1. Chapter 9 Production Functions
Nicholson and Snyder, Copyright ©2008 by Thomson South-Western. 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
Marginal physical product is the additional output that can be produced by employing one more unit of that input
holding other inputs constant

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

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

6. 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

7. 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

8. 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

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

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

11. RTS and Marginal Productivities
Because MPl and MPk will both be nonnegative, RTS will be positive (or zero)
However, it is generally not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone

12. RTS and Marginal Productivities
To show that isoquants are convex, we would like to show that d(RTS)/dl < 0
Since RTS = fl/fk

13. 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

14. 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

15. 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

17. 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

20. 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
RTS is constant
 = 

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

28. CES variants

29. A Generalized Leontief Production Function
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

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

31. 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

32. 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

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

34. Technical Progress
Dividing by q gives us

35. 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

36. Technical Progress
Since