Usher Political Economy (Blackwell, 2003)

Political Economy

Dan Usher

Copyright © 2003 by Dan Usher

C h a p t e r S i x

TECHNOLOGY

Famine seems to be the last, the most dreadful resource of nature. The power of population is so superior to the power in the earth to produce subsistence for man, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction; and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, advance in terrific array, and sweep off their thousands and ten thousands. Should success still be incomplete; gigantic inevitable famine stalks in the rear, and with one mighty blow, levels the population with the food of the world.

Thomas Malthus, 1798

Among the oldest arguments in political economy is that mankind is destined to perpetual poverty because poverty alone can check the growth of population. Families are biologically programmed to bear about ten children, and most of our great grandparents did just that. If, of these ten births, four children survive to bear children themselves and if the average age of the mother at childbirth is twenty-five, then population doubles every twenty-five years. A million people in 1800 becomes two million in 1825, becomes four million in 1850, and so on. Sooner or later, population outruns the food supply and the standard of living falls to whatever level is necessary to check population growth. Starvation, poverty-born disease and malnutrition may be required to suppress the survival rate and to balance births and deaths. When people become prosperous through good harvests or technical change, their prosperity is soon eaten away by population growth and the old conditions of poverty are restored on a larger scale. A small ruling class might remain prosperous, but the great majority of people can never do so. Only within the last hundred years has technical change outdistanced population growth in much of the world, allowing ordinary people to become prosperous beyond the wildest dreams of our ancestors.

T E C H N O L O G Y

203

The model of technology in chapters 3 and 4 was adequate for explaining how markets work, but it is not adequate for expounding the Malthusian argument. In that model, bread and cheese were produced on lands of different productivities, with no recognition of the role of labor in production or of population growth. These are introduced here. The organization of the chapter is as follows: First, technology is represented as a simple production function in which bread is produced with land and labor. There follows a discussion of the optimal size of firm. The Malthusian story is then retold with the aid of an aggregate production function for the economy as a whole. Several aspects of production are then examined briefly: the determination of wages and rents, a multiplicity of goods and investment. Finally, technical change is introduced and the Malthusian story is modified accordingly.

THE PRODUCTION FUNCTION

Consider the technology of a farm where bread is produced by the application of labor to land. Specifically (supposing for simplicity that farmers grow loaves of bread rather than bushels of wheat), a farm of d acres of land and employing l workers produces b loaves of bread per day in accordance with the farm’s production function, f,

b = f (l, d)

(1)

For a given input of land, d, the production function is illustrated in figure 6.1. If the input of land were greater than d, the entire production function would swing counterclockwise, indicating that a larger output of bread would be obtained with any given input of labor. The principal assumption about the form of the production function can be expressed in two equivalent ways: that the production function bends forward, and that an increase in the input of labor decreases the output of bread per worker. The bending forward of the production function, as shown in figure 6.1, is referred to as concavity. Diminishing output per worker is illustrated in figure 6.2, which is identical to figure 6.1 except for the addition of new information. For any input of labor, such as l1, the output of bread per worker, b1/l1, is shown in figure 6.1 as the slope of the line from the origin to the production function above l1. It is immediately evident that output per worker would be constant if the production function were an upwardsloping straight line, but that output per worker decreases because the production function bends forward. Specifically, if

	l3 > l2			> l1		(2)
then,
	b3	<	b2	<	b1	(3)
	l3		l2		l1

The farm owner’s demand for labor – the number of workers hired at any given wage – and his residual rent on land – output of bread over and above the cost of

204

Output of bread, b (loaves per day)

T E C H N O L O G Y

Input of land fixed at d

f (l, d)

Input of labor, l (workers per day)

Figure 6.1 The production function.

b

b3			f (l, d)
b2
b1
0			1
	l1	l2	l3

Figure 6.2 How an increase in the input of labor lowers the output of bread per worker.

labor – are illustrated in figure 6.3 which is another extension of the production function in figure 6.1. The farm owner hires workers at a market-determined wage, w, graduated in loaves of bread rather than money. He can hire any number of workers at a wage of w loaves of bread per day. [One can equally well think of the wage as $w per day when the price of bread is $1 per loaf, or as $wp per day when the price of bread is $p per loaf.] His options are represented in figure 6.3 by a “cost of labor,” the line, 0β, with slope w, showing total payment to labor, wl, for any given number, l, of workers employed.

