So far we have assumed that the production func-tion for the representative firm takes the form Y = zF(K, Nd), where the function F has some very general properties (constant returns to scale, diminishing marginal products, etc.). When macroeconomists work with data to test theories, or when they want to simulate a macroeconomic model on the com-puter to study some quantitative aspects of a theory, they need to be much more specific about the form the production function takes. A very common production function used in the-ory and empirical work is the Cobb–Douglas production function. This function takes the form
Y= zKa(Nd)1-a,
where a is a parameter, with 0 6 a 6 1. The exponents on K and Nd in the function sum to 1 (a+ 1 - a = 1), which reflects constant returns to scale. If there are profit-maximizing, price-taking firms and constant returns to scale, then a Cobb–Douglas production function implies that a will be the share that capital receives of national income (in our model, the prof-its of firms), and 1 - a the share that labor receives (wage income before taxes) in equi-librium. What is remarkable is that, from the National Income and Product Accounts (NIPA) of the United States, the capital and labor shares of national income have been roughly constant in the United States, which is consistent with the Cobb–Douglas production function. Given this, an empirical estimate of a is the aver-age share of capital in national income, which from the data is about 0.30, or 30%, so a good
approximation to the actual U.S. aggregate pro-duction function is
Y= zK0.30(Nd)0.70. (4-10) In Equation (4-10), the quantities Y, K, and Nd can all be measured. For example, Y can be measured as real GDP from the NIPA, K can be measured as the total quantity of cap-ital in existence, built up from expenditures on capital goods in the NIPA, and Nd can be measured as total employment, in the Current Population Survey done by the Bureau of Labor Statistics. But how is total factor productivity z measured? Total factor productivity cannot be measured directly, but it can be measured indirectly, as a residual. That is, from Equation (4-10), if we can measure Y, K, and Nd, then a measure of z is the Solow residual, which is calculated as
z= Y
K0.30(Nd)0.70. (4-11) This measure of total factor productivity is named after Robert Solow.5 In Figure 4.18, we graph the Solow residual for the United States for the period 1948–2010, calculated using Equation (4-11) and measurements of Y, K, and Nd as described above. Measured total factor productivity grows over time, and it fluc-tuates about trend. In Chapters 7, 8, and 13, we see how growth and fluctuations in total factor productivity can cause growth and fluctuations in real GDP.
5See R. Solow, 1957. “Technical Change and the Aggre-gate Production Function,” Review of Economic Statistics 39, 312–320.
Figure 4.18 The Solow Residual for the United States
The Solow residual is a measure of total factor productivity, and it is calculated here using a Cobb–Douglas production function. Measured total factor productivity has increased over time, and it also fluctuates about trend, as shown for the period 1948–2010.
1940 1950 1960 1970 1980 1990 2000 2010
0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14
Year
Solow Residual
where K is fixed. Here, p is real profit. In Figure 4.19, we graph the revenue function, zF(K, Nd), and the variable cost function, wNd. Profit is then the difference between total revenue and total variable cost. Here, to maximize profits, the firm chooses Nd= N∗. The maximized quantity of profits, p∗, is the distance AB. For future reference, p∗ is the distance ED, where AE is a line drawn parallel to the variable cost function.
Thus, AE has slope w. At the profit-maximizing quantity of labor, N∗, the slope of the total revenue function is equal to the slope of the total variable cost function. The slope of the total revenue function, however, is just the slope of the production function, or the marginal product of labor, and the slope of the total variable cost function is the real wage w. Thus, the firm maximizes profits by setting
MPN= w. (4-12)
To understand the intuition behind Equation (4-12), note that the contribution to the firm’s profits of having employees work an extra hour is the extra output produced
Figure 4.19 Revenue, Variable Costs, and Profit Maximization
Y= zF(K, Nd) is the firm’s revenue, while wNdis the firm’s variable cost. Profits are the difference between the former and the latter. The firm maximizes profits at the point where marginal revenue equals marginal cost, or MPN= w.
