Transcript Lecture 6

EC109 Microeconomics – Term 2, Part 1
Cost Curves
Laura Sochat
26/01/2016
Plan
• Long run total cost curves
– Long run average and marginal cost curves
– Economies and diseconomies of scale
• Short run cost curves
– Relationship with long run total cost curves
– Short run average and marginal cost curves
– Relationship with long run average and marginal cost
curves
• Economies of scope
K, Capital services
Long run total cost curve
𝑇𝐶2
𝑟
𝑇𝐶1
𝑟
𝐾2
𝐾1
B
A
2 million TVs per year
1 million TVs per year
𝐿1
𝐿2
𝑇𝐶1
𝑤
𝑇𝐶2
𝑤
L, Labor services
B
𝑇𝐶2 = 𝑤𝐿2 + 𝑟𝐾2
𝑇𝐶1 = 𝑤𝐿1 + 𝑟𝐾1
𝑇𝐶(𝑄)
A
1 million
2 million
TVs per year
As the level of output varies, holding
input prices constant, the cost
minimizing combination of input
changes
The long run total cost curve shows
how minimized total cost varies with
output, assuming constant input
prices and that the firm chooses the
input combination to minimize its
costs.
The long run total cost curve must be
increasing in Q, and must be equal to
0, when 𝑄 = 0
Finding the total cost curve from a production function
Assume a production function of the form: 𝑄 = 50 𝐾𝐿
a) How does minimized total cost depend on the output level, and the input prices,
for this production function?
b) What is the graph of the long run total cost curve when 𝑤 = 25 and 𝑟 = 100?
K, Capital services
How does the long run cost curve shift when input prices
change?
𝐶1 isocost line before the price of
capital goes up
𝐶2 million isocost line after the price
of capital goes up
𝐶3 million isocost line after the price
of capital goes up
𝐶1
𝐶3
A
𝐶2
𝐶1 = 𝐶2 < 𝐶3
B
1 million TV per year
Labor services per year
Starting from point A, where
the firm produces 1 million
televisions, on isocost line 𝐶1 .
After the price increase, the
cost minimising input
combination occurs at point B,
where total cost is greater
than it was at point A.
Long run total cost curve: Change in the price of inputs
The effect a proportional increase
in the price of both inputs
TC, dollars per year
TC, dollars per year
The effect of an increase in the
price of capital on the TC(Q) curve
𝑇𝐶(𝑄) after the
increase in the price
of capital
𝑇𝐶(𝑄) after the
increase in the price of
both inputs by 10%
B
𝐶3
𝐶1
B
A
1 million
𝑇𝐶(𝑄) before the
increase in the price
of capital
TVs per year
𝑇𝐶2 = 1.10𝑇𝐶1
𝑇𝐶1
A
1 million
𝑇𝐶(𝑄) before the
increase in the price of
both inputs by 10%
TVs per year
TC, dollars
Long run average and marginal cost curves
𝑇𝐶(𝑄)
C
£1,500
A
B
𝑀𝐶,𝐴𝐶, per
unit
0
Q, units per year
50
A’
£30
Long run average cost:
𝑇𝐶(𝑄)
𝐴𝐶 𝑄 =
𝑄
Long run marginal cost:
∆𝑇𝐶(𝑄)
𝑀𝐶 𝑄 =
∆𝑄
The relationship between
the two is such that:
•
𝑀𝐶 𝑄 = slope
of 𝑇𝐶(𝑄)
•
A’’
£10
50
𝐴𝐶 𝑄 slope of ray from O
to 𝑇𝐶(𝑄)
Q, units per year
•
When AC is decreasing in Q,
𝐴𝐶 𝑄 > 𝑀𝐶(𝑄)
When AC is increasing in Q,
𝐴𝐶 𝑄 < 𝑀𝐶(𝑄)
When AC is at a minimum,
𝐴𝐶 𝑄 = 𝑀𝐶(𝑄)
Economies and diseconomies of scale
We saw before that when a firm exhibits increasing returns to scale, output
increases more than proportionally to an proportional increase in both inputs: The
firm’s average cost falls as output increases.
