1. Kt Lt Production Function Production Function Q=ƒ(Kt,Lt) Qt=ƒ(inputst)
Qt=output rate
inputt=input rate
where is technology?
Firms try to be on the surface of the PF.
Inside the function implies there is waste, or technological inefficiency. Kt Lt 10

2. Difference between LR and SR
LR is time period where all inputs can be varied.
Labor, land, capital, entrepreneurial effort, etc.
SR is time period when at least some inputs are fixed.
Usually think of capital (i.e., plant size) as the fixed input, and labor as the variable input. 11

3. LR production function as many SR production functions.
Long Run: Q = f (K,L)
Suppose there are two different sized plants, K1 and K2.
One Short Run:
Q = f ( K1,L) i.e., K fixed at K1
A second Short Run:
Q = f ( K2,L) i.e., K fixed at K2
Show this graphically 13

4. Two Separate SR Production FunctionsQ
Q = f( K2, L )
Q = f( K1, L )
K2 > K1 L 15

5. What Happens when Technology Changes?
This shifts the entire production function, both in the SR and in the LR. 16

6. Technology Changes Q
TP after computer
TP before computer L 17

7. SR Production Function in More Detail
Qt=ƒ(Kfixed,Lt)
I II III
Express this in two dimensions, L and Q, since K is fixed.
Define Marginal Product of Labor.
Slope is MPL=dQ/dL
Identify three ranges
I: MPL >0 and rising
II: MPL >0 and falling
III: MPL<0 and falling Q L 18

8. Where Diminishing Returns Sets In
As you add more and more variable inputs to fixed inputs, eventually marginal productivity begins to fall.
As you move into zone II, diminishing returns sets in!
Why does this occur? Q
I II L 20

9. Why Diminishing Returns Sets In
Since plant size (i.e., capital) is fixed, labor has to start competing for the fixed capital.
Even though Q still increases with L for a while, the change in Q is smaller. Q
I II L 21

10. Define APL and MPL Average Product = Q / L Marginal Product = dQ/dL
output per unit of labor.
frequently reported in press.
Marginal Product = dQ/dL
output attributable to last unit of labor used.
what economists think of. 23

11. Average Productivity Graphically
Take ray from origin to the SR production function.
Derive slope of that ray
Q=Q1
L=L1
Thus,
Q/L =Q1 /L1 Q
Q=f(KFIXED,L) Q1 Q L L1 L 26

12. Average Productivity Graphically
APL rises until L2
Beyond L2 , the APL begins to fall.
That is, the average productivity rises, reaches a peak, and then declines Q
Q=f(KFIXED,L) Q2 L
Q/L L2
APL L2 27

13. Average & Marginal Productivity
There is a relationship between the productivity of the average worker, and the productivity of the marginal worker.
Think of a batting average.
Think of your marginal productivity in the most recent game.
Think of average productivity from beginning of year.
When MP > AP, then AP is RISING
When MP < AP, then AP is FALLING
When MP = AP, then AP is at its MAX 28

14. Average Productivity GraphicallyQ
MPL rises until L1
Beyond L1 , the MPL begins to fall.
Look at AP
i. Until L2, MPL >APL and thus APL rises.
ii. At L2, MPL=APL and thus APL peaks.
iii. Beyond L2, MPL<APL and thus APL falls. L L1 L2
Q/L
MPL
APL L1 L2 30

15. Intuitive explanation
Anytime you add a marginal unit to an average unit, it either pulls the average up, keeps it the same, or pulls it down.
When MP > AP, then AP is rising since it pulls it the average up.
When MP < AP, then AP is falling since it pulls the average down.
When MP = AP, then AP stays the same.
Think of softball batting average example. 33

16. LR Production FunctionQt Kt
Isoquants
(i.e.,constant
quantity) Lt 19

17. Define Isoquant
Different combinations of Kt and Lt which generate the same level of output, Qt. 20

18. Isoquants & LR Production Functions
ISOQUANT MAP
Qt = Q(Kt, Lt)
Output rate increases as you move to higher isoquants.
Slope represents ability to tradeoff inputs while holding output constant.
Marginal Rate of Technical Substitution.
Closeness represents steepness of production hill. K Q3 Q2 Q1 L 21

