It is tempting to view poverty as an easy problem to solve. If one assumed that real GDP was essentially fixed, then it might make sense to redistribute income from the lucky rich to the unlucky poor so that we could be collectively better off. Government income redistribution—through the tax sys-tem and the provision of goods and services such as parks and health care—might be seen as poverty insurance. We might then argue that since the private sector has failed to provide this insurance against poverty, the government should step in to fill the void.
An important means for redistributing income in the United States and other coun-tries is through the taxation of labor. In particular, federal and state income taxes are progressive, in that income taxes represent a smaller fraction of income for the typical poor person than for the typical rich person.
This contrasts with sales taxes, which tend to be regressive—the typical poor person pays a larger fraction of his or her income to the gov-ernment in the form of sales taxes than does a typical rich person.
It is well understood, though, that in-come taxation has incentive effects. In our simple model of consumer behaviour, an easy way to study the effects of income taxation on a consumer is to assume that the con-sumer’s wage income is taxed at a constant rate, t. Then, supposing that the lump-sum tax is zero, or T = 0, the consumer would pay total taxes of tw(h-l), and the consumer’s budget constraint would be
C= w(1 - t)(h - l) + p.
Then, if we wanted to analyze the effects of a change in the income tax rate t on labor supply, given the market real wage w, this would be the same as analyzing the effects of
a change in the real wage, since now w(1-t) is the effective real wage for the consumer, and an increase in t is equivalent to a decrease in the consumer’s effective real wage. From our analysis in this chapter, theory tells us that an increase in the income tax rate t may cause an increase or a decrease in the quantity of labor supplied, depending on the relative strengths of opposing income and substitu-tion effects. For example, an increase in t will reduce labor supply only if the substitution effect is large relative to the income effect.
That is, if the substitution effect is relatively large, then there can be a large disincentive effect on hours of work as the result of an increase in income tax rate.
This disincentive effect might give us pause if we we wanted to think of the income tax as a useful means for redistribut-ing income. If the substitution effect is large, so that the elasticity of labor supply with respect to wages is large, then real GDP should not be considered a fixed pie that can be redistributed at will. An attempt to divide up the pie differently by increasing income tax rates for the rich and reducing them for the poor might actually reduce the pie substantially.
Whether the reduction in real GDP from a redistributive tax system is a serious prob-lem turns on how large the elasticity of labor supply is. Typically, in studying how indi-viduals adjust hours of work in response to changes in wages, labor economists find the effects to be small. Thus, according to microeconomic evidence, the elasticity of labor supply with respect to wages is small, therefore the disincentive effects from income taxation are small, and redistributing the pie will not reduce the size of the pie by very much.
(Continued)
However, for macroeconomists this is not the end of the story. In a recent working paper,1Michael Keane and Richard Rogerson review the evidence on labor supply elas-ticities. For macroeconomists, the key idea is that what matters for aggregate economic activity is total labor input, which is deter-mined by three factors: (i) how many hours each individual works; (ii) how many indi-viduals are working; and (iii) the quality of the labor hours supplied.
Microeconomic evidence on labor ply typically focuses on the labor hours sup-plied by individuals and how this responds to wages. This is the so-called inten-sive margin—how inteninten-sively an individual works. However, as Keane and Rogerson point out, changes in aggregate hours worked over both short and long periods of time are influenced in an important way by choices at the extensive margin, that is, the choices of individuals about whether to work or not.
While a higher market income tax rate might have little effect on any individual worker’s hours of work, it might induce more people to leave the labor force. For example, some people might choose to care for their children at home rather than working in the market.
In fact, if we take into account the extensive margin, the aggregate labor supply elasticity is much higher.
Thus, in our macroeconomic model, it is most useful to think of the representative consumer as a fictitious person who stands in for the average consumer in the econ-omy. Hours of work for this fictitious person should be interpreted as average hours of work in the whole economy. Thus, when aggregate hours of work change in practice, because of changes along the intensive and extensive margins, we should think of this
1M. Keane and R. Rogerson, 2011. “Reconciling Micro- and Macrolabor Supply Elasticities: A Structural Perspective,” NBER working paper 17430.
as changes in the representative consumer’s hours of work in our model.
Another dimension on which labor sup-ply can change, as mentioned above, is in terms of the quality of labor hours sup-plied. This is essentially a long-run effect that occurs through occupational choice. For example, if income tax rates are increased for the rich and reduced for the poor, this reduces the incentive of young people to obtain the education required to perform higher-paying jobs. Fewer people will choose to become engineers, and more will choose to become plumbers. The evidence in Keane and Rogerson’s article suggests that this effect is large.
What would United States look like if we chose to be a much more egalitarian soci-ety, by using the income tax to redistribute income from rich to the poor? For a very rich person, the after-tax wage earned for on an extra hour of work would be much lower than it is now, and for a very poor person, the after-tax wage at the margin would be much higher. Average hours worked among employed people in this egalitarian soci-ety would be somewhat lower than now, but there would be many more people who would choose not to participate in the labor force. As well, over the long run, the aver-age level of skills acquired by the population would be much lower. Real GDP would fall.
