Transcript Lecture 4 - Trent University

Production Theory
LECTURE 4
ECON 340
MANAGERIAL ECONOMICS
Christopher Michael
Trent University
Department of Economics
© 2006 by Nelson, a division of Thomson Canada Limited
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Topics
•
•
•
•
•
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The Production Function
The Short-Run Production Function
Optimal Use of the Variable Input
Long-Run Production Functions
Optimal Combination of Inputs
Returns to Scale
© 2006 by Nelson, a division of Thomson Canada Limited
2006 Thomson Nelson
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Overview
• Managers must decide not only what to produce
for the market, but also how to produce it in the
most efficient or least cost manner.
• Economics offers widely accepted tools for
judging whether the production choices are least
cost.
• A production function relates the most that can
be produced from a given set of inputs.
» Production functions allow measures of the
marginal product of each input.
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The Production Function
• A Production Function is the maximum quantity
from any amounts of inputs or
• Q = f (L,K)
• If L is labour and K is capital, one popular functional
form is known as the Cobb-Douglas Production
Function
• Q = a • L b1• K b2 is a Cobb-Douglas
Production Function
• The number of inputs is often large. But economists
simplify by suggesting some, like materials or labour,
are variable, whereas plant and equipment is fairly
fixed in the short run.
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The Short-Run
Production Function
• Short Run Production Functions:
» MAX output, from any set of inputs
» Q = f ( X1, X2, X3, X4, X5 ... )
FIXED IN SR
VARIABLE IN SR
_
Q=
f ( K, L) for two input case, where K as Fixed
• A Production Function has only one variable
input, labour, is easily analyzed. The one
variable input is labour, L.
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• Total Product = Q * L
• Average Product = Q / L
» output per labour
• Marginal Product = ΔQ/ΔL or Q/L
» output attributable to last unit of labour
applied
• Similar to profit functions, the Peak of MP
occurs before the Peak of average product
• When MP = AP, we are at the peak of the AP
curve
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Elasticities of Production
• The production elasticity of labour,
» EL = MPL / APL = (DQ/DL) / (Q/L) = (DQ/DL)·(L/Q)
» The production elasticity of capital has the identical in
form, except K appears in place of L.
• When MPL > APL, then the labour elasticity, EL > 1.
» A 1 percent increase in labour will increase output by more than
1percent.
• When MPL < APL, then the labour elasticity, EL < 1.
» A 1 percent increase in labour will increase output by less than 1
percent.
• When MPL = APL , then the labour elasticity, EL = 1
» A 1 percent increase in labour will increase output by 1 percent.
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Short-Run Production Function
Numerical Example
L
Q
MP
0
1
2
3
4
5
0
20
46
70
92
110
--20
26
24
22
18
AP
--20
23
23.33
23
22
Labour Elasticity is greater then one,
for labour use up through L = 3 units
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Marginal Product
Average
Product
1
2
3
4
5
L
8
• When MP > AP, then AP is RISING
» IF YOUR MARGINAL GRADE IN THIS CLASS IS
HIGHER THAN YOUR GRADE POINT AVERAGE,
THEN YOUR G.P.A. IS RISING
• When MP < AP, then AP is FALLING
» IF YOUR MARGINAL BATTING AVERAGE IS LESS
THAN THAT OF THE TORONTO BLUE JAYS,
YOUR ADDITION TO THE TEAM WOULD LOWER
THE JAY’S TEAM BATTING AVERAGE
• When MP = AP, then AP is at its MAX
» IF THE NEW HIRE IS JUST AS EFFICIENT AS
THE AVERAGE EMPLOYEE, THEN AVERAGE
PRODUCTIVITY DOESN’T CHANGE
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Law of Diminishing Returns
INCREASES IN ONE FACTOR OF PRODUCTION,
HOLDING ONE OR OTHER FACTORS FIXED,
AFTER SOME POINT,
MARGINAL PRODUCT DIMINISHES.
MP
A Short-Run Law
point of
diminishing
returns
Variable input
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Stages of Production – TP, AP and MP
Stage 1 Ep>1
Stage 2
0<Ep<1
TP Q
• Stage 1: average
Pt of
Marginal
product rising.
Ep=1
Returns
• Stage 2: average Increasing Returns
product declining
Decreasing Returns
(but marginal
L1
L2
L3
product positive).
AP,
MP
• Stage 3: marginal
product is negative,
or total product is
declining.
Stage 3 Ep<0
Ep=0
TP
Negative Returns
L
AP
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L1
L2
L3
MP
L
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Optimal Use of the Variable Input
• HIRE, IF GET MORE
MRPL  MPL • MRQ = W
REVENUE THAN
COST
wage
• HIRE if
DTR/DL > DTC/DL
• HIRE if the marginal
revenue product >
W  MFC
W
•
marginal factor cost:
MRPL
MRP L > MFC L
• AT OPTIMUM,
L
MRPL = W  MFC
optimal labour
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MRPL is the Demand for Labour
• If Labour is MORE
productive, demand
for labour increases
• If Labour is LESS
productive, demand
for labour decreases
• Suppose an
EARTHQUAKE destroys
capital 
• MPL declines with
less capital, wages
and labour are
HURT
SL
W
DL
D’ L
L’ L
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Long-Run Production Functions
• All inputs are variable
» greatest output from any set of inputs
• Q = f (L, K) is two input example
• MP of capital and MP of labour are the
derivatives of the production function
» MPL = DQ / DL or Q /L
• MP of labour declines as more labour
is applied. Also the MP of capital
declines as more capital is applied.