In choosing how many workers to hire, the farm owner seeks to obtain the largest possible rent, defined as the difference between the amount of bread produced and the amount of bread the farm owner must pay out in wages. For any input of labor, rent is represented in figure 6.3 by the vertical distance between the production function and the cost of labor line. As is evident from figure 6.3, this distance is as large as possible when the farm owner hires l workers, where the slope of the production function is parallel to the cost of labor line. Equivalently, the rent-maximizing number of workers is that for which the marginal product of labor is just equal to the wage rate. The marginal product of labor is the slope of the production function. It is the

T E C H N O L O G Y

205

b

γ	α	Production function,
		b = f(l, d)

Payment to labor at l = l*

				b
	δ					Cost of labor, wl
					β
Rent at
l = l*				w	l
0						1
	l l + l
						l*

Figure 6.3 How the farm’s demand for labor responds to the wage rate.

increase	in output per unit increase in labor. It is the ratio b/ l, where	b is the
increase	in the production of bread brought about by a small increase,	l, in the

number of workers. When l workers have been hired, the change in rent from a small increase, l, in the number of workers is b − w l. If the change in rent is positive, the increase in the number of workers is advantageous to the farm owner and the original number of workers, l, must be too small. Similarly, if the change in rent is negative, the original number of workers must be too large. Only if there is no change

– if b − w l = 0 when the number of workers is increased slightly – is the farm owner’s rent maximized. But if b − w l = 0, then ( b/ l) = w which is precisely the condition that the marginal product of labor is equal to the wage. [Notice that the word “rent” is employed here in a somewhat unusual sense. It refers here not to the amount of money one must pay for the use of someone else’s land, but to the amount of money (or bread in this example) one acquires from the usage of one’s own land. In the next section, we shall revert to the ordinary usage of the word. It turns out both usages of the term “rent” are virtually identical within the model of the agricultural economy in this chapter.]

The line δα in figure 6.3 is drawn parallel to the cost of labor line, touching the production function at α and intersecting the vertical axis at δ. The point α has to be directly above l on the horizontal axis. It is immediately evident from the geometry of figure 6.3 that, when the farm owner hires the rent-maximizing number of workers, l , his payment to labor becomes βl , equal to γδ on the vertical axis, and that his rent becomes αβ, equal to δα on the vertical axis. The total output of bread, 0γ, is allocated γδ to labor and 0δ to land.

THE ORGANIZATION OF PRODUCTION BY FARMS

Concentrating, as it has, on the individual farm, the analysis of production has as yet yielded no explanation of the size of farms (or, more generally, of the size of firms no matter what they produce). Suppose an economy contains L workers (each assumed

206	T E C H N O L O G Y

to work a fixed number of hours per day) and D acres of land. We still require an explanation of how the land is divided up into farms. Bread might be produced by a vast number of small farms (as many farms as there are people, so that each farm contains D/L acres of land), by one great collective farm comprising the entire land of the nation, or by something in between. Bread, cheese, and most vegetables are typically produced by family farms. Cars, ships, and TV sets are typically produced by large firms hiring thousands of workers.

