Announcements Last lecture Tonight 5-6 ARTS Main LT Thursday 9-10 Muirhead Main LT: Fazeer Exercise Class on Production WorkSheet Marks on Test 2 hopefully.

Announcements

• Last lecture Tonight 5-6 ARTS Main LT

• Thursday 9-10 Muirhead Main LT: Fazeer Exercise Class on Production WorkSheet

• Marks on Test 2 hopefully by Friday

• CORRECT Version of Solution to Test 2 on Network (some minor 1 am errors in original)

Tests in Last Week• 201 Test 3 on Production Worksheet

– Monday 11th Dec 5-6 Vaughan Jeffries

– see final previous years

• 201 TFU Final

– Thursday 14th Dec 5-6 Vaughan Jeffries

– see final previous years

• 203 Final (Two essays)

– Wednesday 13th 12-1 Vaughan Jeffries

• Will meet in Semester 2 to review test procedure

So have 3 Distinct Problems

1) Maximise profits

Maxx1, x2 = P f (x1, x2) – w1x1 – w2x2

Gives factor demand functions

X1 = x1 (w1, w2)

X2 = x2 (w1, w2)

May not be well defined if there are constant returns to scale

Varian Appendix Ch 18

2) Maximise subject to a constraintCxwx)s.t.wx,f(xQMax 221121xx 21

3) Minimise subject to a constraintQ)x,s.t.f(xxwxwMinC 212211

Problem 3) is the Dual of 2)

Called Duality Theory

Essentially allows us to look at problems in reverse and can often give very important insights.

Varian Appendix Ch19

Varian Appendix Ch 18

Take Cobb-Douglas example of Problem 2:

e.g rK]wLcλ[LKQ ba

λwLbKdL

dQ 1ba (1)

(2)

rKwLcdλdQ (3)

λwL

bQ

λrLaKdK

dQ b1-a λrK

aQ

=0

=0

=0

From 1) + 2)

λr

λw

KaQL

bQ

r

w

L

K

a

b

Lr

w.

b

aK

λwL

bQ λr

K

aQ

Substitute into (3)

]r[wLC

wLb

awL C )wL

b

a(1

)wLb

ba(C

w

C

ba

bL

This is a the factor-demand function for L

Lr

w

b

a

Lr

w.

b

aK

Strictly it is a Cost-constrained factor-demand function

So maximising output subject to a cost or finance constraint

w

C

ba

bL

rK]wLcλ[LKQMAX baLK,

yields a solution for the use of the two factors:

r

C

ba

aK

Similar to demand theory

So Production subject to a cost or a finance constraint is just like the usual consumption problem

• And a Cost-constrained factor-demand function is just a like normal Marshallian demand function in consumer theory

• …. with all its associated properties, adding-up, conditions, Cournot etc.

But what about the third problem… That of minimising

cost subject to an output constraint?

• Again we will work through a Cobb-Douglas example to give you a flavour of the story.

Cobb-Douglas Example of problem 3

dL

dC)1(

d

dC)3(

dK

dC)2(

1baLbKw

b1-a LaKr

L

Qbw

K

Qar

)LK(rKwLC MIN baLK, Q

baLKQ

=0

=0

=0

From (1) + (2)

KQλa

LQλb

r

w

L

K

a

b

r

w

Lr

w.

b

aK

L

Qbw

K

Qλar

Which is precisely the same condition as before, So both problems tell us that we should use K and L according to the same rule

(4)

But now comes the different bit!baLKQThe constraint (3) was

ba L Q Lr

w.

b

a

and Lr

w.

b

aK

baa

Lr

w

b

aQ

(4)

But if have CRS a+b=

baa

Lr

w

b

aQ

Lr

w

b

aQ

a

1

Qw

r

a

bL

a

[note we invert the bracket when we bring it to the other side]

This is a different factor demand function to the ‘normal’, ‘Marshallian-like’ demand function we had earlier.

What does it correspond to?

Qw

r

a

bL

a

L

K

Q

We are adjusting Costs to get to best point on target isoquant given w/r

If have different w/r, have differently sloped line, tangent to

Q

w0/r0

So picking out points of tangency along isoquant

This is a different factor demand function to the ‘normal’, ‘Marshallian-like’ demand function we had earlier.

