1 production function q t =ƒ(inputs t ) q t =output rate input t =input rate where is...
TRANSCRIPT
1
Production FunctionProduction Function
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.
Production Function
Q=ƒ(Kt,Lt)
Qt
Kt
Lt
2
Difference between LR and SRDifference 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.
3
Long Run: Q = f (K,L)Long Run: Q = f (K,L) Suppose there are two different sized Suppose there are two different sized
plants, Kplants, K11 and K and K22..
One Short Run: One Short Run:
Q = f ( KQ = f ( K11,L) ,L) i.e., K fixed at Ki.e., K fixed at K11
A second Short Run:A second Short Run:
Q = f ( KQ = f ( K22,L) ,L) i.e., K fixed at Ki.e., K fixed at K22
Show this graphicallyShow this graphically
LR production function as many SR production LR production function as many SR production functions.functions.
4
Two Separate SR Production FunctionsTwo Separate SR Production Functions
LL
Q = f( KQ = f( K22, L ), L )
Q = f( KQ = f( K11, L ), L )
KK2 2 > K> K11
What Happens when Technology Changes?What Happens when Technology Changes?
This shifts the entire production function, both in the SR and in the LR.
6
Technology ChangesTechnology Changes
LL
TP before computerTP before computer
TP after computerTP after computer
7
SR Production Function in More DetailSR 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 I: MPL >0 and rising
II: MPL >0 and falling
III: MPL<0 and falling
I II III
L
Qt=ƒ(Kfixed,Lt)
8
Where Diminishing Returns Sets InWhere 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?
L
I III II
9
Why Diminishing Returns Sets InWhy 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.
L
I III II
10
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.
Define APDefine APLL and MP and MPLL
11
Average Productivity GraphicallyAverage 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
QQ11
LL
Q=f(KQ=f(KFIXEDFIXED,L),L)
LL11LL
12
Average Productivity GraphicallyAverage 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
QQ22
LL
Q=f(KQ=f(KFIXEDFIXED,L),L)
LL22
LL22
Q/LQ/L
APAPLL
13
Average & Marginal ProductivityAverage & 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
14
Average Productivity GraphicallyAverage Productivity Graphically
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.
LL22
LLLL22
Q/LQ/L
APAPLL
LL11
LL11
MPMPLL
15
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.
Intuitive explanationIntuitive explanation
16
LR Production FunctionLR Production Function
Qt
Lt
Kt
IsoquantsIsoquants(i.e.,constant(i.e.,constant quantity)quantity)
Define IsoquantDefine Isoquant
Different combinations of Kt and Lt which generate the same level of output, Qt.
18
Isoquants & LR Production FunctionsIsoquants & LR Production Functions
QQtt = Q(K = Q(Ktt, L, Ltt))
Output rate increases as you move to higher Output rate increases as you move to higher isoquants.isoquants.
Slope represents ability to tradeoff inputs while Slope represents ability to tradeoff inputs while holding output constant.holding output constant. Marginal Rate of Technical SubstitutionMarginal Rate of Technical Substitution ..
Closeness represents steepness of production Closeness represents steepness of production hill.hill.
ISOQUANT MAPISOQUANT MAP
Q1
Q2
Q3
K
L
19
Slope of IsoquantSlope 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
KKtt
LLtt
20
Extreme CasesExtreme Cases
No Substitutability Perfect Substitutability
LL
KK
QQ11
QQ22
Inputs used in fixedInputs used in fixed proportions!proportions!
KK
QQ11
QQ22
LL
Tradeoff is constantTradeoff is constant
21
SubstitutabilitySubstitutability
Low Substitutability High Substitutability
LL
KK
QQ11
KK
QQ11
LL
Slope of Isoquant Slope of Isoquant changes very littlechanges very little
Slope of Isoquant Slope of Isoquant changes a lotchanges a lot
Isoquants and Returns to ScaleIsoquants and Returns to Scale
Returns to scale are cost savings associated with a firm getting larger.
23
Increasing Returns to ScaleIncreasing Returns to Scale
Production hill is rising quickly. Lines on the contour map get
closer with equal increments in Q.KK
LLQ=10Q=10
Q=20Q=20Q=30Q=30
Q=40Q=40
24
Decreasing Returns to ScaleDecreasing Returns to Scale
Production hill is rising slowly. Lines on the contour map get
further apart with equal increments in Q.
KK
LLQ=10Q=10
Q=20Q=20
Q=30Q=30
Q=40Q=40
25
How Can You Tell if a PF has IRS, DRS, or CRS?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 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 FunctionsCobb-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 FunctionsCobb-Douglas Production Functions
This is a Constant Elasticity Function Elasticity of substitution = 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 LRTechnical 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 changeTechnical change
Labor Saving Capital Saving
K
L
K
L
Lets now turn to the Cost SideLets now turn to the Cost Side
What is Goal of Firm?
32
Define Isocost LineDefine 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
KK
LL
Slope=-w/rSlope=-w/rTC/rTC/r
33
TC constant along Isocost line.TC constant along Isocost line.
KK
LL
TCTC11/r/r
TCTC11/w/w
34
in TC parallel shifts Isocostin TC parallel shifts Isocost
KK
LL
TCTC11/r/r
TCTC11/w/w
TCTC22/r/r
TCTC22/w/w
TCTC22 > TC> TC11
35
Change in input price rotates IsocostChange in input price rotates Isocost
KK
LL
TC/rTC/r
TC/wTC/w11TC/wTC/w22
ww2 2 < w< w11
36
Optimal Input Levels in LROptimal Input Levels in LR
Suppose Optimal Output level is determined (Q1).
Suppose w and r fixed. What is least costly way to
produce Q1?
KK
LL
QQ11
37
Optimal Input Levels in LROptimal 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.