There is some evidence that higher taxa-tion of labor explains differences between the United States and Europe in labor supply and real GDP per capita.2Similarly, the Canadian income tax system is more progressive than the U.S. system. Thus, a higher degree of income redistribution in Canada than in the United States could in part explain why US real GDP per capita is higher than in Canada.
2E. Prescott, 2004. “Why Do Americans Work More Than Europeans?” Minneapolis Federal Reserve Bank Quarterly Review 28, No. 1, 2–13.
The firms in this economy own productive capital (plant and equipment), and they hire labor to produce consumption goods. We can describe the production technology available to each firm by a production function, which describes the technological possibilities for converting factor inputs into outputs. We can express this relationship in algebraic terms as
Y = z F(K, Nd), (4-9)
where z is total factor productivity, Y is output of consumption goods, K is the quantity of capital input in the production process, Ndis the quantity of labor input measured as total hours worked by employees of the firm, and F is a function. Because this is a one-period or static (as opposed to dynamic) model, we treat K as being a fixed input to production, and Nd as a variable factor of production. That is, in the short run, firms cannot vary the quantity of plant and equipment (K) they have, but they have flexibility in hiring and laying off workers (Nd). Total factor productivity, z, captures the degree of sophistication of the production process. That is, an increase in z makes both factors of production, K and Nd, more productive, in that, given factor inputs, higher z implies that more output can be produced.
For example, suppose that the above production function represents the technol-ogy available to a bakery. The quantity of capital, K, includes the building in which the bakery operates, ovens for baking bread, a computer for doing the bakery accounts, and other miscellaneous equipment. The quantity of labor, Nd, is total hours worked by all the bakery employees, including the manager, the bakers who operate the ovens, and the employees who work selling the bakery’s products to customers. The variable z, total factor productivity, can be affected by the techniques used for organizing pro-duction. For example, bread could be produced either by having each baker operate an individual oven, using this oven to produce different kinds of bread, or each baker could specialize in making a particular kind of bread and use the oven that happens to be available when an oven is needed. If the latter production method produces more bread per day using the same inputs of capital and labor, then that production process implies a higher value of z than does the first process.
For our analysis, we need to discuss several important properties of the production function. Before doing this, we need the following definition.
DEFINITION 4 The marginal product of a factor of production is the additional output that can be produced with one additional unit of that factor input, holding constant the quantities of the other factor inputs.
In the production function on the right side of Equation (4-9), there are two factor inputs, labor and capital. Figure 4.12 shows a graph of the production function, fixing the quantity of capital at some arbitrary value, K∗, and allowing the labor input, Nd, to vary. Some of the properties of this graph require further explanation. The marginal product of labor, given the quantity of labor N∗, is the slope of the production function
Figure 4.12 Production Function, Fixing the Quantity of Capital, and Varying the Quantity of Labor The marginal product of labor is the slope of the production function at a given point. Note that the marginal product of labor declines with the quantity of labor.
Slope = MPN
F(K*, Nd)
A
N*
Labor Input, Nd
Output, Y
at point A; this is because the slope of the production function is the additional out-put produced from an additional unit of the labor inout-put when the quantity of labor is N∗ and the quantity of capital is K∗. We let MPN denote the marginal product of labor.
Next, in Figure 4.13 we graph the production function again, but this time we fix the quantity of labor at N∗and allow the quantity of capital to vary. In Figure 4.13, the marginal product of capital, denoted MPK, given the quantity of capital K∗, is the slope of the production function at point A.
The production function has five key properties, which we will discuss in turn.
1. The production function exhibits constant returns to scale. Constant returns to scale means that, given any constant x7 0, the following relationship holds:
zF(xK, xNd)= xzF(K, Nd).
That is, if all factor inputs are changed by a factor x, then output changes by the same factor x. For example, if all factor inputs double (x = 2), then output also doubles. The alternatives to constant returns to scale in production are increas-ing returns to scale and decreasincreas-ing returns to scale. Increasincreas-ing returns to scale
Figure 4.13 Production Function, Fixing the Quantity of Labor, and Varying the Quantity of Capital The slope of the production function is the marginal product of capital, and the marginal product of capital declines with the quantity of capital.
Slope = MPK
F(K, N*)
A
K*
Capital Input, K
Output, Y
implies that large firms (firms producing a large quantity of output) are more efficient than small firms, whereas decreasing returns to scale implies that small firms are more efficient than large firms. With constant returns to scale, a small firm is just as efficient as a large firm. Indeed, constant returns to scale means that a very large firm simply replicates how a very small firm produces many times over. Given a constant-returns-to-scale production function, the economy behaves in exactly the same way if there were many small firms producing con-sumption goods as it would if there were a few large firms, provided that all firms behave competitively (they are price-takers in product and factor markets). Given this, it is most convenient to suppose that there is only one firm in the economy, the representative firm. Just as with the representative consumer, it is help-ful to think of the representative firm as a convenient stand-in for many firms, which all have the same constant-returns-to-scale production function. In prac-tice, it is clear that in some industries decreasing returns to scale are important.