19. Slope of Isoquant Kt Q Lt Slope is typically not constant.
Tradeoff between K and L depends on level of each.
Can derive slope by totally differentiating the LR production function.
Marginal rate of technical substitution is –MPL/MPK Kt Q Lt 22

20. Extreme Cases K K Q2 Q2 Q1 Q1 L L Inputs used in fixed proportions!
No Substitutability
Perfect Substitutability K K Q2 Q2 Q1 Q1 L L
Inputs used in fixed
proportions!
Tradeoff is constant 25

21. Substitutability K K Q1 Q1 L L Slope of Isoquant changes a lot
Low Substitutability
High Substitutability K K Q1 Q1 L L
Slope of Isoquant
changes a lot
Slope of Isoquant
changes very little 25

22. Isoquants and Returns to Scale
Returns to scale are cost savings associated with a firm getting larger. 31

23. Increasing Returns to Scale
Production hill is rising quickly.
Lines on the contour map get closer with equal increments in Q. K
Q=40
Q=30
Q=20
Q=10 L 33

24. Decreasing Returns to Scale
Production hill is rising slowly.
Lines on the contour map get further apart with equal increments in Q. K
Q=40
Q=30
Q=20
Q=10 L 35

25. How Can You Tell if a PF has IRS, DRS, or CRS?
It is possible that it has all three, along various ranges of production.
However, you can also look at a special kind of function, called a homogeneous function.
Degree of homogeneity is an indicator returns to scale.

26. Homogeneous Functions of Degree 
A function is homogeneous of degree k
if multiplying all inputs by , increases the dependent variable by
Q = f ( K, L)
So,  • Q = f(K,  L) is homogenous of degree k.
Cobb-Douglas Production Functions are homogeneous of degree  + 

27. Cobb-Douglas Production Functions
Q = A • K  • L  is a Cobb-Douglas Production Function
Degree of Homogeneity is derived by increasing all the inputs by 
 Q = A • ( K) • ( L) 
 Q = A •   K •   L
 Q =   A • K • L

28. Cobb-Douglas Production Functions
This is a Constant Elasticity Function
Elasticity of substitution s = 1
Coefficients are elasticities
 is the capital elasticity of output, EK
 is the labor elasticity of output, E L
If Ek or L <1 then that input is subject to Diminishing Returns.
C-D PF can be IRS, DRS or CRS
if  +  1, then CRS
if  + < 1, then DRS
if  + > 1, then IRS

29. Technical Change in LR
Technical change causes isoquants to shift inward
Less inputs for given output
May cause slope to change along ray from origin
Labor saving
Capital saving
May not change slope
Neutral implies parallel shift

30. Technical change
Labor Saving
Capital Saving K L K L

31. Lets now turn to the Cost Side
What is Goal of Firm?

32. Define Isocost Line K Slope=-w/r L TC/r Isocost Line
Put K on vertical axis, and L on horizontal axis.
Assume input prices for labor (i.e., w) and capital (i.e., r) are fixed.
Define: TC=w*L + r*K
Solve for K:
r*K= TC-w*L
K=(TC/r) - (w/r)*L K
TC/r
Slope=-w/r L

33. TC constant along Isocost line.K
TC1/r L
TC1/w

34. in TC parallel shifts IsocostK
TC2 > TC1
TC2/r
TC1/r L
TC2/w
TC1/w

35. Change in input price rotates IsocostK
w2 < w1
TC/r L
TC/w2
TC/w1

36. Optimal Input Levels in LR
Suppose Optimal Output level is determined (Q1).
Suppose w and r fixed.
What is least costly way to produce Q1? K Q1 L

37. Optimal Input Levels in LR
Suppose Optimal Output level is determined (Q1).
Suppose w and r fixed.
What is least costly way to produce Q1?
Find closest isocost line to origin!
Optimal point is point of allocative efficiency. K K1 Q1 L1 L

38. Cost Minimizing Condition
Slopes of Isoquant and Isocost are equal
Slope of Isoquant=MRTS=- MPL/ MPK
Slope of Isocost=input price ratio=-w/r
At tangency, - MPL/ MPK = -w/r
Rearranging gives: MPL/w= MPK /r
In words:
Additional output from last $ spent on L = additional output from last $ spent on K.

