Transcript Cost Functions - Faculty Directory | Berkeley-Haas

MBA201a: Economic Costs & Costs of Production
Economic categorizations of costs: cost functions
• We have seen how economic costs include opportunity costs
and exclude sunk costs.
• Once we’ve got the right costs, what do we do with them?
 Cost functions
– Cost functions relate some cost to the quantity a firm
produces.
– Cost functions represent ideal rather than actual costs.
– A firm’s cost function may change over time.
Professor Wolfram
MBA201a - Fall 2009
Page 1
T-shirt factory example
You run a t-shirt factory for Fruit-of-the-Loom.
You’re currently producing 45 t-shirts a week, and selling them to
headquarters for $1.80 each, for total revenues of $80/week.
Your division manager tells you that if you’re making 45 t-shirts a
week, your total costs are $71/week, so your average cost per tshirt is $1.58.
Are you happy?
Professor Wolfram
MBA201a - Fall 2009
Page 2
Categorizing the costs
Upon further investigation, you learn that your cost structure looks
like this:
To produce t-shirts:
• You must lease one machine at $20 / week.
• The machine requires one worker.
• The machine, operated by the worker, produces one t-shirt per
hour.
• Worker is paid $1/hour on weekdays (up to 40 hours), $2/hour
on Saturdays (up to 8 hours), $3 on Sundays (up to 8 hours).
Are you still happy?
Professor Wolfram
MBA201a - Fall 2009
Page 3
The Average Cost Fallacy
• The first 40 t-shirts cost:
C(n) = F + V(n) = $20 + $1*40 = $60
fixed
variable
At a price of $1.80, profits on these is $12.
• The next 5 t-shirts cost:
$2*5 = $10
At a price of $1.80, they generate revenue of $9.
You’re losing $1 by producing up to 45! You should have stopped
at 40.
Professor Wolfram
MBA201a - Fall 2009
Page 4
We will talk about 6 types of cost functions
Total cost (C): total cost of inputs the firm needs to produce
output q. Denoted C(q).
Fixed cost (FC): the cost that does not depend on the output
level, C(0) [or really C(0.00001)]
Variable cost (VC): that cost which would be zero if the output
level were zero, C(q) – C(0) [or really C(q) – C(0.00001)].
Average total cost (ATC) (aka simply “average cost” (AC)): total
cost divided by output level, C(q)/q.
Average variable cost (AVC): variable cost divided by output
level, VC(q)/q.
Marginal cost (MC): the unit cost of a small increase in output.
– Derivative of cost with respect to output, dC/dq
– Approximated by C(q)-C(q-1), e.g. C(40)-C(39)
Professor Wolfram
MBA201a - Fall 2009
Page 5
A total cost function graphically
An example: C(Q) = 10 + .5Q
30
25
C(Q)
20
15
10
5
0
1
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9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Q
C(Q)
Professor Wolfram
MBA201a - Fall 2009
Page 6
The average total cost function
ATC(Q) = C(Q)/Q = 10/Q + .5
12
10
ATC(Q)
8
6
4
2
0
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10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Q
ATC(Q)
Professor Wolfram
MBA201a - Fall 2009
Page 7
Marginal costs
MC(Q) = dC(Q)/dQ = .5
12
10
ATC(Q)/MC(Q)
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6
4
2
0
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10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Q
ATC(Q)
Professor Wolfram
MC(Q)
MBA201a - Fall 2009
Page 8
T-shirts: costs
Suppose output level is 40 t-shirts per week. Then,
– Fixed cost: FC = $20.
– Variable cost: VC = 40 x $1 = $40.
– Average total cost: ATC = (20+40)/40 = $1.5
– Average variable cost: AVC = (40)/40 = $1
– Marginal cost: MC = $1.
(Note that producing an extra T-shirt would imply working on
Saturday, which costs more: MC(41) = $2.)
