the set using all the capital allocated to that set; then agent may move on the set if or if And so on, until time is up. This is one way of scheduling the specialization of capital; many other schedules may work as well. Suppose are the specialized capital allocated to each of

the set of -tasks by each agent. If there are no transaction/coordination costs for using the specialization of capital, then the effective amount of capital the representative agent may use in the set as a result of the specialization of capital is
(18)

Equation (18) implies that the effective per-capita capital input is

(19)

And the production function (16) is transformed to

(20)

Equation (20) implies that as the number of sets of specialized capital increases, the productivity parameter of the constant-returns-to-scale production function (16) increases; we thus derive the Ethier (1982) formulation. But of course the production relationship (14) between output and effective capital input has not changed. Therefore, both Dennison (1966) and Jorgenson and Griliches (1967) are right — if is used, then there is a total productivity parameter; but if is used, then the total productivity parameter disappears. Also note that the effective capital input may “embody” the specialization of capital effect that enables different sets of agents to time-share a set of specialized capital.

There are various transaction/coordination costs for using the specialization of capital; and they need to be incorporated into the model. For example, it is costly to draw up a timetable for time-sharing a set of capital by different shifts of workers, even something as simple as the time schedule discussed in the previous paragraph; or to schedule a rental timetable for a set of renters. It may be costly to search for a set of agents whose time needs for a rental good fit well together. It is also costly to monitor how the workers/renters use the capital goods and make sure that they pay the required maintenance costs. In summary, it is usually costly to find shiftworkers or renters, to schedule, to monitor, to measure, and to enforce rental contracts or shiftwork. And information costs are a part of all these costs. When there are sets of -tasks and sets of specialized capital, it may also be costly to determine how

they are to be put together to produce the final good. Whether the coordination of different users and of different types of capital is done within the firm or in rental markets, there are transaction/coordination costs involved. And some sort of hierarchy may be used; see Williamson (1975, 1985) for more discussions on hierarchies and organizations. Williamson (1975, 1985) and North (1990) also discuss how coordination works in general; and the institutions and the transaction costs involved. See also Coase (1937), Cheung (1969), and Williamson (1971) for earlier discussions on transaction costs and the organization of contracts.

We assume that a coordination-cost aggregate decreases the effective amount of the capital aggregate used in the production of the good when there is the specialization of capital (i.e. ).⁷ The per-capita coordination-cost aggregate
is derived from the coordination costs each agent pays in each set
, using a linearly homogenous coordination-cost aggregator function


(21)

For simplicity, we assume that coordination costs are homogeneous and that

(22)

Further assume that each agent pays the amount in each set for using the specialization of capital when there are sets of -tasks and specialized capital:

(23)

7. 
   Our model of coordination costs is similar to Samuelson's (1954) "iceberg" model of transportation costs and the Becker and Murphy (1992) model. See also Greenwood and Jovanovic (1990) for a model in which each agent pays a once-and-for-all fee in return for financial intermediation services.
   

This assumption is consistent with the idea that as the number of sets of specialized capital and the number of agents sharing each set increase, the coordination costs each agent pays for using each set of specialized capital increase.

Equations (22) and (23) imply that


(24)

After subtracting the coordination-cost aggregate, the effective per-capita capital

input is


(25)
when

The representative agent chooses the stage of the specialization of capital in order to maximize the effective per-capita capital input . We allow to be real numbers larger than or equal to 1 because this allows changes in the specialization of capital to be continuous. This formulation is consistent with North's (1990, p.6) observation that “institutions typically change incrementally rather than in discontinuous fashion.” By choosing the stage of the specialization of capital, the representative agent also chooses the institutions and the amount of transaction costs to pay. The representative agent may thus endogenize the institutions, the transaction costs, and the net benefits from the specialization of capital. By choosing in the following maximization problem:

(26)

the representative agent derives

(27)

The optimal number of sets of specialized capital is determined by the per-capita capital endowment and the transaction cost . The number of sets of -tasks and of sets of specialized capital increases with and decreases with . By combining equations (25) and (27) we derive the effective per-capita capital input as:

(28)
Equation (28) implies that, if , then ; and the set of agents derive benefits from using the specialization of capital. When , transaction costs are too high and the set of agents do not use the specialization of capital.⁸

When the effective per-capita capital input (28), the production function (16) becomes

as is in equation

(29)

Since , equation (29) may be written to look as if there are increasing returns. However, the term represents the effects of the specialization of capital and the fact that specialized capital provide rental services that are quasi- public goods. Note that the effective capital input is an aggregate of the total amount of capital services used as inputs in production; and it depends on the underlying organization of work. Therefore, a measure of by the Griliches- Jorgenson method captures innovations in the organization of work in the economy;

7. 
   Our result is related to Adam Smith's hypothesis that the division of labor is limited by the extent of the market, which he says is “in proportion to the riches and populousness” of the economy. Stigler (1951) and Rosen (1982) have analyzed this hypothesis. Becker and Murphy (1992) point out that transaction costs may be important for this hypothesis. Williamson (1971) has also pointed out that there may be transaction costs reasons for vertical integration.
   

just as their adjustment for quality changes captures vintage effects in the capital inputs. Equation (29) also implies that if a measure of is used instead in growth accounting, then organizational change would appear in the residual.

