Transcript Modelling the producer: Costs and supply decisions

Modelling the producer:
Costs and supply
decisions
Production function
Production technology
The supply curve
Modelling the producer

Up until now, we have focused on how
consumers choose bundles of good



But we have not examined how these goods are
produced
We have implicitly assumed that they just “exist”
But, clearly, a theory of decision should also
explain the decision to produce goods.

We shall see that the framework of consumer
choice can also be used to understand producer
choices
Modelling the producer
The production function
Isoquants and Isocosts
Costs and supply
The production function

The production function is the relation
between inputs to production and the
amount of output produced for a given
technology
Y  f  I1, I2 ,..., I n 

For the moment, let us assume that there is
a single input to production (simplification)

A farm using labour to produce wheat
Tons of wheat per year
The production function of a farm
N˚ of
employees Output
0
0
3
1
10
2
24
3
36
4
40
5
42
6
42
7
40
8
Number of employees
The production function of a farm
Y
Tons of wheat per year
Impossible
Production frontier
Feasible
Number of employees
The production function of a farm
Tons of wheat per year
Y
b
Decreasing returns
starting from b
Number of employees
The production function of a farm
Tons of wheat per year
Y
Maximum Output
b
Number of employees
The production function

Total output (TP):
Y  f  I1, I2 ,..., I n 

The average output (AP):
Y
Ii

for i 1, 2,..., n
The marginal product (mP):
Y
I i
for i  1, 2,..., n
mP, AP
Tons of wheat per year
The production function of a farm
TP
DY = 14
DL = 1
Number of
employees
mP = DY / DL = 14
Number of
employees
TP
DY = 14
DL = 1
Number of
employees
mP, AP
Tons of wheat per year
The production function of a farm
mP
Number of
employees
Tons of wheat per year
The production function of a farm
TP
mP, AP
Number of
employees
AP = TP / L
mP
Number of
employees
TP
Decreasing returns
(inflexion point)
Number of
employees
mP, AP
Tons of wheat per year
The production function of a farm
AP = TP / L
mP
Number of
employees
TP
Maximum
output
Number of
employees
mP, AP
Tons of wheat per year
The production function of a farm
AP = TP / L
mP
Number of
employees
The production function

Relation between the average and marginal
products



The average product is maximal when it is
equal to the marginal product
If mP>AP , then the average product must be
increasing
If mP<AP , then the average product must be
decreasing
Modelling the producer
The production function
Isoquants and Isocosts
Costs and supply
Isoquants and Isocosts

Lets go back to the case with several inputs to
production. Imagine a case with 2 inputs
Y  f  L, K 
…which are labour (L) and capital (K)…

We define an isoquant as the set of
combinations of inputs that are just sufficient
to produce the same level of output.

This is where the analogy with consumer choice
will become obvious
Isoquants and Isocosts
Units of capital (K)
Isoquants are a 2-D mapping of the 3-D
production function
Just like:
Z
Indifference curves are a 2-D mapping of
the 3-D utility function
Y
Y= 150
X
A
B
Y= 100
Y= 50
Units of labour (L)
Units of capital (K)
Isoquants and Isocosts
DK
The technical rate of substitution
7
6
5
X
TRS = - (Slope of the Isoquant)
4
K
 Rate of substitution of factors
L
3
2
1
0
2
3
4
DL
5
6
7
8
9
10
Units of labour (L)
Isoquants and Isocosts

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Reminder : The marginal product of a factor is the
increase in total output (TP) following a marginal
increase in that factor (∂L or ∂ K)
On any given Isoquant :
K  mPK  L  mPL
TRS 

mP
K
 L
L
mPK
Note the similarity with the marginal rate of
substitution
mU x2
x1
MRS 

x2
mU x1
Isoquants and Isocosts

The overall aim of the firm is to maximise
profits, i.e. the difference between revenue
and production costs


However, for a given price of output, the
combination of inputs that maximises profits is
also the one that minimises costs
Therefore, when choosing the best
combination of inputs, the aim of the firm is
to minimise the cost of production for any
level of output
Isoquants and Isocosts
Imagine 5 combinations A, B, C, D, E
Cost = (L x pL)+ (K x pK)
Units
Units
Combination of capital (K) of labour (L)
If pL = 1€
& pK = 1€
If pL = 5€
& pK = 1€
A
2
10
12 €
52 €
B
3
6
9€
33 €
C
4
4
8€
24 €
D
6
3
9€
21 €
E
10
2
12 €
20 €
The best combination depends on the price of
the inputs
Isoquants and Isocosts
Units of capital (K)
Isocost: Set of combination of inputs available for a given cost
of production
All the spending on a single input
C  KpK  LpL
pL
C
K