This is essentially a Hicksian Factor demand curve

It is a Quantity-Constrained CONDITIONAL factor-demand curve

Qw

r

a

bL

a

baLKQThe constraint (3) was

baK Q

and Kw

r.

a

bL

bba

w

r

a

bKQ

Similarly we can solve for K. The easiest way to do this is to go back to the FOC

Kw

r

a

b

Inverting (4) above

b

w

r

a

bKQ

Q

r

w

b

aK

b

[note we again invert the bracket when we bring it to the other side]

And with CRS a + b=1 as before,

So Minimising Cost subject to an output constraint

yields a solution for the use of the two factors:

Conditional on the level of output.

Similar to Hicksian demand in consumer theory

)LK(rKwLC MIN baLK, Q

Qw

r

a

bL

a

Q

r

w

b

aK

b

Note, since

w

Q

aw

bra

dw

dLa

So conditional factor-demand functions always slope down

Q Constant – so no ‘income’ type effect

Qw

r

a

bL

a

< 0

Cost Function

• But the objective of this exercise is not just to derive conditional factor demands, but rather to derive a cost function

• To do this we return to the original

• C= wL + rK

Note can now formalise the cost function for the item C = wL +rK

rwC

bbb1

aaa1

b

aQwr

a

bQrw

b

br

awQ

a

aw

brQ

b

b

ba

a

a

b

aQ

r

wr

a

bQ

w

rw

Qrw C ab

Qr w2 C 1/21/21/2b1/2,a e.g.

bba

aab

b

aQwr

a

bQrw

baab

b

a

a

bQrw

Qr w2 C 1/21/2

L

K

1/21/2LK Q

Notice C-D production function takes the

form:

w

r

While its cost function for a given quantity Q

takes the form:

ba

1ba

a

ba

b

Qrk w C

Now we assumed earlier that there were CRS and that a+b=1

No reason why it should be of course, and more generally the Cobb-Douglas version of the cost function is:

Where k is a constant that depends on a and b

TC

AC

Q

Q

ba

1ba

a

ba

b

Qrkw C

ba

b-a-1ba

a

ba

b

Qrkw AC

Long Run total and average cost functions

Qrkw C ab

TC

AC abrkw AC

If a + b =1, CRS and constant LR average costs

Q

Q

TC

AC

If a + b <1, DRS and increasing LR average costs

Q

Q

ba

1ba

a

ba

b

Qrkw C

ba

b-a-1ba

a

ba

b

Qrkw AC

TC

AC

If a + b >1, IRS and decreasing LR average costs

Q

Q

ba

1ba

a

ba

b

Qrkw C

ba

b-a-1ba

a

ba

b

Qrkw AC

What about the Short-Run?

• Derivation of short-run costs from an isoquant map

– Recall in SR Capital stock is fixed

• Derivation of short-run costs from an isoquant map

– Recall in SR Capital stock is fixed

fig

Un

its o

f ca

pita

l (K

)

O

Units of labour (L)

TC1

TC4

TC7

100

400

700

Deriving a SDeriving a SRACRAC curve from an isoquant map curve from an isoquant map

L0

K0

TC

AC

SR TC and AC curve lies above LR costs

Q

Q

The Short Run Cost Functionba

LKQ Note

So the actual L employed at any point is :

b

1

aK

QL

So short run costs are:

KrwLSRTC KrK

Q w

b

1

a

The Short Run Cost Function for a=b=1/2

Since

KrK

Q w

b

1

a

C

KrK

Qw

2

1/2

KrQ

K

w 2

The Short Run Marginal Cost Function for a=b=1/2

Since

We take the derivative to find MC

KrQK

w 2 C

QK

w2

dQ

dC

Under Perfect Competition MC=MR=P

Which gives us the short run supply curve of the firm i:

Kw

P

2

1QS

PQK

w2

Q

P MC=Qs

Equilibrium :

To find an equilibrium then we have to set D =S

i

iMCS

D

P

Q

The remaining aspects of any production problem ask you to apply first year ideas to a specific problem

Gereralised version of Problem 3

Minimise cost : wL + w2 K

s.t. f(x1, x2) =Q

)]x,f(xQλ[xwxwCMin 212211xx21

0)x,f(xQdλdC

0dxdf

λwdxdC

0dxdf

λwdxdC

21

2

2

2

1

1

1

3 EQNS – 3 unknowns x1, x2,

So solve for ‘Quantity Constrained’

Conditional factor demands

X1 = x1 (w1, w2, )

X2 = x2 (w1, w2, )