Usher Political Economy (Blackwell, 2003)

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To establish this proposition, rewrite equation (16) as
	Y(t) = AL(t)αD(1−α)	(20)

where Y replaces B in designating the total output of bread (because Y is the usual symbol in economics for the national income), where Y and L are dependent on time, and where A (representing the state of technology in the economy) and D (the total output of land) are assumed to be invariant. The implications of technical change, allowing A to vary over time, will be examined later in the chapter. Now suppose the labor force grows steadily at a rate of n, or 100n percent per year, so that the number of workers, L(t), in each year t becomes

L(t) = L(0)ent

(21)

where L(0) is the input of labor in some reference year 0 which may be thought of as the present.4

Dividing both sides of equation (20) by L(t), we acquire an indicator of output of bread per worker, Y(t)/L(t), each year

B(t)/L(t) = A[D/L(t)](1−α) = A[D/L(0)](1−α)e−(1−α)nt

(22)

Since A, D, and L(0) are invariant, it follows that output of bread per person must decline over time at a rate of (1 − α)n per year.

The rate of growth of output per head = −(1 − α)n

(23)

If the labor force grows at 3 percent per year (i.e. n = 0.03) and if labor’s share of the national income is two-thirds (i.e. if α = 2/3), then the output of bread per person declines at a rate of 1 percent per year (i.e. –(1 − α)n = −(1/3)(0.03) = −0.01). Of course output per head cannot decline at a constant rate forever. Eventually, the fall in output per head checks population growth because people reduce birth rates voluntarily or because death rates rise as a consequence of impoverization.

With this machinery in hand, we can now tell the Malthusian story numerically. The numbers are chosen to represent an imaginary agricultural Canada where the entire national income consists of one great pile of bread produced with land and labor. Designate the year 2000 as the current year for which the value of t in equation (20) is equal to 0. There are 30 million people, of whom half are in the labor force. The national income is $900 billion. The area of land is 300,000 square miles (a strip of land 3,000 miles long and 100 miles high, on the assumption that all land more than 100 miles from the US border is barren). Labor’s share of the national income is 2/3, but people’s income derives in part from the ownership of land. Thus, Y(0) = 900, L(0) = 15, D = 3, and α = 2/3 where Y is expressed in billions of dollars, L is expressed in millions of workers, and D is expressed in hundreds of thousands of square miles. To be consistent with these numbers,5 the value of A in equation (20) must be 102.6. It follows immediately that, with half the population at work, a doubling of the population from 30 million to 60 million leads simultaneously to a rise in the national income from $900 billion to $1,429 billion and to a fall in income per worker from $60,000 to $47,622, or a fall in income per head from $30,000 to $23,811.

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Obviously, population cannot grow forever. Eventually, population growth at any given rate per year drives income per head down to the subsistence level where people are so impoverished that population growth stops, either because the birth rate falls or because the death rate from starvation or disease increases to match the existing population growth. Suppose current population growth is 3 percent per year and the subsistence income is $15,000 per head (or $30,000 per worker), exactly half the income per head in the year 2000. Designating t as the year when income per head falls to the subsistence level, it follows immediately from equation (20) that L(t) equals 120 million workers and population in the year t must have grown to 240 million people.6 When the year 0 is 2000, the year t must be 2069 because t = 69 is the solution to the equation

L(t) = L(0)e(0.03)t	(24)
where L(0) = 15, 000 and L(t) = 120, 000.	If the production function in

equation (20) were the true representation of the technology of the Canadian economy and if the assumed parameters were strictly correct, population growth would continue for the next 69 years and then stop, while income per head would decline steadily over this period until it becomes half what it is today.

It is interesting to consider what happens when population growth is not invariant, but varies in accordance with the standard of living. The greater the income per head the higher the rate of population growth up to some critical limit beyond which population growth remains constant. Realistic demographic models can be quite complicated because account must be taken of birth and death rates at every age of a person’s life. It is sufficient for our purposes to adopt the radically simplified demography incorporated in the following assumptions:

1Everybody works, so that population and labor force are the same.

2Property is equally distributed, so that everybody’s income, y, is the same.

3On surviving to age 30, half the population gives birth to quintuplets. Childbearing

occurs at no other time in a person’s life.

4The proportion of the population surviving to age 30 is as indicated by the curve

in figure 6.5, showing that one’s probability of living to age 30 increases steadily up to 100 percent at a standard of living of y1 and that only two-fifths of the population survives to age 30 if the standard of living falls as low as y0. [A survival rate of two-fifths is just sufficient to keep the population constant when half the population surviving to age 30 gives birth to 5 children and when no births occur at any other age.]

