Microeconomics
Lecture 4
Jona Linde
https://sites.google.com/site/jonalinde/
j.linde@maastrichtuniversity.nl
Course structure
Part
I Supply and demand
Week
1
2
II Consumer theory
3
III Producer theory
IV Perfect competition
4
5
6
V Market power and
market structure
7
Task
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Subject
Supply and demand
Applying the supply and demand model
Consumer choice
Applying the model of the consumer
Firms and production
Short-run costs
Long-run costs
Perfect competition
Markets and welfare
Monopoly
Government and welfare
Pricing
Oligopoly and monopolistic competition
Game theory
Chapter
1, 2
3
4
5
6
7
7
8
9
11
9, 11
12
13
14
Last week: Short run production
•
•
120
100
output Q
Total production
- Maximum output produced
with a specific amount of labour
ഥ
- TPL = 𝑞 = 𝑓 𝐿, 𝐾
80
60
40
20
Average product of labour
- Average output per unit
of labour
- APL = 𝑞Τ𝐿
0
0
2
4
6
8
10
12
14
labour L
24
20
16
Marginal Product of labour
- “extra output from using
one extra unit of labor”
- MPL = Δ𝑞ൗΔ𝐿 = 𝑑𝑓 𝑞 ൗ𝑑𝐿
APL, MPL
•
12
MPL
8
APL
4
0
-4
0
2
4
6
8
labour L
10
12
14
Last week: Long run production
•
•
Both labour and capital are variable
- 𝑞 = 𝑓 𝐿, 𝐾
(no bar over K!)
Isoquant connecting input combinations
that yield the same output (level curve)
- 𝑞ത = 𝑓 𝐿, 𝐾
Marginal rate of technical substitution
- Slope of isoquants
Δ𝐾
𝑀𝑃
- MRTS= Δ𝐿 = − 𝑀𝑃 𝐿
-
•
𝐾
diminishing marginal returns
imply diminishing MRTS
Returns to scale
- Decreasing, increasing, or constant
- 𝑓 2𝐿, 2𝐾 <? > 2𝑓 𝐿, 𝐾
7
6
capital input K
•
5
4
3
2
1
0
0
1
2
3
4
labour input L
5
6
7
From production function to costs function
•
7.1) What are “costs”?
•
7.2) How do costs depend on output in the short run?
•
7.3) How do costs depend on output in the long run?
•
7.4) Lower costs in the long run
•
(skip section 7.5)
7.1 What are costs
•
Opening quote Task 6:
“Me and my wife, we own the place, so we don’t pay any rent. We don’t need staff,
so we don’t have any labour costs. We cannot complain, financially.”
(innkeeper of a “pannenkoekenhuis” near Maastricht, spring 2001)
•
Economic costs  accounting costs  Economic profit  accounting profit
- Sometimes no conflict: buying materials which are used in production directly
- Sometimes big difference: buying materials as stock that could be resold
Economic costs = opportunity costs
- “value of the best alternative use of an input”
- “what you have to give up to use that input”
Basic rule: for each input, ask yourself: “what is its value in its next most valuable
use?”
•
•
7.1 What are costs
•
Labour costs
- Hiring labour
- Simple
- but permanent contracts make accounting and opportunity costs diverge
- Partly sunk costs
- Sunk costs irrelevant to your decisions today
- “bygones are bygones”
- Own labour
- Accounting: some payment in salary, some in profit
- Economic/opportunity costs: best alternative use of time
- Earn money working elsewhere
- Value of leisure
7.1 What are costs
•
Capital costs
- Rent
- Simple, but can be (partly) sunk if in a long-term lease
- Own
- Accounting costs: Depreciated over several years (according to tax rules)
- Economic costs: imputed rents
- Next best use: rent it out, alternative use
- Sell and invest money
- Can be very different
- Real estate boom
- Specialized tools
- Resale value and therefore opportunity costs 0
- Initial expense sunk cost
7.2 Short-run costs
•
•
Not all inputs can be varied
- Capital (K) fixed, price per unit r rental rate
- Labour (L) variable, price per unit w wage rate
If capital is fixed, are capital costs sunk?
