Transcript スライド 1 - GRIPS

Chapter 7: Costs
• Firms use a two-step procedure to decide how
much to produce.
– Technological efficiency: summarized in
production functions
– Economical efficiency: summarized in cost
functions
• In this chapter we learn cost functions.
• We study LR cost curves first before SR cost
curves.
Measuring Costs
• Explicit Costs:
– Explicit expenditure on inputs that firms made.
• Opportunity Cost (Economic Cost):
– The value of the best alternative use of a resource.
– e.g.) your own labor, land, capital, time studying
• Sunk Cost:
– An expenditure that has been incurred and cannot be
recovered (zero opportunity cost).
– e.g.) equipment that cannot be used for other purpose
– Different from fixed costs as fixed costs can be
recovered (for example, by selling the factory).
Deriving Cost Curves
Firm’s Cost-Minimization:
• Given the input prices (w and r), firms choose the
combination of inputs (L and K) which allows them
to produce the amount of output they desire with
the least cost using their production function
• In the LR, all inputs are variable. In the SR, K is
fixed, so firms vary L to change the amount of
output.
• LR: both K and L are variable
• Suppose w=5 and r =10. Isocost lines are:
C  wL  rK
C w
K  L
r r
Slope = -w/r
• Suppose the firm wants to produce q=100.
• At the cost minimizing point (x), the isoquant is
tangent to the isocost line.
Optimum condition:
MPL
MPL MPK
w
MRTS  



MPK
r
w
r
– Last dollar spent on labor adds as much extra output
as the last dollar spent on capital.
• Mathematically,
Min C  wL  rK
s.t. q  f ( L, K )
• Repeating this
analysis for different q,
firms know how their
cost varies with output.
• Expansion path: costmin combination of L
and K for each output
level
• Use this path to draw
cost curves.
• Cost function: C = f(q)
– Curve that relates cost of production to output (tells how
much it costs to produce q)
• Cost equation (line): C = wL + rK
– w=wage, L=labor, r=rental rate, K=capital.
– Shows the total costs of using inputs L and K
• Isocost equation (line): C = wL + rK
– All the combinations of inputs that require the same total
expenditure
– Similar to budget line in consumer theory
• It is clear that optimum L and K are functions of w, r,
and q:
L* =L*(w, r, q), K* =K*(w, r, q)
• Substitute L* and K* to the cost equation:
C*(w, r, q) = wL*(w, r, q) + rK*(w, r, q),
where C* is the cost function, which is derived from
the cost minimizing behavior.
• Assuming that factor prices are constant, we often
define cost function as C=C(q).
Exercise: Choice of Appropriate Technology:
Why is labor-using technology appropriate for
low income countries?
Capital
Same output, q, can be produced
by different technologies, where q1
is capital intensive and q2 is labor
using technology.
Slope of isocost
lines = -w/r
q1
q2
Labor
Possible Shapes of TC Curves
• Given constant input prices, production function
determines the shape of cost curves.
TC
(a)
TC
(b)
q
TC
q
TC
(c)
q
(d)
q
Average and Marginal Costs
• Average cost: Total cost per unit of output. Slope
of the line from the origin to the corresponding
point on the TC.
AC = TC/q
• Marginal Cost: Additional cost of producing one
more unit of output. Slope of TC.
MC = dTC/dq
• AC is closely related to returns to scale.
Relationship between TC and MC
TC
MC
D
C
B
A
FC
B
A
1
2
3
4
1
D
C
2
3
4
Shapes of AC & MC Curves
AC,
MC
(a)
AC,
MC
(b)
MC
AC
AC, MC
q
AC,
MC
q
AC,
MC
(c)
(d)
AC
MC
AC
MC
q
q
Relationship between AC and MC
By definition: AC  TC (q)  TC (q)q 1
q
To see the change in AC due to q, take derivative
of AC with respect to q:
AC MC TC 1

 2   MC  AC 
q
q
q
q
Thus, AC is decreasing when MC<AC, and
increasing when MC>AC.
Short-run Costs
• In the SR, at least one factor of production is fixed
(normally, K).
• Total Cost (TC)=
Variable Cost (VC) + Fixed Cost (FC)
– VC: Cost that varies with the quantity of output
produced
– FC: Cost that does not vary with quantity of output
produced
• Marginal cost (MC) = △TC/△q = △VC/△q
• Average cost (AC) = TC/q = VC/q + FC/q
Deriving SR cost curve
• Because K, firms vary only L to produce different
amount of q.
K
Isoquants
Firms can only
choose L to vary
output level
K
q = 350
q = 250
q = 100
0
10
20
30
L
Output
q
100
Input
K
1
L
10
Input cost
FC
10
VC
50
SR Cost Curve:
C = f(q)
C
60
C
200
250
1
20
10
100 110
150
350
1
30
10
150 160
100
50
0
100
200
300
400
q
Shapes of SR Cost Curves
Shape of VC:
• Due to diminishing marginal returns to labor, VC rises
more than in proportion as q increases.
q: Output
SR Production
Curve
L: Labor
Cost= wL
SR Cost Curve
C
Shapes of
Short-Run Cost,
Marginal Cost, and
Average Cost Curves
SR Cost Curve
1200
750
600
500
0
10
20
30
40
q
MC,
AC
MC
50
AC
30
25
q
0
10
20
30
40
Shape of MC:
• Since K is fixed, VC = wL.
VC
L
w
MC 
w

q
q MPL
• MC moves in the opposite direction of MPL.
• Due to diminishing marginal returns to labor, we
know that MPL tends to rise initially and, after
some point, fall with q.
• Thus, MC tends to fall and rise with q.
Shape of AC Curves:
• AC = AVC + AFC
• AFC strictly falls with q because FC is spread over q.
• For AVC:
VC wL
w
AVC 


q
q
APL
• AVC moves in the opposite direction of APL, and we know
that APL tends to initially rise and then fall with q.
• Thus, AVC tends to fall and rise with q, and so does AC
(because AFC strictly falls).
Lower Costs in the LR
• In the LR planning, firms choose K that minimizes
costs given their target output level. In the SR, K
is fixed. At some point, firms may invest more in
K to produce more efficiently, affecting its SR
costs. Thus, LR costs are always equal to or
lower than SR costs.
• LRAC connects the lowest point of SRACs that
have different levels of K.
Discrete levels of plant size
AC
SRAC curves
LRAC curves
q
LRAC as the envelope of SRAC
LR vs. SR expansion path
Economies of Scope
• Economies of scope exists if it is less expensive
to produce goods jointly than separately.
C (q1 , 0)  C (0, q2 )  C (q1 , q2 )
SC 
C (q1 , q2 )
– Economies of scope if SC>0
– No economies of scope if SC=0
– Diseconomies of scope if SC<0
Some Exercises
Q: What would happen to TC, AC, and MC
a. when factor prices increase?
b. when per-unit tax is imposed on output?
c. when “franchise tax” (lump-sum) is
imposed?
d. when technological change takes place?
e. when firms gain experience from learning by
doing?
a. Factor Price Change
b. Per-Unit Tax on Output
c. Franchise (lump-sum) Tax