chap. 22 cost curves. cost... · 2019. 9. 5. · cost curves (geometry of costs) §total cost...
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Engineering Economic Analysis2019 SPRING
Prof. D. J. LEE, SNU
Chap. 22
COST CURVES
Average cost & Marginal cost
§ Total cost function can be derived from the cost min. problem: c(w1,…,wn, y)
§ Various costs can be defined based on the total cost function• variable cost, fixed cost, • average cost, marginal cost, • long-run cost, short-run cost
1
Average cost & Marginal cost
§ Let
2
( ): vector of fixed inputs, : vector of variable factors
,
f v
v f
x x
w w w=
§ Short-run cost function ( ) ( ), , , ,f v v f f fc w y x w x w y x w x= × + ×
• Short-run average cost (SAC): ( ), , fc w y x
y
• Short-run average variable cost (SAVC): ( ), ,v v fw x w y x
y
×
• Short-run average fixed cost (SAFC): f fw xy×
• Short-run marginal cost (SMC): ( ), , fc w y w
y
¶
¶
Average cost & Marginal cost
3
§ Long-run cost function ( ) ( ) ( ), , ,v v f fc w y w x w y w x w y= × + ×
• Long-run average cost (LAC): ( ),c w y
y
• Long-run marginal cost (LMC): ( ),c w y
y
¶
¶
• Note that ‘long-run average cost’ equals ‘long-run average variable cost’ and ‘long-run fixed costs’ are zero’
There is no fixed input
Average cost & Marginal cost
4
§ Example: Short-run Cobb-Douglas cost function • In a short-run, 2x k=
• Cost-min. problem: 1 1 21
1
min
. . a a
w x w k
s t y x k -
+
=
1 1
1
aa ax k y-
= ×
• SR Cost function: ( )1 1
1 2, ,aa ac w y k w k y w k-
= × +
1
21
1
1
2
1
11
aa
aa
aa
w kySAC wk y
ySAVC wk
w kSAFCy
ySMC wa k
-
-
-
æ ö= +ç ÷è ø
æ ö= ç ÷è ø
=
æ ö= ç ÷è ø
Cost curves (Geometry of costs)
§ Total cost function can be derived from the cost min. problem: c(w1,…,wn, y)
§ Assume that the factor prices to be fixed, then c(y).
§ Total cost, c(y), is assumed to be monotonic in y.§ Various cost curves: Average cost curve, Marginal
cost curve, LR cost curve, SR cost curve etc.§ How are these cost curves related to each other?
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Cost curves (Geometry of costs)
§ SR Average cost• SR total cost = VC + FC:• SAC = SAVC + SAFC
6
( ) ( )vc y c y F= +
( ) ( ) ( ), ,v v f f fvw x w y xc y c y F w x
y y y y y
× ×= + = +
U-shapedSAC
Cost curves (Geometry of costs)
§ Marginal cost vs. Average cost
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( ) ( ) ( )2
AC( ) c y c y y c yd y ddy dy y y
¢ -æ ö= =ç ÷
è ø
( ) ( ) ( )1 1 MC( ) AC( )c y
c y y yy y yæ ö¢= - = -ç ÷
è ø
( ) ( )( ) ( )( ) ( )
AC is increasing (AC ( ) 0)
AC is decreasing (AC ( ) 0)
AC is minimum (AC ( ) 0)
y MC y AC y
y MC y AC y
y MC y AC y
¢ > Û >
¢ < Û <
¢ = Û =
• Thus,
• MC for the first small unit of amount equals AVC for a single unit of output
( ) ( ) ( ) ( ) ( )1 0 1(1) (0)1 1
1 1 1v v vc F c F cTC TCMC AVC
+ - --= = = =
Cost curves (Geometry of costs)
§ Marginal cost vs. Average cost
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AC decreasing AC increasing
Cost curves (Geometry of costs)
§ Marginal cost vs. Variable cost• Since
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( ) ( ) vdc y
MC ydy
=0
( ) ( )y
vc y MC z dz= ò
• The area beneath the MC curve up to y gives us the variable cost of producing y units of output
MC(y)
y0
Area is the variablecost of making y� units
Cost
y¢
Cost curves (Geometry of costs)
§ Example• SR total cost function:
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( ) 2 1c y y= +
• Variable cost:• Fixed cost:
( ) 2vc y y=
( ) 1fc y =
( )( )
2AVC /
AFC 1/
y y y y
y y
= =
=
( )AC 1/y y y= +
( )MC 2y y=
Cost curves (Geometry of costs)
§ Example: C-D Technology• Recall that
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( )
( )
1
1
a
aa
c y Ky F
FAC y Kyy
-
= +
= +
( )1 1
1 2 1 2, ,b a
a ba b a ba b a b a b a ba ac w w y A w w y
b b
-- + ++ + + +é ùæ ö æ öê ú= +ç ÷ ç ÷ê úè ø è øë û
• For a fixed w1, w2, ( )1
, 1a bc y Ky a b+= + £
( )
( )
1
1
a ba b
a ba b
AC y Ky
KMC y ya b
- -+
- -+
=
=+
• In the short-run, recall that ( )1 1
1 2, ,aa ac w y k w k y w k-
= × +
Cost curves (Geometry of costs)
§ Example: MC curves for two plant
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• How much should you produce in each plant?
