Specialization

A Simple Model of The Specialization of Capital

Consider a single-period economy in which there are two intrinsically identical individuals, and ; and two tasks, farming and cattle raising. Each individual is endowed with one unit of labor (labor is interpreted as the amount of human time) and one unit of capital Labor and capital may be allocated to either farming and/or cattle raising.

Farm products are produced using labor and capital allocated to farming, according to the following constant-returns-to-scale technology:
(1)

Cattle products are produced using labor and capital allocated to cattle raising, according to the following constant-returns-to-scale technology:

(2)

Each individual has preferences for farm and cattle products, represented by the utility function

(3)

Suppose that each individual produces and consumes in autarky. Then each solves the following utility-maximization problem:

(4)
The solution to problem (4) indicates that each individual allocates 1/2 unit of capital and 1/2 unit of labor to farming; and 1/2 unit of capital and 1/2 unit of labor to cattle raising. Because there are two sets of capital and labor in farming and in cattle raising, there are no capital specialization and no labor specialization. In this case, each individual produces and consumes 1/2 unit of farm products and 1/2 unit of cattle products. The total output of farm products is 1 unit; and the total output of cattle products in the economy is also 1 unit. Notice that, in the above autarky situation, each individual is using his half unit of farming capital for only half of the time and his half unit of cattle-raising capital for the other half of the time. So when an individual is farming, his cattle-raising capital is not being used; and similarly when he is raising cattle.
Now, suppose that individuals and may coordinate with each other when carrying out their production and consumption activities. Then they may coordinate so that, when he is farming, an individual uses all the capital allocated to farming; and when he is raising cattle, an individual uses all the capital allocated to cattle

raising. We will call the coordination technology that eliminates the overlaps in the allocation of capital to the same tasks and that allows different individuals to use the same set of specialized capital the specialization of capital. This kind of coordination may be done within the firm or it may be done in rental markets. We now show how the specialization of capital may work in a rental market.5 Suppose that when farms, he rents his cattle-raising capital to in return for ’s farming capital; and when he raises cattle, he rents his farming capital to in return for 's cattle-raising capital. will accept this arrangement because it is Pareto optimal. This rental arrangement implies that and will have to raise cattle at different times and farm at different times. In the next section, we will incorporate coordination/transaction costs into the model; assume for now that there are no such costs.

To see the effects of the specialization of capital, we let , denote the amount of farm products individual produces and consumes; and let denote the amount of cattle products individual produces and consumes. Let denote the
amount of labor allocates to farming; the amount of labor allocates to farming; the amount of labor allocates to cattle raising; and the amount of labor allocates to cattle raising. Let be the effective capital input in farming;
and let be the effective capital input in cattle raising.
⁵ Deardoff and Stafford (1975) and Williamson (1975, 19S5) discuss how coordination be carried out within the firm. See also Foss (1997) and Marris (1964) for the early studies on shiftwork.


Individuals and jointly solve the following Pareto optimal problem:


(5)

The solution to problem (5) indicates that and together allocate 1 unit of capital to farming ( 1) and 1 unit of capital to cattle raising ( 1). Each

individual allocates 1/2 unit of labor to farming and 1/2 unit of labor to cattle raising
— there is no labor specialization. The total output of farm products is units; and the total output of cattle products in the economy is also units. Since 1 <
the total output in the economy has increased. See Table 1.

Although farming capital and cattle-raising capital are not public goods, we have shown that they are quasi-public goods — rivalrous if used at the same time but non-rivalrous if used at different times. When is not using a piece of specialized capital, may use it without affecting ’s use of it.

By the specialization of capital, the technologies (1) and (2) may also be interpreted as having been transformed to, respectively,


(6)

Note that using the new technologies in (6) the marginal product of labor increases

(7)
for each of the tasks. And the marginal product of capital also increases:


(8)
for each of the tasks. Therefore, the specialization of capital is labor-augmenting and capital-augmenting at the same time. It is labor-augmenting because the specialization of capital without labor specialization reduces job fatigue.⁶ It is capital-augmenting because specialized capital goods are quasi-public goods.



⁶ The solution to problem (5) shows that there is no labor specialization. But suppose that there is; for example, suppose that and ; and and Then it is easy to show that
capital specialization (which means that uses 1 unit of farming capital all the time instead of 1/2 unit and uses 1 unit of cattle-raising capital all the time instead of 1/2 unit) improves welfare. We can further show that capital and labor specialization is Pareto inferior to capital specialization and no labor specialization. This is because, with capital and labor specialization, the total output of farm products is 1 unit; and the total output of cattle products in the economy is also 1 unit. These allocations are the same as those in autarky. The reason for this result is that labor specialization leads to more job fatigue; and capital specialization without labor specialization clearly avoids this problem. The fact that labor fatigue may be eliminated by spending time in a variety of tasks may be the main difference between labor and machine. Marshall (1949, Chapter IX) says, when discussing the monotony of work, that when a task has been reduced to a routine, it may be taken over by the machine; and labor may move on to other, perhaps supervisory, tasks.

4.