39. The LR Expansion Path K expansion path Q2 Q1 L K2 K1 L1 L2
Costs increase when output increases in LR!
Look at increase from Q1 to Q2.
Both Labor and Capital adjust.
Connecting these points gives the expansion path. K
expansion path K2 K1 Q2 Q1 L L1 L2

40. We can show that LR adjustment along the expansion path is less costly than SR adjustment holding K fixed!

41. Start at an original LR equilibrium (i.e., at Q1).K K1 Q1 L L1

42. LR Adjustment K Q2 Q1 L K2 K1 L1 L2 LR adjustment:
K increases (K1 to K2)
L increases (L1 to L2)
TC increases from black to blue isocost. K K2 K1 Q2 Q1 L L1 L2

43. SR Adjustment K Q2 Q1 L K1 L1 L3 SR adjustment. K constant at K1.
L increases (L1 to L3)
TC increases from black to white isocost. K K1 Q2 Q1 L L1 L3

44. LR Adjustment less Costly
White Isocost (i.e., SR) is further from the origin than the Blue Isocost (LR).
Thus, the more flexible LR is less costly than the less flexible SR. K K2 K1 Q2 Q1 L L1 L2 L3

45. Impact of Input Price Change
Start at equilibrium.
Recall slope of isocost=K/L= -w/r
Suppose w and optimal Q stays same (i.e., Q1)
Rotate budget line out, and then shift back inward! K Q1 K1 L1 L

46. Decrease in wage leads to substitution
Firms substitute away from capital (K1 to K2).
Firms substitute toward labor (L1 to L2)
Pure substitution effect: a to b
Maps out demand for labor curve K L Q1 L1 K1 K2 L2 a b

47. Derivation of Labor Demand from Substitution Effect
Wage falls w K w1 a K1 b w2 K2 Q1
DL1 L1 L2 L L1 L2 L

48. There is also a scale effect
Scale effect is increase in output that results from lower costs
Scale effect: b-c K Q1 Q2 a c K1 b L1 L

49. Scale Effect Shifts Demand
Wage falls w K w1 c a K1 w2 b K2
DL2 Q1
DL1 L1 L2 L3 L L1 L2 L L3

50. Recall the Isocost Line TC=w*L + r*K
Thus, TC=TVC+TFC
Lets relate the cost relationships to the production relationships.
Recall the Law of Diminishing Returns.

51. Law of Diminishing Marginal Returns
As you add more and more variable inputs (L) to your fixed inputs (K), marginal additions to output eventually fall (i.e., MPL= Q/L falls)
What does this say about the shape of cost curves?

52. Marginal Productivity (MPL) and Marginal Cost (MC)
Look at how TC changes when output changes.
Assume w and r are fixed.
Since TC=w*L+r*K
then TC = w*L + r*K
How does K change in SR?

53. Changes in TC in SR must come from changes in Labor.
TC = w* L
Divide through by change in Q (ie. Q)
TC/Q = w* (L/Q)
TC/Q = Marginal Cost = MC
What is MPL?
MPL=(Q/L)
Thus: TC/Q = w* 1/(Q/L)
This gives: MC=w/MPL

54. Look at where Diminishing Returns sets in.
MC=w/MPL
Look at where Diminishing Returns sets in. MC
MPL
MPL L Q L1

55. Substitute L1 into SR Production Function Q1=f(KFIXED,L1)
MC=w/MPL
Substitute L1 into SR Production Function
Q1=f(KFIXED,L1) MC
MPL MC
MPL L Q L1 Q1