- Maybe, maybe not
- Perhaps can avoid by stopping production
- There can be alternative uses, but plausibly less valuable in the short run than
in the long run
- If so, economic fixed costs lower in the short run
- Conversely, no costs are fixed in the long run
7.2 Short-run costs
•
•
•
•
Three basic cost-level concepts and associated curves
ഥ
Fixed Costs
costs of fixed inputs
e.g. 𝐹𝐶 = 𝑟𝐾
FC does not depend on q
Variable Costs
costs of variable inputs
e.g. 𝑉𝐶 = 𝑤𝐿
VC rises as q rises
Total Costs
costs of all inputs
𝑇𝐶 = 𝐹𝐶 + 𝑉𝐶
TC rises as q rises, through VC
Four spin-offs: cost-per-unit concepts → curves
Δ𝑇𝐶
Δ𝑉𝐶
Marginal Costs:
extra costs of producing one extra unit 𝑀𝐶 = Δ𝑞 = Δ𝑞
•
Aver. Fixed Costs:
fixed costs per unit produced
•
Aver. Variable Costs:
variable costs per unit produced
•
Average Costs:
total costs per unit produced
•
•
𝐹𝐶
𝑞
𝑉𝐶
𝐴𝑉𝐶 = 𝑞
𝑇𝐶
𝐴𝐶 = 𝑞 = 𝐴𝐹𝐶 + 𝐴𝑉𝐶
𝐴𝐹𝐶 =
7.2 Short-run costs
q
FC
VC
TC
MC
AFC
AVC
AC
0
48
0
48
X
X
X
X
1
48
25
73
25
48
25
73
2
48
46
94
21
24
23
47
3
48
66
114
20
16
22
38
4
48
82
130
16
12
20.5
32.5
5
48
100
148
18
9.6
20
29.6
6
48
120
168
20
8
20
28
7
48
141
189
21
6.9
20.1
27
8
48
168
216
27
6
21
27
9
48
198
246
30
5.3
22
27.3
7.2 Short-run costs
80
400
70
350
60
250
FC
200
VC
TC
150
Cost-per-unit
Cost level
300
50
MC
AFC
40
AVC
AC
30
20
100
10
50
0
0
0
2
4
6
Output Q
8
10
12
0
2
4
6
Output Q
8
10
12
7.2 Short-run costs
• Specifics of / relations between these seven cost concepts
•
FC horizontal line
•
VC rising curve, starting at origin
•
TC vertical sum of FC and VC  parallel to VC
•
MC slope of tangent TC = slope of tangent VC
•
AFC declines monotonically (“spreading out”)
•
AVC falls when MC < AVC, rises when MC > AVC
 MC intersects AVC at its minimum (compare APL  MPL, last week)
•
AC falls when MC < AC, rises when MC > AC  MC intersects AC at its minimum
vertical sum of AFC and AVC  AC ever closer to AVC
 minimum AC lies beyond minimum AVC
7.2 Short-run costs
From production function to cost curves
ഥ
- Production function: 𝑞 = 𝑓 𝐿, 𝐾
ഥ (𝑟=rental rate), e.g. 𝑟 = 5, 𝐾
ഥ = 4→ FC=20
- Fixed costs: 𝑟𝐾
- Variable costs: w𝐿 (w=wage rate), more complex because L is a choice
ഥ we want 𝐿 as a function of 𝑞 i.e. the inverse of 𝑓
- Have 𝑞 = 𝑓 𝐿, 𝐾
100
120
80
80
If w = 10
60
40
Variable Cost VC
100
output Q
•
60
40
20
20
0
0
2
4
6
labour L
8
10
0
0
20
40
60
Output Q
80
100
120
7.2 Short-run costs
•
•
•
Short-run cost curve determined by short-run production function
Relation between AVC and APL?
𝑉𝐶
𝑤𝐿
𝑤
𝑤
- 𝐴𝑉𝐶 = 𝑞 = 𝑞 = 𝑞ൗ = 𝐴𝑃
𝐿
𝐿
Relation between MC and MPL?
𝛥𝑉𝐶
𝑤𝛥𝐿
𝑤
𝑤
- 𝐴𝑉𝐶 = 𝛥𝑞 = 𝛥𝑞 = 𝛥𝑞 = 𝑀𝑃
ൗ𝛥𝐿
•
𝐿
Mathematical example
ഥ=4
- Cobb-Douglas: 𝑞 = 𝐿0.5 𝐾 0.5 , with 𝑟 = 5 & 𝐾
- 𝑞 = 2𝐿0.5 → 𝐿 = 0.25𝑞 2
7.2 Short-run costs
Mathematical example continued
ഥ = 5 ∗ 4 = 20
- FC: 𝑟𝐾
- VC: 𝑤𝐿 = 10 ∗ 0.25𝑞 2 = 2.5𝑞 2
- TC: FC+VC= 20 + 2.5𝑞 2
- AFC: 20Τ𝑞; AVC: 2.5𝑞
20
→ATC: 2.5𝑞 +
𝑞
•
- MC: 5𝑞
Note:
- decreasing MPL → rising MC
- MC > AVC → AVC rises
- falling AFC and rising AVC generate
U-shaped AC
- MC hits AC at its minimum
50
40
Cost-per-unit
•
AFC
30
AVC
AC
20
MC
10
0
0
1
2
3
4
5
6
Output Q
7
8
9 10
7.3 Long-run costs
•
Long run:
•
all inputs can be varied practically
both capital and labour are variable inputs
 q = f(L,K)
no “bar” over K !
