1. Production

2. Technology
The physical laws of nature and limits of material availability and human understanding that govern what is possible in converting inputs into output.

3. Inputs, Factors of Production
Land (incl. raw materials)
Labor (including human capital)
Capital (physical capital, like machinery and buildings)

4. Production Function
A 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)
Producing less than the maximum is always possible and all levels of output below the maximum are feasible and define the “production set.”

5. Production Function q
q = f(K, L) K L

6. All points “under” the production function
Production set
q = f(K, L) q K
All points “under” the production function L

7. Production Function and Isoquants
q = f(K, L) q
In the long run, all combinations of inputs are possible K
Isoquants are horizontal cross sections of the production function projected on the base plane. L

8. Short Run, Long Run
Long Run, quantities of ALL inputs used in production can be varied.
Short Run, the quantity of at least one input used in production is fixed.
ALL production takes place in a short run environment.
You can think of the long run as the ability to move from one short run environment to another.
Actual time it takes to make this move depends on many factors, technical, economic and regulatory.

9. The model
Standard basic model to think of production as a function of K and L.
L variable in the short run while K is fixed.

10. Short run, hold K fixed. L q = f(K, L) q
In the short run, K is fixed and only L can vary K
The cross section of the production function at a fixed K is the short run production function L

11. More, fixed K
q = f(K, L) q
In the short run, K is fixed and only L can vary K
The cross section of the production function at a fixed K is the short run production function L

12. Three levels of K q = f(K=K3, L) q q = f(K=K2, L)
In the short run, we assume, the quantity of at least one input used --but not all -- is fixed.
q = f(K=K1, L) L

13. L constant
q = f(K,L=L3) q
L and K are just names for inputs. Either one could be fixed in the short run.
Just intuitive that K is fixed and L variable in the SR.
q = f(K,L=L2)
q = f(K,L=L1) K

14. SR and then LR
First we’ll think about the short run, and then turn to the long run.

15. Marginal Physical Product
Marginal Product is the additional output that can be produced by employing one more unit of that input
holding other inputs constant, so a short run concept

16. Marginal Productivity Assumptions
We assume managers are not going to allow employees in the building if they bring total output down.
However, over the range where profit is maximized, marginal products are positive.

17. Increasing and Diminishing Marginal Product (assumes something is fixed)
Empirically, economists find that most production processes exhibit (as L increases from zero):
Increasing Marginal Returns – each worker added causes output to increase by more than the previous worker (workers are not able to gain from specialization, K is fixed)
And then…
Decreasing Marginal Returns –workers added to production add less to output than the previous worker (workers crowd each other as they try to share a fixed amount of capital)

18. Marginal Productivity Assumptions
Because of IMR and DMR, these are possible:
Whether MP is always diminishing or whether it first increases and then diminishes depends on the context of the economic discussion.
In economics classes, we think of increasing marginal returns and then diminishing marginal returns (need this for a U-shaped MC curve).

19. MP Assumptions
As revenue or profit max means producing where MC is rising (MPL is falling), theoretically, we tend to ignore IMR and assume DMR

20. Malthus and Diminishing Marginal Productivity
He argued that population growth meant declining marginal labor productivity
His mistake was holding all else (except labor, i.e. population) constant.
Ignored technological growth.
Productivity was actually growing exponentially, but at such a slow rate that he did not see it.
Per Capita Output
Watts’s Steam Engine
Economic growth of IR first noticed in the 1830s
Essay on the Principle of Population, 1st ed (1798)
Malthus Dies, 1834
Year
1800
1840
1880

21. Effect of Technology
If we think of higher technology as being like having MORE capital, then you can think of the industrial revolution the result of fLK > 0 and a rapid expansion of K.