Maximized profits are the distance AB, or the distance ED.
zF(K, Nd) wNd
A
B
D E
N*
Labor Input, Nd
Revenue, Variable Costs
minus what the extra input costs—that is, MPN- w. Given a fixed quantity of capital, the marginal product of labor is very high for the first hour worked by employees, and the way we have drawn the production function in Figure 4.12, MPN is very large for Nd = 0, so that MPN- w 7 0 for Nd = 0, and it is worthwhile for the firm to hire the first unit of labor, as this implies positive profits. As the firm hires more labor, MPN falls, so that each additional unit of labor is contributing less to revenue, but contributing the same amount, w, to costs. Eventually, at Nd = N∗, the firm has hired enough labor so that hiring an additional unit implies MPN- w 6 0, which in turn means that hiring an additional unit of labor only causes profits to go down, and this cannot be optimal. Therefore, the profit-maximizing firm chooses its labor input according to Equation (4-12).
In our earlier example of the accounting firm, suppose that there is one photocopy machine at the firm, and output for the firm can be measured in terms of the clients the firm has. Each client pays $20,000 per year to the firm, and the wage rate for an accountant is $50,000 per year. Therefore, the real wage is 50,00020,000 = 2.5 clients. If the firm has 1 accountant, it can handle 5 clients per year, if it has 2 accountants it can handle 9 clients per year, and if it has 3 accountants it can handle 11 clients per year.
What is the profit-maximizing number of accountants for the firm to hire? If the firm hires Sara, her marginal product is 5 clients per year, which exceeds the real wage of 2.5 clients, and so it would be worthwhile for the firm to hire Sara. If the firm hires
Figure 4.20 The Marginal Product of Labor Curve Is the Labor Demand Curve of the Profit-Maximizing Firm.
This is true because the firm hires labor up to the point where MPN= w.
MPN or Labor Demand Curve
Quantity of Labor Demanded, Nd
Real Wage, w
Sara and Paul, then Paul’s marginal product is 4 clients per year, which also exceeds the market real wage, and so it would also be worthwhile to hire Paul. If the firm hires Sara, Paul, and Julia, then Julia’s marginal product is 2 clients per year, which is less than the market real wage of 2.5 clients. Therefore, it would be optimal in this case for the firm to hire two accountants, Sara and Paul.
Our analysis tells us that the representative firm’s marginal product of labor sched-ule, as shown in Figure 4.20, is the firm’s demand curve for labor. This is because the firm maximizes profits for the quantity of labor input that implies MPN= w. Therefore, given a real wage w, the marginal product of labor schedule tells us how much labor the firm needs to hire such that MPN = w, and so the marginal product of labor schedule and the firm’s demand curve for labor are the same thing.
Chapter Summary
• In this chapter, we studied the behavior of the representative consumer and the representative firm in a one-period, or static, environment. This behavior is the basis for constructing a macroeconomic model that we can work with in Chapter 5.
• The representative consumer stands in for the large number of consumers that exist in the economy as a whole, and the representative firm stands in for a large number of firms.
• The representative consumer’s goal is to choose consumption and leisure to make himself or herself as well off as possible while respecting his or her budget constraint.
• The consumer’s preferences have the properties that more is always preferred to less and that there is preference for diversity in consumption and leisure. The consumer is a price-taker in that he or she treats the market real wage as given, and his or her real disposable income is real wage income plus real dividend income, minus real taxes.
• Graphically, the representative consumer optimizes by choosing the consumption bundle where an indifference curve is tangent to the budget constraint or, what is the same thing, the marginal rate of substitution of leisure for consumption is equal to the real wage.
• Under the assumption that consumption and leisure are normal goods, an increase in the representative consumer’s income leads to an increase in consumption and an increase in leisure, implying that labor supply goes down.
• An increase in the real wage leads to an increase in consumption, but it may cause leisure to rise or fall, because there are opposing income and substitution effects. The consumer’s labor supply, therefore, may increase or decrease when the real wage increases.
• The representative firm chooses the quantity of labor to hire so as to maximize profits, with the quantity of capital fixed in this one-period environment.
• The firm’s production technology is captured by the production function, which has constant returns to scale, a diminishing marginal product of labor, and a diminishing marginal product of capital. Further, the marginal products of labor and capital are positive, and the marginal product of labor increases with the quantity of capital.
• An increase in total factor productivity increases the quantity of output that can be produced with any quantities of labor and capital, and it increases the marginal product of labor.
