ECO 365

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ECO 365 Intermediate Microeconomics

• Lecture Notes

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Cost Minimization

• Before looked at maximizing Profits (p) TR TC
  or
• p 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?

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• 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 gt what would happen to K to keep Y
  constant?
• Thus, isoquants are downward sloping and convex
  (why?)

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• 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 C100 r10 and
  w20?

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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
Slope -w/r -20/10 -2
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• 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.

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• 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.

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• 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 gt must also be minimizing
  costs.
• If minimizing costs are you necessarily
  maximizing profit?
• No. Why not?

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• 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 gt
  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
• ?w?L ?r?K 0 implications?
• If ?r 0 and ?w gt 0 gt ?L 0 or derived D for
  labor must be downward sloping.
• Same is true of the derived D for Kapital.

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• 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?

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? returns
? returns
? returns
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• Short-Run Costs
• L may vary but K is fixed .
• C CS (Y, K) with K fixed.
• Or choose L to min CwL rK, again with K fixed.
• Simpler problem (also assumes w and r are fixed).
• Short run factor demand functions are given by
• Short-run Costs are given by
• note that long-run costs
• What does this mean?

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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)

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• What do the curves look like?
• costs are increasing at an increasing rate. Why?
  

C(y)
CV(y)
C
F
y
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• Because of the fixed factor k (i.e. as L ? more
  and more gt MPL must decline )
• Law of diminishing MP
• What do cost curves like this imply about ACs?

B
A
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Since ACAFC AVC A B imply C A is easy B
follows from assumption about MPL in the SR.
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• Now suppose MPL ? at first as L increases due to
  specialization and decreases as L increases past
  some point gt now what does the cost curve look
  like?

Why?
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• Marginal Costs
• MC(y) ?CSR (y)/ ?y ? Cv(y)/ ? y ? F / ?y
• Total or variable cost curve or rate of change of
  costs
• Also note that MCAVC for 1st unit of output
• MC(?y) (Cv(? y) F Cv(0) F) / ?y
• Cv(? y) / ?y AVC(?y)
• Since variable costs 0 when y0

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• 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) MCAC at min AC why?

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Area under MC up to Y total variable costs of
producing y why?
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• Example
• C(y) y3 4
• Cv(y) y3
• Cf(y) 4
• AVC y2
• ACy2 4/y
• MC3y2

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• 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 gt
• C(y) lt CSR(y, K(y)) why?

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• Now, what if not at optimal K in SR?
• i.e. y changes in the SR
• gt C(y) lt CSR(y, K(y)
• Why? K is not chosen optimally
• Relationship between SR and LR AC must be

SRAC
SRAC2
SRAC1
LRAC
Y2 y
y
y1
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• This follows since C(y) ltCSR(y, K(y))
• gtACs (y, K) gt 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 gt
  say only three

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• Long-Run Marginal Cost
• Discrete Plant Sizes

LRAC SRAC until move to new one. LRMC SRMC
until move to new one. gtLRMC SRMC as long as
LRACSRAC for 1,2,3, etc
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• Continuous Plant Sizes
• Same idea is true for LRMC here but continuous