1. Chapter 9
Production Functions

2. Chapter Overview Marginal and Average Products Isoquants
1. The Production Function
Marginal and Average Products
Isoquants
The Marginal Rate of Technical Substitution
2. Returns to Scale
Technical Progress
Some Special Functional Forms

3. Key Concepts
Productive resources, such as labor and capital equipment, that firms use to manufacture goods and services are called inputs or factors of production.
The amount of goods and services produces by the firm is the firm’s output.
Production transforms a set of inputs into a set of outputs
Technology determines the quantity of output that is feasible to attain for a given set of inputs.

4. Key Concepts Production Function: Q = output K = Capital L = Labor
The production function tells us the maximum possible output that can be attained by the firm for any given quantity of inputs.
Production Function:
Q = output
K = Capital
L = Labor

5. The Production Function & Technical Efficiency
Technically efficient: Sets of points in the production function that maximizes output given input (labor)
Technically inefficient: Sets of points that produces less output than possible for a given set of input (labor)

6. The Production Function & Technical Efficiency

7. Short run versus Long run
Short run - a period of time so brief that at least one factor of production cannot be varied practically.
Fixed input - a factor of production that cannot be varied practically in the short run.
Variable input - a factor of production whose quantity can be changed readily by the firm during the relevant time period.
Long run - a lengthy enough period of time that all inputs can be varied.

8. Short-Run Production
In the short run, the firm’s production function is
q = f(L, K)
where q is output, L is workers, and K is the fixed number of units of capital.

9. Total Product, Marginal Product, and Average Product of Labor with Fixed Capital

10. Total Product of Labor
Total product of labor- the amount of output (or total product) that can be produced by a given amount of labor.

11. Marginal Product of Labor
Marginal product of labor (MPL ) - the change in total output, Dq, resulting from using an extra unit of labor, DL, holding other factors constant:
MPL = Q/L
(holding constant all other inputs)
MPK = Q/K

12. Marginal Physical Product
Marginal physical product is the additional output that can be produced by employing one more unit of that input
holding other inputs constant

13. Diminishing Marginal Productivity
Marginal physical product depends on how much of that input is used
In general, we assume diminishing marginal productivity

14. Average Product of Labor
Average product of labor (APL ) - the ratio of output, q, to the number of workers, L, used to produce that output:
APL = Q/L
APK = Q/K

15. Production Relationships with Variable Labor
110
Output, q, Units per day 90 B 56 A
Diminishing Marginal Returns sets in! 4 6 11
L, Workers per day
(b) L a MP 20 , L AP b 15
Average product, APL
Marginal product, MPL c 4 6 11
L, Workers per day

16. Total Product

17. Law of Diminishing Marginal Returns
If a firm keeps increasing an input, holding all other inputs and technology constant, the corresponding increases in output will become smaller eventually.
That is, if only one input is increased, the marginal product of that input will diminish eventually.

18. Example: Short run Production Function
Suppose the production function for a firm 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

19. Long-Run Production
In the long run both labor and capital are variable inputs.
It is possible to substitute one input for the other while holding output constant.

20. Isoquants
Isoquant - a curve that shows the efficient combinations of labor and capital that can produce a single (iso) level of output (quantity).
Equation for an isoquant:
q = f (L, K).

21. 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

22. Properties of Isoquants
The farther an isoquant is from the origin, the greater the level of output.
Isoquants do not cross.
Isoquants slope downward

23. Substituting Inputs
Marginal rate of technical substitution (MRTS) - the number of extra units of one input needed to replace one unit of another input that enables a firm to keep the amount of output it produces constant.
Slope of Isoquant!

24. How the Marginal Rate of Technical Substitution Varies Along an Isoquant16
, Units of capital D K
= –6 b 10 D L
= 1 K –3 1 c 7 –2 d 1 5 e –1 4 1 q
= 10 1 2 3 4 5 6 7 8 9 10 L , W o r k
Ers

25. Production Function With Two Variable Inputs: Moving from short run to long run
Q = f(L, K)

26. Production Function With Two Variable Inputs

27. Production With Two Variable Inputs
Isoquants show combinations of two inputs that can produce the same level of output.
Firms will only use combinations of two inputs that are in the economic region of production.