• When the firm optimizes, it sets the marginal product of labor equal to the real wage. This implies that the firm’s marginal product of labor schedule is its demand curve for labor.
Key Terms
Static decision A decision made by a consumer or firm for only one time period. (p. 96)
Dynamic decision A decision made by a consumer or firm for more than one time period. (p. 96) Consumption good A single good that represents an aggregation of all consumer goods in the economy.
(p. 97)
Leisure Time spent not working in the market.
(p. 97)
Representative consumer A stand-in for all con-sumers in the economy. (p. 97)
Utility function A function that captures a con-sumer’s preferences over goods. (p. 97)
Consumption bundle A given consumption–leisure combination. (p. 97)
Normal good A good for which consumption increa-ses as income increaincrea-ses. (p. 99)
Inferior good A good for which consumption decre-ases as income incredecre-ases. (p. 99)
Indifference map A set of indifference curves repre-senting a consumer’s preferences over goods; has the same information as the utility function. (p. 99) Indifference curve A set of points that represents consumption bundles among which a consumer is indifferent. (p. 99)
Marginal rate of substitution Minus the slope of an indifference curve, or the rate at which the consumer is just willing to trade one good for another. (p. 101) Competitive behaviour Actions taken by a consum-er or firm if market prices are outside its control.
(p. 102)
Barter An exchange of goods for goods. (p. 103) Time constraint Condition that hours worked plus leisure time sum to total time available to the con-sumer. (p. 103)
Real wage The wage rate in units of the consumption good. (p. 103)
Numeraire The good in which prices are denomi-nated. (p. 103)
Dividend income Profits of firms that are distributed to the consumer, who owns the firms. (p. 103) Lump-sum tax A tax that is unaffected by the actions of the consumer or firm being taxed. (p. 103) Budget constraint Condition that consumption eq-uals wage income plus nonwage income minus taxes.
(p. 104)
Rational Describes a consumer who can make an informed optimizing decision. (p. 106)
Optimal consumption bundle The consumption bundle for which the consumer is as well off as possi-ble while satisfying the budget constraint. (p. 106) Relative price The price of a good in units of another good. (p. 108)
Pure income effect The effect on the consumer’s optimal consumption bundle due to a change in real disposable income, holding prices constant. (p. 110) Substitution effect The effect on the quantity of a good consumed due to a price change, holding the consumer’s welfare constant. (p. 112)
Income effect The effect on the quantity of a good consumed due to a price change, as a result of having an effectively different income. (p. 113)
Labor supply curve A relationship describing the quantity of labor supplied for each level of the real wage. (p. 113)
Perfect complements Two goods that are always consumed in fixed proportions. (p. 114)
Perfect substitutes Two goods with a constant mar-ginal rate of substitution between them. (p. 116) Production function A function describing the tech-nological possibilities for converting factor inputs into output. (p. 119)
Total factor productivity A variable in the produc-tion funcproduc-tion that makes all factors of producproduc-tion more productive if it increases. (p. 119)
Marginal product The additional output produced when another unit of a factor of production is added to the production process. (p. 119)
Constant returns to scale A property of the produc-tion technology whereby if the firm increases all inputs by a factor x this increases output by the same factor x. (p. 120)
Increasing returns to scale A property of the pro-duction technology whereby if the firm increases all inputs by a factor x this increases output by more than the factor x. (p. 120)
Decreasing returns to scale A property of the pro-duction technology whereby if the firm increases all inputs by a factor x this increases output by less than the factor x. (p. 120)
Representative firm A stand-in for all firms in the economy. (p. 121)
Cobb-Douglas production function A particular mathematical form for the production function that fits U.S. aggregate data well. (p. 128)
Solow residual A measure of total factor productivity obtained as a residual from the production function, given measures of aggregate output, labor input, and capital input. (p. 128)
Questions for Review
All questions refer to the elements of the macroeconomic model developed in this chapter.
1. What goods do consumers consume in this model?
2. How are a consumer’s preferences over goods represented?
3. What three properties do the preferences of the representative consumer have? Explain the importance of each.
4. What two properties do indifference curves have? How are these properties associated with the properties of the consumer’s preferences?