22. Average Physical Product
Labor productivity is often measured by average productivity.

23. Specific Function
Suppose the production function for tennis balls can be represented by
To construct MPL and APL, we must assume a value for K
let K = 10
The production function becomes

24. SR Production Function (K = 10)q L

25. Marginal Product The marginal product function is
When MPL = 0, total product is maximized at L = 80.

26. SR Production Function (K = 10)q
Slope of function is MPL at that level of L L

27. Inflection Point
Output where MPL goes from increasing to decreasing (inflection point)

28. SR Production Function (K = 10)
At inflection point, MPL is at its highest q LI L

29. Average Product
To find average productivity, we hold K=10 and solve

30. SR Production Function (K = 10)
Slope of ray from origin to curve at any L is = APL
Slope of this ray =36,000
So APL =36,000 when L= 60 q LA L

31. MPL and APL In fact, when L = 60, both APL and MPL are equal to 36,000
Thus, when APL is at its maximum, APL and MPL are equal
So long as a worker hired has a MPL higher than the overall APL, the APL will continue to rise.
If the MPL = APL,
But if a worker hired has a MPL below the overall APL, the APL will fall.

32. MPL and APL LI LA

33. MPL and APL
Where the ray is also tangent,
MPL = APL

34. Long Run All mixes of K and L are possible.
Daily decisions about production always have some fixed inputs, so the long run is a planning time horizon.

35. Isoquant Map
Each isoquant represents a different level of output, q0 = f(K0,L0), q1 = f(K1,L1) K
q1 = 30
q0 = 20 L

36. Marginal Rate of Technical Substitution (TRS, RTS, MRTS)
The slope of an isoquant shows the rate at which L can be substituted for K, or how much capital must be hired to replace one Laborer. K A KA B KB
q0 = 20 L LA LB

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

38. Alternatively: Implicit Function Rule

39. Diminishing TRS
Again, for demand (this time of inputs) to be well behaved, we need production technology (akin to preferences) to be convex. K
Which means, the slope rises, gets closer to zero as L increases.
And means the TRS falls as L increases. L

40. Diminishing TRS
To show that isoquants are convex (that dK/dL increases – gets closer to zero) along all isoquants)
That is, either:
The level sets (isoquants) are strictly convex
The production function is strictly quasi-concave

41. Convexity (level curves)
dK/dL increases along all indifference curves
We can use the explicit equation for an isoquant, K=K(L, q0) and find
to demonstrate convexity.
That is, while negative, the slope is getting larger as L increases (closer to zero).
But we cannot always get a well defined equation for an isoquant.
Binger and Hoffman, page 115

42. Alternatively (level curves)
As above, starting with q0 =f(K,L),
So convexity if

43. Convexity (level curves)
And, that is
*Note that fK3 > 0
What of:
fL > 0, monotonacity
fK > 0, monotonacity
fLL < 0, diminishing marginal returns
fKK < 0, diminishing marginal returns
fLK = ?
Binger and Hoffman, page 115

44. Strict Quasi-Convexity (production function)
Also, convexity of technology will hold if the production function is strictly quasi-concave
A function is strictly quasi-concave if its bordered Hessian
is negative definite
Binger and Hoffman, page 115

45. Negative Definite (production function)
So the production function is strictly quasi-concave if
1. –fLfL < 0
2. 2fLfKfLK-fK2fLL -fL2fKK > 0
Related to the level curve result:
Remembering that a convex level set comes from this
We can see that strict convexity of the level set and strict quasi-concavity of the function are related, and each is sufficient to demonstrate that both are true.

46. TRS and Marginal Productivities
Intuitively, it seems reasonable that 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
assuming diminishing TRS means that MPL and MPK diminish quickly enough to compensate for any possible negative cross-productivity effects

47. TRS and MPL and MPK Back to our sample production function:
For this production function

48. IMR and DMR vs. NMR Pull out a few terms
If K = 10, then MPL = 0 at L=80

49. IMR vs. DMR Because If K = 10, then inflection point at L=40
fLL> 0 and fKK > 0 if K*L < 400
fLL< 0 and fKK < 0 if K*L > 400
If K = 10, then inflection point at L=40

50. Cross Effect
Cross differentiation of either of the marginal productivity functions yields
fLK > 0 if KL < 533
fLK < 0 if KL > 533
If K = 10
fLK> 0 when L < 53.3
fLK< 0 when L > 53.3

51. A Diminishing TRS? Strictly Quasi-Concave if
Lots of parts that have different signs depending on K and L. + ?
? ? ? + ?