The production function in equation (1) has been looked upon as representing the technology of an isolated farm with a fixed acreage of land. It may also be looked upon as the technology of farming available to anybody and everybody who wishes to establish a farm. Think of an economy where bread, b, is produced with labor, l, and land, d, in accordance with the production function in equation (1), and where land and labor are for hire at market-determined factor prices: labor at a wage of w and land at a rent of r, both denominated in loaves per day. In this economy, entrepreneurs hire labor and land to maximize profit, expressed as a quantity of bread and defined as the difference between the quantity of bread produced and the quantity of bread paid out as wages and rents. Specifically,

cost of production = wl + rd

(4)

and

profit = b − (wl + rd) = f (l, d) − (wl + rd)

(5)

Our problem in this section is to explain the formation of wages, rents, and the size of farms in an agricultural economy with fixed supplies of labor and land. Profit maximization may be considered in two stages: the choice of l and d for any given b, and the choice of b on the understanding that l and d are optimal for any given b. The production function allows a given quantity of bread to be produced with many different combinations of labor and land. The first stage of profit maximization is to find the least expensive combination of labor and land for the production of a given amount of bread at a given wage and a given rent. The second stage is to choose the best size of farm on the understanding that the mix of land and labor is best for any given size output.

In the first stage, l and d are chosen to minimize the cost of production when output is fixed at b loaves of bread. Though the b loaves of bread could be produced with various combinations of labor and land (more land and less labor, or more labor and less land), there must be some best combination of l and d at which the average cost of bread (given w and r) is as low as possible. A profit-maximizing entrepreneur seeks to minimize the average cost of production, T(b, w, r), where the average cost must, of course, be dependent on the wage, w, and the rent, d, and would normally be dependent on the output of bread, b, as well. The best proportion is whatever minimizes the cost of production in equation (4). As it is advantageous for the farm to minimize its cost of production at each and every possible quantity of bread produced, there may be defined a minimum average cost, T(b, w, r), of producing b loaves of

T E C H N O L O G Y

207

T (b, w, r)

Average cost, T

b*

Output of bread, b

Figure 6.4 The average cost curve.

bread when the wage is w and the rent is r. Specifically, for any, b, w, and r,

wl + rd

T(b, w, r) = min where f (l, d) = b (6) l,d f (l, d)

and where the term minl,d means “the combination of l and d that minimizes the value of the term in square brackets.”

Since average cost is minimized for each and every combination of w, r, and b, all cost-minimizing combinations of l and d can be represented by a pair of functions,

l = l(w, r, b)	(7)
and
d = d(w, r, b)	(8)
with the property that
T(b, w, r) = [wl(w, r, b) + rd(w, r, b)]/b	(9)

It could happen that the average cost of bread is independent of scale so that T(b, w, r) is the same regardless of whether the firm’s output of bread, b, is large or small. If so, there would be no technological explanation of the size of firms. Otherwise, for any w and r, there is some optimal size of farm, b , for which

T(b , w, r) < T(b, w, r)

(10)

where b is any other quantity of bread. As illustrated in figure 6.4, the average cost of bread for given values of w and r would be U-shaped with a minimum at b = b .

The postulated shape of the average cost curve in figure 6.4 requires some explanation. There is reason for supposing it might be flat rather than U-shaped. The reason is that experiments can be replicated. If you can produce 5 tons of bread with 1 acre of land and 2 workers, then you should be able to produce 10 tons of bread with 2 acres

208	T E C H N O L O G Y

and 4 workers. On the other hand, if cost curves of all products were flat, there would be no explanation of the size of firms and no reason why all the nation’s goods and services are not produced together by one huge firm. To explain why average cost curves might be U-shaped is to explain why increases in output lower average cost up to some optimal output b , but raise average cost thereafter. The fall in average cost below b is typically explained by “indivisibilities.” A farm requires at least one farmer, just as an automobile company needs a complete assembly line whether it produces one car or ten thousand. The rise in average cost beyond some b is typically explained by limitations in the span of control. The larger the farm, the larger the hierarchy, and the greater the cost of communication between the center and the periphery and the greater the risk of malfeasance at all ranks in the hierarchy. Optimal scale – the exact position of b – depends on the technology of production. Once food was distributed by many thousands of small independent grocery stores. Now it is distributed by a few large grocery chains.

