Cost Constraint/Isocost Line

COST CONSTRAINT

COST CONSTRAINT

C= wL + rK C wL rK

(m = p₁x₁ +p₂x₂)

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

r: rental rate of capital

R i

Rearranging:

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

COST CONSTRAINT

COST CONSTRAINT

EQUILIBRIUM

EQUILIBRIUM

Y

rK

wL

C





EQUILIBRIUM

QU

U

We can either We can either

Minimise cost subject to

e

and

C

Y

Y





Maximise output subject to Maximise output subject to

e

and

Y

C

C



C



Y

and

e

C





EQUILIBRIUM

EQUILIBRIUM

Minimise cost subject to output

Lagrangian method







Y

f

K

L



rK

wL

L

*











,

Minimise cost subject to output







f

,



or or

Maximise output subject to costs



C

wL

rL



K

L

f

Lagrange Method

Lagrange Method

 Set up the problem (same as with utility

maximisation subject to budget j g constraint).

 Find the first order conditions.  Find the first order conditions.

 It is easier to maximise output subject to a

cost constraint than to minimise costs cost constraint than to minimise costs subject to an output constraint. The

answer will be the same (in essence) answer will be the same (in essence) either way.

Example: Cobb-Douglas Equilibrium

Derive a demand function for Capital and Labour by maximising output subject to a cost y g p j

constraint. Let L* be Lagrange, L be labour and K be capital.

Set up the problem p

 





L AK

L*  a 1a     Multiply out the part in brackets

  _{C wL rK}

L AK

Cobb Douglas: Equilibrium

Cobb-Douglas: Equilibrium

differentiation with respect to K, L and 

* L _{ } 0 1 1      _{ } r L AaK K L _{a a } 1st _FOC









Cobb Douglas: Equilibrium

Cobb-Douglas: Equilibrium

Rearrange the 1_g st _{FOC and the 2}nd _{FOC so that} 

is on the left hand side of both equations

  L A K a 1 1 a r L AaK a a   1 1  ₁st _FOC





Cobb-Douglas: Equilibrium

Cobb

oug s: qu b u

We now have two equations both equal to - so we can get rid of 

we can get rid of 

  L AaK a1 1a A









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

Cobb-Douglas: Equilibrium

Cobb

oug s: qu b u

























Cobb-Douglas: Equilibrium

Cobb

oug s: qu b u









Return to the 3rd _{FOC and replace rk}

Cobb-Douglas: Equilibrium

Cobb

oug s: qu b u

0







wL

rK

C

Cobb-Douglas: Equilibrium

Cobb

oug s: qu b u

Cobb-Douglas: Equilibrium

Cobb

oug s: qu b u









Rearrange so that wL is on the left hand side Rearrange so that wL is on the left hand side

Now go back to th 3rd FOC d rK a a wL  (1  )

the 3rd FOC and replace wl and f ll th

a _{follow the same}

procedure as b f t l

C a

Demand _{before to solve}

for the demand f C it l r C a K  Demand for

capital _{for Capital}

EQUILIBRIUM

QU

U

Slope of isoquant = Slope of isocost

MRTS = (-) w/r MRTS ( ) w/r

or

Aside: General Equilibrium

Aside: General Equilibrium

K MRTS A _{= – w/r} Fi A Firm A L L

Aside: General Equilibrium

Aside: General Equilibrium

K MRTS B _{= – w/r} Fi B Firm B L L

Aside: General Equilibrium

s de: Ge e

qu b u

MRTSA _{= – w/r}

MRTSB _{= – w/r}

MRTSA _{= MRTS}B

We will return to this later when doing g 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.p

 c(y) denotes the firm’s smallest

possible total cost for producing y possible total cost for producing y units of output.

The Cost Minimization Problem

The Cost-Minimization Problem

 Consider a firm using two inputs to make one output.p

 The production function is

y = f(K L) y = 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

The Cost-Minimization Problem

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

minimization problem is to solve

)

(

min

rK

wL



_f

_f

₍

₍

_K

_,

_,

_L

₎

₎



_y

_y

Here we are minimising costs subject to a t t t i t U ll ill t b output constraint. Usually you will not be

asked to work this out in detail, as the working t i t di

The Cost Minimization Problem

The Cost-Minimization Problem

 The levels K*(r w y) and L*(r w y) in the  The levels K (r,w,y) and L (r,w,y) in the

least-costly input bundle are referred

t th fi ’ diti l d d

to as the firm’s conditional demands for inputs 1 and 2.

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

)

(

)

(

)

(

r

w

y

rK

r

w

y

wL

r

w

y

c

(

r

,

w

,

y

)

rK

(

r

,

w

,

y

)



wL

(

r

,

w

,

y

)

c





Conditional Input Demands

Conditional Input Demands

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

costly input bundle located?

 And how is the total cost function d?

A Cobb-Douglas Example of

Cost Minimization

 A firm’s Cobb-Douglas production function is _{1/ 3 2/ 3} function is I i d 3 / 2 3 / 1

)

,

(

K

L

K

L

f

y





 Input prices are r and w.

 What are the firm’s conditional input  What are the firm s conditional input

demand functions?

K*( ) L*( )

A Cobb-Douglas Example of

Cost Minimization

At the input bundle (K*,L*) which minimizes the cost of producing y output units:p g y p

(a) * 1/ 3 * 2/ 3

)

(

)

(

K

L

y



A Cobb-Douglas Example of

Cost Minimization

*)

(

)

(

K

L

y



A Cobb-Douglas Example of

Cost Minimization

₍

₎

₍

_*)

*)

(

)

(

K

L

y



* 3 / 2 3 / 2 * 3 / 1 * 1/3 2r *   2r  * * 2 2 ) ( K w r K w r K y               y w K 3 / 2 * 2      

So is the firm’s conditional

d d f C it l

y r

2 

A Cobb-Douglas Example of

Cost Minimization

is the firm’s conditional demand for input 2.

y r w  2  p

r

2





y

w

r

L

2















w





A Cobb-Douglas Example of

Cost Minimization

So the cheapest input bundle yielding y output units is output units is



K

(

r

,

w

,

y

),

L

(

r

,

w

,

y

)































w

r

y

y

2

)

,

,

(

),

,

,

(







































y

w

r

y

r

w

2

,

2

_











Conditional Input Demand Curves

p

Total Price Effect

Total Price Effect

 Recall the income and substitution effect of a price change.p g

 We can also apply this technique to find out what happens when the

find out what happens when the price of a factor of production changes e g wage falls

changes, e.g. wage falls

 Substitution and output (or scale ) ff t

Total Price Effect

Total Price Effect

Total Price Effect

Total Price Effect

K _{T t l ff t i t}

K _{Total effect is a to c,} more labour is used L t L L₁ to L₃ c a Y2 a Y L L Y₁ L L₁ L₃

Total Price Effect

Total Price Effect

K _{L t L i th}

K _L

1 to L2 is the

substitution effect. M l b i b i

More labour is being used. c a _Y b Y₂ Y L_{1 L} Y₁ L L_{1 L} 2

Total Price Effect

Total Price Effect

K _{L t L i th t t} K _L 2 to L3 is the output (scale) effect. c a _Y b Y₂ Y L L Y₁ L L₂ L₃

Total Price Effect

Total Price Effect

 What about perfect substitutes and perfect complements? (Homework) perfect complements? (Homework)