52. 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 Adam Smith’s pin factory

53. Returns to Scale Two forces that occur as inputs are scaled upwards
greater division of labor and specialization of function
loss in efficiency (bureaucratic inertia)
management may become more difficult
fall of the Roman Empire?
General Motors?

54. Returns to Scale
Starting at very small scale and then expanding, firms tend to exhibit increasing returns to scale at small scale, which changes to constant returns over a range, and then when they get larger, face decreasing returns to scale.
Obviously, the scale at each transition can vary.
Vacuum Cleaner Repair Shops
Steel Mills
Doughnut Shops
Automobile manufacture
Empirical analysis reveals that established firms tend to operate at a CRS scale.

55. 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

56. Returns to Scale Constant Returns to Scale
q = K.5L.5
What if we increase all inputs by a factor of t?
(tK).5(tL).5 = ?
t(K).5(L).5 = tq
For t > 1, increase all inputs by a factor of t and output increases by a factor of t
I.e. increase all inputs by x% and output increases by x%

57. Returns to Scale Decreasing Returns to Scale
q = K.25L.25
What if we increase all inputs by a factor of t?
(tK).25(tL).25 = ?
t.5(K).25(L).25 = t.5q, which is < tq
For t > 1, increase all inputs by a factor of t and output increases by a factor < t
I.e. increase all inputs by x% and output increases by less than x%

58. Returns to Scale Increasing Returns to Scale
q = K1L1
What if we increase all inputs by a factor of t?
(tK)1(tL)1 = ?
tq < t2(K)1(L)1 = t2q, which is > tq
For t>1, increase all inputs by a factor of t and output increases by a factor > t
I.e. increase all inputs by x% and output increases by more than x%

59. Returns to Scale
Using the usual homogeneity notation, alternatively, it is notated, for t > 0.
That is, production is homogeneous of degree k.

60. Returns to Scale, Example
Solve for k
q = K.4L.4
tkq = (tK).4(tL).4 = t.8(K).4(L).4
k ln(t) + ln(Q) = .8ln(t)+.4ln(K)+.4ln(L)
k ln(t) = .8ln(t)+.4ln(K)+.4ln(L) - ln(Q)
k ln(t) = .8ln(t)+.4ln(K)+.4ln(L)-.4ln(K)-.4ln(L)
k ln(t) = .8ln(t)
k ln(t) = .8ln(t)/ln(t)
k = .8, production is Homogeneous of degree .8
k < 1 so DRS

61. Returns to Scale by Elasticity
What is the % change in output for a t% increase in all inputs?
Generally evaluated at t = 1
CRS: q,t =1
DRS: q,t < 1
IRS: q,t > 1

62. Returns to Scale by Elasticity
What is the % change in output for a t% increase in all inputs? Evaluated at t = 1.
In this example, RTS varies by K and L.

63. Constant Returns to Scale is Special
Empirically, firms operate at a CRS scale.
If a function is HD1, then the first partials will be HD0. If
Then

64. Constant Returns to Scale is Special
Obviously, if CRS, we can scale by any t > 0
But let’s pick a specific scale factor, 1/L: If
Then
Which tells us that if production is CRS, then it is also homothetic. Isoquants are radial expansions with the RTS the same along all linear expansion paths.