5. What is the representative consumer’s goal?
6. When the consumer chooses his or her optimal consumption bundle while respecting his or her budget constraint, what condition is satisfied?
7. How is the representative consumer’s behavior affected by an increase in real dividend income?
8. How is the representative consumer’s behavior affected by an increase in real taxes?
9. Why might hours worked by the representative consumer decrease when the real wage increases?
10. What is the representative firm’s goal?
11. Why is the marginal product of labor diminishing?
12. What are the effects on the production function of an increase in total factor productivity?
13. Explain why the marginal product of labor curve is the firm’s labor demand curve.
Problems
1. Using a diagram show that if the consumer prefers more to less, then indifference curves cannot cross.
2. In this chapter, we showed an example in which the consumer has preferences for consump-tion with the perfect complements property.
Suppose, alternatively, that leisure and con-sumption goods are perfect substitutes. In this case, an indifference curve is described by the equation
i= al + bC,
where a and b are positive constants, and u is the level of utility. That is, a given indifference curve has a particular value for u, with higher indifference curves having higher values for u.
(a) Show what the consumer’s indifference cur-ves look like when consumption and leisure are perfect substitutes, and determine graphically and algebraically what con-sumption bundle the consumer chooses.
Show that the consumption bundle the con-sumer chooses depends on the relationship between a/b and w, and explain why.
(b) Do you think it likely that any consumer would treat consumption goods and leisure as perfect substitutes?
(c) Given perfect substitutes, is more preferred to less? Do preferences satisfy the diminish-ing marginal rate of substitution property?
3. Consider the consumer choice example in this chapter, where consumption and leisure are
perfect complements. Assume that the con-sumer always desires a consumption bundle where the quantities of consumption and leisure are equal, that is, a= 1.
(a) Suppose that w= 0.75, p = 0.8, and T = 6.
Determine the consumer’s optimal choice of consumption and leisure, and show this in a diagram.
(b) Now suppose that w = 1.5, p = 0.8, and T = 6. Again, determine the consumer’s optimal choice of consumption and leisure, and show this in your diagram. Explain how and why the consumer’s optimal con-sumption bundle changes, with reference to income and substitution effects.
4. Suppose that the government imposes a pro-portional income tax on the representative con-sumer’s wage income. That is, the concon-sumer’s wage income is w(1- t)(h - l) where t is the tax rate. What effect does the income tax have on consumption and labor supply? Explain your results in terms of income and substitution effects.
5. Suppose, as in the federal income tax code for the United States, that the representative con-sumer faces a wage income tax with a standard deduction. That is, the representative consumer pays no tax on wage income for the first x units of real wage income, and then pays a pro-portional tax t on each unit of real wage income greater than x. Therefore, the consumer’s budget constraint is given by C = w(h - l) + p if
w(h- l) … x, or C = (1 - t)w(h - l) + tx + p if w(h- l) Ú x. Now, suppose that the government reduces the tax deduction x. Using diagrams, determine the effects of this tax change on the consumer, and explain your results in terms of income and substitution effects. Make sure that you consider two cases. In the first case, the consumer does not pay any tax before x is reduced, and in the second case, the consumer pays a positive tax before x is reduced.
6. Suppose that the representative consumer’s div-idend income increases, and his or her wage rate falls at the same time. Determine the effects on consumption and labor supply, and explain your results in terms of income and substitution effects.
7. Suppose that a consumer can earn a higher wage rate for working overtime. That is, for the first q hours the consumer works, he or she receives a real wage rate of w1, and for hours worked more than q he or she receives w2, where w2 7 w1. Suppose that the consumer pays no taxes and receives no nonwage income, and he or she is free to choose hours of work.
(a) Draw the consumer’s budget constraint, and show his or her optimal choice of consumption and leisure.
(b) Show that the consumer would never work q hours, or anything very close to q hours.
Explain the intuition behind this.
(c) Determine what happens if the overtime wage rate w2increases. Explain your results in terms of income and substitution effects.
You must consider the case of a worker who initially works overtime, and a worker who initially does not work overtime.
8. Show that the consumer is better off with a lump-sum tax rather than a proportional tax
8. Show that the consumer is better off with a lump-sum tax rather than a proportional tax