Similar calculations can be made for other output levels, leading to
the cost functions …
Professor Wolfram
MBA201a - Fall 2009
Page 9
T-shirt factory cost functions
Cost ($)
MC
3
2
ATC
1.5
1
48
10
Professor Wolfram
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30
40
MBA201a - Fall 2009
50
T-shirts
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Marginal and average cost curves: generic shape
Cost ($)
MC
p2
ATC
p1
q1 q2
Marginal cost always crosses average cost at its minimum.
Professor Wolfram
MBA201a - Fall 2009
Page 11
More average cost and marginal cost in Excel
C(Q) = 10 + .2Q2; ATC = 10/Q +.2Q; AVC= .2Q; MC = .4Q
14
12
ATC(Q)/MC(Q)
10
8
6
4
2
0
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10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Q
ATC(Q)
Professor Wolfram
MC(Q)
MBA201a - Fall 2009
Page 12
Economies of scale
Economies of scale describe how the firm’s average costs change
as output increases.
– ATC  with quantity = “diseconomies of scale”
– ATC  with quantity = “economies of scale”
Note: a cost function can exhibit economies of scale at some
output levels and diseconomies of scale at other output
levels.
Professor Wolfram
MBA201a - Fall 2009
Page 13
T-shirt factory profits
Cost ($)
MC
3
Profits
2
ATC
1.8
1.5
1
48
10
Professor Wolfram
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40
MBA201a - Fall 2009
50
T-shirts
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What if Fruit-of-the-LoomTM offers a lower price?
Scenario B: Fruit-of-the-Loom™ offers p = $1.3 per t-shirt.
No matter how much factory produces, price is below per-unit cost;
i.e., no matter how much factory produces, it will lose money:
p < AC implies q x p < q x AC
implies Revenue < Cost
Optimal decision is not to produce at all.
Professor Wolfram
MBA201a - Fall 2009
Page 15
General lessons on output decisions
There is a general lesson from the factory example.
What to produce: Firms should produce every unit for which the
income on that unit (in this case the price) is greater than the
cost.
What not to produce: Firms should not produce any unit for which
the income on that unit is less than the cost.
Note that AC(q) is not playing a role in determining how much to
produce, only whether to produce at all.
Marginal cost: how much to produce
Average Cost: whether to produce
Professor Wolfram
MBA201a - Fall 2009
Page 16
Output decisions in the generic case
Cost ($)
MC
p2
ATC
p1
q1
Professor Wolfram
MBA201a - Fall 2009
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Output decisions in the generic case
Cost ($)
MC
p2
=
lost profits
q1
Professor Wolfram
MBA201a - Fall 2009
q2
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Supply curve
Supply curve: how much a firm produces at each price.
Generalizing from previous example:
Firm can sell all it wants at given price (we say market is
“perfectly competitive”).
If price is below minimum average cost, p0, then firm is better
off by shutting down.
If price is greater than P0, say P’, then firm should sell output q’
such that MC=P’.
Supply curve is given by MC curve for values of P
greater than the minimum of AC, zero for values of p
below minimum of AC.
Professor Wolfram
MBA201a - Fall 2009
Page 19
Supply curve for a competitive (price-taking) firm
Price
S
p’
AC
p0
MC
q0
Professor Wolfram
MBA201a - Fall 2009
q’
Quantity
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Next week
– Next week, we’ll talk about how to take a supply curve for a
single firm producing a perfectly competitive industry (what
we just saw for the t-shirt example)…
– and derive the aggregate industry supply curves that we’ve
been drawing since the first lecture.
– We’ll also begin talking about monopoly pricing, which you’ll
see is an extension of the ideas we’ve introduced today.
Professor Wolfram
MBA201a - Fall 2009
Page 21
Takeaways
We defined a number of economic cost functions.
We discussed how fixed costs do not include sunk costs.
Similarly, fixed and variable costs do include opportunity costs.
We saw how cost functions help firms make decisions about how
much to produce and whether to produce at all.
If a firm is facing a fixed price, it will:
– Use ATC to decide whether to produce or shut down
(produce if p > ATC, otherwise shut down).
– Use MC to decide how much to produce (produce as long as
p > MC). In fact, MC defines the firm’s supply function if it is
producing.
Professor Wolfram
MBA201a - Fall 2009
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