Chapter 3: National Income: Where it Comes From and Where it Goes Continued CHAPTER 3 National Income

Demand for labor The firm tries to maximize its profits: Profit = Total Revenue – Total Cost = P.Y – W.L – R.K Profit=P. F (K, L) – W.L – R.K The competitive firm takes P, W, R as given and chooses L and K to maximize its profits this is the basic force behind the demand function for the factors of production CHAPTER 3 National Income

A firm hires each unit of labor if the cost does not exceed the benefit. cost = real wage benefit = marginal product of labor CHAPTER 3 National Income

Marginal product of labor (MPL ) definition: The extra output the firm can produce using an additional unit of labor (holding other inputs fixed): MPL = F (K, L +1) – F (K, L) CHAPTER 3 National Income

MPL and the production function Y output 1 MPL As more labor is added, MPL  1 MPL (Figure 3-3 on p.51) Production function is upwards sloping (output increases with the additional labor): MPL is positive The slope is declining: Diminishing MPL. Slope of the production function equals MPL MPL 1 L labor CHAPTER 3 National Income

Diminishing marginal returns As a factor input is increased, its marginal product falls (other things equal). Intuition: Suppose L while holding K fixed  fewer machines per worker  lower worker productivity Tell class: Many production functions have this property. This slide introduces some short-hand notation that will appear throughout the PowerPoint presentations of the remaining chapters: The up and down arrows mean increase and decrease, respectively. The symbol “” means “causes” or “leads to.” Hence, the text after “Intuition” should be read as follows: “An increase in labor while holding capital fixed causes there to be fewer machines per worker, which causes lower productivity.” Many instructors use this type of short-hand (or something very similar), and it’s much easier and quicker for students to write down in their notes. CHAPTER 3 National Income

Example Suppose there is a firm with one computer, one Xerox machine, and one employee. The worker’s job is to answer e-mail and make photocopies. When a second worker is hired, together they can produce more because while one is answering e-mail, the other can make copies. They hire another worker. Now output still increases (because they can take breaks on rotation) but not as much as the case after 2nd worker. Hence, holding the amount of capital fixed, as we increase the amount of labor, MPL decreases. That is, output does increase for each additional unit of labor, but the change in output is smaller and smaller for each additional worker. CHAPTER 3 National Income

The firm checks the change in profits by hiring an additional worker: How does the firm use this analysis to decide on how many workers to hire? The firm checks the change in profits by hiring an additional worker: ∆Profits = ∆Revenue - ∆Cost ∆Profits/∆L = ∆Revenue/∆L - ∆Cost/∆L = ∆ (P.Y)/ ∆L - ∆ (W.L)/ ∆L = P (∆Y/∆L) – W (∆L/∆L) = P.MPL – W => The firm will hire more workers as long as ∆Profits/∆L > 0 P.MPL > W CHAPTER 3 National Income

As L increases, MPL decreases up to the point where P.MPL = W MPL=W/P The firm hires additional workers until MPL is equal to the real wage ( i.e. the nominal wage as a ratio of the price level: W/P) Note: price level= price of output CHAPTER 3 National Income

Exercise L Y MPL 0 0 n.a. 1 10 10 2 19 9 3 27 8 Suppose W/P = 6. 1 10 10 2 19 9 3 27 8 4 34 7 5 40 6 6 45 5 7 49 4 8 52 3 9 54 2 10 55 1 Suppose W/P = 6. If L = 3, should firm hire more or less labor? Why? If L = 7, should firm hire more or less labor? Why? If L=3, then the benefit of hiring the fourth worker (MPL=7) exceeds the cost of doing so (W/P = 6), so it pays the firm to increase L. If L=7, then the firm should hire fewer workers: the 7th worker adds only MPL=4 units of output, yet cost W/P = 6. The point of this slide is to get students to see the idea behind the labor demand = MPL curve. CHAPTER 3 National Income

MPL and the demand for labor Units of output Units of labor, L Each firm hires labor up to the point where MPL = W/P. MPL, Labor demand Real wage Quantity of labor demanded It’s easy to see that the MPL curve is the firm’s L demand curve. Let L* be the value of L such that MPL = W/P. Suppose L < L*. Then, benefit of hiring one more worker (MPL) exceeds cost (W/P), so firm can increase profits by hiring one more worker. Instead, suppose L > L*. Then, the benefit of the last worker hired (MPL) is less than the cost (W/P), so firm should reduce labor to increase its profits. When L = L*, then firm cannot increase its profits either by raising or lowering L. Hence, firm hires L to the point where MPL = W/P. This establishes that the MPL curve is the firm’s labor demand curve. CHAPTER 3 National Income

Notes The firm takes the real wage as given and decides on how much labor to demand. Real wage (measured in units of output) is located on the vertical axis. The demand curve (MPL) is positive but decreasing (due to law of diminishing returns) CHAPTER 3 National Income

The equilibrium real wage Units of output Units of labor, L Labor supply The real wage adjusts to equate labor demand with supply. MPL, Labor demand equilibrium real wage The labor supply curve is vertical: We are assuming that the economy has a fixed quantity of labor, Lbar, regardless of whether the real wage is high or low. Combining this labor supply curve with the demand curve we’ve developed in previous slides shows how the real wage is determined. CHAPTER 3 National Income