On these assumptions, the only mortality rate that matters is the probability of not surviving to age 30. Since all births occur at age 30 and since each and every birth is of quintuplets, the population would grow two and a half times per generation – or at an annual rate of 3 percent per year – if the standard of living were high enough that people’s survival rate to age 30 were 100 percent. But the standard of living cannot remain above subsistence permanently. As shown in figure 6.2, a steady rise in population with a fixed supply of land causes a steady fall in the standard of living.

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1

Proportions

2/5

T E C H N O L O G Y

Proportion of population surviving to age 30

Proportion of survivors

required to keep population stable

y0

y1

Income per person, y

Figure 6.5 Births, deaths, and the standard of living.

Eventually, the standard of living drops to y0, causing a high enough mortality rate to stop population growth. From then on, the standard of living remains low enough that the birth rate and death rate are the same. Actual population dynamics can be quite complex, but our simple assumptions require the economy to evolve into a steady state where only two-fifths of all children born survive until age 30 to give birth to children of their own. As shown in chapter 1, the world was somewhat like that until about a hundred years ago. If y1 were $25,000 per year and y0 were $15,000, then the population would eventually stabilize at the lower income. The Malthusian story is not that population keeps growing forever but that it stops growing at some level of income low enough to increase the death rate or reduce the birth rate sufficiently that they become the same.

The pessimistic account of the human condition in this chapter stands in sharp contrast to the glowing account of economic progress in chapter 1, especially the historic rise in both population and national income per head as exemplified by the Canadian time series in table 1.9. A ten-fold rise in national income per head was accompanied by a ten-fold increase in population. Can we therefore dismiss the Malthusian story as unrealistic fiction that need not trouble us today? There are several reasons why the story may still be interesting. It does seem relevant for many countries where potential prosperity has been thwarted by population growth. It may point to environmental risks down the road where the scarcity may be minerals, water, or air as well as land. By focussing on a very simple but internally consistent model of production, it supplies a framework for the introduction and analysis of additional considerations that may account for the observed coexistence of population growth and increase in the standard of living over a long period of time. The rest of this chapter is an examination and extension of the simple Malthusian model.

Why aren’t Canadians starving? There are at least four possible answers. The first is that Canadians have carved out for themselves a very large and sparsely populated chunk of the world’s territory and are actively keeping others out. The second is that

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people have learned to control population, reducing the average size of family even when the standard of living is high. The third is that investment can compensate for population growth. The fourth is that productivity per worker has been buoyed up by technical change. The first answer is undoubtedly right, but may not be the whole story. The second may be very much more important, not just to Canada but to many countries close to the margin of subsistence. One of the reasons why couples have traditionally wanted large families was to ensure that some of their children survive to care for them in their old age or to preserve the family name. As discussed in chapter 1, very large families were once necessary to preserve communities from extinction and to ensure parents that some children would survive to care for them in their old age. The rapid decline in mortality rates as a consequence of prosperity and of advances in medical care brings forth huge increases in population until such time as people learn to reduce the size of their families, as they may be inclined to do in order to afford to educate their children well. The third and fourth considerations will be discussed below.

PRODUCED MEANS OF PRODUCTION: CAN SOCIETY INVEST

ITS WAY OUT OF CATASTROPHE?

We have reasoned so far as though production and consumption occurred timelessly and simultaneously. In chapter 3, each day’s bread and cheese were produced by land on that very day. In this chapter, each day’s bread is produced with current supplies of labor and land. Nothing done yesterday could augment the area of land today. Nothing done today can augment the area of land tomorrow. To reason from a model of goods produced with labor and land is to abstract from the many factors of production that are themselves produced. Even land is augmentable in that fertilization today affects the productivity of land tomorrow. Other factors of production – such as factories and machines – are not supplied by nature, must themselves be built, and depreciate or become obsolete in the course of time.

To capture this aspect of technology we introduce a new factor of production called capital defined as the summation of all non-human resources: land, factories, machines, trains, ships, roads, airplanes, and so on. What differentiates capital from land in our model of production is reproducibility. Capital can be produced: land cannot. Think of capital as a number of machines, though the nation’s stock of capital is in reality much more comprehensive. The stock of capital in the year t is designated as K(t).7 The role of capital in the economy can be modeled by three key assumptions.

