This document discusses production functions and the relationship between inputs and outputs. [1] It defines production functions as describing the technological relationship between inputs used in production and the output produced. It examines concepts like marginal product and how adding more of an input affects output while holding other inputs fixed. [2] Diminishing marginal returns are explained as the phenomenon where the additional output from an added unit of an input declines as more of that input is used, given fixed amounts of other inputs. [3] Isoquants, marginal rate of technical substitution, and elasticity of substitution are introduced as ways to characterize the ease with which inputs can be substituted in the production process. Whether production exhibits increasing,

1. Production
Dr. Andrew McGee
Simon Fraser University

2. Productions funtions
• Production functions: a recipe. For given levels
of inputs, how much output you get.
– P.f. characterize the production technology used
by the firm
– The technology describes what a firm can do in
terms of converting input into output
Inputs Production process Output
(Technology)

3. Production functions

4. Marginal product

5. Marginal product
• MP is evaluated holding other inputs constant
(fixed)
• Imagine the MPL of the 51st farm worker with
and without a tractor:
– Without a tractor:
• 50L → 1000 bushels of wheat
• 51L → 1002 bushels of wheat
– With a tractor:
• 51L → 1050 bushels of wheat
• Notice how different your calculation of the MPL
of the 51st worker would be if you did not hold K
(tractor) fixed

6. Diminishing Marginal Product
• Assume diminishing MP → Holding other
inputs constant, the extra output from using
more of an input goes down as you use more
of the input after a given usage level of the
input
Workers Output MPL
1 100 100 (100-0)
2 300 200 (300-200)
3 550 250 (550-300)
4 600 50 (600-550)

7. Diminishing Marginal Product
• As the usage of an input increases, there are
benefits from specialization resulting in a region
of increasing marginal product
• After a point, it becomes inefficient to increase
the usage of an input holding constant the
other inputs as the marginal product declines
Ouput
per
unit
input
(MPL)
Gains from Diminishing MPL
specialization MP
L (input)
L’

8. Diminishing Marginal Product
Total fLL
output
Q=f(L)
L 0 L
L’ L’
L’’
Output
per unit
input
(MPL)
MPL
L
L’ L’’

9. Marginal Product & Malthus
• Thomas Malthus (1766-1834) argued that
Britain’s population would be constrained by
its ability to feed people. If the population got
too large, famine and disease would check an
underfed population
• Problem: Britain’s population today is many
times larger than when Malthus wrote. What
was his mistake?
• He never took ECON201/301!

10. Marginal Product & Malthus
Malthus argued that because the quantity of
Q arable land in Britain was fixed, its potential
(food) food output was capped at Q’. This would act as
an effective constraint on population growth.
Max food Q=f(L,K’) or
production? Q=g(L,K)
Q=f(L,K)
Land
Malthus failed to appreciate his own ceteris paribus All the land in Britain
assumptions. In particular, he assumed that both (a)
the capital stock would remain fixed and (b) the
technology used to produce food would remain the
same. In the end, neither proved to be true.

11. Definitions & Terminology

12. Relationship between Average &
Marginal Products
Relationship between MPL & APL:
Output (1) MPL>APL → APL increasing
per unit (2) MPL<APL → APL decreasing
input (L) (3) MPL=APL → APL at its maximum value
MPL This is a purely mathematical
property having nothing to do
with economics. Quiz
example.
APL
L

13. Example

14. Isoquants
• Given a production technology Q=f(K,L), an
isoquant depicts all combinations of inputs K
& L that yield the same level of output
While isoquants might remind you of indifference
K
curves, note that there is absolutely no connection
between the two. The similarity arises because we
adopt the same technique for collapsing a 3-
dimensional function into 2-dimensions for the sake of
depicting it.
Isoquant Q=Q0
L

15. Isoquant Map
K
Direction of increasing quantity produced
Q=Q3
Q=Q2
Q=Q1
Q=Q0
L

16. Slope of an Isoquant

17. Marginal Rate of Technical Substitution

18. Why are isoquants negatively sloped?

19. Diminishing MRTS

20. MRTS
K K K
L L L
Diminishing MRTS Constant MRTS Increasing MRTS

21. Example: Diminishing MRTS
• Imagine a bakery that employs labor (bakers but
also security guards, accountants, etc.) and
capital (whisks, rolling pins, industrial-sized
mixers, ovens, etc.)
• You can replace an electric mixer with a person
with a wooden spoon, you could even take away
the wooden spoon and mix with your bare hands,
but how do you replace the oven with labor?
• Similarly, you can automate all of the mixing,
pouring into pans, and baking, but who monitors
this automated process?