28. Production With Two Variable Inputs
Isoquants

29. Economic Region of Production
The firm should not use certain combinations of outputs in the long run no matter how cheap they are.
These input combinations are represented by the portion of the isoquant curve that has a positive slope.
A positively sloped isoquant means that merely to maintain the same level of production, the firm must use more of both the inputs if it increases its use of one of the inputs.
Ridge Lines:
The lines connecting the points where the marginal product of an input is equal to zero (one line for each input) in the isoquant map and forming the boundary for the economic region of production.

30. Production With Two Variable Inputs
Economic Region of Production

31. The Economic Region of Production
The economic region of production is the range in an isoquant diagram where both inputs have a positive marginal product.
It lies inside the ridge lines.
A profit maximizing firm will never try to produce using input combinations outside the ridge lines.

32. MARGINAL RATE OF TECHNICAL SUBSTITUTION
Isoquants are downward sloping and convex to the origin.
The slope of the isoquant (dK / dL) defines the degree of substitutability of the factors of production.
The slope of the isoquant decreases (in absolute terms) as we move downwards along the isoquant, showing the increasingdifficulty in substituting in substituting K for L.

33. Production Functions—Two Special Cases
Two extreme cases of production functions show the possible range of input substitution in the production process:
1) the case of perfect substitutes and
2) the fixed proportions production function
The fixed-proportions production function describes situations in which methods of production are limited.

34. Production With Two Variable Inputs
Perfect Substitutes
Perfect Complements
When the isoquants are straight lines, the MRTS is constant.
The rate at which K and L can be substituted for each other is the same no matter what level of inputs is being used.
When the isoquants are L-shaped, only one combination of labor and capital can be used to produce a given output.
Adding more labor alone does not increase output, nor does adding more capital alone.

35. Substitutability of Inputs

36. Elasticity of Substitution
A measure of how easy is it for a firm to substitute labor for capital.
It is the percentage change in the capital-labor ratio for every one percent change in the MRTSL,K along an isoquant.

37. Elasticity of Substitution
The elasticity of substitution () measures the proportionate change in K/L relative to the proportionate change in the MRTSx,y along an isoquant
The value of  will always be positive because K/L and MRTS move in the same direction.

38. Elasticity of Substitution
Both MRTS and K/L will change as we move from point A to point B A B
 is the ratio of these
proportional changes
K per period
 measures the
curvature of the
isoquant
RTSA
RTSB
(k/l)A
(k/l)B
q = q0
L per period

39. Elasticity of Substitution
If  is high, the MRTS will not change much relative to K/L
the isoquant will be relatively flat
If  is low, the MRTS will change by a substantial amount as K/L changes
the isoquant will be sharply curved
It is possible for  to change along an isoquant or as the scale of production changes

40. Elasticity of SubstitutionK
"The shape of the isoquant indicates the degree of substitutability of the inputs…"
 = 0
 = 1
 = 5
 =  L

41. Elasticity of Substitution
If we define the elasticity of substitution between two inputs to be the proportionate change in the ratio of the two inputs to the proportionate change in MRTS, we need to hold output and the levels of other inputs constant

42. Returns to Scale
How much will output increase when ALL inputs increase by a particular amount?
suppose that all inputs are doubled, would output double?

43. Returns to Scale
If the production function is given by q = f(k,l) and all inputs are multiplied by the same positive constant (t >1), then

44. Returns to Scale RTS = [%Q]/[% (all inputs)]
How much will output increase when ALL inputs increase by a particular amount?
RTS = [%Q]/[% (all inputs)]
If a 1% increase in all inputs results in a greater than 1% increase in output, then the production function exhibits increasing returns to scale.
If a 1% increase in all inputs results in exactly a 1% increase in output, then the production function exhibits constant returns to scale.
If a 1% increase in all inputs results in a less than 1% increase in output, then the production function exhibits decreasing returns to scale.