Generalizing from the farm to the firm in any industry whatsoever, the principal source of economies of scale is the division of labor. A thousand men working together can produce more cars per year than if each of them attempted to build an entire car all by himself. Modern production requires more knowledge and skill than any one person can be expected to master. The accountant, the lawyer, the marketing agent, the machinist, and specialized workers attending to each of the many stages in the making of a car must somehow coordinate their activity to produce cars in the right amounts and at the lowest possible cost. Sometimes stages in production can be separated by prices, as when a car company buys auto parts or material from other firms. The greater the importance of detailed coordination among stages in production, the greater the advantage of placing labor in one large firm that owns the means of production. Typically though not invariably, the optimal size of firm is substantially less than the total output of the good in the economy as a whole. Excessively large firms become unwieldy as initiative and flexibility are lost in a vast administrative bureaucracy. We shall have more to say about the organization of the economy in the next chapter.

Competition destroys profit. As long as every farm’s technology is the same and as long as everybody is free to start up a farm by renting land and hiring labor at the going market prices, the only possible outcome in the market is for wages and rents to adjust to eliminate profit altogether, directing the entire output of the economy to workers and to owners of land. If profit, as defined in equation (5), were anything in excess of zero, more and more new entrepreneurs would establish new farms until such time as the scarcity of land and labor drives up wages and rent in a process that must continue until all profit is wiped out. Profit greater than zero is inconsistent with a competitive economy. Profit less than zero would drive farms out of business. Profit must be zero, and that can only occur when farms are producing at minimum average cost. Of course, profit can only be zero when l and d are chosen to minimize average cost of production in equation (6). Otherwise profit would have to be negative.

To say that pure profit (as defined in equation (4)) is wiped out in a competitive economy, is to say that

b = wl(w, r, b ) + rd(w, r, b )

(11)

T E C H N O L O G Y	209
but
b < wl(w, r, b) + rd(w, r, b)	(12)

for any b other than b . Competition among entrepreneurs requires w and r to be such that the cost (when wages and rents are denominated in loaves of bread) of producing b loaves of bread must be exactly b loaves of bread.

The reader may well balk at this implication of our model of the competitive economy. Surely, the entrepreneurs cannot be left with zero profits because businessmen who make no money would not remain in business at all. The reason why this story seems so peculiar is that real live entrepreneurs exert effort in contributing to the success of their firms, and they are compensated accordingly. They contribute a special kind of labor. By contrast, entrepreneurs in this description of the market contribute nothing, earn nothing, and are only useful as a rhetorical device to explain how the cost-minimizing farm emerges in the market.

The word “profit” is used here in a very special sense. Frequently the word profit is used to refer to the return to capital (so far not discussed) and entrepreneurship; on that definition, farms must earn profit. Here profit refers to the return to the farm over and above the earnings of all inputs; on this definition the equilibrium profit must be zero, and a non-zero profit portends some change in prices or production. The rock bottom meaning of the zero profit condition is that competition governs the returns to the talents of businessmen, just as it governs prices of goods and the returns to ordinary factors of production, with nothing left over as a free gift to the entrepreneur. A more realistic model of production would allow entrepreneurs to differ in ability; in effect, each entrepreneur would have his own production function linking output to all other inputs. In that case, all but the least efficient among the entrepreneurs who are actually operating firms would earn some profit. This case is usually covered in ordinary textbooks of microeconomics.

Finally, the market determines wages and rents to equate the sums of all farms’ demands for labor and land to the available supplies.

Nl = L	(13)
and
Nd = D	(14)

where N is the number of farms, l and d are each farm’s usage of labor and land, and L and D are the total supplies of labor and land in the economy as a whole.

Pulling all this together, we can represent this agricultural economy by six equations:

technology:

b = f (l, d)

(1)

the farm’s demand for labor:

l = l(w, r, b)

(7)

210	T E C H N O L O G Y
the farm’s demand for land:
	d = d(w, r, b)	(8)
disposition of the supply of bread:
	b = wl(w, r, h) + rd(w, r, h)	(11)
market for labor:
	Nl = L	(13)
market for land:
	Nd = D	(14)

These six equations determine market-clearing values of six unknowns: b, w, r, l, r, and N. Markets and technology determine wages, rents and the size of farms, just as they determined the outputs and prices of bread and cheese in chapter 3.