22. What Isoquants Tell Us about
Technology
• The shape of an isoquant tells us both how
easily capital is substituted for labor and vice
versa as well as how important an input is to
the production process:
K K
Capital (machine) intensive
technology Labor-intensive technology
Q=Q0
Q=Q0
L L

23. Relationship between MRTS and MP
• Does diminishing MRTS follow from assumption of
diminishing MPL and diminishing MPK? No.
• Diminishing MPL and diminishing MPK speak to what
happens to output when you increase usage of one input
holding the other input constant. Moving along an isoquant
necessarily changes both inputs at the same time.
• It turns out that with an additional assumption, diminishing
MPL and diminishing MPK will guarantee diminishing MRTS:
if fKL>0, then diminishing MPL and diminishing MPK
guarantee diminishing MRTS.
• If fKL<0, then whether you have diminishing MRTS will
depend on how “rapidly” the MP of K & L diminish.
• Note that it is hard to imagine why fKL<0. It is far more
reasonable to assume that fKL>0, but this means that the
assumption of diminishing MP will indeed guarantee
diminishing MRTS.
– Important because it means we don’t need to check SOC for
firm’s cost minimization problem

24. Returns to Scale in Production
• Diminishing MPL and MPK refer to what
happens to output when you change one
input holding the other constant
• Returns to scale refer to how output changes
when you change all inputs by the same
proportion (t>0)

25. Types of Returns to Scale (RTS)

26. Seeing RTS in an Isoquant Map
K
100
Q=16
DRS
8
Q=12
CRS
4
2 IRS Q=6
Q=2
L
2 4 8 100
The isoquant map reveals a lot about the production technology. A production
function can exhibit all 3 types of returns to scale: IRS, CRS, & DRS.

27. A conundrum
• Economists are skeptical of the existence of DRS
in production technologies.
– If one can combine a given level of capital and labor
(or any combination of inputs) to produce a given
level of output, why shouldn’t one be able to exactly
replicate this production process with exactly the
same amounts of the inputs and get exactly the same
amount of output? That is, shouldn’t we always
expect to see at least constant returns to scale in
production?
• Fact: DRS is observed in real-world production
data.
• Why the disconnect between theory and reality?

28. DRS: Myth or Reality?
• Suppose you combine 3 workers, a storefront,
a cash register, a fry machine, and a grill and
get 500 Happy Meals per day. What would
prevent you from simply getting another
storefront, hiring 3 more workers, getting a
cash register, fry machine and grill and then
producing 500 Happy Meals per day at this
other location?

29. DRS: Resolution
• Economists’ explanation for DRS observed in
data: there must be some unobserved input not
increasing by the same proportion as other inputs
– What we see is not really DRS; it only looks like it
• Most likely unobserved input?
– Management
• 2 McDonalds require 2 managers
• Economists generally expect CRS
– CRS production functions have nice properties—
namely constant marginal costs as we shall see

30. DRS and diminishing MP
• Diminishing MPL (or MPK) does not imply DRS.
Indeed, there is no connection between the
two concepts. MP is evaluated holding all
other inputs constant, while RTS necessarily
speaks to how output changes when all inputs
are varied.

31. RTS with more than 2 inputs

32. Homothetic Production Functions
K
K/L
MRTS is the same at all input
combinations where the capital-labor
ratio is the same
q1 All CRS production functions
q0 are homothetic
L

33. Ease of substitution &MRTS

34. “Ease of substitution” measure
• Why do we need a units-free measure of
substitutability and one that reflects how
substitutability changes with the capital labor
K
ratio? K dK/dL=-1
dk/dL=-5
q0 q0
L L
Is it really any harder to substitute labor for capital in the graph on the left? It might
just reflect units of measure. In both cases, you can always substitute a fixed amount
of capital for a fixed amount of labor.

35. “Ease of substitution” measure
K MRTS changing
MRTS=constant K Somewhere between
MRTS=straight line: easy to perfectly easy to
substitute K for L; can always substitute and impossible
substitute K for L in some fixed to substitute K for L
ratio
q0
L q0
K L
MRTS undefined
Impossible to substitute K
q0 for L: K and L must always be
used in fixed proportions to
one another
L

36. Elasticity of substitution between
inputs

37. Elasticity of substitution
(K/L)
∆(K/L)
(K’/L’)
MRTS1
∆MRTS
MRTS2

38. Types of Production Functions
K
What sort of input mix (K/L) will industries with
q0/a perfect substitutes in production use?
-b/a --Never use both inputs. Only use the relatively
cheaper input.
--Looking ahead to the cost minimization
problem firms solve, this type of production
technology will lead to corner solutions.
L
q0/b

39. Types of Production Functions
Ratio of K to L is fixed at b/a
when firms solve cost
K b/a
minimization problem.
q0
L

40. Types of Production Functions
K
q0
L

41. Types of Production Functions

42. Finding the Elasticity of Substitution

43. Finding the Elasticity of Substitution