L
pK pK
Units of labour (L)
Isoquants and Isocosts
Units of capital (K)
The optimal combination of inputs minimises the
production cost for a given level of output
The isocost curve is tangent to the isoquant

C
Definition of the
technical rate of
substitution at C !!!
Optimal combination
Units of labour (L)
Isoquants and Isocosts


The optimal combination is at the tangency
of the isoquant and the isocost
Therefore :
mPL
pL
TRS  

mPk
pK
mPL mPk

pL
pK

The ratio between the marginal output of an
input and its price (marginal cost of the
input) is the same for all inputs ...
Modelling the producer
The production function
Isoquants and Isocosts
Costs and supply
Costs and supply

There are different types of costs to
consider

Depending on the type of input
 Fixed / Variable costs

Depending on the time horizon
 Short / Long term
Costs and supply

Important note: Economic costs take into
account the existence of an opportunity cost


The opportunity cost is the cost of giving up the
next-best alternative.
What is the cost of a year at university ?


Objective costs: fees, books, laptop, food, rent,
etc.
Opportunity cost: The year’s worth of (minimum)
wages you are forgoing whilst you are at
university. In France, that’s 12,000 € !!
Costs and supply

Fixed and variable costs

Fixed costs are the incompressible costs that
the firm incurs regardless of the level of
production.
 Example: lighting of a factory floor, setup cost
of a new production line, etc.

Any other production cost is part of the
variable cost, because their size increases
with the level of production.
Costs and supply


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The time horizon is important in determining the
fixed/variable nature of production costs.
In the short run, the firm cannot change the
production technology (the method of production)
or the combination of inputs (the size of the
production plant is fixed)
In the long run, all the inputs are theoretically
adjustable. Most of the inputs that are fixed in the
short run become variable in the long run.
Costs and supply

The total cost curve gives the total
expenditure on inputs required for any
given level of output.


It is the minimal cost of production for that level
It is obtained through the cost-minimisation
process described in the previous section

For each level of output (isoquant), the firm
chooses the combination on the lowest (tangent)
isocost curve.
Costs and supply
Output TFC
(Y)
(€)
0
1
2
3
4
5
6
7
12
12
12
12
12
12
12
12
The total cost of a firm is
obtained by adding the total
fixed cost …
TFC
Costs and supply
Output
(Y)
0
1
2
3
4
5
6
7
TVC
(€)
0
10
16
21
28
40
60
91
TVC
… and total variable cost
Costs and supply
Output TFC
(Y)
(€)
0
12
1
12
2
12
3
12
4
12
5
12
6
12
7
12
TVC
(€)
0
10
16
21
28
40
60
91
TC
TVC
TFC
Costs and supply

The average cost curve gives the unit cost of
production for each level of output.

It obtained by dividing total cost (TC) by the level
of output (Y)
TC
AC 
Y

The average fixed cost falls with the level of
output

An increasing production means that the total
fixed cost can be spread over more units
Costs and supply


The marginal cost curve gives the increase
in total cost for a one-unit increase in
output.
The marginal cost curve at a given level of
output gives the slope of the total cost
curve for that level of output
TC
mC 
Y
Costs and supply
Working out the marginal cost
DTC=5
DY=1
TC
mC
Costs and supply
Costs(€)
General form of the
marginal cost
mC
Output (Y )
Costs and supply
AC
AVC
Costs(€)
Average and
marginal costs
mC
z
y
x
AFC
Output (Y )
Costs and supply

The marginal cost curve cuts the average
cost curve at its minimum point



If the marginal cost is lower than the average
cost, the average cost is decreasing
If the marginal cost is higher than the average
cost, the average cost is increasing
If the marginal cost is equal to the average cost,
the average cost does not change
Costs and supply

This is important as it tells us about the
level of returns to scale


If the average cost is decreasing, then total costs
are increasing more slowly than output
⇒increasing returns to scale
If the average cost is increasing, then total costs
are increasing faster than output ⇒ decreasing
returns to scale
Costs and supply

The profit maximising condition

A firm’s profit is given by total revenue minus
total cost :
  TR  TC

The firm chooses its output such that profit is
maximised (marginal profit is zero)

0
q

TR TC

0
q
q
mR  mC  0
mR  mC
Costs and supply

On a perfectly competitive market, the price
p is given by the market.



We will see next week that in order to maximise
its profits, the firm will choose its output q such
that the marginal cost of production equals the
price ⇒ p = mC
This condition gives the supply curve of the firm
Note: if the market price is less than the
average variable cost, the firm will prefer to
produce nothing (shutdown condition)
Costs and supply
Price
mC
AVC
Supply curve
z
pz
ps
AC
s
qs
qz
Output (Y )