KK
LL
QQ11
KK11
LL11
38
Cost Minimizing ConditionCost 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 PathThe 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
L
Q1
Q2
L1 L2
K1
K2
expansion path
We can show that LR adjustment along the We can show that LR adjustment along the expansion path is less costly than SR adjustment expansion path is less costly than SR adjustment
holding K fixed!holding K fixed!
41
Start at an original LR equilibrium (i.e., at QStart at an original LR equilibrium (i.e., at Q11).).
K
L
Q1
L1
K1
42
LR AdjustmentLR Adjustment
LR adjustment: K increases (K1 to K2)
L increases (L1 to L2)
TC increases from black to blue isocost.
K
L
Q1
Q2
L1 L2
K1
K2
43
SR AdjustmentSR Adjustment
SR adjustment. K constant at K1.
L increases (L1 to L3)
TC increases from black to white isocost.
K
L
Q1
Q2
L1
K1
L3
44
LR Adjustment less CostlyLR 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
L
Q1
Q2
L1 L2
K1
K2
L3
45
Impact of Input Price ChangeImpact 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
L
Q1
L1
K1
46
Decrease in wage leads to substitutionDecrease 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
ab
47
Derivation of Labor Demand from Substitution Derivation of Labor Demand from Substitution EffectEffect
Wage falls w
K
L
Q1
L1
K1
K2
L2
ab
LL1 L2
w1
w2
DL1
48
There is also a scale effectThere is also a scale effect
Scale effect is increase in output that results from lower costs
Scale effect: b-c K
L
Q1
L1
K1a
bc
Q2
49
Scale Effect Shifts DemandScale Effect Shifts Demand
Wage falls w
K
L
Q1
L1
K1
K2
L2
ab
LL1 L2
w1
w2
c
L3
L3
DL1
DL2
50
Recall the Isocost LineRecall the Isocost LineTC=w*L + r*KTC=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 ReturnsLaw 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 (MPMarginal Productivity (MPLL) and Marginal Cost (MC)) 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 Changes in TC in SR must come from changes in Labor.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
MC=w/MPMC=w/MPLL
MPL
LL1
MC
Q
Look at where Diminishing Returns sets in.
MPL
55
MC=w/MPMC=w/MPLL
MPLMC
Substitute L1 into SR Production Function
Q1=f(KFIXED,L1)
LL1Q
MC
Q1
MPL
56
Alternatively: TC and TPAlternatively: TC and TP
Q TC
Substitute L1 into SR Production Function
Q1=f(KFIXED,L1)
LL1Q
TC
Q1
MPL
57
Relationship between APRelationship between APLL and AVC 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 APAVC and APLL
AVC=w*L/Q Rearranging: AVC=w/(Q/L) Since Q/L=APL
AVC=w/APL
Diagram is similar.
59
AVC=w/APAVC=w/APLL
APL
Substitute L2 into SR Production Function
Q2=f(KFIXED,L2)
LL2
AVC
Q
AVC
Q2
APL
Put SR Cost Curves TogetherPut SR Cost Curves Together
61
Average Cost CurvesAverage Cost Curves
$
Q
ATC
AVC
AFC
62
Short Run Average Costs and Marginal CostShort Run Average Costs and Marginal Cost
$
Q
ATC
AVCMC
63
Cost Curve ShiftersCost Curve Shifters(Variable Cost Increases)(Variable Cost Increases)
A change in the wage shifts the AVC and MC curves.
Thus, the ATC curve also shifts upward.
$
Q
ATC
MC
ATC’MC’
AVC
AVC’
64
Cost Curve ShiftersCost Curve Shifters(Fixed Cost Increases)(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.
$
Q
AVC
MC ATCATC’
65
Costs in the LRCosts 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 GraphicallyLR Total Costs Graphically
TCTC
IRSIRSDRSDRS
CostCost CRSCRS
67
Why are there Economies of Scale?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?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 ScaleEconomies and Diseconomies of Scale
Assume Q increases 10 units for each isoquant
IRS
K
L
70
Economies and Diseconomies of ScaleEconomies and Diseconomies of Scale
Assume Q increases 10 units for each isoquant
IRS
K
L
DRS
71
Economies and Diseconomies of ScaleEconomies and Diseconomies of Scale
Assume Q increases 10 units for each isoquant LRAC curve
IRS
K
L
DRS$
Q
IRSDRS
CRS
QMES
CRS
LRMC and LRAC CurvesLRMC and LRAC Curves
73
LRAC and LRMCLRAC and LRMC
$
Q
LRACLRMC
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.
Relating SR to LR curvesRelating SR to LR curves
75
Relationship between SR ATC and LRAC curvesRelationship between SR ATC and LRAC curves..
At Q1, the SR plant size which gives
ATC1 is least costly.
$
Q
LRACATC1
Q1
76
Relationship between SR ATC and LRAC curves.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.$
Q
LRACATC1
Q1
77
Relationship between SR ATC and LRAC curves.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.
$
Q
LRACATC1
Q1
78
Adjustments in SR are still more costly than LRAdjustments in SR are still more costly than LR
At Q2, the SR plant size which gives
ATC1 is no longer least costly.
$
Q
LRACATC1
Q2
atc1
lrac1
79
Adjustments in SR are still more costly than LRAdjustments 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!$
Q
LRACATC1
Q2
atc1
lrac1
80
LRAC is lower “envelope” of family of SRATC LRAC is lower “envelope” of family of SRATC curvescurves
$
Q
LRAC
ATC1ATC3ATC2
Q1 Q2=QMES Q3
81
SRMC and LRMCSRMC and LRMC
q1 q2 q3
SRATC1
SRATC2
SRATC3
SRMC1
SRMC2
SRMC3 LRAC
LRMC$
q