• 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.

• 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).

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
K

C
 w r r
L
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:
MRTS
 
MP
MP
K
L
  w r

MP w
L

MP r
K
– Last dollar spent on labor adds as much extra output as the last dollar spent on capital.
• Mathematically,
Min C
 wL
 rK
. .

( , )

• 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 q
1 is capital intensive and q
2 is labor using technology.
Slope of isocost lines = -w/r q
1
Labor q
2

• Given constant input prices, production function determines the shape of cost curves.
(a)
TC TC (b)
TC (c) q
TC
(d) q q q

• 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.

TC
MC
D
C
FC
A
B
1 2 3 4
D
C
A
B
1 2 3 4

AC,
MC
(a)
AC,
MC
(b)
MC
AC
AC, MC q
AC,
MC
(c)
AC,
MC q
(d)
MC
AC
AC
MC q q

By definition:
AC
  
1 q
To see the change in AC due to q, take derivative of AC with respect to q:

AC


MC

TC

1
2 q q q q

MC

AC

Thus, AC is decreasing when MC<AC, and increasing when MC>AC.

• 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

• Because K, firms vary only L to produce different amount of q.
K Isoquants
K
0
10 20
Firms can only choose L to vary output level
30 q = 350 q = 250 q = 100
L

Output Input q K L FC
Input cost
VC C
100 1 10 10 50 60
250 1 20 10 100 110
C
200
150
350 1 30 10 150 160
100
SR Cost Curve:
C = f(q)
50
0 100 200 300 400 q

Shape of VC:
• Due to diminishing marginal returns to labor, VC rises more than in proportion as q increases.
q: Output
SR Production
Curve
Cost= wL
SR Cost Curve
L: Labor

Shapes of
Short-Run Cost,
Marginal Cost, and
Average Cost Curves
C
1200
750
600
500
SR Cost Curve
MC,
AC
50
0 10 20 30 40
MC
30
25 q
AC q
0 10 20 30 40

Shape of MC:
• Since K is fixed, VC = wL.
MC


VC
 q
 w


L
 w q MP
L
• MC moves in the opposite direction of MP
L
.
• Due to diminishing marginal returns to labor, we know that MP
L 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:
AVC

VC
 wL
 w q q AP
L
• AVC moves in the opposite direction of AP
L
, and we know that AP
L 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).

• 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.

AC
SRAC curves
LRAC curves q

• Economies of scope exists if it is less expensive to produce goods jointly than separately.
SC

C q
1

C (0, q
2
)

C q q
1 2
)
C q q
1 2
)
– Economies of scope if SC>0
– No economies of scope if SC=0
– Diseconomies of scope if SC<0

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

c. Franchise (lump-sum) Tax b. Per-Unit Tax on Output