(a) Capital replaces land in the economy-wide production function, so that

Y(t) = F(L(t), K (t))

(25)

where L(t) is the input of labor in the year t, K(t) is the input of capital in the year t, and Y(t) is income in the year t, best thought of in this context as the quantity of some all-purposes good.

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	(b) Income may be consumed or invested in new capital.
	Y(t) = C(t) + I(t)	(26)

where C(t) is consumption in the year t and I(t) is investment in the year t. Equation (26) is the production possibility curve for consumption goods and invest-

ment goods together. With bread as the only consumption good and machines as the only capital good, equation (26) shows what combinations of bread and machines can be produced in the year t. This production possibility frontier is comparable to the production possibility frontier for bread and cheese in chapter 3, but linear rather than curved. The critical assumption in equation (26) is that the relative price of bread and machines is independent of how many of each are produced, that the supply curve of machines in terms of bread is flat. There is an additional assumption that, with Y measured in dollars, the prices of bread and machines are both equal to 1; this additional assumption is innocuous because units of bread and cheese can be appropriately defined. If a loaf of bread costs $2 and a new machine costs $100, then a loaf of bread counts as two units of income and a machine counts as 100 units of income.

(c) Investment this year augments capital next year through the equation
K (t + 1) = K (t)(1 − d) + I(t)	(27)

where d is the rate of depreciation per year. The capital stock available for use in the year t + 1 is the sum of the undepreciated portion of the capital stock from the previous year plus the extra capital produced in that year.

The replacement of irreducible land with reproducible capital alters the Malthusian story significantly. Inevitable impoverishment is replaced by a possibility of consumption per head remaining permanently above the subsistence level, but not of permanent growth. Impoverishment can sometimes be warded off indefinitely, and consumption per person held permanently above the subsistence level when the rate of population growth is not too large.

To show this, replace the general production function in equation (25) with the specific function

Y(t) = AL(t)αK (t)1−α

(28)

and assume that (1) population grows at a rate of n per year, (2) the rate of investment is constant at i year after year. In other words,

L(t)/L(t) = n	(29)
and
I(t)/Y(t) = i	(30)

Equation (29) is the same as equation (24) except for the representation of time. In equation (24), the variable t is continuous. In equation (29), t refers to a chunk of time such as a day or a year and L(t) refers to the changing population of the labor force during that year. Together, equations (29) and (30) ensure that society

Growth rates

n

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217

Of labor, L/L

Of capital, K/K

k* Capital–output ratio, k

Figure 6.6 Growth rates of labor and capital.

evolves automatically to a steady state at which consumption per head does not decline, but remains the same, year after year, forever. The steady state is characterized by a permanent capital–output ratio, k , at which Y and K grow at the same rate as L. From equations (27) and (30), it follows immediately that

K (t)/K (t) = I(t)/K (t) − d = i/k(t) − d

(31)

where K (t) is defined as K (t + 1) − K (t), and k(t) is the capital–output ratio, K (t)/Y(t). The meaning of equation (31) is that the rate of growth of capital is high when the capital–output ratio is low, and is low when the capital–output ratio is high, as illustrated in figure 6.6. One can see from the figure that as there is no upper or lower limit to K (t)/K (t), there must be some value, k , of the capital–output ratio for which K (t)/K (t) = L(t)/L(t) = n, signifying that the growth rates of labor and capital are the same. From equation (28) it follows that if L and K both grow at a rate n, then Y must grow at that rate as well, ensuring that consumption per head remains constant.

The permanent, steady-state value of the capital–output ratio must be
k = i/(n + d)	(32)
and the steady-state value of consumption per head becomes8
C/L = A1/α[(1 − i)i(1−α)/α]/[(n + d)](1−α)/α	(33)

The steady-state value of consumption per person (the value of C/L that can be maintained year after year) depends on the parameters i (the rate of investment), n (the rate of population growth), d (the rate of depreciation), α (labor’s share of the national income when the aggregate production function is as shown in equation (28)), and A, the constant term in the production function. The lower the rate of depreciation, the lower the rate of population growth and the higher the rate of investment

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(as long as the rate of investment remains below some critical level), the higher the steady-state consumption per head will turn out to be. Depending on the magnitudes of the parameters, the steady-state consumption per head may or may not exceed the subsistence level, and, if not, disease or malnutrition can be expected to lower the rate of population growth. But under no circumstances can consumption per head increase indefinitely. Eventually, some steady-state consumption per head is attained, beyond which there can be no additional growth in consumption per head. For no values of the parameters is consumption per head permanently increasing. For long-term economic growth, there must be technical change.