45. Returns to Scale K 2K
Q = Q1 K
Q = Q0 L
L L

46. Returns to Scale

47. 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
the degree of returns to scale is generally defined within a fairly narrow range of variation in input usage

48. Constant Returns to Scale
Constant returns-to-scale production functions are homogeneous of degree one in inputs
f(tk,tl) = t1f(k,l) = tq
The marginal productivity functions are homogeneous of degree zero
if a function is homogeneous of degree k, its derivatives are homogeneous of degree k-1

49. Constant Returns to Scale
Along a ray from the origin (constant K/L), the MRTS will be the same on all isoquants
k per period
q = 3
q = 2
q = 1
The isoquants are equally
spaced as output expands
l per period

50. f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq
Returns to Scale
Returns to scale can be generalized to a production function with n inputs
q = f(x1,x2,…,xn)
If all inputs are multiplied by a positive constant t, we have
f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq
If k = 1, we have constant returns to scale
If k < 1, we have decreasing returns to scale
If k > 1, we have increasing returns to scale

51. The Linear Production Function
Suppose that the production function is
q = f(k,l) = ak + bl
This production function exhibits constant returns to scale
f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l)
All isoquants are straight lines

52. The Linear Production Function
Capital and labor are perfect substitutes
k per period
MRTS is constant as K/L changes
slope = - b/a q1 q2 q3
 = 
l per period

53. Fixed Proportions Suppose that the production function is
q = min (ak,bl) a,b > 0
Capital and labor must always be used in a fixed ratio
the firm will always operate along a ray where K/L is constant
Because K/L is constant,  = 0

54. 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
 = 0
l per period

55. 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(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+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

56. Cobb-Douglas Production Function
The Cobb-Douglas production function is linear in logarithms
ln q = ln A + a ln k + b ln l
a is the elasticity of output with respect to k
b is the elasticity of output with respect to l

57. CES Production Function
Suppose that the production function is
q = f(k,l) = [k + l] /   1,   0,  > 0
 > 1  increasing returns to scale
 < 1  decreasing returns to scale
For this production function
 = 1/(1-)
 = 1  linear production function
 = -  fixed proportions production function
 = 0  Cobb-Douglas production function

58. A Generalized Leontief Production Function: A special case
Suppose that the production function is
q = f(k,l) = k + l + 2(kl)0.5
Marginal productivities are
fk = 1 + (k/l)-0.5
fl = 1 + (k/l)0.5
Thus,

59. A Generalized Leontief Production Function: A special case
This function has a CES form with  = 0.5 and  = 1
The elasticity of substitution is

60. Technological Progress
Definition:
Technological progress (or invention)
Shifts the production function by allowing the firm
to achieve more output from a given combination of inputs
or the same output with fewer inputs.

61. Neutral Technological Progress
Technological progress that decreases the amounts of labor and capital needed to produce a given output.
Affects MRTSK,L

62. Technological Progress
Labor saving technological progress results in a fall in the MRTSL,K along any ray from the origin
Capital saving technological progress results in a rise in the MRTSL,K along any ray from the origin.

63. Labor Saving / Capital deepening Technological Progress
Technological progress that causes the marginal product of capital to increase relative to the marginal product of labor

64. Capital Saving / Labour deepening Technological Progress
Technological progress that causes the marginal product of labor to increase relative to the marginal product of capital

65. 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

66. Technical Progress Suppose that the production function is
q = A(t)f(k,l)
where A(t) represents all influences that go into determining q other than k and l
changes in A over time represent technical progress
A is shown as a function of time (t)
dA/dt > 0

67. Technical Progress
Differentiating the production function with respect to time we get
Dividing by q gives us

68. Technical Progress
For any variable x, [(dx/dt)/x] is the proportional growth rate in x
denote this by Gx
Then, we can write the equation in terms of growth rates

69. Technical Progress
Since