Chapter 3 Profit and costs 1

CHAPTER 3

Profit maximisation, input demand, output supply and

duality

Chapter 3 Profit and costs 2

Before going into detail

1. Producers/firms make decisions that fit them best. That also holds for producers in rural areas.

2. Decisions result in output supply, input demand and reward for the quasi-fixed inputs (labour, capital, land)

3. If producers realize relative low rewards for quasi-fixed inputs, then it is to be expected that those quasi-fixed inputs shift to other production processes (may take time!).

Chapter 3 Profit and costs 3

Overview and basic concepts

• Cost minimisation (output given)– Cost function– Compensated demand functions– (Marginal cost function, supply and

producer surplus)• Profit maximisation

– Profit function– Uncompensated supply and demand

functions

Chapter 3 Profit and costs 4

Profit maximisation

• Profits = total revenues of output – total costs of variable inputs

• Marginal revenue = Marginal costs

Chapter 3 Profit and costs 5

Profit maximisation: overview• Multiple outputs (1,..,m)• Multiple inputs:

− Variable inputs 1,..,r− Quasi fixed inputs 1,..,s

• Short run optimisation of profits under technology constraint (transformation function) given level of quasi-fixed inputs:

i

r

iij

m

jj

xyxwyp

ij

,max

0),...,,,...,,,...,( 111 srm zzxxyyF

Chapter 3 Profit and costs 6

Cost minimisation

• Cost minimisation:

subject to:

ixminimise

r

i ii

C w x

)...,,,...,( 11 sn zzxxfy

Chapter 3 Profit and costs 7

Cost minimisation

• Optimisation under restrictions (Lagrange):

• Graphical analysis (next slide)

/

/i i i

ijj j j

f x f wMRS

f x f w

Chapter 3 Profit and costs 8

x 1 ’ x 1 *

C * /w 2

C ’ /w 2

Input x1

Input x 2

0

x 2 *

x 21 ’

Optimal choice

•

• Isocost lines slope = -w1/w2

(Low w1)

Isoquant f(x1, x2) = y

Isocost lines slope = -w1/w2

(High w1)

Effect of a price increase of input x1

Optimal input changes from

(x1*,x2

*) to (x1’,x2

’)

Cost minimisation

Chapter 3 Profit and costs 9

Cost minimisation

• Compensated demand function:

• Restrictions of demand functions– Homogeneous of degree zero in prices– Decreasing in own price– Symmetry restriction

),..,,,...,,(ˆˆ 11 snii zzwwyxx

Chapter 3 Profit and costs 10

Cost function

Minimum cost to produce a certain level of y, given input prices and quasi fixed inputs

),..,,,..,,(),..,,,..,,(ˆ 1111 snsnii zzwwyczzwwyxwC

Input price w1 w0 0

Costs

C0

C1 C2

Chapter 3 Profit and costs 11

Cost function

Minimum cost to produce a certain level of y, given input prices and quasi fixed inputs

),..,,,..,,(),..,,,..,,(ˆ 1111 snsnii zzwwyczzwwyxwC

/ 0C w

•Characteristics cost function

• Increasing in input prices • Linear homogeneous in input prices

Chapter 3 Profit and costs 12

Cost function

• Concave in input prices

• Increasing in output

• Symmetry condition

• Shephard’s Lemma:

(downward sloping input demand!!)

2 2/ 0C w

/ 0C y 2 2

i j j i

C C

w w w w

ˆii

Cx

w

02

2

iw

C

Chapter 3 Profit and costs 13

Profit maximisation with a cost function

Apply the principle marginal costs = marginal revenue (=price)

),....,,,....,,(max 11 sry

zzwwycpy

MCpy

cp

y

0

Chapter 3 Profit and costs 14

Profit maximisation with cost function

• Solve for y (inverse supply curve):

• Marginal cost function = inverse supply

• Area below supply curve: Total variable costs

• Area between supply curve and price line: producer surplus (also called restricted or variable profit)

),..,,,....,,( 11**

sn zzwwpyy

Chapter 3 Profit and costs 15

Profit maximisation with cost function Illustration by means of a figure

Total variable costs

Producersurplus

Producer surplus

2p

1p

MC

1y 2y

Pric

e

Output

Δ v

aria

ble

cost

s

Chapter 3 Profit and costs 16

One stage profit maximisation

• subject to:

Uncompensated demand and supply functions:

i

r

iij

m

jj

xyxwyp

ij

,max

0),..,,,...,,,...,( 111 srm zzxxyyF

kjF

F

p

p

k

j

k

j , qiF

F

w

w

q

i

q

i , jiF

F

w

p

i

j

i

j ,

),..,,,...,,...,( 111**

srmii zzwwppxx * *

1 1 1( ,..., , ,... , ,.., )j j m r sy y p p w w z z

Chapter 3 Profit and costs 17

Profit function

Substitution of demand and supply equation in profit equation leads to profit function:

is maximum profits, given prices and quasi fixed inputs

),.,,,.,,,.,( 111****

srm

r

iii

m

jjj zzwwppxwyp

*

Chapter 3 Profit and costs 18

Profit function• Characteristics

• Convex in all prices• Increasing in output prices • Decreasing in input prices

0/ p0/0/ 2222 wp

0/ w

Chapter 3 Profit and costs 19

Profit function

• Linear homogeneous in all prices

• Symmetry conditions

*1***** )()()(),( r

iii

m

jjj

r

iii

m

jjj xwypxwypwp

ijjiijji ppppwwww

2222

Chapter 3 Profit and costs 20

Profit function

• Hotelling’s Lemma

• The shadow prices of quasi-fixed inputs

),...,,,...,,,...,(),...,,,...,,,...,(

111111

*

srmkk

srm zzwwppsz

zzwwpp

),...,,,...,,,...,(~),...,,,...,,,...,(111

*111*

srmjj

srm zzwwppyp

zzwwpp

),...,,,...,,,...,(~),...,,,...,,,...,(111

*111*

srmii

srm zzwwppxw

zzwwpp

Chapter 3 Profit and costs 21

Dual approach profit function

It is also possible to START with a specification of a profit function that meets the theoretical requirements, and subsequently derive demand and supply equations. This is known as the DUAL APPROACH.

(= uncompensated supply)0*

*

j

j py

Chapter 3 Profit and costs 22

Dual approach profit function

(= uncompensated input demand)

(upward sloping supply curve)

(downward sloping demand curve)

shadow price quasi fixed input k

0*

*

i

i wx

0*

2

*2

j

j

j p

y

p

0*

2

*2

i

i

i w

x

w

kk

sz

*

Chapter 3 Profit and costs 23

Shadow prices of quasi-fixed input (VMPL)

• Land– Agriculture– Forestry– Nature preservation– Recreation– Residential area

How to approach?

• Labour– Agriculture– Handicraft work– Non-farm work– Migration

Same framework?

),...,,,...,,,...,(),...,,,...,,,...,(

111111

*

srmkk

srm zzwwppsz

zzwwpp

Chapter 3 Profit and costs 24

Dual approach profit function

Continuing with shadow prices (with two processes with profit functions 1 and 2):

and

and

1 11 1 1 1 2

1

( , , , )s p w z zz

1 12 1 1 1 2

2

( , , , )s p w z zz

2 21 2 1 1 2

1

( , , , )s p w z zz

2 22 2 1 1 2

2

( , , , )s p w z zz

Chapter 3 Profit and costs 25

Dual approach profit function

This provides information on the directions where quasi-fixed inputs move between processes.

Direct calculation of welfare effects:

*1 1 1 1 1 2 1 1 1 1 2

*2 2 1 1 1 2 2 2 1 1 2

, , , , , ,

, , , , , ,

*2

*2

W (p w z z z z ) - (p w z z )

(p w z z z z ) - (p w z z )

Chapter 3 Profit and costs 26

Conclusions• Cost and profit functions are powerful tools• They contain a lot of information about the

behaviour (not about location!)• Several functions can be derived from the

cost function or the profit function: they make sense in economic analysis

• Theory provides many restrictions which should hold under the assumptions of cost minimisation or profit maximisation

• Shadow prices might be the most relevant concept; in particular of land

Chapter 3 Profit and costs 27

Main Characteristics

Production function

increasing in inputs

concave in inputs

Cost function

Input demand

increasing in input prices

decreasing in input prices

concave in input prices

increasing in output

linear homogeneous in input prices

homogeneous of degree zero in prices

symmetry

symmetry

Profit function

Input demand

Output supply

decreasing in input prices

decreasing in input prices

convex in all prices

increasing in output prices

increasing in output prices

linear homogeneous in all prices

homogeneous of degree zero in prices

homogeneous of degree zero in prices

symmetry

symmetry

symmetry

)(xfy

),,( zwycc

),,( zwp

),,( zwyxx

),,( zwpxx

),,( zwpyy