56. Alternatively: TC and TP
Substitute L1 into SR Production Function
Q1=f(KFIXED,L1) TC TC Q
MPL L Q L1 Q1

57. Relationship between APL and AVC
TC=TVC + TFC
TC = w*L + r*K
Divide equation by Q to get average cost formula.
TC/Q = w*L/Q + r*K/Q
ATC = AVC AFC
Thus, AVC=w*L/Q

58. AVC and APL AVC=w*L/Q Rearranging: AVC=w/(Q/L) Since Q/L=APL AVC=w/APL
Diagram is similar.

59. Substitute L2 into SR Production Function Q2=f(KFIXED,L2)
AVC=w/APL
Substitute L2 into SR Production Function
Q2=f(KFIXED,L2)
APL
AVC
AVC
APL L Q L2 Q2

60. Put SR Cost Curves Together

61. Average Cost Curves
ATC $
AVC
AFC Q

62. Short Run Average Costs and Marginal Cost$
ATC MC
AVC Q

63. Cost Curve Shifters (Variable Cost Increases)
A change in the wage shifts the AVC and MC curves.
Thus, the ATC curve also shifts upward.
ATC’
MC’ $
AVC’
ATC
AVC MC Q

64. Cost Curve Shifters (Fixed Cost Increases)
An increase in price of capital increases fixed costs, but not variable costs.
Thus, AVC and MC are fixed, but ATC increases. $
ATC’ MC
ATC
AVC Q

65. Costs in the LR Why did SR cost curves have the shape they did?
Why do LR cost curves have the shape they do?

66. LR Total Costs GraphicallyTC
Cost
CRS
DRS
IRS Q

67. Why are there Economies of Scale?
Specialization in use of inputs.
Less than proportionate materials use as plant size increase.
Some technologies are not feasible at small scales.

68. Why do Diseconomies of Scale Set In?
Eventually, large scale operations become more costly to operate (i.e., they use more resources) due to problems of coordination and control.
e.g., red tape in the bureaucracy.
Graphical Representation

69. Economies and Diseconomies of Scale
Assume Q increases 10 units for each isoquant K
IRS L

70. Economies and Diseconomies of Scale
Assume Q increases 10 units for each isoquant K
DRS
IRS L

71. Economies and Diseconomies of Scale
LRAC curve
Assume Q increases 10 units for each isoquant $ K
DRS
DRS
IRS
CRS
CRS
IRS L Q
QMES

72. LRMC and LRAC Curves

73. LRAC and LRMC $ LRMC LRAC Q
LRMC is TC/Q (i.e., change in TC from a change in Q) when all inputs are variable inputs.
When LRMC is above LRAC, it pulls the average up, and vice-versa. $
LRMC
LRAC Q

74. Relating SR to LR curves

75. Relationship between SR ATC and LRAC curves.
At Q1, the SR plant size which gives ATC1 is least costly. $
ATC1
LRAC Q Q1

76. Relationship between SR ATC and LRAC curves.
At Q1, the SR plant size which gives ATC1 is least costly.
SR ATC is tangent to LRAC at one point. $
ATC1
LRAC Q Q1

77. Relationship between SR ATC and LRAC curves.
At Q1, the SR plant size which gives ATC1 is least costly.
SR ATC is tangent to LRAC at one point.
Tangency is not at minimum point of ATC1. $
ATC1
LRAC Q Q1

78. Adjustments in SR are still more costly than LR
At Q2, the SR plant size which gives ATC1 is no longer least costly. $
ATC1
LRAC
atc1
lrac1 Q Q2

79. Adjustments in SR are still more costly than LR
At Q2, the SR plant size which gives ATC1 is no longer least costly.
Optimal move would be to larger plant size! $
ATC1
LRAC
atc1
lrac1 Q Q2

80. LRAC is lower “envelope” of family of SRATC curves$
ATC3
ATC1
ATC2
LRAC Q Q1
Q2=QMES Q3

81. SRMC and LRMC $ LRMC LRAC q q1 q2 q3 SRMC1 SRMC3 SRMC2 SRATC3 SRATC1