Intermediate Microeconomics WESS16
Cost and cost minimization
Laura Sochat
Sunk versus non sunk costs
Another important concept for the firm’s decision making. It will enable the firm to
assess the costs that the decision will actually affect.
Sunk costs refer to the costs which the firm has already incurred. They therefore
cannot be avoided by the firm.
Non sunk costs refer to the costs which would only be incurred should a particular
decision was taken. They can therefore be avoided by not taking such a decision.
Cost minimization in the long run
Let’s go back to an important decision the firm needs to take- Out of all possible
combinations of inputs possible to produce a given amount of output, which one
will be the cost minimizing combination?
The long run here refers to the situation where the firm is able to vary the
quantities of all the inputs it uses. What can we say of the costs associated with a
long run decision for the firm?
The short run will refer to a situation where the firm faces some constraints as it will
not be able to vary the quantities of all inputs.
Suppose the firm uses both Labor and capital . The price of labor is the wage rate 𝑤
and the price of capital is 𝑟. The firm has to produce amount 𝑄 of output over the
next year. The firms costs of production are such that:
𝑇𝐶 = 𝑤𝐿 + 𝑟𝐾
Isocost lines
K, Capital
An isocost line represents a set of combination of labor and capital that have the
same total cost for the firm.
For a cost level of £1m, and assuming that the
price of labor, 𝑤 = £10 and the price of
capital is such that 𝑟 = £20.
What is the equation describing the £1m
isocost line?
More generally, what is the equation for an
isocost line.
𝑇𝐶3
𝑟
Slope: −
𝑇𝐶2
𝑟
𝑤
𝑟
𝑇𝐶1
𝑟
𝑇𝐶1
𝑤
𝑇𝐶2
𝑤
𝑇𝐶3
𝑤
L, Labor
K, Capital
Graphical representation of the long run cost minimization
problem
• Point D is off the isoquant
• Point B and C are technically efficient,
but not cost minimizing
• Point A is technically efficient and cost
minimizing
𝑇𝐶2
𝑟
B
𝑇𝐶1
𝑟
D
• At point A: The slope of the isoquant =
The slope of the isocost.
A
C
𝑄 isoquant
𝑇𝐶1
𝑤
𝑇𝐶2
𝑤
L, Labor
𝑤
𝑀𝑅𝑇𝑆𝐿,𝐾 =
𝑟
𝑀𝑃𝐿 𝑤
=
𝑀𝑃𝐾
𝑟
Cost minimization using Lagrangean method
We can re write the cost minimization problem of the firm chosing Labor and
Capital to minimize its cost such as:
min 𝑤𝐿 + 𝑟𝐾
𝐿,𝐾
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑓 𝐿, 𝐾 = 𝑄
Defining the Lagrangean function such as
Λ 𝐿, 𝐾, 𝜆 = 𝑤𝐿 + 𝑟𝐾 − 𝜆(𝑓 𝐿, 𝐾 − 𝑄)
𝑀𝑃𝐿 𝑤
=
𝑀𝑃𝐾
𝑟
;
f L, K = Q
Corner point solutions
K, Capital
The cost minimizing input combination happens at a point where the firm uses no
Capital:
𝑀𝑃𝐿
𝑀𝑃𝐾
𝑄 isoquant
=
𝑤
𝑟
does not hold at any point along
the isoquant. More particularly, the
isocost line is flatter than the isoquant at
every point.
The situation is such that:
𝑀𝑃𝐿 𝑤
>
𝑀𝑃𝐾
𝑟
𝑀𝑃𝐿 𝑀𝑃𝐾
>
𝑤
𝑟
• Every dollar spent on labor is more
productive than every dollar spent on
Capital.
C
B
A
L, Labor
Comparative statics: A change in input prices
K, Capital
Suppose that the price of capital, 𝑟 , and the quantity of output, 𝑄, are both held
fixed. The figure below shows the effect of an increase in the price of labor, 𝑤.
𝐾2
B
𝐾1
𝐼1
A
𝑄 isoquant
𝐼2
𝐿2
𝐿1
L, Labor
• As the price of labor increases, the isocost line
becomes steeper.
• With diminishing marginal rate of technical
substitution of labor for capital, the new
optimal point is farther up the isoquant.
• The firm now uses more capital and less labor,
as the relative price of labor has increased.
• Note two important assumptions needed for
those results:
• The firm was already using positive
amounts of both inputs
• The isoquants are smooth
Comparative statics: A change in output
K, Capital
Suppose that now the price of both inputs stays unchanged but that the level of
output the firm wants to produce first changes to 𝑄 = 200, from an initial level of
𝑄 = 100 and subsequently increase further, to 𝑄 = 300.
• The optimal combinations of inputs move
north east as the quantity of output
increases.
• The line linking the three different
optimal bunbles is called the expansion
path.
• Both capital and labor are normal goods.
Expansion path
𝐾3
C
𝐾2
𝐾1
B
A
𝑄 = 300
𝑄 = 200
𝑄 = 100
𝐿1 𝐿2 𝐿3
L, Labor
Using comparative statics to derive input demand curves
K, Capital
•
B
A
C
•
𝑤, dollar per unit of Labor
L, Labor
•
£2
£1
B’
C’
A’
L, Labor
The top graph shows both the
effect of a change in the price
of labor, and the effect of a
change in output
The bottom graph summarises
the implications of the re
optimisation of the firm: The
labor demand curve
As the quantity of output
changes (and both the price of
labor and capital stay
unchanged), the labor demand
curve shifts upwards.
Costs in the short run: Fixed and variable costs
In the short run, the firm faces constraints in its ability to vary the quantity of some
inputs. We will consider a case where the amount of capital the firm can use is fixed
in the short run. We can rewrite the firm’s total cost such that:
𝑇𝐶 = 𝑤𝐿 + 𝑟𝐾
Where 𝐾 represents the fixed amount of capital.
The cost of labor constitues the firm’s total variable cost- It will vary as the firm
chooses to produce more or less output.
The cost of capital constitutes the firm’s total fixed cost- It will not vary as the firm
produces more or less output.
K, Capital
Cost minimization in the short run
The firm’s only technically efficient combinaition of inputs occurs at point B, where
it uses the minimum amount of labor which allows the firm to produce an amount
𝑄 of output, in combination to the fixed quantity of capital.
In the short run, the firm cannot substitute
between the two inputs: The optimal
combination of inputs does not involve a
tangency condition.
A
B
𝐾
𝑄 Isoquant
L, Labor
Cost minimization in the short run
K, Capital
The short run combination of inputs usually differs from the combination of inputs
used in the long run: The firm typically operates at higher costs in the short run.
Short run expansion path
C
𝐾
D
B
A
Consider point B in the figure to
the left. In the long run, the firm
will choose to produce at point B
(it chooses freely between Labor
and Capital). In the short run, the
firm can only use an amount 𝐾 of
capital to produce a level 𝑄2 of
output: In this case, the firm incurs
the same cost in the long run and
in the short run.
E
𝑄3 Isoquant
𝑄2 Isoquant
𝑄1 Isoquant
L, Labor