Determining the rental rate We have just seen that MPL = W/P. The same logic shows that MPK = R/P : diminishing returns to capital: MPK  as K  The MPK curve is the firm’s demand curve for renting capital. Firms maximize profits by choosing K such that MPK = R/P . In our model, it’s easiest to think of firms renting capital from households (the owners of all factors of production). R/P is the real cost of renting a unit of K for one period of time. In the real world, of course, many firms own some of their capital. But, for such a firm, the market rental rate is the opportunity cost of using its own capital instead of renting it to another firm. Hence, R/P is the relevant “price” in firms’ capital demand decisions, whether firms own their capital or rent it. CHAPTER 3 National Income

Notes The marginal product of capital (MPK) is the amount of extra output the firm gets from an additional unit of capital, holding the amount of labor constant. MPK = F (K + 1, L) – F (K, L) What’s the increase in profits due to the additional unit of capital? ∆Profits/∆K = (∆Revenue/∆K) – (∆Cost/∆K) = (∆ (P. Y)/ ∆K) – (∆ (R.K)/ ∆K) = (P (∆Y/∆K)) – (R (∆K/∆K)) = P.MPK – R To maximize profits, the firm rents more capital until P.MPK = R MPK = R/P CHAPTER 3 National Income

The equilibrium real rental rate Units of output Units of capital, K Supply of capital The real rental rate adjusts to equate demand for capital with supply. MPK, demand for capital equilibrium R/P The previous slide used the same logic behind the labor demand curve to assert that the capital demand curve is the same as the downward-sloping MPK curve. The supply of capital is fixed (by assumption), so the supply curve is vertical. The real rental rate (R/P) is determined by the intersection of the two curves. CHAPTER 3 National Income

The Neoclassical Theory of Distribution states that each factor input is paid its marginal product is accepted by most economists When I teach this theory, after saying “accepted by most economists” I append “at least, as a starting point.” This theory is fine for macro models with only one type of labor. But taken literally, it implies that people who earn low wages have low marginal products. Thus, this theory would attribute the entire observed wage gap between white males and minorities to productivity differences, a conclusion that most would find objectionable. CHAPTER 3 National Income

How income is distributed: total labor income = total capital income = If production function has constant returns to scale, then (we will see the proof later) The last equation follows from Euler’s theorem, discussed in text on p. 54. national income labor income capital income CHAPTER 3 National Income

Assuming fixed amounts of L and K (and full employment), we determine the relative shares of these factors based on marginal products. Looking at the actual data, one observes that the shares of labor and capital income are relatively constant over time. CHAPTER 3 National Income

The ratio of labor income to total income in the U.S. Labor’s share of total income Labor’s share of income is approximately constant over time. (Hence, capital’s share is, too.) This graph appears in the textbook as Figure 3-5 on p.57. This and the next two slides cover the Cobb-Douglas production function. In the textbook’s previous edition, this material was covered in an appendix. Source: www.bea.gov CHAPTER 3 National Income

Question What type of a production function generates constant factor shares under our regular assumptions of perfect competition and profit maximization? That is: Capital income: MPK. K = αY Labor income: MPL.L = (1-α) Y α determines the relative shares of factors. CHAPTER 3 National Income

The Cobb-Douglas Production Function The Cobb-Douglas production function has constant factor shares: α = capital’s share of total income: capital income = MPK x K = α Y labor income = MPL x L = (1 – α )Y The Cobb-Douglas production function is: where A represents the level of technology. CHAPTER 3 National Income

MPL = ∆Y/∆L = ∂F (K, L)/∂L = A (1- α) K α L-α As K increases, MPL increases As L increases, MPL decreases As A increases, MPL increases CHAPTER 3 National Income

MPK = ∆Y/∆K = ∂F (K, L)/∂K = A α K α-1 L 1-α As L increases, MPK increases As K increases, MPK decreases As A increases, MPK increases CHAPTER 3 National Income

We can write the marginal product of labor as: MPL = A (1- α) K α L –α = A (1- α) K α L –α (L/L) =[A (1- α) K α L 1-α]/L = (1- α) [A K α L 1-α]/L MPL= (1-α) Y/L CHAPTER 3 National Income

We can write the marginal product of capital as: MPK = A α K α-1 L 1-α = α A K α L 1-α K-1 = (A α K α L 1-α )/K = α (A K α L 1-α )/K MPK= α (Y/K) CHAPTER 3 National Income

The Cobb-Douglas Production Function Each factor’s marginal product is proportional to its average product: These formulas can be derived with basic calculus and algebra. CHAPTER 3 National Income

Total labor income: MPL.L = [(1-α) Y/L] L = (1-α) Y Total capital income: MPK. K = [α Y/K] K = α Y Ratio: (1-α) Y/ α Y = (1-α)/ α (a constant) Y = MPL. L + MPK. K = (1-α) Y + α Y = Y CHAPTER 3 National Income