For example, high-quality restaurant food seems to be produced most efficiently on a small scale. Alternatively, increasing returns to scale are important in the automobile industry, where essentially all production occurs in very large-scale firms, such as General Motors (which of course is not as large-scale as it once
was). This does not mean, however, that it is harmful to assume that there exists constant returns to scale in production at the aggregate level, as is the case in our model. This is because even the largest firm in the U.S. economy produces a small amount of output relative to U.S. GDP, and the aggregate economy can exhibit constant returns to scale in aggregate production, even if this is not literally true for each firm in the economy.
2. The production function has the property that output increases when either the capital input or the labor input increases. In other words, the marginal products of labor and capital are both positive: MPN 7 0 and MPK 7 0. In Figures 4.12 and 4.13, these properties of the production function are exhibited by the upward slope of the production function. Recall that the slope of the production function in Figure 4.12 is the marginal product of labor and the slope in Figure 4.13 is the marginal product of capital. Positive marginal products are quite natural properties of the production function, as this states simply that more inputs yield more output. In the bakery example discussed previously, if the bakery hires more workers given the same capital equipment, it will produce more bread, and if it installs more ovens given the same quantity of workers, it will also produce more bread.
3. The marginal product of labor decreases as the quantity of labor increases. In Figure 4.12 the declining marginal product of labor is reflected in the concav-ity of the production function. That is, the slope of the production function in Figure 4.12, which is equal to MPN, decreases as Nd increases. The following example helps to illustrate why the marginal product of labor should fall as the quantity of labor input increases: Suppose accountants work in an office build-ing that has one photocopy machine, and suppose that they work with pencils and paper but at random intervals need to use the photocopy machine. The first accountant added to the production process, Sara, is very productive—that is, she has a high marginal product—as she can use the photocopy machine when-ever she wants. Howwhen-ever, when the second accountant, Paul, is added, Sara on occasion wants to use the machine and she gets up from her desk, walks to the machine, and finds that Paul is using it. Thus, some time is wasted that could otherwise be spent working. Paul and Sara produce more than Sara alone, but what Paul adds to production (his marginal product) is lower than the marginal product of Sara. Similarly, adding a third accountant, Julia, makes for even more congestion around the photocopy machine, and Julia’s marginal product is lower than Paul’s marginal product, which is lower than Sara’s. Figure 4.14 shows the representative firm’s marginal product of labor schedule. This is a graph of the firm’s marginal product, given a fixed quantity of capital, as a function of the labor input. That is, this is the graph of the slope of the production function in Figure 4.12. The marginal product schedule is always positive, and it slopes downward.
4. The marginal product of capital decreases as the quantity of capital increases. This property of the production function is very similar to the previous one, and it is illustrated in Figure 4.13 by the decreasing slope, or concavity, of the production function. In terms of the example above, if we suppose that Sara, Paul, and Julia are the accountants working in the office and imagine what
Figure 4.14 Marginal Product of Labor Schedule for the Representative Firm
The marginal product of labor declines as the quantity of labor used in the production process increases.
MPN
Labor Input, Nd Marginal Product of Labor, MPN
happens as we add photocopy machines, we can gain some intuition as to why the decreasing-marginal-product-of-capital property is natural. Adding the first photocopy machine adds a great deal to total output, as Sara, Paul, and Julia now can duplicate documents that formerly had to be copied by hand.
With three accountants in the office, however, there is congestion around the machine. This congestion is relieved with the addition of a second machine, so that the second machine increases output, but the marginal product of the second machine is smaller than the marginal product of the first machine, and so on.
5. The marginal product of labor increases as the quantity of the capital input increases.
To provide some intuition for this property of the production function, return to the example of the accounting firm. Suppose that Sara, Paul, and Julia initially have one photocopy machine to work with. Adding another photocopy machine amounts to adding capital equipment, and this relieves congestion around the copy machine and makes each of Sara, Paul, and Julia more productive, includ-ing Julia, who was the last accountant added to the workforce at the firm.
Therefore, adding more capital increases the marginal product of labor, for each quantity of labor. In Figure 4.15 an increase in the quantity of capital from K1 to K2 shifts the marginal product of labor schedule to the right, from MP1N to MP2N.
Figure 4.15 Adding Capital Increases the Marginal Product of Labor
For each quantity of the labor input, the marginal product of labor increases when the quantity of capital used in
For each quantity of the labor input, the marginal product of labor increases when the quantity of capital used in