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Isoquants &
LR Production Functions
• In the LONG RUN, ALL factors
are variable
• Q = f (L, K)
• ISOQUANTS -- locus of input
combinations which produces the
same output (A & B or on the
same isoquant)
» MAGNITUDE of SLOPE of
ISOQUANT is ratio of
Marginal Products, called the
MRTS, the marginal rate of
technical substitution
» MRTS = (K1-K2)/(L1-L2) =
<>K/<>L
» MRTS = MPL/MPK
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ISOQUANT
MAP
K
Q3
C
B
Q2
A
Q1
L
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Optimal Combination of Inputs
Equimarginal Criterion:
• The objective is to
minimize cost for a given Produce where
MPL/CL = MPK/CK
output
where marginal products
• ISOCOST lines are the
per dollar are equal
combination of inputs for a
given cost, C0
Figure 4.9
• C0 = CL·L + CK·K
• K = C0/CK - (CL/CK)·L
• Optimal where:
» MPL/MPK = CL/CK·
» Rearranged, this becomes the
equimarginal criterion
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K
D
at D, slope of
isocost = slope
of isoquant
C(1)
L
Q(1)
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Example – Isocost / Isoquant
• Deep Creek Mining Co.
» Cost per worker is $50 per period CL
» Mining Equipment can be leased at $0.2 per brake
per horse power.
» Recall C0 = CL·L + CK·K
» C = 50L + 0.2K, rearranging K = C/.2 – (250/.2)L
» K = C/0.2 – 250L
K
Q= f(L,K)
K=C/0.2 – CL
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L
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Use of the
Equimarginal Criterion
• Q: Is the following firm
EFFICIENT?
• A dollar spent on labour
produces 3, and a dollar
spent on capital produces 2.
• Suppose that:
USE RELATIVELY
» MP L = 30
MORE LABOUR!
» MPK = 50
• If spend $1 less in capital,
» CL = 10 (labour cost)
output falls 2 units, but rises
3 units when spent on labour
» CK = 25 (capital cost)
• Shift to more labour until the
• Labour: 30/10 = 3
equimarginal condition
• Capital: 50/25 = 2
holds.
• A: No!
• That is peak efficiency.
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Allocative & Technical Efficiency
• Allocative Efficiency – asks if the firm is using the
least cost combination of inputs
» It satisfies: MPL/CL = MPK/CK
• Technical Efficiency – asks if the firm is
maximizing potential output from a given set of
inputs
» When a firm produces at point T
rather than point D on a lower
isoquant, they firm is NOT
producing as much as is
technically possible.
© 2006 by Nelson, a division of Thomson Canada Limited
D
T
(1)
Q
Q(0)
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Scale Efficiency
• Scale Efficiency – asks if the firm is using
the lowest possible minimum average cost
for all production processes – defined as
the ratio of this lowest cost to the potential
average cost of production process
chosen
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Overall Production Efficiency
• Overall Production Efficiency – the
product of allocative, technical, and scale
efficiencies
» (Overall Production Efficiency) =
(Allocative Efficiency) x (Technical Efficiency) x
(Scale Efficiency)
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Returns to Scale
• A function is homogeneous of degree n
» if multiplying all inputs by  (lambda) increases
the dependent variable byn
» Q = f (L, K)
» So, f ( L, K) = n • Q
• Constant Returns to Scale is homogeneous of
degree 1.
» 10% more all inputs leads to 10% more output.
• Cobb-Douglas Production Functions are
homogeneous of degree
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b1 + b2
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Cobb-Douglas Production Functions
• Q = α • L b1 • Kb2
is a Cobb-Douglas Production
Function
• IMPLIES:
» Can be CRS, DRS, or IRS
if b1 + b2  1, then constant returns to scale
if b1 + b2 < 1, then decreasing returns to scale
if b1 + b2 > 1, then increasing returns to scale
• Coefficients are elasticities
b1 is the labour elasticity of output, often about 0.33
b2 is the capital elasticity of output, often about 0.67
which are E L and EK
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Most firms have some slight increasing returns to scale
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Problem
Suppose: Q = 1.4 L 0.70 K 0.35
1. Is this function homogeneous?
2. Is the production function constant
returns to scale?
3. What is the production elasticity of
labour?
4. What is the production elasticity of
capital?
5. What happens to Q, if L increases
3% and capital is cut 10%?
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Answers
1. Yes. Increasing all inputs by , increases
output by 1.05. It is homogeneous of
degree 1.05.
2. No, it is not constant returns to scale. It is
increasing Returns to Scale, since 1.05 >
1.
3. 0.70 is the production elasticity of labour
4. 0.35 is the production elasticity of capital
5. %DQ = EL• %DL+ EK • %DK =
0.70(+3%) + 0.35(-10%) = 2.1% -3.5%
=
-1.4%
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