Lecture Notes
Cost Minimization
 Before looked at maximizing Profits (π) = TR – TC or
 π =pf(L,K) – wL – rK
 But now also look at cost minimization
 That is choose L and K to minimize costs = wL + rK
subject to Y = f(L, K).
 From this problem derive a cost function C = C(w, r, Y).


Minimum cost of producing output Y given input prices w and
r.
How do we get these minimum costs?
 Recall the definition of an IsoQuant
 Shows the relationship between two inputs, L and K,
holding output (Y) constant.
 What would an isoquant look like?


If use more L => what would happen to K to keep Y constant?
Thus, isoquants are downward sloping and convex (why?)
K
Y=Y*
L
 Isoquants show a given output, Y*, that the firm wants
to produce. How to minimize costs of producing this
output?
 Isocost curve = shows combinations of L and K
keeping cost constant.
 Recall C = total costs = wL + rK or
 K = C/r – w/rL




This is an isocost line.
Intercept = C/r
Slope = -w/r
What does the line look like for C=100 r=10 and w=20?
Isocost curve is given by K = C/r – w/rL
K
Everywhere on isocost curve total cost = 100
As Costs Increase
Move to a higher Isocost
Intercept = C/r = 10
Slope = -w/r = -20/10 = -2
Intercept = C/w = 5
L
 Problem is to choose L and K to produce a given
output, Y* (on fixed isoquant), so that costs are
minimized (on lowest isocost possible.)
 Where is the point of minimum cost on C1?

Tangency point between isocost and isoquant.
K
C1
C2
K*
Y=Y*
L*
L
 Tangency between isocost and isoquant occurs where
slopes are equal or
 Slope of isoquant = technical rate of substitution
= - MPL /MPK.
 Slope of isocost = -w/r
 Therefore cost minimization requires that:





- MPL /MPK = -w/r or
- MPL /w = MPK/r
Does this look familiar at all?
These are the conditions required for long-run profit
maximization.
Therefore, cost minimization and profit maximization occur
simultaneously.
 Let L* and K* define optimal (cost minimizing) L and
K
 L* = f(Y*, w, r)
 K* = f(Y*, w, r)
 These are the conditional or derived factor demand
curves.
 Derived from what?
 How are profit maximization and cost minimization
different?


If maximizing profit => must also be minimizing costs.
If minimizing costs are you necessarily maximizing profit?
 No. Why not?
 Revealed Cost Minimization
 Similar idea to revealed profit maximization
 Observe choices in two time periods, t and s, where firm
choose L and K to minimize costs => must be true that:


(1) wt Lt + rt Kt ≤ wt Ls + rt Ks - why?
(2) ws Ls + rs Ks ≤ ws Lt + rs Kt - why?
 WACM = Weak Axiom of Cost Minimization
 To be minimizing costs the costs from actual choices must be
≤ the costs from other possible choices at that time.
 Follow the same steps to transform (1) and (2) to get:
 ΔwΔL + ΔrΔK ≤ 0 – implications?
 If Δr = 0 and Δw > 0 => ΔL ≤ 0 or derived D for labor must be
downward sloping.
 Same is true of the derived D for Kapital.
 Returns to Scale and Cost Functions
 Define Average Costs = AC = (C(w, r, Y*))/Y* or:
 AC = C(Y*)/Y* - (assuming w and r are constant).
 AC and returns to scale





Constant Returns to Scale
 AC is constant as Y increases
Increasing Returns to Scale
 AC is decreasing as Y increases
Decreasing Returns to Scale
 AC is increasing as Y increases
Why?
What does the AC and C look like with the three types of
returns to scale?
↑ returns
$
→ returns ↓ returns
AC
Y
$
C1 ↑ returns
C2
→ returns
C3
↓ returns
Y
 Short-Run Costs
 L may vary but K is fixed .
 C = CS (Y, K) with K fixed.
 Or choose L to min C=wL +rK, again with K fixed.
 Simpler problem (also assumes w and r are fixed).
 Short run factor demand functions are given by:

L*  LS ( w, r , K , Y )

K*  K
 Short-run Costs are given by:



CSR (Y , K )  wLS (w, r, K )  r K
note that long-run costs = C (Y )  CSR (Y , K * (Y ))
What does this mean?
Cost Curves
 First, examine the Short-Run Cost Curves
 CSR (y) = Cv(y) + F or
 TC = TVC +TFC
 So that ACSR (y) = CSR(y) / y = CV(y)/y + F/y
 Or ACSR(y) = AVC(y) + AFC(y)
 What do the curves look like?
C
C(y)
CV(y)
F
y
 costs are increasing at an increasing rate. Why?
 Because of the fixed factor k (i.e. as L ↑ more
and more => MPL must decline )

Law of diminishing MP
 What do cost curves like this imply about
AC’s?
B
A
AVC
$
$
Why?
Why?
AFC
y
y
C
$
Why
?
y
Since AC=AFC+ AVC A & B imply C
A is easy; B follows from assumption about MPL in the SR.
 Now suppose MPL ↑ at first as L increases due to
specialization and decreases as L increases past some
point => now what does the cost curve look like?
$
AC
AVC
Why?
y
 Marginal Costs
 MC(y) = ΔCSR (y)/ Δy = Δ Cv(y)/ Δ y + Δ F / Δy
 Total or variable cost curve or rate of change of costs
 Also note that MC=AVC for 1st unit of output
 MC(∆y) = (Cv(Δ y) + F – Cv(0) –F) / Δy
 = Cv(Δ y) / Δy = AVC(Δy)
 Since variable costs = 0 when y=0
 Recall…
 (1) AVC may initially fall as y increases (not necessary)
but must eventually rise due to fixed factors.
 (2) AC initially falls due to decreacng AFC but eventually
rises de to increased AVC.
 (3)MC= AVC for 1st unit produced
 (4) MC= AVC at min AVC why?
 (5) MC=AC at min AC why?
MC
AC
AVC
MC
Area under MC up to Y*= total
variable costs of producing y* why?
Y*
 Example
 C(y) = y3 + 4
 Cv(y)= y3
 Cf(y)= 4
 AVC = y2
 AC=y2 + 4/y
 MC=3y2
MC
AC
AVC
TVC
TC
 Long-Run Costs
 (1) No fixed factors: K can vary
 (2)Can think of costs associated with different plant
sizes

For any given LR output, y, there will be some optimal K or
plant size
 (3)Once K is chosen in the LR, K becomes fixed in the
SR
 Long Run AC is the envelope of SR AC curves

Recall: LR Costs or C(y*)
 C(y*) =CSR(y*, K*(y*))
 Why?
 If not at optimal K in short-run =>
 C(y) < CSR(y, K(y)) – why?
 Now, what if not at optimal K in SR?
 i.e. y changes in the SR
 => C(y*) < CSR(y*, K*(y*)


Why? …K is not chosen optimally
Relationship between SR and LR AC must be…
SRAC*
SRAC2
SRAC1
LRAC
y*
y1
Y2
y
 This follows since C(y*) <CSR(y*, K*(y*))
 =>ACs (y, K*) > AC(y) since AC (y) = C(y)/y
 And ACs (y, K*) = Cs(y, K*)/(y)
 LRAC is the lower envelope of all SRAC curves (only true for
continuous plant sizes)
 NOTE: if only discrete levels of plant sixe => say only three:
SRAC1
SRAC3
SRAC2
 Long-Run Marginal Cost
 Discrete Plant Sizes
SRAC
1
SRAC2
SRAC3
y
LRAC = SRAC until move to new one.
LRMC = SRMC until move to new one.
=>LRMC =SRMC as long as LRAC=SRAC for 1,2,3, etc…
 Continuous Plant Sizes
 Same idea is true for LRMC here but continuous
SRMC
1
SRAC
$
LRMC
LRAC
1
y*
y