1. Production Function
The firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (K) and labor (L)
q = f(K,L)

2. Marginal Physical Product
To study variation in a single input, we define marginal physical product as the additional output that can be produced by employing one more unit of that input while holding other inputs constant

3. Diminishing Marginal Productivity
Because of diminishing marginal productivity, 19th century economist Thomas Malthus worried about the effect of population growth on labor productivity
But changes in the marginal productivity of labor over time also depend on changes in other inputs such as capital

4. Average Physical Product
Labor productivity is often measured by average productivity
Note that APL also depends on the amount of capital employed

5. A Two-Input Production Function
Suppose the production function for flyswatters can be represented by
q = f(K,L) = 600K 2L2 - K 3L3
To construct MPL and APL, we must assume a value for K
Let K = 10
The production function becomes
q = 60,000L L3

6. A Two-Input Production Function
The marginal productivity function is
MPL = q/L = 120,000L L2
which diminishes as L increases
This implies that q has a maximum value:
120,000L L2 = 0
40L = L2
L = 40
Labor input beyond L=40 reduces output

7. A Two-Input Production Function
To find average productivity, we hold K=10 and solve
APL = q/L = 60,000L L2
APL reaches its maximum where
APL/L = 60, L = 0
L = 30

8. A Two-Input Production Function
In fact, when L=30, both APL and MPL are equal to 900,000
Thus, when APL is at its maximum, APL and MPL are equal

9. Isoquant Maps
To illustrate the possible substitution of one input for another, we use an isoquant map
An isoquant shows those combinations of K and L that can produce a given level of output (q0)
f(K,L) = q0

10. Isoquant Map Each isoquant represents a different level of output
output rises as we move northeast
q = 30
q = 20
K per period
L per period

11. Marginal Rate of Technical Substitution (MRTS)
The slope of an isoquant shows the rate at which L can be substituted for K LA KA KB LB A B
K per period
- slope = marginal rate of technical substitution (MRTS)
MRTS > 0 and is diminishing for
increasing inputs of labor
q = 20
L per period

12. Marginal Rate of Technical Substitution (MRTS)
The marginal rate of technical substitution (MRTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant

13. Returns to Scale
How does output respond to increases in all inputs together?
Suppose that all inputs are doubled, would output double?
Returns to scale have been of interest to economists since the days of Adam Smith

14. Returns to Scale
Smith identified two forces that come into operation as inputs are doubled
greater division of labor and specialization of labor
loss in efficiency because management may become more difficult given the larger scale of the firm

15. Returns to Scale
It is possible for a production function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels
economists refer to the degree of returns to scale with the implicit notion that only a fairly narrow range of variation in input usage and the related level of output is being considered

16. The Linear Production Function
Capital and labor are perfect substitutes
K per period
RTS is constant as K/L changes
slope = -b/a q1 q2 q3
L per period

17. Fixed Proportions
No substitution between labor and capital is possible
K/L is fixed at b/a
q3/b
q3/a
K per period q1 q2 q3
L per period

18. Cobb-Douglas Production Function
Suppose that the production function is
q = f(K,L) = AKaLb A,a,b > 0
This production function can exhibit any returns to scale
f(mK,mL) = A(mK)a(mL) b = Ama+b KaLb = ma+bf(K,L)
if a + b = 1  constant returns to scale
if a + b > 1  increasing returns to scale
if a + b < 1  decreasing returns to scale

19. Cobb-Douglas Production Function
Suppose that hamburgers are produced according to the Cobb-Douglas function
q = 10K 0.5 L0.5
Since a+b=1  constant returns to scale
The isoquant map can be derived
q = 50 = 10K 0.5 L0.5  KL = 25
q = 100 = 10K 0.5 L0.5  KL = 100
The isoquants are rectangular hyperbolas

20. Cobb-Douglas Production Function
The MRTS can easily be calculated
The MRTS declines as L rises and K falls
The MRTS depends only on the ratio of K and L

21. Technical Progress Methods of production change over time
Following the development of superior production techniques, the same level of output can be produced with fewer inputs
the isoquant shifts in