–
If a firm’s average cost decreases as output increases, the firm is said to enjoy Economies of Scale.
when a firm exhibits decreasing returns to scale, output increases less than
proportionally to an proportional increase in both inputs: The firm’s average cost
increase as output increases.
–
If a firm’s average cost increases as output increases, the firm is said to enjoy Diseconomies of
Scale.
when a firm exhibits constant returns to scale, output increases proportionally to an
proportional increase in both inputs: The firm’s average cost stays unchanged as
output increases.
–
If a firm’s average cost neither increases or decreases as output increases, the firm does not enjoy
economies, or diseconomies of scale.
AC, per unit
Economies and diseconomies of scale
𝐴𝐶(𝑄)
Economies of
scale: Average
cost falls as
output increases
Diseconomies of
scale: Average cost
increases as output
increases
The smallest quantity at which the
long run average cost curve attains
its minimum efficient scale (MES).
The size of the MES relative to the
size of the market indicates the
significance of economies of scale
in particular industries.
The largest MES-market size ratio
represent significant economies of
scale.
The lowest MES-market size ratio
represent weaker economies of
scale.
𝑄1
𝑄2
Q units per year
Some examples of production functions
Production functions
𝐐 = 𝐋𝟐
L(Q)
𝐿=
𝑄
𝐐= 𝐋
𝐐=𝐋
𝐿 = 𝑄2
𝐿=𝑄
TC(Q)
𝑇𝐶 𝑄 = 𝑤 𝑄
𝑇𝐶 𝑄 = 𝑤𝑄 2
𝑇𝐶 𝑄 = 𝑤𝑄
AC(Q)
𝐴𝐶 𝑄 = 𝑤
𝐴𝐶 𝑄 = 𝑤𝑄
AC(Q)=𝑤
𝑄
How does long run average
cost vary with output
Decreasing
Increasing
Constant
Economies/ diseconomies
of scale?
Economies of scale
Diseconomies of scale
Neither
Returns to scale?
Increasing
Decreasing
Constant
The output elasticity of total cost as a measure to the
extent of Economies of scale
Output elasticity of total cost is the percentage change in total cost per 1 percent
change in output.
∆𝑇𝐶
∆𝑇𝐶 𝑄
𝑇𝐶
𝜖 𝑇𝐶,𝑄 =
=
∆𝑄
∆𝑄 𝑇𝐶
𝑄
Recall that 𝐴𝐶 𝑄 =
𝑇𝐶(𝑄)
;
𝑄
𝑀𝐶 𝑄 =
∆𝑇𝐶(𝑄)
∆𝑄
We can therefore rewrite the output elasticity of total cost such as:
𝑀𝐶
𝜖 𝑇𝐶,𝑄 =
𝐴𝐶
The output elasticity of total cost as a measure to the
extent of Economies of scale
Taking account of the relationship between long run average and marginal cost
corresponds with the way average cost varies with output. We can tell the extent of
economies of scale, using the output elasticity of total cost.
Value of 𝝐𝑻𝑪,𝑸
MC versus AC
How AC varies as Q
increases
Economies/
diseconomies of scale
𝜖 𝑇𝐶,𝑄 < 1
𝑀𝐶 < 𝐴𝐶
Decreases
Economies of scale
𝜖 𝑇𝐶,𝑄 > 1
𝑀𝐶 > 𝐴𝐶
Increases
Diseconomies of scale
𝜖 𝑇𝐶,𝑄 = 1
𝑀𝐶 = 𝐴𝐶
Constant
Neither
Short run total cost curve
We have seen when looking at the firm’s cost minimization problem, that in the
short run the firm faces both fixed and variable costs. The firm’s short run total cost
will be the sum of those two components.
