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