Chapter 2: Basic Microeconomic Tools 1 Basic Microeconomic Tools.
ECON6021 Microeconomic Analysis Production I. Definitions.
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Transcript of ECON6021 Microeconomic Analysis Production I. Definitions.
ECON6021 Microeconomic Analysis
Production I
APMP if 1
APMP if 1
if 1
labor of elasticityOutput
labor ofproduct marginal
labor ofproduct average
),( product total
L
LL
;
LL
L
LLQ
L
L
APMP
AP
MP
Q
L
L
Q
LdLQdQ
E
MPL
Q
APL
Q
LKfQ
Definitions
L
Q
pt of inflexion
Short-run production),( LKfQ
),( LKf
)Kgiven (I
IaIb II
IIIMPL
APLL )Kgiven (
L1 L2 L3
Short Run Production
Stage I: 0 & ,
Stage II: 0 but ,
Stage III: 0 & , both &
L L L L
L L L L
L L L L
MP MP AP AP
MP MP AP AP
MP MP AP AP Q
Law of diminishing Marginal Productivity—eventually, if a variable input is combined with a fixed input,its marginal product will, beyond some point decline, i.e., beyond L1,
02
2
L
MP
L
Q L
Short Run Production
LL
LL
L
LLL
LLL
APMP
APMP
APMP
APMPLL
AP
L
APLAP
L
QMP
if 0
if 0
if 0
)(1
L
Short Run Production
),( BB LKf
)2,2( BB LKf
2KB
KB
LB 2LB
Isoquant (the locus of (K,L) that yields the same quantity of good)
1. Constant returns to scale: a doubling of inputs doubles outputs2. Decreasing returns to scale: a doubling of inputs less than doubles output.3. Increasing returns to scale: a doubling of inputs more than double output
Isoquants
1. Cardinal—each isoquant represents a certain Q whose value is objective.
2. Coverage—for any point, there is always an isoquant passing through it
3. Negative Slope—because MPL>0, MPK>0 (assuming not in Region III)
4. Can’t cross5. Bending towards the origin6. Farther away from the origin, the greater the quantity.
Properties of Isoquants
( , )
( , ) ( , )
( , ) ( , )
Supp. 0 ( we are moving along an isoquant).
KSlope of Isoquant , -ve
L
, th
K L
L
K
L
Q Q K
Q f K L
K L K K L L
f K L f K LQ K L
K LMP K MP L
Q
MP
MP
dK MPMRTS
dL MP
e maximum amount of K
a producer would willingly forgo for one more unit of L,
holding output level constant.
Isoquants and Slopes
K
KKQ
L
LLQ
KQLQ
AP
MPE
AP
MPE
K
dKE
L
dLE
K
dK
Q
K
K
f
L
dL
Q
L
L
f
K
K
Q
dK
K
f
L
L
Q
dL
L
f
Q
dQ
dKK
fdL
L
fdQ
;;
;;
, where
Output Elasticities
, ,
Suppose , same proportional change in inputs.
/ % change in output
/ % change in all input
1 i.r.t.s.
1 c.r.t.s.
1 d.r.t.s.
Q L Q K
dL dK d
L K
dQ QR
d
MR E E
L K
L K
P MP
AP AP
Output Elasticities
1 1
2 2
11 1
212 2
11 1
212 2
1
2
1
2
( , ) 10
110 5
2
110 5
2
5 /
5 /
L
K
L
K
Q f K L K L
Q KMP K L
L L
Q LMP K L
K K
K LMP KMRTS
MP LL K
An Example: Cobb-Douglas Production Function
1 1 12 2 2
1
2
1
2
; 1
2
1
2
; 1
2
1010
10
5 / 1
210 /
5 / 1
210 /
L
K
LQ L
L
KQ K
K
Q K L KAP
L L L
Q LAP
K K
K LMPE
AP K L
L KMPE
AP L K
An Example: Cobb-Douglas Production Function
1 1
1
;
;
( , ) 10
10 10
10
/
/
/
/
and
L
K
L
K
LQ L
L
Q K
Q f K L K L
Q QMP K L K L L
L LQ Q
MP K LK KMP Q L K
MRTSMP Q K L
MP Q LE
AP Q L
E R
Cobb-Douglas production functionIn general,
. .
1
1
1
L
K
e g Q K L
MP
MP
MRTS
L
K
AL
K
BL
K
Linear Production Function
a
b
MP
MPMRTS
a
bL
a
QKorbLaKQ
K
L
b
Q0
a
Q0
a
bSlope=
L
K
Linear Production Function
LKQge
bLaKQ
,2min ..
,min
L
K
2K=L
2
1
(or aK=bL, in general)
Leontief Production Function
0 0
Question: what is the min cost that yields Q?
min
. . ( , ) where
An example
min
. . Q L where
and K, L, TC endegenous variables
w,r, Q exogenous variables
, , par
TC wL rK
s t Q f K L Q Q
TC wL rK
s t K Q Q
ameters
Cost Minimization:
The optimal input mix
* *
1. tangency condition: slope of iso-cost line slope of isoquant
2. Q
, known
L K
L K TC
L
K
A
B
C
TC
r'TC
r
D
Isoquant,
Lrelative price of
K
wslope
r
LKK
L MRTSMP
MPslope
Q
constrKwL
O
Cost minimization: Long Run Problem
'
Suppose w , v const,
A B
total cost TC * *OC v OD v
Cost minimization: Long Run Problem
Optimal choice of (K,L) that yields Qo
with min. cost.
Optimal Input Choice
L
K Locus of equal MRTSLK
(output-expansion path for given input prices)
Iso-cost linewL+rK=const
L
K
Qw
TC '
'TC
r
TC
r
w
TC
w
TC '
'Q
Q
Output expansion path
Output Expansion Path
K
L
output expansion path
output expansion path
Output Expansion Path
* *
* *
*
*
min ,
Hence to min cost,
more general, min ,
Q K L
K L Q
TC wK rL w r Q
Q aK bL
QK
aQ
Lb
Q Q w vTC w r Q
a b a b
Leontief Production Function
min ,
to min cost,
,
Q aK bL
Q aK bL
Q QK L
a bw r w r
TC wL rK Q Q Qb a b a
From now on, we use cost function, rather than production function.
outcome of cost min. problem
Leontief Production Function
The End