𝑆𝑇𝐶 𝑄 = 𝑇𝑉𝐶 𝑄 + 𝑇𝐹𝐶 𝑄
Assuming the firm is constrained by the amount of capital it can use, 𝐾, and that the
price of capital is 𝑟, we can rewrite the expression for the short run total cost of the
firm as:
𝑆𝑇𝐶 𝑄 = 𝑇𝑉𝐶 𝑄 + 𝑟𝐾
TC, per year
Short run total cost curve
𝑆𝑇𝐶(𝑄)
𝑇𝑉𝐶(𝑄)
𝑟𝐾
𝑇𝐹𝐶 (𝑄)
Q, units per year
Let’s go back to the
production function we have
been using:
𝑄 = 50 𝐾𝐿
Assume again that 𝑤 = 25
and that 𝑟 = 100. If capital is
fixed at a level 𝐾
What is the short run total
cost curve? What are the
total variable and total fixed
cost curves?
𝑄2
𝑆𝑇𝐶 𝑄 =
+ 100𝐾
100𝐾
TC, per year
K, Capital
Relationship between the long run and short run total cost
curves
𝑆𝑇𝐶(𝑄)
Long run expansion path
B
𝑇𝐶(𝑄)
𝐾2
Short run expansion path
C
A
𝐾1
A
C
B
𝑄2 = 2 million
TVs isoquant
𝑄1 = 1 million
TVs Isoquant
𝑟𝐾1
L, Labor
1 million
2 million
Q, units per year
Cost per unit
Short run average and marginal cost curves
𝑆𝑀𝐶(𝑄)
𝑆𝐴𝐶(𝑄)
𝐴𝑉𝐶(𝑄)
𝐴𝐹𝐶(𝑄)
Q, units per year
Short run average cost:
𝑆𝑇𝐶(𝑄)
𝑆𝐴𝐶 𝑄 =
𝑄
Short run marginal cost:
∆𝑆𝑇𝐶(𝑄)
𝑆𝑀𝐶 𝑄 =
∆𝑄
Average fixed and variable
cost:
𝑇𝐹𝐶
𝑇𝐴𝐶
𝐴𝐹𝐶 =
; 𝐴𝑉𝐶 =
𝑄
𝑄
Where we can write that:
𝑆𝐴𝐶 𝑄 = 𝐴𝐹𝐶 + 𝐴𝑉𝐶
Cost, per year
The long run average cost curve as an envelope curve
𝐴𝐶 𝑄
𝑆𝐴𝐶1 𝑄 , 𝐾 = 𝐾1
𝑆𝐴𝐶3 𝑄 , 𝐾 = 𝐾3
𝑆𝐴𝐶2 𝑄 ,𝐾 = 𝐾2
£60
A
£50
C
£35
B
1 million
2 million
3 million Q, TVs per year
The long run average cost curve
forms a boundary around the set of
shot run average cost curves
corresponding to different levels of
output and fixed input.
Each short run average curve
corresponds to a different level of
fixed capital.
Point A is optimal for the firm to
produce 1 million TVs per year, with
fixed level of capital 𝐾1 .
Economies of scope
We have so far been looking at firms producing a single product. Consider now a
firm which produces two different products. The firm’s total cost will depend on the
quantity of product 1 it manufactures (𝑄1 ), and on the quantity of product 2 (𝑄2 ).
When it is less costly for a single firm to produce both products, relative to two
separate firms manufacturing the product separately, that is, when we have that:
𝑇𝐶 𝑄1 , 𝑄2 < 𝑇𝐶 𝑄1 , 0 + 𝑇𝐶(𝑄2 , 0)
Efficiencies have arisen, which are called economies of scope
The additional cost of producing 𝑄2 units of the second product, when the firm is
already producing 𝑄1 units of the first product is lower that the additional cost of
producing 𝑄2 when the firms does not manufacture product 1.
𝑇𝐶 𝑄1 , 𝑄2 − 𝑇𝐶(𝑄1 , 0) < 𝑇𝐶 0, 𝑄2 + 𝑇𝐶(0,0)