 all costs are avoidable/variable: (T)C = wL + rK
-
only one cost-level concept:
-
only two cost-per-unit concepts:
TC = VC, say C
AC and MC
7.3 Long-run costs
Isoquants
- Combinations of L & K that yield the
same output level 𝑞ത = 𝑓 𝐿, 𝐾
Δ𝐾
𝑀𝑃
- Slope: Δ𝐿 = 𝑀𝑅𝑇𝑆 = − 𝑀𝑃 𝐿
7
6
𝐾
-
“how much K can you replace by an
extra L, with q constant”
diminishing marginal returns imply
diminishing MRTS
 convex isoquants
capital input K
•
5
4
3
2
1
0
0
1
2
3
4
labour input L
5
6
7
7.3 Long-run costs
Isocost line: combinations of L and K that cost the same amount
- Example: w=10 & r=5
Find isocost line of 30
Note:
7
- 6K, 0L or 0K, 3L 0r 1K, 2L
• At given factor prices, isocost
6
- Mathematically: 𝑤𝐿 + 𝑟𝐾 = 𝐶ҧ
lines are parallel
5
- Here: 10𝐿 + 5𝐾 = 30
• lines further from origin
Δ𝐾
𝑤
mean higher cost
4
- Slope: Δ𝐿 = − 𝑟 →here: -2
• (e.g. try = 20 and = 40)
3
- “how much K do you have to
give up for an extra L,
2
while costs C remain the same”
capital input K
•
1
0
0
1
2
3
4
labour input L
5
6
7
7.3 Long-run costs
•
•
•
What does this look like
- Isoquants (see indifference curves)
- Isocost lines (see budget lines)
- But important difference: cost level is not fixed
Cost minimization
- Given factor prices find combination of inputs that minimizes costs required to
produce a certain level of output
- Graphically: hit relevant isoquant with lowest possible isocost line
Note:
- Cost minimization is necessary condition for profit maximization
- All combinations on an isoquant are technologically efficient
- But only the cheapest combination (given factor prices), i.e. combination on the
lowest isocost line, is economically efficient
7.3 Long-run costs
•
Example: : w=10 & r=5; find the cheapest way to produce 24
Find the lowest possible cost line
- C=30 clearly too low
7
q=24
- Shift right until you hit the isoquant
6
At optimal point
5
- Equal slopes: MRTS=ratio of prices
- “rate at which inputs can be
4
substituted = rate at which they can
3
be traded”
𝑀𝑃
𝑤
𝑀𝑃
𝑀𝑃
2
- Formally: 𝐿 = → 𝐿 = 𝐾
capital input K
•
•
-
𝑀𝑃𝐾
𝑟
𝑤
𝑟
‘Last euro rule’
Counter example: 𝑀𝑃𝐿 = 𝑀𝑃𝐾
safe 5 by replacing one L by 1 K
1
C=35
C=30
0
0
1
2
3
4
labour input L
5
6
7
7.3 Long-run costs
•
Conclusion: our analysis gives the cheapest way to produce a specific output level,
given specific factor prices
•
Three obvious questions:
1. What happens when factor prices change?
2. What is the cheapest way of producing other outputs?
3. What determines the shape of the long-run cost curve(s)?
7.3 Long-run costs
capital input K
1. What happens when factor prices change?
• with w = 10 and r = 5, use L = 2 and K = 3 to produce q = 24, at min. cost of C = 35
