Cost Constraint/Isocost Line

COST CONSTRAINTC= wL + rK

(m = p1x1 +p2x2)

w: wage rate (including fringe benefits, holidays, PRSI, etc)

r: rental rate of capital

Rearranging:

K=C/r-(w/r)L

COST CONSTRAINT

L

K

Represents society’s willingness to trade

the factors of production

C=wL+rK

C/w

C/r

Slope of the isocost = K/L=K/L= (-) w/r

(-) w/r

EQUILIBRIUM

L

K

Y

erKwLC

EQUILIBRIUM

We can either

Minimise cost subject to

e and CYY

or

Maximise output subject to

e and YCC

equilibrium

equilibrium

EQUILIBRIUM

LKfYrKwLL ,*

rLwLCKLfL ),(*

Minimise cost subject to output

or

Maximise output subject to costs

Lagrangian method

Lagrange Method

Set up the problem (same as with utility maximisation subject to budget constraint).

Find the first order conditions. It is easier to maximise output subject to a

cost constraint than to minimise costs subject to an output constraint. The answer will be the same (in essence) either way.

Example: Cobb-Douglas Equilibrium

Set up the problem

rKwLCLAKL aa 1*

Multiply out the part in brackets

rKwLCLAKL aa 1*

Derive a demand function for Capital and Labour by maximising output subject to a cost constraint. Let L* be Lagrange, L be labour and K be capital.

Cobb-Douglas: EquilibriumFind the First Order Conditions by differentiation with respect to K, L and

0 11*

rLAaKK

L aa 1st FOC

0 1*

wLKaAL

L aa 2nd FOC

0*

rKwLCL

3rd FOC

Cobb-Douglas: EquilibriumRearrange the 1st FOC and the 2nd FOC so that is on the left hand side of both equations

r

LAaK aa

11

1st FOC

1

w

LKaA aa 2nd FOC

Cobb-Douglas: EquilibriumWe now have two equations both equal to - so we can get rid of

r

LAaK aa 11

1

w

LKaA aa

(Aside: Notice that we can find Y nested within these equations.)

Cobb-Douglas: Equilibrium

r

LAaK aa 11

1

w

LKaA aa

r

LKAaK aa 11

1 1

wL

LLKaA aa

We have multiplied by L/L=1

r

LKAaK aa 11

r

aK 1

1 1

wL

LKaA aa

1

wL

a

Cobb-Douglas: Equilibrium

rK

a

1

wL

a Note: K-1 =1/K

)1( a

awLrK

Return to the 3rd FOC and replace rk

Cobb-Douglas: Equilibrium

)1( a

awLrK

0 rKwLC

Ca

awLwL

)1(

Ca

awL

)1(

1

Cobb-Douglas: EquilibriumSimplify

again.Ca

awL

)1(

1

)1( a

wLC

Demand function for

Labourw

CaL

)1(

Cobb-Douglas: Equilibrium

rK

a

1

wL

a

Rearrange so that wL is on the left hand side

rKa

awL

)1(

Now go back to the 3rd FOC and replace wl and

follow the same procedure as

before to solve for the demand

for Capitalr

CaK

Demand for

capital

EQUILIBRIUM

Slope of isoquant = Slope of isocost

MRTS = (-) w/r

or

MPL/MPK = (-) w/r

Aside: General Equilibrium

L

KMRTSA = – w/r

Firm A

Aside: General Equilibrium

L

KMRTSB = – w/r

Firm B

Aside: General Equilibrium

MRTSA = – w/r

MRTSB = – w/r

MRTSA = MRTSB

We will return to this later when doing general equilibrium.

The Cost-Minimization Problem

A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost.

c(y) denotes the firm’s smallest possible total cost for producing y units of output.

c(y) is the firm’s total cost function.

The Cost-Minimization Problem

Consider a firm using two inputs to make one output.

The production function isy = f(K,L)

Take the output level y 0 as given. Given the input prices r and w, the

cost of an input bundle (K,L) is rK + wL.

The Cost-Minimization Problem

For given r, w and y, the firm’s cost-minimization problem is to solve

)(min0,

wLrKLK

subject to yLKf ),(

Here we are minimising costs subject to a output constraint. Usually you will not be asked to work this out in detail, as the working out is tedious.

The Cost-Minimization Problem The levels K*(r,w,y) and L*(r,w,y) in the

least-costly input bundle are referred to as the firm’s conditional demands for inputs 1 and 2.

The (smallest possible) total cost for producing y output units is therefore

),,(),,(),,( ** ywrwLywrrKywrc

Conditional Input Demands

Given r, w and y, how is the least costly input bundle located?

And how is the total cost function computed?

A Cobb-Douglas Example of Cost Minimization

A firm’s Cobb-Douglas production function is

Input prices are r and w. What are the firm’s conditional input

demand functions?

K*(r,w,y), L*(r,w,y)

3/23/1),( LKLKfy

A Cobb-Douglas Example of Cost Minimization

At the input bundle (K*,L*) which minimizesthe cost of producing y output units:(a)

(b)

3/2*3/1* )()( LKy and

*

*

3/2*3/2*

3/1*3/1*

2

)())(3/1(

)())(3/2(

/

/

L

K

LK

LK

Ky

Ly

r

w

A Cobb-Douglas Example of Cost Minimization

3/23/1* *)()( LKy *

*2

L

K

r

w(a) (b)

A Cobb-Douglas Example of Cost Minimization

(a) (b)

From (b),** 2

Kw

rL

Now substitute into (a) to get

*3/23/2

*3/1* 22)( K

w

rK

w

rKy

yr

wK

3/2*

2

So is the firm’s conditional

demand for Capital.

3/23/1* *)()( LKy *

*2

L

K

r

w

A Cobb-Douglas Example of Cost Minimization

is the firm’s conditional demand for input 2.

Since and

yr

w

w

rL

3/2*

2

2

yr

wK

3/2*

2

** 2

Kw

rL

yw

rL

3/1* 2

A Cobb-Douglas Example of Cost Minimization

So the cheapest input bundle yielding y output units is

y

w

ry

r

w

ywrLywrK3/13/2

**

2,

2

),,(),,,(

Fixed r and w

Conditional Input Demand Curves

outputexpansionpath

Cond. demandfor Labour

Cond.demand

forCapital

y

y

L*(y1)

L*(y2)

L*(y3)

K*(y2)K*(y1) K*(y3)

y2y3

y3

y1

y2

y3

y1

y2

y1

L*(y1)L*(y2)

L*(y3) L*

K*K*(y1)

K*(y2)

K*(y3)

NB

Total Price Effect

Recall the income and substitution effect of a price change.

We can also apply this technique to find out what happens when the price of a factor of production changes, e.g. wage falls

Substitution and output (or scale ) effects

Total Price EffectK

L

a

Q1

Total Price EffectK

LL1

c

a

L3

Total effect is a to c, more labour is used L1 to L3

Y2

Y1

Total Price EffectK

LL1

b

c

a

L2

L1 to L2 is the substitution effect.

More labour is being used.

Y2

Y1

Total Price EffectK

L

b

c

a

L2 L3

L2 to L3 is the output (scale) effect.

Y2

Y1

Total Price Effect

What about perfect substitutes and perfect complements? (Homework)