In our model, there is a continuum of agents borrowing and lending specialized capital. There is also a continuum of agents performing the coordination functions. Suppose the number of types of specialized capital is Then in a competitive equilibrium each agent pays the competitive price to enter a rental market.⁹ Because each agent will enter rental markets, the total amount he pays is The total net payoff from entering the rental markets is getting the effective capital input ; and entering the rental markets is better than not
entering when What are the competitive equilibrium rental prices? Suppose
= l/ is the amount of time each agent allocates to set and is the amount of specialized capital each agent has in set . We show that the rental prices
/ for renting capital good , , will work in equilibrium. Taking these equilibrium prices as given, the agent who is working on task may simply rent out , for the amount = 1/ . For each agent, there are
sets of specialized capital to rent out, giving the agent a total of

The agent may use this sum to rent sets of capital Therefore, each

agent rents out ( / )


( / )
amounts of capital in return for

amounts of capital. This is true for each of

the continuum of agents and the rental markets clear.

7. 
   See Greenwood and Jovanovic (1990) for another competitive equilibrium in which coordination cost is competitively determined. In their model, each individual pays an once- and-for-all fee for ñ return on capital each period. In our model, each individual pays a competitive fee in each period that depends on the extent of the specialization of capital.
   

5. 
   The Dynamic Model and Long-Run Growth
   

In this section we show that the specialization of capital is a mechanism for endogenous technological change in the neoclassical-growth model and that this mechanism generates long-run growth. We let the production of the good at time depend on the effective capital input at time , according to

(30)

where aggregate form:
Let denote the rate of change of the stock of per-capita capital Let the capital accumulation equation take the following simple
(31)

The set of human agents has the following representative utility function over the consumption stream of the good:

(32)


where and ; and is endowed with capital aggregate The representative agent maximizes intertemporal utility given by equation (32) subject to the constraint (31) and
Incorporating equation (28) and referring to equation (29), the production function (30) may be written as


(33)

where is the degree of the specialization of capital or the number of
types of specialized capital at time ¹⁰ And equation (31) becomes


(34)
Since This means that the rate of
change in the degree of the specialization of capital is equal to the rate of change in the stock of per-capita capital aggregate If we let , then we can write
(33) as We can also show that


(35)

Equation (35) says that the rate of technological change
the rate of change in the degree of the specialization of capital
is proportional to

Suppose that we are only interested in the balanced growth path. Let

, and let denote the rate of growth of per-capita consumption,
, on a balanced growth path. Then it is well known [see Lucas (1988) for a derivation] that, on a balanced growth path,


( 36)


Equation (36) says that, on a balanced growth path, per-capita capital aggregate and per-capita consumption grows at the constant rate given by which is also the rate of change in the degree of the specialization of capital.
Equations (35) and (36) together imply that or Putting and into equation (33), we may show that

'⁰ Equation (33) can be written as

(37)

and that the balanced-growth case of our model is the “ " model which is well- known [see Barro and Sala-i-Martin (1995)] as the simplest growth model without diminishing returns. The steady state growth rate equals:

— (38)


Therefore, if or then That is, if coordination cost
is small enough, then the specialization of capital can get started, resulting in positive steady state growth.
This model has many more implications and rich dynamics that we may further investigate. For example, if we allow the coordination cost to vary, then we may vary the growth rate. If coordination cost goes down, then the growth rate will increase. If coordination cost goes up, then the growth rate will decrease. We may also further study dynamics that are out of the balanced growth path, by studying cases where or . If we allow to vary with time, then from equation (33) we can derive


(39)
And in the steady state where a constant , then


(40)
If then steady state growth is possible with , as shown in equation

(38). But if and , then equation (40) shows that steady state growth is not possible. Growth is explosive if and If , then

if ; but growth is possible if If and , then growth is explosive. But if and then there may be non-explosive growth. These results imply that a more capital-intensive economy may benefit more from capital specialization and sharing; that institutional and structural changes have effects on economic growth.

6.