The same general principles govern the allocation of the various resources of the nation into firms producing the entire spectrum of goods that people consume. It is essential for a competitive economy that the cost-minimizing size of firms in any industry be small relative to industry as a whole. Small firms are price-takers, like the owners of the five plots of land in the latter part of chapter 3. Firms too small for their behavior to affect prices, wages, or rents look upon all such prices as marketdetermined and invariant, and they make their production decisions accordingly. By contrast, large firms exercise a degree of monopoly power and must bargain with one another over prices of inputs and outputs.

THE AGGREGATE PRODUCTION FUNCTION AND

THE IMPOVERISHMENT OF MANKIND

When the productive resources of the economy are allocated appropriately among farms, one can pass from the production function of the farm – b = f (l, d) in equation (1) – to an aggregate production function for the economy as a whole

B = F(L, D)

(15)

where B is total output of bread, L is the total supply of labor, D is the total acreage of land, and the function F for the entire economy is a magnification of the function f for the farm. The aggregate function F inherits the general shape of the function f, so that output per worker, B/L, diminishes steadily as L increases when D remains constant, and the slope of the function F(L, D) – like that of the farm in figure 6.3 – becomes the wage of labor in the economy as a whole. In short, figures 6.1, 6.2, and 6.3 can be reinterpreted as pertaining to F, L, and D rather than f, l, and d, on the understanding that the scale of production is appropriately chosen.

T E C H N O L O G Y

211

It is often helpful to give the production function a specific algebraic form. A useful specification for the purposes of this chapter is

B = F(L, D) = ALαD(1−α)

(16)

where A is a parameter representing the efficiency of the economy. This specification of the production function has three very convenient properties: constant returns to scale, constant share of output accruing to labor, and a simple specification of the rate of growth of output per worker. These will be considered in turn.1

Constant returns to scale means that a proportional increase in the inputs, L and D, generates the same proportional increase in output, B. If inputs of L and D yield an output B in accordance with equation (16), then, for any x whatsoever, inputs of xL and xD yield an output of xB.2 If it just so happens that 10 workers and 5 acres of land can produce 1 million loaves of bread per year, then 30 workers and 15 acres of land can produce 3 million loaves of bread per year. Note that there is no contradiction between constant returns to scale in the aggregate production function and variable returns to scale – increasing returns to scale up to some output b and decreasing returns to scale afterwards – for the farm. The reason is that farms can be replicated. A U-shape cost curve for the firm is consistent with a flat cost curve for the entire industry to which the firm belongs.

The parameter α in equation (16) turns out to be the share of output accruing to labor.3 This remains true regardless of whether the input of labor is large or small. Specifically, it follows from equation (16) that

wL = αB	(17)
and
rD = (1 − α)B	(18)

If the input of labor is large, the equilibrium wage of labor must be small. If the input of labor is small, the equilibrium wage of labor must be large. Adjustments in w within a competitive economy are just sufficient to keep wL constant and equal to αB. From equations (17) and (18) together, it follows that

wL + rD = B

(19)

Constant returns to scale in the production function guarantees that the returns to land and labor exhaust the product with nothing left over.

Malthus’ pessimistic prediction in the quotation at the beginning of this chapter can be translated into the language of the production function. The prediction is that there is no escape from perpetual poverty because all income over and above bare subsistence is eaten away by population growth. It is that, if population grows steadily at some fixed percentage per year whenever output per worker, B/L, exceeds the subsistence level, then output per worker can never exceed the subsistence level for any length of time. Technical change or a great epidemic, such as the Black Death, may lift output per worker temporarily, but population growth always returns output per worker to a biologically predetermined level.