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PRODUCTION, COST,AND SUPPLY FUNCTIONS
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Production function
• Firm: transform inputs into outputs
• Production funciton:
q = f (x1, ..., xn)
• In what follows, two inputs: K , L
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Isoquants
• Contour lines that connect points with same in (K , L) spaceproducing same output level.
• Similarly to indifference curves, generally convex (diminishingmarginal returns).
• The more convex, the the more complementary the inputs; theflatter, the closer substitutes.
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Production functions: two extreme cases
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2
3
1 2 3
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# planes
# pilots
q = 1 q = 2
1
2
3
1 2 3
Nebraska beef (tons)
Texas beef (tons)
q = 3
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Isoquants and cost minimization
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...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......1
2
3
4
1.333
1 2 3 4
K
L
q = 1 q = 2 q = 3
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••
K∗=1.6
L∗=2.5
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..........−w/r
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•
K
L
q = 2
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Demand for inputs
• For given input prices r ,w , and for a given output level q, findoptimal input mix K , L.
• The resulting L, for example, is demand for labor: Ld
• How does Ld depend on w , p and especially r?
• Example 1: desktop computer and demand for labor
• Example 2: Compare two industries (hydroelectric damconstruction; aircraft construction) in two countries (U.S. andIndia)
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Productivity
• Cobb-Douglas production function
qi = ωi Kαi Lβi
• Labor productivity: qi/Li
• Total factor productivity (TFP): ωi
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Estimating TFP
• Estimate coefficients from (e.g.) Cobb-Douglas productionfunction:
α̂ =r Ki
qiβ̂ =
w Liqi
where r ,w is cost of capital, labor
• Take logarithms and solve production function w.r.t. ωi :
ln ω̂i = ln qi − α̂ lnKi − β̂ ln Li
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Cost functions
• For given input prices r ,w , and for a given output level q, findoptimal input mix K , L
• Determine cost r K + w L
• C (q): minimum cost required to achieve output level q
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Cost concepts
• Fixed cost (FC): the cost that does not depend on the outputlevel, C (0)
• Variable cost (VC): that cost which would be zero if the outputlevel were zero, C (q) − C (0)
• Average cost (AC) (a.k.a. “unit cost”): total cost divided byoutput level, C (q)/q
• Marginal cost (MC): the unit cost of a small increase in output
− Definition: derivative of cost with respect to output, d C/d q
− Approximated by C(q) − C(q − 1)
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Examples
• Bagels: modest fixed cost (space), relatively constant marginalcost (labor and materials)
• Electricity generation: large fixed cost (plant), initially decliningmarginal cost (large plants are more efficient, and many plantshave startup costs)
• Music CDs: large fixed cost (recording), small marginal cost(production and distribution)
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Example: the T-shirt factory
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T-shirt factory example
To produce T-shirts:
• Lease one machine at $20/week
• Machine requires one worker, produces one T-shirt per hour
• Worker is paid $1/hour on weekdays (up to 40 hours), $2/hour onSaturdays (up to 8 hours), $3 on Sundays (up to 8 hours)
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T-shirt factory costs
Suppose output level is 40 T-shirts per week. Then,
• Fixed cost: FC = $20. Variable cost: VC = 40 × $1 = $40
• Average cost: AC = ($20+$40)/40 = $1.5
• Marginal cost: MC = $2
(Note that producing an extra T-shirt would imply working onSaturday, which costs more.)
Similar calculations can be made for other output levels, leading tothe cost function . . .
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T-shirt factory cost function
0
1
2
3
0 40 48 56
p
q
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...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ............................................................................................................................................................
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AC
MC
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Cost functions: more general case
p◦
p′
q◦ q′
p
q
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AC
MC
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T-shirt factory output choice
• Scenario A: BenettonTM, sole buyer of T-shirts, offers pricep = $1.8 per T-shirt (for any number of T-shirts)
• Should factory increase output beyond 40 T-shirts/week, thusoperating on Saturdays?
• p = 1.8, AC = 1.5, MC = 2.
• Although factory is making money at q=40 (because p > AC),profits would be lower if it produced more (because p < MC); itwould lose money at the margin. (Verify this: compute profit atq=40, 41.)
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T-shirt factory output choice
• Scenario B: BenettonTM, sole buyer of T-shirts, offers pricep = $1.3 per T-shirt (for any number of T-shirts)
• No matter how much factory produces, price is below per-unitcost; i.e., no matter how much factory produces, it will losemoney:
p < AC implies q × p < q × ACimplies Revenue < Cost
• Optimal decision is not to produce at all
Use marginal cost to decide how much to produce.Use average cost to decide whether to produce.
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T-shirt factory supply function
0
1
2
3
0 40 48 56
p
q
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AC
MCSLR
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T-shirt factory supply function
• Suppose fixed cost has already been paid for the week;then it’s a sunk cost
• Define Average Variable Cost (AVC) as average costexcluding fixed cost
• Short-run supply switches to zero at min AVC
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T-shirt factory supply function
0
1
2
3
0 40 48 56
p
q
AVC
MCSSR
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Supply by price taking firm
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•minAVC
minAC
q◦SR q◦LR
p
q
AVC
AC
MCSLRSSR.......................................................................................................................................................
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Takeaways
• Use marginal cost when deciding how much to produce, averagecost when deciding whether to produce. In other words, marginalcosts for marginal decisions.
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