TECHNICAL CHANGE

As sketched in chapter 1, the twentieth century has witnessed a vast increase in the standard of living for much of the world’s population coupled with an equally vast increase in population itself, a combination that would have been impossible if technology had conformed to the production functions we have described so far. Even with full allowance for the role of reproducible capital in the production function, the increase in population per acre of arable land would have reduced the standard of living drastically, or starvation would have choked off population growth long before the present population was attained.

In practice, technical change and capital formation interact in complex ways, but, to introduce the economics of technical change, it is expeditious to abstract from capital, reverting to the simple production function in equation (16) where income is produced with land and labor. There are several ways to incorporate technical change into this aggregate production function. The simplest, and perhaps most instructive, is to convert the term A in the production from a constant to a variable that increases over time reflecting the accumulation of knowledge, spontaneously or as a consequence of deliberate research. The production function becomes

Y(t) = A(t)L(t)αD(1−α)

(34)

where D is constant, L(t) grows steadily over time at a rate n, there are constant returns to scale in L and D (doubling L and D leads to a doubling of Y as well), and technical change is at a steady rate g, that is

A(t) =

Aegt

(35)

On this specification of technology, the standard of living may rise or fall over time as the outcome of a balance between population growth and technical change. Other things being equal, population growth reduces output per head. Other things being equal, technical change increases output per head. The balance could go either way, depending on the rate of technical change, the rate of population growth, and the precise form of the production function.

Let y(t) represent income per person in the year t, that is y(t) = Y(t)/L(t). When population grows at a rate n, income per person in the year t becomes

y(t) = Y(t)/L(t) =	Aegt [D/L(t)]1−α =		[D/L(0)]1−αe[g−(1−α)n]	(36)
		A

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219

where L(t) grows steadily at a rate g in accordance with equation (21) and where A[D/L(0)]1−α is invariant. It follows immediately from equation (36) that

Growth rate of income per head = g − n(1 − α)

(37)

Equation (37) shows that income per head may be positive or negative depending on the balance between population growth and technical change. In our example, where population grows at 3 percent and labor’s share of the national income is 23 , a rate of technical change of 2 percent generates a rate of growth of income per head of 1 percent per year. There is no guarantee that technical change is sufficient to offset population growth in the determination of output per head. The balance depends on the magnitudes of g, n, and α. Presumably, the Canadian parameters have been sufficient to maintain a steady rise in output per head.

INVESTMENT, TECHNICAL CHANGE, AND RETURNS TO SCALE

Increasing or decreasing returns to scale are easily incorporated into the aggregate production function. It is sufficient to rewrite equation (16) as

B = ALαDβ

(38)

where the parameters α and β need not add up to 1. Returns to scale depend on the sum of α and β. The aggregate production function shows

constant returns to scale if α and β = 1,

increasing returns to scale if α and β > 1

and

decreasing returns to scale if α and β < 1.9

Except where α is greater than 1, increasing returns to scale cannot, all by itself, stave off the decline in output per head, but it can reinforce or dampen other effects.10 We saw above how investment can stabilize output per head in the presence of population growth. Some growth of output per head can be attained when there are increasing returns to scale. In that case, a steady rate of investment preserves the capital–labor ratio, allowing increasing returns to scale to buoy up output per head. The mechanics of the interaction among investment, technical change and returns to scale are beyond the scope of this book, but would be covered in texts on economic growth.

Investment and technical change reinforce one another, investment preserving the capital–labor ratio and technical change lifting output per unit of input of labor and capital together. Nor is it necessary, as we have so far supposed, that technical change be attached to the constant term, A, in the aggregate production function. To attach technical change to the constant term is to envision technical change as falling like

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manna from heaven regardless of what people do or how they behave. It might instead be assumed that technical change attaches itself to capital, investment, or even population. To attach technical change to investment is to suppose not only that new technology is incorporated in new machines, but that, the more society invests, the greater the impact of technology must be. That assumption might be incorporated into the model of investment above by allowing A to remain constant and replacing the term I(t) in equation (27) with the term I(t)egt where g is the rate of technical change. To attach technical change to labor or population is to suppose each person is as likely as any other to discover a way to increase productivity, so that the more people there are, the greater the overall change in productivity will be.