c1(y)
Plant 1
y1 c2(y)
Plant 2
y2
( ) ( )* *1 1 2 2MC y MC y=
{ }( ) ( )
1 21 1 2 2,
1 2
min
. . y y
c y c y
s t y y y
+
+ =
2 1y y y= -( ) ( )
11 1 2 1min
yc y c y y+ -
( ) ( )1 1 2 2 2 2
1 1 2 1 1
0 and 1dc y dc y dy dyc
y dy dy dy dy¶
= + = = -¶
• Therefore, the optimality condition is
Long-run vs. Short-run Cost Curves
§ In the long run, all inputs are variable• LR problem: Planning the type and scale investment
• SR problem: Optimal operation
§ Given a fixed factor: Plant size k• SR cost function: ( ),sc y k• In LR, let the optimal plant size to produce y be k(y)
Øfirm’s conditional factor demand for plant size!
§ Then LR cost function is ( ) ( )( ),sc y c y k y= How this looks graphically?
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Long-run vs. Short-run Cost Curves
• For some given level of output y*
• Since SR cost min problem is just a constrained version of the LR cost min problem, SR cost curve must be at least as large as the LR cost curve for all y
( )( )
( )( )
* * *
* *
: optimal plant size for
, : SR cost function for a given
, : LR cost function
s
s
k k y y
c y k k
c y k y
Þ =
Þ
Þ
( ) ( )( ) ( ) ( )
*
* * * * *
, for all level of
, when
s
s
c y c y k y
c y c y k k k y
£
= =
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Long-run vs. Short-run Cost Curves
• Also
• Hence SR and LR cost curves must be tangent at y*
( ) ( )( ) ( ) ( )
*
* * * * *
LAC SAC , for all level of
LAC SAC , when
y y k y
y y k k k y
£
= =
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ymin
( ) ( )( ) ( )
*min min
*min min
Note that LAC SAC , ,
that is, LAC SAC ,
y y k
y y k
¹
£LAC(ymin)
( )*minargmin SAC , = y k y
SAC(ymin)
Long-run vs. Short-run Cost Curves
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Long-run vs. Short-run Cost Curves
17
• Discrete levels of plant size: 1 2 3 4, , ,k k k k
• LR average cost curve is the lower-envelop of SR average cost curves
Long-run vs. Short-run Cost Curves
§ LR marginal cost• When there are discrete levels of the fixed factor, the firm will
choose the amount of the fixed factor to minimize costs.
• Thus the LRMC curve will consist of the various segments of the SRMC curves associated with each different level of the fixed factor.
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Long-run vs. Short-run Cost Curves
§ LR marginal cost• This has to hold no matter how many different plant sizes
there are !
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LAC(y)
Cost
y
SACs
Short-Run & Long-Run Marginal Cost Curves
LAC(y)
LMC(y)Cost
y
SRMCs
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Long-run vs. Short-run Cost Curves
§ LR marginal cost
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• LR cost function
• Differentiating LR cost function w.r.t. y
• Since k* is the optimal at y=y*,
• Thus LRMC at y* equals to SR marginal cost at (k*, y*)