Transcript Production Function - National Bureau of Economic Research

Production Function
 Qt=ƒ(inputst)
 Qt=output rate
 inputt=input rate
 where is technology?
Production Function
Q=ƒ(Kt,Lt)
Qt
Kt
 Firms try to be on the surface of the PF.
 Inside the function implies there is
waste, or technological inefficiency.
Lt
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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.
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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
3
Two Separate SR Production Functions
Q
Q = f( K2, L )
Q = f( K1, L )
K2 > K1
L
4
What Happens when Technology Changes?
This shifts the entire production function, both in the SR and in
the LR.
Technology Changes
Q
TP after computer
TP before computer
L
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SR Production Function in More Detail
 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
Qt=ƒ(Kfixed,Lt)
I
II
III
Q
 I: MPL >0 and rising
 II: MPL >0 and falling
 III: MPL<0 and falling
L
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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
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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
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Define APL and MPL
 Average Product = Q / L
 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.
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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
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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
Q/L
L
L2
APL
L2
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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
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Average Productivity Graphically
Q
 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.
L1
L
L2
Q/L
MPL
L1 L2
APL
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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.
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LR Production Function
Qt
Kt
Isoquants
(i.e.,constant
quantity)
Lt
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Define Isoquant
Different combinations of Kt and Lt which generate the
same level of output, Qt.
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.

K
Marginal Rate of Technical Substitution.
 Closeness represents steepness of production
hill.
Q3
Q2
Q1
L
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Slope of Isoquant
 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
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Extreme Cases
 No Substitutability
 Perfect Substitutability
K
K
Q2
Q
Q1
2
Q1
L
Inputs used in fixed
proportions!
L
Tradeoff is constant
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Substitutability
Low Substitutability
High Substitutability
K
K
Q1
Q1
L
Slope of Isoquant
changes a lot
L
Slope of Isoquant
changes very little
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Isoquants and Returns to Scale
Returns to scale are cost savings associated with a firm
getting larger.
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
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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
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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.
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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 
+
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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 
= A • ( K) • ( L) 
 Q = A •   K •   L
 Q =   A • K • L
  Q


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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
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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
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Technical change
 Labor Saving
 Capital Saving
K
K
L
L
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Lets now turn to the Cost Side
What is Goal of Firm?
Define 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
Isocost Line
K
TC/r
Slope=-w/r
L
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TC constant along Isocost line.
K
TC1/r
L
TC1/w
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in TC parallel shifts Isocost
K
TC2/r
TC2 > TC1
TC1/r
L
TC1/w TC2/w
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Change in input price rotates Isocost
K
w2 < w1
TC/r
L
TC/w1 TC/w2
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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
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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!
K
 Optimal point is point of allocative
efficiency.
K1
Q1
L1
L
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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.
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The LR Expansion Path
 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
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We can show that LR adjustment along the
expansion path is less costly than SR adjustment
holding K fixed!
Start at an original LR equilibrium (i.e., at Q1).
K
K1
Q1
L
L1
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LR Adjustment
 LR adjustment:
 K increases (K1 to K2)
 L increases (L1 to L2)
 TC increases from black to blue isocost.
K
K2
K1
Q2
Q1
L1 L2
L
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SR Adjustment
 SR adjustment.
 K constant at K1.
 L increases (L1 to L3)
 TC increases from black to white
isocost.
K
K1
Q2
Q1
L1
L3
L
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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
L1 L2 L3
L
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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
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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
a
K1
b
K2
Q1
L1
L2
L
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Derivation of Labor Demand from Substitution
Effect
 Wage falls
w
K
w1
a
K1
w2
b
K2
Q1
L1
L2
DL1
L
L1
L2
L
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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
K1
c
b
L1
L
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Scale Effect Shifts Demand
 Wage falls
w
K
w1
K1
c
a
w2
b
K2
DL2
Q1
L1
L2 L3
DL1
L
L1
L2 L3
L
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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.

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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?
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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?
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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





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MC=w/MPL
Look at where Diminishing Returns sets in.
MC
MPL
MPL
L1
L
Q
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MC=w/MPL
Substitute L1 into SR Production Function
Q1=f(KFIXED,L1)
MC
MPL
MC
MPL
L1
L
Q1
Q
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Alternatively: TC and TP
Substitute L1 into SR Production Function
Q1=f(KFIXED,L1)
TC
Q
TC
MPL
L1
L
Q1
Q
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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
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AVC and APL
 AVC=w*L/Q
 Rearranging: AVC=w/(Q/L)
 Since Q/L=APL
AVC=w/APL
 Diagram is similar.
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AVC=w/APL
Substitute L2 into SR Production Function
Q2=f(KFIXED,L2)
APL
AVC
AVC
APL
L2
L
Q2
Q
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Put SR Cost Curves Together
Average Cost Curves
ATC
$
AVC
AFC
Q
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Short Run Average Costs and Marginal Cost
$
ATC
MC
AVC
Q
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Cost Curve Shifters
(Variable Cost Increases)
 A change in the wage shifts the
AVC and MC curves.
 Thus, the ATC curve also shifts
upward.
MC’ ATC’
$
AVC’
ATC
AVC
MC
Q
63
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
64
Costs in the LR
 Why did SR cost curves have the shape they did?
 Why do LR cost curves have the shape they do?
65
LR Total Costs Graphically
TC
Cost
CRS
DRS
IRS
Q
66
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.
67
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
68
Economies and Diseconomies of Scale
Assume Q increases 10 units for each isoquant
K
IRS
L
69
Economies and Diseconomies of Scale
Assume Q increases 10 units for each isoquant
K
DRS
IRS
L
70
Economies and Diseconomies of Scale
LRAC curve
Assume Q increases 10 units for each isoquant
K
$
DRS
DRS
IRS
CRS
CRS
IRS
L
QMES
Q
71
LRMC and LRAC Curves
LRAC and LRMC
 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 viceversa.
$
LRMC LRAC
Q
73
Relating SR to LR curves
Relationship between SR ATC and LRAC curves.
 At Q1, the SR plant size which gives
ATC1 is least costly.
$
ATC1
LRAC
Q
Q1
75
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
76
Relationship between SR ATC and LRAC curves.
$
ATC1
LRAC
 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.
Q
Q1
77
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
78
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
79
LRAC is lower “envelope” of family of SRATC
curves
$
ATC1
ATC2
ATC3
LRAC
Q
Q1
Q2=QMES
Q3
80
SRMC and LRMC
$
LRMC
SRMC1
SRMC3
SRMC2
LRAC
SRATC3
SRATC1
SRATC2
q1
q2
q3
q
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