THE ENVIRONMENT OF TECHNICAL CHANGE

From a distance, aggregate technical change can be looked upon as a steady, almost autonomous, increase in productivity represented by the growth over time in the “constant term” of the aggregate production function. Up close, aggregate technical change consists of a thousand small, deliberate, man-made changes in the economy. People create technical change by scientific research into the laws of nature, by tinkering with products to improve them, and by developing new kinds of goods and services. Recognition that technical change is deliberate raises questions about whether the right amount of resources is being devoted to technical change, about whether the competitive market deploys those resources expeditiously and about the proper role, if any, of the collectivity in sponsoring and directing discovery.

Research is like investment in that expenditure of money and effort today yields a stream of benefits tomorrow. Research differs from most ordinary investment because knowledge is a public good, like television, roads, or guns, yielding benefits to large numbers of people simultaneously. No two people can eat the same slice of bread or own the same machine; but any two people can watch the same television program, drive on the same road or possess the same knowledge of the laws of nature without one person’s activities interfering with the activities of the other. Your understanding of a theorem does not block me from understanding that theorem as well. Reverting to the example in chapter 4, imagine a society where everybody has a plot of land and where, until people learn how to make cheese, every plot yields 20 loaves of bread and nothing else. Once people discover how to make cheese, every plot yields 20 loaves of bread or 10 pounds of cheese or any linear combination of the two, such as 10 loaves and 5 pounds of cheese, or 16 loaves and 2 12 pounds of cheese. Each person chooses a combination of bread and cheese to make himself as well off as possible, and my choice of a combination of bread and cheese imposes no restriction on yours. Any restriction on the production of cheese is inefficient in the sense that an appropriate reorganization of production and distribution could make everybody better off.

An unfortunate attribute of public goods – not excluding knowledge – is that nobody produces them voluntarily and at his own expense. Though everybody gains from being defended by the army, people must be compelled to contribute

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a share of the cost. People vote to finance the army through taxation, though nobody would be prepared to pay any tax unless compelled by the community to do so. The same is true of the acquisition of knowledge. If the cost of learning how to make cheese is more than any one person’s benefit from the discovery but less than the sum of everybody’s benefits from the discovery, and without a collective arrangement for sharing the cost or rewarding the inventor, the secret of how to make cheese may never be discovered, though everybody would gain from the discovery.

The acquisition of knowledge confronts society with a trade-off between efficiency in the acquisition of knowledge and efficiency in use. As knowledge is a public good, efficiency in use warrants that information be made freely available to everybody; but if knowledge were free there would be no private incentive to create it. There are two ways around this dilemma. Like roads and the army, the acquisition of knowledge might be administered by the government and financed by taxation. Alternatively, creators of knowledge might be awarded property rights to their discoveries, in the belief that society’s gain from providing an incentive to create knowledge outweighs its loss from restricting subsequent use. The first procedure is followed when research is conducted in publicly funded research labs or universities. The second is followed when patents are awarded for invention.

A patent is a grant to the inventor by the state of a monopoly on the use of his invention. By making invention profitable, patents draw forth inventions from every corner of society, mobilizing talent and ideas that no centrally directed agency would be able to identify and coordinate. Among the virtues of the competitive economy listed at the end of chapter 5 was that markets economize on knowledge. Prices are signals to each and every participant in the market about how his resources may be employed not just to enhance his own income, but to produce what the rest of society values and is prepared to pay for. Profit-seeking in response to prices induces people with special knowledge of local conditions to act as though coordinated by an all-seeing central planner, though no actual flesh-and-blood planner could ever recognize and employ all the information that traders collectively command. This is so not just for local opportunities (such as where to locate and how to design a restaurant or grocery store), but, much more importantly, for invention. Invention is almost always problematic. People disagree about whether this or that prospective invention is likely to materialize. Companies with solid records of achievement make what is subsequently recognized as huge mistakes in judgment. No planner could ever be expected to know in advance what is worth developing and which routes lead to a deadend. Here, above all, it is important to cull ideas from everywhere and to draw upon a diversity of experience and expectations. Patents do this, albeit imperfectly, by enabling a person with an idea for invention to proceed all by himself if he has the funds or if he can persuade a few rich people or firms that his idea has merit, as exemplified by the now-famous garages of Menlo Park. Patents draw forth invention that no planner could be expected to recognize.

There are social costs to the patent system. Among these costs is the restriction on the use of the newly invented product. A patent is a monopoly, and, like any monopoly, can only supply revenue to its possessor by enabling him to raise the price of the monopolized product above what the price would be in a competitive market.