• Imagine w falls to 2.5
- same isoquant
7
- shallower isocost line 6
- Now tangency at
5
L=3, K=2, C=17.5
4
• Two reasons for lower C
3
1. Old bundle cheaper
2
2. Substitute to relatively
Q=24
1
cheaper L
0
• Changing prices change
0
1
2
3
4
5
6
7
8
9
10
11
12
13
costs and input mix
labour input L
14
7.3 Long-run costs
q=48
capital input K
2. What is the cheapest way of producing other output levels?
• Assume again w=10 and r=5
know for q=24 L=2, K=3, C=35
7
q=24
• Add new isoquant, e.g. q=48
6
- L=4, K=6, C=70
5
• Connect tangency points to find
expansion path
4
- Cheapest input combinations for
3
different levels of q
2
• Plot associated C against q →
long-run cost curve
1
C=35
- Minimum costs per output level
0
ceteris paribus: factor prices
0
1
2
3
C=708
4
labour input L
5
6
7
7.3 Long-run costs
3. What determines the shape of the long-run cost curve(s)?
• Returns to scale
• Ex.: take w = 10, r = 10 and draw some typical isoquants
Constant r2s
Increasing r2s
6
4
2
8
6
6
capital input K
capital input K
capital input K
8
4
2
0
2
4
6
0
8
2
4
6
0
8
80
80
60
40
20
0
15
Output Q
20
25
30
4
6
8
100
60
40
20
80
60
40
20
0
0
10
2
labour input L
Long-run cost C
100
Long-run cost C
100
5
2
labour input L
labour input L
0
4
0
0
0
Long-run cost C
Decreasing r2s
8
0
5
10
15
Output Q
20
25
30
0
5
10
15
Output Q
20
25
30
7.3 Long-run costs
•
Conclusion:
-
Constant returns: output rises in proportion with inputs
 costs rise in proportion to output
AC constant
-
Increasing returns: output rises faster than inputs
 costs rise less than output
AC falling, “economies of scale”
-
Decreasing returns: output rises less than inputs
 costs rise faster than output
 AC rising, “diseconomies of scale”
7.3 Long-run costs
•
•
350
60
300
50
250
Cost-per-unit
•
Typical shape long-run cost curve for competitive firms:
initially increasing returns, then decreasing returns

U-shaped AC- and MC-curves
Same geometrical relations as
in the short run:
MC: slope of tangent to C
= derivative of C
AC: falls when MC < AC,
rises when MC > AC
→ MC intersects AC at its
minimum
Cost level
•
•
200
150
40
MC
30
AC
20
100
10
50
0
0
2
4
6
Output Q
8
10
0
12
0
2
4
6
Output Q
8
10
12
7.3 Long-run costs
•
•
Mathematical approach
I) App. 7C: the Lagrange method; “constrained maximization”
II) Alternative: translate the graph into math
mathematically equivalent: this is what we’ll do
Ex.: derive the long-run cost function when 𝑞 = 10𝐿0.5 𝐾 0.5, with w = 20 and r = 5
- Note: this Cobb-Douglas production function has constant returns to scale
𝐾0.5
𝐿0.5
𝑀𝑃𝐿 = 5 𝐿0.5 , 𝑀𝑃𝐾 = 5 𝐾0.5
•
Long-run cost function gives minimum costs for any level of q
•
This is achieved by choosing the cheapest input bundle (L,K) that can produce this q
•
•
- 𝐶 = 𝑤𝐿 + 𝑟𝐾 → 𝐶 𝑞 = 𝑤𝐿 𝑞 + 𝑟𝐾 𝑞
i) The cheapest input bundle for producing amount q must lie on the q-isoquant
ii) Isocost line through the cheapest input bundle for q must be tangent to the q-isoquant
7.3 Long-run costs
•
•
i) The cheapest input bundle for producing amount q must lie on the q-isoquant
- 𝑓 𝐿, 𝐾 = 𝑞 → 𝑞 = 10𝐿0.5 𝐾 0.5
ii) Isocost line through the cheapest input bundle for q must be tangent to the qisoquant
𝐾0.5
𝐿
𝐿0.5
5 0.5
𝐾
5 0.5
-
𝑀𝑃𝐿
𝑤
=
→
𝑀𝑃𝐾
𝑟
20
𝐾
20
-
10𝐿0.5 (4𝐿)0.5 = 𝑞
20𝐿 = 𝑞
𝐿 𝑞 = 0.05𝑞, K 𝑞 = 0.20𝑞
𝐶 𝑞 = 𝑤𝐿 𝑞 + 𝑟𝐾 𝑞 = 20 ∗ 0.05𝑞 + 5 ∗ 0.2𝑞 = 2𝑞
- Constant returns to scale with MC=AC=2
= 5 → 𝐿 = 5 →𝐾 = 4𝐿
(expansion path)
7.4 Lower costs in the long-run
•
•
•
Relation between short-run and long-run costs
Basic message: LRAC<SRAC at any q
- K fixed in the short run
- Equal if (short-run) K is at the optimal level
- In the long-run firm chooses K so K is always at it’s optimal level
- Note the ‘envelope property’ (Fig. 7.9)
- Additional reasons
- Technological progress
- Learning by doing
Read this at home
Application
•
•
Opportunity costs of government policy
- Spending more on policy x either means higher taxes (which?) or less spending
elsewhere
- Costs of taxation usually larger than the tax revenue
- Deadweight loss and bureaucracy
- Benefits of spending equally difficult to measure
- Not equal to the effect on GDP
- Conversely cutting spending can have more costs than the raw number suggests
- E.g. government spending may have induced others to spend
Businesses might face similar difficulties
- E.g. effect of reducing employee benefits on future recruitment
Limitations and extensions
•
•
Time dimension
- Spend money now to make money later
- Matters when costs occur
- Interest rate as opportunity costs of spending now rather than later
- Therefor work with net present value
With more than 2 inputs we could get more than 1 optimum