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AP ECON 2300 F2012 Session 11 Instructor: Dr. David K. Lee
Department of Economics York University
Topic: Cost and Supply Curve
Reading: Chapters 21 and 22
Announcement: Review materials for the final exam will be posted by the next week.
Ch 21: Cost:
Fixed, Variable & Total Cost Functions
• F is the total cost to a firm of its short-run fixed inputs. F, the firm’s fixed cost, does not vary with the
firm’s output level.
• cv(y) is the total cost to a firm of its variable inputs when producing y output units. c (y) vs the firm’s
variable cost function.
• cv(y) depends upon the levels of the fixed inputs.
• c(y) is the total cost of all inputs, fixed and variable, when producing y output units. c(y) is the firm’s
total cost function;
c( y) = F + c v y).
$
c(y)
c (y)
v
F
F
y
© 2010 W. W. Norton & Company, Inc. 10
Av. Fixed, Av. Variable & Av. Total Cost Curves
• The firm’s total cost function is
c( y) = F + c v y).
• For y > 0, the firm’s average total cost function is
Page 1 of 14 F cv(y)
AC(y)= +
y y
= AFC(y)+AVC(y).
• What does an average fixed cost curve look like?
F
AFC ( y) =
y
• AFC(y) is a rectangular hyperbola so its graph looks like ...
• In a short-run with a fixed amount of at least one input, the Law of Diminishing (Marginal) Returns must
apply, causing the firm’s average variable cost of production to increase eventually.
• And ATC(y) = AFC(y) + AVC(y)
$/output unit Since AFC(y) → 0 as y →∞ ,
ATC(y) → AVC(y) as y →∞.
And since short -run AVC(y) must
eventually increase, ATC(y) must
eventually increase in a short -run.
ATC(y)
AVC(y)
AFC(y)
0 y
© 2010 W. W. Norton & Company, Inc. 21
Marginal Cost Function
• Marginal cost is the rate-of-change of variable production cost as the output level changes. That is,
∂ cv( y)
MC ( y) = ∂ y .
• The firm’s total cost function is
c( y) = F + cv( y)
• and the fixed cost F does not change with the output level y, so
∂ cv( y) ∂ c( y)
MC ( y) = = .
∂ y ∂ y
• MC is the slope of both the variable cost and the total cost functions.
• Since MC(y) is the derivative of cv(y), v (y) must be the integral of MC(y). That is,
∂ cv( y) y
MC ( y) = ⇒ c v y) = ∫MC (z)dz .
∂ y 0
Page 2 of 14 Marginal and Variable Cost Functions
$/output unit
MC(y)
Area is the variable
cost of making y’ units
0 y
© 2010 W. W. Norton & Company, Inc. 25
• How is marginal cost related to average variable cost?
cv(y) ∂ AVC (y) y × MC ( y) −1× c (v)
AVC (y) = , = 2 .
y ∂ y y
> > >
∂ AVC ( y )= 0 y × MC ( y ) = c ( y ). MC ( y ) =c v y )= AVC ( y ).
∂ y v y
< < <
>
MC ( y ) = AVC ( y ).
<
$/output unit
The short-run MC curve intersects
the short-run AVC curve from
MC(y)
below at the AVC curve’s
minimum.
AVC(y)
© 2010 W. W. Norton & Company, Inc. y 35
Page 3 of 14 c( y) ∂ ATC ( y) y × MC ( y) −1× c( y)
ATC ( y) = y , = 2 .
∂ y y
>
∂ ATC ( y = 0 > >
∂ y y × MC ( y ) = c ( y ).MC ( y ) = c ( y= ATC ( y ).
< y
< <
$/output unit
as
MC(y)
ATC(y)
y
© 2010 W. W. Norton & Company, Inc. 39
• The short-run MC curve intersects the short-run AVC curve from below at the AVC curve’s minimum.
• And, similarly, the short-run MC curve intersects the short-run ATC curve from below at the ATC curve’s
minimum.
Page 4 of14 $/output unit
MC(y)
ATC(y)
AVC(y)
y
© 2010 W. W. Norton & Company, Inc. 41
Short-Run & Long-Run Total Cost Curves
• A firm has a different short-run total cost curve for each possible short-run circumstance.
• Suppose the firm can be in one of just three short-runs;
x2= x2’
or x = x ’’ x ’ < x ’’ < x ’’’.
2 2 2 2 2
or x2= x2’’’.
$
c sy;x ′2
F′ = w x 2 2
F′′ = w x 2 2
A larger amount of the fixed
c sy;x ′2)
input increases the firm’s
fixed cost.
F′′
F′
© 2010 W. W. Norton & Company, Inc. y
45
Page 5 of14 • MP i1 the marginal physical productivity of the variable input 1, so one extra unit of input 1 gives MP 1
extra output units.
• Therefore, the extra amount of input 1 needed for 1 extra output unit is 1 / MP 1units of input 1.
• Each unit of input 1 costs w1, so the firm’s extra cost from producing one extra unit of output is
w1
MC = . is the slope of the firm’s total cost curve.
MP 1
• If input 2 is a complement to input 1 then MP 1s higher for higher x2. Hence, MC is lower for higher x2.
• That is, a short-run total cost curve starts higher and has a lower slope if2x is larger.
• The firm has three short-run total cost curves.
• In the long-run the firm is free to choose amongst these three since it is free to selec2 x equal to any of
x 2, 2 ’’, o2 x ’’’.
• How does the firm make this choice?
$
For 0 ≤ y ≤ y′, choose x 2 = x ′2 c (y;x ′)
s 2
For y ′ ≤ y ≤ y′′, choose x 2 = x ′2.
For y ′′ < y, choose x 2 = x ′2′ .
c sy;x ′′2
c (y;x ′′′ )
s 2 c(y), the
F′′′
firm’s long -
run total
F′′
F′ cost curve.
© 2010 W. W. Norton & Company, Inc. y′′ y 61
• The firm’s long-run total cost curve consists of the lowest parts of the short-run total cost curves. The
long-run total cost curve is the lower envelope of the short-run total cost curves.
• If input 2 is available in continuous amounts then there is an infinity of short-run total cost curves but
the long-run total cost curve is still the lower envelope of all of the short-run total cost curves.
• For any output level y, the long-run total cost curve always gives the lowest possible total production
cost.
• Therefore, the long-run av. total cost curve must always give the lowest possible av. total production
cost.
• The long-run av. total cost curve must be the lower envelope of all of the firm’s short-run av. total cost
curves.
• The firm’s long-run average total cost curve is the lower envelope of the short-run average total cost
curves ...
Q: Is the long-run marginal cost curve the lower envelope of the firm’s short-run marginal cost curves?
A: No.
Page 6 of14 $/output unit MC (y;x ′) MC (y;x ′′)
s 2 s 2
AC (y;x ′′′ )
s 2
AC (y;x ′)
s 2
AC (s;x ′′2

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