1. BASIC PRINCIPLES AND EXTENSIONS
Chapter 11
PRODUCTION FUNCTIONS
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

2. 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)

3. 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

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

5. 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
we need to consider fLK which is often > 0

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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. Marginal Rate of Technical Substitution (RTS)
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 (RTS)
RTS > 0 and is diminishing for
increasing inputs of labor
q = 20
L per period

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

15. RTS and Marginal Productivities
Take the total differential of the production function:
Along an isoquant dq = 0, so

16. RTS and Marginal Productivities
Because MPL and MPK will both be nonnegative, RTS will also be nonnegative
However, it is not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone

17. RTS and Marginal Productivities
To show that isoquants are convex, we would like to show that d(RTS)/dL < 0
Since RTS = fL/fK

18. RTS and Marginal Productivities
Using the fact that dK/dL = -fL/fK along an isoquant and Young’s theorem (fKL = fLK)
Because we have assumed fK > 0, the denominator is positive
Because fLL and fKK are both assumed to be negative, the ratio will be negative if fKL is positive

19. RTS and Marginal Productivities
Intuitively, it seems reasonable that fKL=fLK should be positive
if workers have more capital, they will be more productive
But some production functions have fKL < 0 over some input ranges
Thus, when we assume diminishing RTS we are assuming that MPL and MPK diminish quickly enough to compensate for any possible negative cross-productivity effects

20. A Diminishing RTS Suppose the production function is
q = f(K,L) = 600K 2L2 - K 3L3
For this production function
MPL = fL = 1200K 2L - 3K 3L2
MPK = fK = 1200KL2 - 3K 2L3
These marginal productivities will be positive for values of K and L for which KL < 400

21. A Diminishing RTS Because
fLL = 1200K 2 - 6K 3L
fKK = 1200L2 - 6KL3
this production function exhibits diminishing marginal productivities for sufficiently large values of K and L
fLL and fKK < 0 if KL > 200

22. A Diminishing RTS
Cross differentiation of either of the marginal productivity functions yields
fKL = fLK = 2400KL - 9K 2L2
which is positive only for KL < 266

23. A Diminishing RTS
Thus, for this production function, RTS is diminishing throughtout the range of K and L where marginal productivities are positive
for higher values of K and L, the diminishing marginal productivities are sufficient to overcome the influence of a negative value for fKL to ensure convexity of the isoquants

24. 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

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

26. Returns to Scale
If the production function is given by q = f(K,L) and all inputs are multiplied by the same positive constant (m > 1), then

27. 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

28. Constant Returns to Scale
Constant returns-to-scale production functions have the useful theoretical property that that the RTS between K and L depends only on the ratio of K to L, not the scale of operation
Geometrically, all of the isoquants are “radial blowups” of the unit isoquant

29. Constant Returns to Scale
Along a ray from the origin (constant K/L), the RTS 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

30. f(mX1,mX2,…,mXn) = mkf(X1,X2,…,Xn)=mkq
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 m, we have
f(mX1,mX2,…,mXn) = mkf(X1,X2,…,Xn)=mkq
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

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

32. Elasticity of Substitution
Both RTS 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

33. Elasticity of Substitution
If  is high, the RTS will not change much relative to K/L
the isoquant will be relatively flat
If  is low, the RTS 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

34. 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(mK,mL) = amK + bmL = m(aK + bL) = mf(K,L)
All isoquants are straight lines
RTS is constant
 = 

35. 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

36. 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

37. 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

38. 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

39. 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

40. Cobb-Douglas Production Function
The RTS can easily be calculated
The RTS declines as L rises and K falls
The RTS depends only on the ratio of K and L
Because the RTS changes exactly in proportion to changes in K/L,  = 1

41. 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

42. 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

43. 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

44. 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

45. Technical Progress
Differentiating the production function with respect to time we get

46. Technical Progress
Dividing by q gives us

47. 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

48. Technical Progress
Since

49. Technical Progress in the Cobb-Douglas Function
Suppose that the production function is
q = 10e 0.05t K 0.5 L0.5
Taking logarithms yields
ln q = ln t ln K ln L
Differentiating with respect to t gives the growth equation

50. Technical Progress in the Cobb-Douglas Function
We can put this in terms of growth rates
Gq = GK + 0.5GL
When K and L are constant, output grows at 5 percent per period
GK = GL = 0
Gq = 0.05

51. Important Points to Note:
If all but one of the inputs are held constant, a relationship between the single variable input and output can be derived
the marginal physical productivity is the change in output resulting from a one-unit increase in the use of the input
the marginal physical productivity of an input is assumed to decline as use of the input increases

52. Important Points to Note:
The entire production function can be illustrated by an isoquant map
The slope of an isoquant is the marginal rate of technical substitution
RTS measures how one input can be substituted for another while holding output constant
RTS is the ratio of the marginal physical productivities of the two inputs
Isoquants are assumed to be convex
they obey the assumption of a diminishing RTS

53. Important Points to Note:
The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputs
if output increases proportionately with input use, there are constant returns to scale
if there are greater than proportionate increases in output, there are increasing returns to scale
if there are less than proportionate increases in output, there are decreasing returns to scale

54. Important Points to Note:
The elasticity of substitution () provides a measure of how easy it is to substitute on input for another in production
a high  implies nearly straight isoquants
a low  implies that isoquants are nearly L-shaped
Technical progress shifts the entire production function and isoquant map
may arise from the use of more productive inputs or better economic organization