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

2. 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

3. 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

4. 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

5. MPL and the production functionY
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

6. 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

7. Example
Suppose there is a firm with one computer, one Xerox machine, and one employee.
The worker’s job is to answer and make photocopies.
When a second worker is hired, together they can produce more because while one is answering , 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

8. 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

9. 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

10. Exercise L Y MPL 0 0 n.a. 1 10 10 2 19 9 3 27 8 Suppose W/P = 6.
2 19 9
3 27 8
4 34 7
5 40 6
6 45 5
7 49 4
8 52 3
9 54 2
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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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:
CHAPTER 3 National Income

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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