65. Constant Returns to Scale
The marginal productivity of any input depends on the ratio of capital and labor
not on the absolute levels of these inputs
Therefore the TRS between K and L depends only on the ratio of K to L, not the scale of operation
That is, increasing all inputs by x% does not affect the TRS
The production function will be homothetic (TRS constant along ray from origin)
Geometrically, this means all of the isoquants are radial expansions of one another

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

67. Economies of Scale (not Returns to Scale)
In the real world, firms rarely scale up or down all inputs (e.g. management does not typically scale up with production).
Economies of scale: %ΔLRAC/%ΔQ
Economies of scale if < 0
Diseconomies of scale if > 0

68. Elasticity of Substitution
The elasticity of substitution () measures the proportionate change in K/L relative to the proportionate change in the TRS along an isoquant
And as was demonstrated earlier, elasticity is the effect of a change in one log on another.
The value of  will always be positive because K/L and TRS move in the same direction

69. 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
 measures the
curvature of the
isoquant
TRSA
TRSB
(K/L)A
q = q0
(K/L)B L

70. Elasticity of Substitution
If  is low, the K/L will not change much relative to TRS
the isoquant will be relatively flat
If  is high, the K/L will change by a substantial amount as TRS changes
the isoquant will be sharply curved
More interesting when you remember that to minimize cost, TRS = pL/pK so TRS changes with input prices.

71. Elasticity of SubstitutionK
q=g(K,L)
It is possible for  to change along an isoquant or as the scale of production changes
g >  f
q=f(K,L) L

72. Elasticity of Substitution
Solving for σ can be tricky, but, we can employ this calculus trick (especially useful for homothetic production functions):
This allows us to turn this problem
Into the (sometimes) easier

73. Elasticity of Substitution CRS is Special Again
For CRS production functions only we have this option too
Let q = f(K,L)

74. Common Production Functions
Linear (inputs are perfect substitutes)
Fixed Proportions (inputs are perfect compliments)
Cobb-Douglas
CES
Generalized Leontief

75. The Linear Production Function (inputs are perfect substitutes)
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

76. Linear Production Function

77. The Linear Production Function
Capital and labor are perfect substitutes K
TRS is constant as K/L changes q1 q2 q3
slope = -b/a
 =  L

78. 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

79. Fixed Proportions
No substitution between labor and capital is possible
K/L is fixed at b/a
q3/b
q3/a q1 q2 q3 K
 = 0 L

80. 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

81. Cobb-Douglas Production Function

82. 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
Statistically, this is how we estimate production functions via regression analysis.

83. CES Production Function
Suppose that the production function is
 > 1  increasing returns to scale
 = 1  constant returns to scale
 < 1  decreasing returns to scale

84. CES Production Function
TRS
Note, not a function of scale, γ

85. CES Production Functionσ

86. CES Production Function
For CES
At limit as  → 1, σ → ∞, linear production function
At limit as  → -, σ → ∞, fixed proportions production function
When  = 0, Cobb-Douglas production function

87. A Generalized Leontief Production Function
Suppose that the production function is
TRS

88. A Generalized Leontief Production Functionσ

89. 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 inward

90. Technical Progress Suppose that the production function is
q = A(t)f(K(t),L(t))
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

91. Technical Progress Differentiating the production function
with respect to time we get
Which simplifies to

92. Technical Progress
Since
And so

93. Technical Progress
Dividing by q gives us

94. Technical Progress
Expand by strategically adding in K/K and L/L

95. 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

96. Technical Progress Note the elasticities Yielding
Growth is a function of technical change and growth in the use of inputs.

97. Solow, US Growth 1909-1949 Solow estimated the following Plug these in
Gq = 2.75%
GL = 1.00%
GK = 1.75%
eq,L = .65
eq,K = .35
Plug these in
And GA = 1.5%
Conclusion, technology grew at a 1.5% rate from % of GDP growth in the period.

98. Appendix
Full derivations of TRS and convexity in production.

99. RTS and Marginal Productivities: Implicit Function Rule

100. Substitute

101. And get to…

102. And get to…

103. Convexity, Increasing dK/dL

104. Diminishing TRS TRS diminishing if this < 0
Which is the same thing.

105. Alternatively, the Bordered Hessian
Strictly Quasi-Concave if
and
which looks a lot like the negative of this: